Heriot-Watt University - Reservoir Simulation part1

CONTENTS

1 WHAT IS A SIMULATION MODEL? 1.1 A Simple Example of a Simulation Model 1.2 A Note on Units

2 WHAT IS A RESERVOIR SIMULATION MODEL? 2.1 The Task of Reservoir Simulation 2.2 What Are We Trying To Do and How Complex Must Our Model Be?

3 FIELD APPLICATIONS OF RESERVOIR SIMULATION 3.1 Reservoir Simulation at Appraisal and in Mature Fields 3.2 Introduction to the Field Cases 3.3 Case 1: The West Seminole Field Simulation Study (SPE10022, 1982) 3.4 Ten Years Later - 1992 3.5 Case 2: The Anguille Marine Simulation Study (SPE25006, 1992) 3.6 Case 3: Ubit Field Rejuvenation (SPE49165,1998) 3.7 Discussion of Changes in Reservoir Simulation; 1970s - 2000 3.8 The Treatment of Uncertainty in Reservoir Simulation

4 STUDY EXAMPLE OF A RESERVOIR SIMULATION

5 TYPES OF RESERVOIR SIMULATION MODEL 5.1 The Black Oil Model 5.2 More Complex Reservoir Simulation Models 5.3 Comparison of Field Experience with Various Simulation Models

6 SOME FURTHER READING ON RESERVOIR SIMULATION

APPENDIX A - References

APPENDIX B - Some Overview Articles on Reservoir Simulation

1. Reservoir Simulation: is it worth the effort? SPE Review, London Section monthly panel discussion November 1990.

2. The Future of Reservoir Simulation - C. Galas, J. Canadian Petroleum Technology, 36, January 1997.

3. What you should know about evaluating simulation results - M. Carlson; J. Canadian Petroleum Technology, Part I - pp. 21-25, 36, No. 5, May 1997; Part II - pp. 52-57, 36, No. 7, August 1997.

11Introduction and Case Studies

2 Institute of Petroleum Engineering, Heriot-Watt University 3


LEARNING OBJECTIVES:

Having worked through this chapter the student should:

• Be able to describe what is meant by a simulation model, saying what analytical models and numerical models are.

• Be familiar with what specifically a reservoir simulation model is.

• Be able to describe the simplifications and issues that arise in going from the description of a real reservoir to a reservoir simulation model.

• Be able to describe why and in what circumstances simple or complex reservoir models are required to model reservoir processes.

• Be able to list what input data is required and where this may be found.

• Be able to describe several examples of typical outputs of reservoir simulations and say how these are of use in reservoir development.

• Know the meaning of all the highlighted terms - or terms referred to in the Glossary - in Chapter 1 e.g. history matching, black oil model, transmissibility, pseudo relative permeability etc.

• Be able to describe and discuss the main changes in reservoir simulation over the last 40 years from the 60's to the present - and say why these have occurred.

• Know in detail and be able to compare the differences between what reservoir simulations can do at the appraisal and in the mature stages of reservoir development.

• Have an elementary knowledge of how uncertainty is handled in reservoir simulation.

• Know all the types of reservoir simulation models and what type of problem or reservoir process each is used to model.

• Know or be able to work out the equations for the mass of a phase or component in a grid block for a black oil or compositional model.

CONTENTS

1 WHAT IS A SIMULATION MODEL? 1.1 A Simple Example of a Simulation Model 1.2 A Note on Units

2 WHAT IS A RESERVOIR SIMULATION MODEL? 2.1 The Task of Reservoir Simulation 2.2 What Are We Trying To Do and How Complex Must Our Model Be?

3 FIELD APPLICATIONS OF RESERVOIR SIMULATION 3.1 Reservoir Simulation at Appraisal and in Mature Fields 3.2 Introduction to the Field Cases 3.3 Case 1: The West Seminole Field Simulation Study (SPE10022, 1982) 3.4 Ten Years Later - 1992 3.5 Case 2: The Anguille Marine Simulation Study (SPE25006, 1992) 3.6 Case 3: Ubit Field Rejuvenation (SPE49165,1998) 3.7 Discussion of Changes in Reservoir Simulation; 1970s - 2000 3.8 The Treatment of Uncertainty in Reservoir Simulation


5 TYPES OF RESERVOIR SIMULATION MODEL 5.1 The Black Oil Model 5.2 More Complex Reservoir Simulation Models 5.3 Comparison of Field Experience with Various Simulation Models


APPENDIX A - References










• Be able to describe what is meant by a simulation model, saying what analytical models and numerical models are.

• Be familiar with what specifically a reservoir simulation model is.

• Be able to describe the simplifications and issues that arise in going from the description of a real reservoir to a reservoir simulation model.

• Be able to describe why and in what circumstances simple or complex reservoir models are required to model reservoir processes.

• Be able to list what input data is required and where this may be found.

• Be able to describe several examples of typical outputs of reservoir simulations and say how these are of use in reservoir development.

• Know the meaning of all the highlighted terms - or terms referred to in the Glossary - in Chapter 1 e.g. history matching, black oil model, transmissibility, pseudo relative permeability etc.

• Be able to describe and discuss the main changes in reservoir simulation over the last 40 years from the 60's to the present - and say why these have occurred.

• Know in detail and be able to compare the differences between what reservoir simulations can do at the appraisal and in the mature stages of reservoir development.

• Have an elementary knowledge of how uncertainty is handled in reservoir simulation.

• Know all the types of reservoir simulation models and what type of problem or reservoir process each is used to model.

• Know or be able to work out the equations for the mass of a phase or component in a grid block for a black oil or compositional model.



BRIEF DESCRIPTION OF CHAPTER 1

A brief overview of Reservoir Simulation is first presented. This module then develops this introduction by going straight into three field examples of applied simulation studies. This is done since this course has some reservoir engineering pre-requisites which will have made the student aware of many of the issues in reservoir development. In these literature examples, we introduce many of the basic concepts that are developed in detail throughout the course e.g. gridding of the reservoir, data requirements for simulation, well controls, typical outputs from reservoir simulation (cumulative oil, watercuts etc.), history matching and forward prediction etc. After briefly discussing the issue of uncertainty in reservoir management, some calculated examples are given. Finally, the various types of reservoir simulation model which are available for calculating different types of reservoir development process are presented (black oil model, compositional model, etc.).

PowerPoint demonstrations illustrate some of the features of reservoir simulation using a dataset which the student can then run on the web (with modification if required) and plot various quantities e.g. cumulative oil, watercuts etc.

This module also contains a Glossary which the student can use for quick reference throughout the course.

1 WHAT IS A SIMULATION MODEL?

1.1 A Simple Example of a Simulation Model

A simulation model is one which shows the main features of a real system, or resembles it in its behaviour, but is simple enough to make calculations on. These calculations may be analytical or numerical . By analytical we mean that the equations that represent the model can be solved using mathematical techniques such as those used to solve algebraic or differential equations. An analytic solution would normally be written in terms of “well know” equations or functions (x2, sin x, ex etc).

For example, suppose we wanted to describe the growth of a colony of bacteria and we denoted the number of bacteria as N. Now if our growth model says that the rate of increase of N with time (that is, dN/dt) is directly proportional to N itself, then:

dNdt

N

= α.

(1)

where α is a constant. We now want to solve this model by answering the question: what is N as a function of time, t, which we denote by N(t), if we start with a bacterial colony of size No. It is easy to show that, N(t) is given by:



N t N eot( ) . .= α (2)

which is the well-known law of exponential growth. We can quickly check that this analytical solution to our model (equation 1), is at least consistent by setting t = 0 and noting that N = No, as required. Thus, equation 1 is our first example of a simulation model which describes the process - bacterial growth in this case - and equation 2 is its analytical solution. But looking further into this model, it seems to predict that as t gets bigger, then the number N - the number of bacteria in the colony - gets hugely bigger and, indeed, as t →∞, the number N also →∞. Is this realistic ? Do colonies of bacteria get infinite in size ? Clearly, our model is not an exact replica of a real bacterial colony since, as they grow in size, they start to use up all the food and die off. This means that our model may need further terms to describe the observed behaviour of a real bacterial colony. However, if we are just interested in the early time growth of a small colony, our model may be adequate for our purpose; that is, it may be fit-for-purpose. The real issue here is a balance between the simplicity of our model and the use we want to make of it. This is an important lesson for what is to come in this course and throughout your activities trying to model real petroleum reservoirs.

In contrast to the above simple model for the growth of a bacterial colony, some models are much more difficult to solve. In some cases, we may be able to write down the equations for our model, but it may be impossible to solve these analytically due to the complexity of the equations. Instead, it may be possible to approximate these complicated equations by an equivalent numerical model. This model would commonly involve carrying out a very large number of (locally quite simple) numerical calculations. The task of carrying out large numbers of very repetitive calculations is ideally suited to the capabilities of a digital computer which can do this very quickly. As an example of a numerical model, we will return to the simple model for colony growth in equation (1). Now, we have already shown that we have a perfectly simple analytical solution for this model (equation 2). However, we are going to “forget” this for a moment and try to solve equation 1 using a numerical method. To do this we break the time, t, into discrete timesteps which we denote by Δt. So, if we have the number of bacteria in the colony at t = 0, i.e. No, then we want to calculate the number at time Δt later, then we use the new value and try to find the number at time Δt later and so on. In order to do this systematically, we need an algorithm (a mathematical name for a recipe) which is easy to develop once we have defined the following notation:

Notation: the value of N at the current time step n is denoted as Nn

the value of N at the next time step, n+1 is denoted as Nn+1

Clearly, it is the Nn+1 that we are trying to find. Going back to the main equation that defines this model (equation 1), we approximate this as follows:

N N

tN

n nn

+ − ≈1

∆α.

(3)

where we use the symbol, "≈", to indicate that equation 3 is really an approximation, or that it is only exactly true as Δt → 0. Equation 3 is now our (approximate) numerical



model which can be rearranged as follows to find Nn+1 (which is the “unknown” that we are after):

N t Nn n+ = +1 1( . ).α ∆ (4)

where we have gone to the exact equality symbol, “=”, in equation 4 since, we are accepting the fact that the model is not exact but we are using it anyway. This is our numerical algorithm (or recipe) that is now very amenable to solution using a simple calculator. More formally, the algorithm for the model would be carried out as shown in Figure 1.

Set, t = 0

Choose the time step size, ∆t

Specify the initial no. of bacteria at t = 0i.e.No and set the first value (n=0) of Nn to No

No = No

Print n, t and N (Nn)

Set Nn+1 = (1 + α.∆t). Nn

Set Nn = Nn+1

n = n+1t = t + ∆t

Time to stop ?e.g. is t > tmax mn ax or n >

No

Yes

End

The above example, although very simple, explains quite well several aspects of what a simulation model is. This model is simple enough to be solved analytically. However, it can also be formulated as an approximate numerical model which is organised into a numerical algorithm (or recipe) which can be followed repetitively. A simple calculator is sufficient to solve this model but, in more complex systems, a digital computer would generally be used.

Figure 1Example of an algorithm to solve the simple numerical “simulation” model in the text



1.2 A Note on UnitsThroughout this course we will use Field Units and/or SI Units, as appropriate. Although the industry recommendation is to convert to SI Units, this makes discussion of the field examples and cases too unnatural.

EXERCISE 1.Return to the simple model described by equation 1. Take as input data, that we start off with 25 bacteria in the colony. Take the value α = 1.74 and take time steps Δt = 0.05 in the numerical model.

(i) Using the scale on the graph below, plot the analytical solution for the number of bacteria N(t) as a function of time between t = 0 and t = 2 (in arbitrary time units).

(ii) Plot as points on this same plot, the numerical solution at times t = 0, 0.5, 1.0, 1.5 and 2.0. What do you notice about these ?

(iii)Using a spreadsheet, repeat the numerical calculation with a Δt = 0.001 and plot the same 5 points as before. What do you notice about these?

Time0

500

1000

1 2

N(t)

(i)

(ii)



2 WHAT IS A RESERVOIR SIMULATION MODEL?

In the previous section, we introduced the idea of a simulation model applied to the growth of a bacterial colony. Now let us consider what we want to model - or simulate - when we come to developing petroleum reservoirs. Clearly, petroleum reservoirs are much more complex than our simple example since they involve many variables (e.g. pressures, oil saturations, flows etc.) that are distributed through space and that vary with time.

In 1953, Uren defined a petroleum reservoir as follows:“ ... a body of porous and permeable rock containing oil and gas through which fluids may move toward recovery openings under the pressure existing or that may be applied. All communicating pore space within the productive formation is properly a part of the rock, which may include several or many individual rock strata and may encompass bodies of impermeable and barren shale. The lateral expanse of such a reservoir is contingent only upon the continuity of pore space and the ability of the fluids to move through the rock pores under the pressures available.”

L.C. Uren, Petroleum Production Engineering, Oil Field Exploitation, 3rd edn., McGraw-Hill Book Company Inc., New York, 1953.

This fine example of old fashioned prose is not so easy on the modern ear but does in fact “say it all”. And, whatever it says, then it is precisely what the modern simulation engineer must model!

2.1 The Task of Reservoir Simulation

Let us consider the possible magnitude of the task before us when we want to model (or simulate) the performance of a real petroleum reservoir. Figure 2 shows a schematic of reservoir depositional system for the mid-Jurassic Linnhe and Beryl formations in the UK sector of the North Sea. Some actual reservoir cores from the Beryl formation are shown in Figure 3. It is evident from the cores that real reservoirs are very heterogenous. The air permeabilities (kair) range from 1mD to almost 3000 mD and it is evident that the permeability varies quite considerably over quite short distances. It is common for reservoirs to be heterogeneous from the smallest scale to the largest as is evident in these figures. These permeability heterogeneities will certainly affect both pressures and fluid flow in the system. By contrast, a reservoir simulation model which might be used to simulate waterflooding in a layered system of this type is shown schematically in Figure 4. This model is clearly hugely simplified compared with a real system. Although the task of reservoir simulation may appear from this example to be huge, it is still one that reservoir engineers can - and indeed must - tackle. Below, we start by listing in general terms the activities involved in setting up a reservoir model.

One way of approaching this is to break the process down into three parts which will all have to appear somewhere in our model:

(i) Choice and Controls: Firstly, there are the things that we have some control over. For example:



• Where the injectors and producer wells are located• The capability that we have in the well (completions & downhole equipment)• How much water or gas injection we inject and at what rate • How fast we produce the wells (drawdown)

We note that certain quantities such as injection and production rates are subject to physical constraints imposed on us by the reservoir itself.

(ii) Reservoir Givens: Secondly, there are the givens such as the (usually very uncertain) geology that is down there in the reservoir. There may or may not be an active aquifer which is contributing to the reservoir drive mechanism. We can do things to know more about the reservoir/aquifer system by carrying out seismic surveys, drilling appraisal wells and then running wireline logs, gathering and performing measurements on core, performing and analysing pressure buildup or drawdown tests, etc.

(iii) Reservoir Performance Results: Thirdly, there is the observation of the results i.e the reservoir performance. This includes well production rates of oil, water and gas, the field average pressure, the individual well pressures and well productivities etc.

Barrier

Fluvial/Floodplain

Estuarine BaySSWFluvial

mud/sandsupply

Fluvial/FloodplainFacies AsociationFC: Fluvial channel sandstonesCRS: Crevasse channel/splay sandstonesOM/L: Overbank/lake mudstoneCS: Coal swamp/marsh mudstone and coal

Estuarine Bay-FillFacies AssociationTC: Tidal channel sandstonesTF: Lower intertidal flat sandstonesTS: Tidal shoal sandstoneSM: Salt marsh/upper intertidal flat mudstonesBM: Brackish bay mudstonesFTD: Flood tidal delta

Tidal Inlet-Barrier ShorelineFacies AssociationTCI: Tidal inlet/ebb channel sandstonesSS: Barrier shoreline sandstoneETD: Ebb tidal delta

Block diagram illustrates the gradual infilling of theBeryl Embayment by fluvial/floodplain (Linnhe l),estuarine-bay fill (Linnhe ll) and tidal inlet-barriershoreline facies sequences (Beryl Formation).

Shoreface

CoalFluvial/crevasse channel-fillsTidal channel-fillsTidal inlet-fillsShoal/barsFlood-oriented currentsEbb-oriented currentsLongshore currents

FC

CRS

OM/CS

OM/CSTC

TC

TC TC

TFTF

TF

TSSM

SMSM

SM

BMFTD

TCI

SS

SSSS

ETD

L L

L

12.15 km

Figure 2Conceptual depositional model for the Linnhe and Beryl formations from the middle Jurassic period (UK sector of the North Sea). (G. Robertson in Cores from the Northwest European Hydrocarbon Provence, edited by C D Oakman, J H Martin and P W M Corbett, Geological Society, London. 1997).



Medium-grainedCarbonate cemented

sandstone (φ =14%, ka = 2mD)- some thin clay and

carbonate rich lamination

Medium-grainedripple-laminated and

bioturbated carbonatecemented sandstone

(φ =10%, ka = 1mD)

Pyritic mudstone (pm)→fine-grained bioturbated

sandstone(φ =16%, ka = 29mD)

Medium to coarse-grainedcross-stratified

sandstone(φ =21%, ka =1440mD)

- in fining-up units

Coarse-grainedcarbonaceous sandstone(φ =20%, ka =2940mD)

- in cross-stratified,fining-up units

1 m

Slab 1Top

15855 ft

Slab 2Top

15852 ft

Slab 3Top

14591 ft

Slab 4Top

14361 ft

Slab 5Top

14358 ft

15858 ftBase

15855 ftBase

14594 ftBase

14364 ftBase

14361 ftBase

∆y∆x

∆z

Input:φ, crock, net to gross

kx, k

y, kz,

Swi, k

rw(Sw), k

rw(Sw),

Pc(S

w)

Water Injector

Producer

Approximate Size of Core vs. Grid Size

Figure 3 Cores from the mid-Jurassic Beryl formation from UK sector of the North Sea. φ is porosity and ka is the air permeability. (G. Robertson in Cores from the Northwest European Hydrocarbon Provence, edited by C D Oakman, J H Martin and P W M Corbett, Geological Society, London. 1997).

Figure 4A schematic diagram of a waterflood simulation in a 3D layered model with an 8x8x5 grid. The information which is input for a single grid block is shown. Contrast this simple model with the detail in a geological model (Figure 2) and in the actual cores themselves (Figure 3).



2.2 What Are We Trying To Do and How Complex Must Our Model Be?

Therefore, at its most complex, our task will be to incorporate all of the above features (i) - (iii) in a complete model of the reservoir performance. But we should now stop at this point and ask ourselves why we are doing the particular study of a given reservoir? In other words, the level of modelling that we will carry out is directly related to the issue or question that we are trying to address. Some engineers prefer to put this as follows:

• What decision am I trying to make? • What is the minimum level of modelling - or which tool can I use - that

allows me to adequately make that decision?

This matter is put well by Keith Coats - one of the pioneers of numerical reservoir simulation - who said:

“The tools of reservoir simulation range from the intuition and judgement of the engineer to complex mathematical models requiring use of digital computers. The question is not whether to simulate but rather which tool or method to use.” (Coats, 1969).

Therefore, we may choose a very simple model of the reservoir or one that is quite complex depending on the question we are asking or the decision which we have to make. Without giving technical details of what we mean by simple and complex, in this context, we illustrate the general idea in Figure 5 which shows three models of the same reservoir. The first (Figure 5a), shows the reservoir as a tank model where we are just concerned with the gross fluid flows into and out of the system. In Chapter 2, we will identify models such as those in Figure 5a as essentially material balance models and will be discussed in much more detail later. The particular advantage of material balance models is that they are very simple. They can address questions relating to average field pressure for given quantities of oil/water/gas production and water influx from given initial quantities and initial pressure (within certain assumptions). However, because the material balance model is essentially a tank model, it cannot address questions about why the pressures in two sectors of the reservoir are different (since a single average pressure in the system is a core assumption). The sector model in Figure 5b is somewhat more complex in that it recognises different regions of the reservoir. This model could address the question of different regional pressures. However, even this model may be inadequate if the question is quite detailed such as: in my mature field with a number of active injector/producer wells where should I locate an infill well and should it be vertical, slanted or horizontal ? For such complicated questions, the model in Figure 5c would be more appropriate since it is more detailed and it contains more spatial information. This schematic sequence of models illustrates that there is no one right model for a reservoir. The simplicity/complexity of the model should relate to the simplicity/complexity of the question. But there is another important factor: data. It is clear that to build models of the types shown in Figure 5, we require increasing amounts of data as we go from Figure 5a→5b→5c. It is also evident that we should think carefully before building a very detailed model of the type shown in Figure 5c, if we have almost no data. There are some circumstances where we might build quite



a complicated model with little data to test out hypotheses but we will not elaborate on this issue at this point.

The simplicity/complexity of the model should relate to the simplicity/complexity of the question, and be consistent with the amount of reliable data which we have.

So, Sw and Sg

Average Pressure =Average Saturations =

Wells Offtake(a) "Tank" Model of the Reservoir

(b) Simple Sector Model

(c) Fine Grid Simulation Model of a Waterflood

Aquifer

Oil Leg

Aquifer

Producer - West Flank Producer - East Flank

Injector Producer

200ft

2000ft

P

We are now aware that various levels of reservoir model may be used and that the reservoir engineer must choose the appropriate one for the task at hand. We will assume at this point that building a numerical reservoir simulation model is the correct approach for what we are trying to achieve. If this is so, we now address the issue: What do we model in reservoir simulation and why do we model it ? There are, as we have said, a range of questions which we might answer, only some of which require a full numerical simulation model to be constructed. Let us now say what a numerical reservoir simulation model is and what sorts of things it can (and cannot) do.

Definition: A numerical reservoir simulation model is a grid block model of a petroleum reservoir where each of the blocks represents a local part of the

Figure 5Schematic illustrations of reservoir models of increasing complexity. Each of these may be suitable for certain types of calculation (see text).



reservoir. Within a grid block the properties are uniform (porosity, permeability, relative permeability etc.) although they may change with time as the reservoir process progresses. Blocks are generally connected to neighbouring blocks are fluid may flow in a block-to-block manner. The model incorporates data on the reservoir fluids (PVT) and the reservoir description (porosities , permeabilities etc.) and their distribution in space. Sub-models within the simulator represent and model the injection/producer wells.

An example of numerical reservoir simulation gridded model is shown in Figure 6, where some of the features in the above definition are evident. We now list what needs to be done in principle to run the model and then the things which a simulator calculate, if it has the “correct” data. To run a reservoir simulation model, you must:

(a) Gather and input the fluid and rock (reservoir description) data as outlined above;

(b) Choose certain numerical features of the grid (number of grid blocks, timestep sizes etc);

(c) Set up the correct field well controls (injection rates, bottom hole pressureconstraints etc.); it is these which drive the model;

(d) Choose which output (from a vast range of possibilities) you would like to haveprinted to file which you can then plot later or - in some cases - while thesimulation is still running.

The output can include the following (non-exhaustive) list of quantities:

• The average field pressure as a function of time • The total field cumulative oil, water and gas production profiles with time• The total field daily (weekly, monthly, annual) production rates of each

phase: oil, water and gas• The individual well pressures (bottom hole or, through lift curves, wellhead)

over time• The individual well cumulative and daily flowrates of oil, water and gas

with time• Either full field or individual well watercuts, GORs, O/W ratios with time• The spatial distribution of oil, water and gas saturations throughout the

reservoir as functions of time i.e. So(x,y,z;t), Sw(x,y,z;t) and Sg(x,y,z;t)

Some of the above quantities are shown in simulator output in Figure 7. This field example is for a Middle East carbonate reservoir where the structural map is shown in Figure 7(d). Figure 7(a) shows the field and simulation results for total oil and water cumulative production over 35 years of field life. Figure 7(b) shows the actual and modelled average field pressure. The type of results shown in Figures 7(a) and 7(b) are very common but the modelling of the RFT (Repeat Formation Tester) pressure shown in Figure 7(c) is less common. The RFT tool measures the local pressure at a given vertical depth and, in this case, it can be seen that the reservoir comprises of three zones each of ~ 100 ft thick and each is at a different pressure. This indicates that



pressure barriers exist (i.e. flow is restricted between these layers). This is correctly modelled in the simulation. This is an interesting and useful example of how reservoir simulation is used in practice.

Note that a vast quantity of output can be output and plotted up and the post-processing facilities in a reservoir simulator suite of software are very important. There is no point is doing a massively complex calculation on a large reservoir system with millions of grid blocks if the output is so huge and complex that it overwhelms the reservoir engineerʼs ability to analyse and make sense of the output. In recent years, data visualisation techniques have played on important role in analysing the results from large reservoir simulations.

Observed Water

Observed Oil

Modelled Water

Observed DataModelled Data

Modelled Oil

00

100

200

300

400

500

600

700

5 10 15 20 25 30 35

Year of Production

Cum

ulat

ive P

rodu

ctio

n (M

MB)

01500

2000

2500

3000

3500

5 10 15 20 25 30 35

Year of Production

Aver

age

Pres

sure

(psi

a)

1000 1500 1000 2500 3000

-300

-200

-100

Datum

Dep

th (f

t.)

(a) Full field history match of cumulative oil and water production

(b) Full field history match of volume weighted pressure

(c) Match of RFT pressure data by reservoir simulation model at Year 30

Observed Modelled

Figure 6An example of a 3D numerical reservoir simulation model. The distorted 3D grid covers the crestal reservoir and a large part of the aquifer which is shown dipping down towards the reader. Oil is shown in red and water is blue and a vertical projection of a cross-section at the crest of the reservoir is shown on the x/z and y/z planes on the sides of the perspective box. Two injectors can be seen in the aquifer as well as a crestal horizontal well. Two faults can be seen at the front of the reservoir before the structure dips down into the aquifer. The model contains 25,743 grid blocks.

Figure 7 (a) to (d) Example of some typical reservoir simulator output. From SPE36540, “Reservoir Modelling and Simulation of a Middle Eastern Carbonate Reservoir”, M.J. Sibley, J.V. Bent and D.W. Davis (Texaco), 1996.



Observed Water

Observed Oil

Modelled Water

Observed DataModelled Data

Modelled Oil

00

100

200

300

400

500

600

700

5 10 15 20 25 30 35

Year of ProductionC

umul

ative

Pro

duct

ion

(MM

B)

01500

2000

2500

3000

3500

5 10 15 20 25 30 35

Year of Production

Aver

age

Pres

sure

(psi

a)

1000 1500 1000 2500 3000

-300

-200

-100

Datum

Dep

th (f

t.)

(a) Full field history match of cumulative oil and water production

(b) Full field history match of volume weighted pressure

(c) Match of RFT pressure data by reservoir simulation model at Year 30

Observed Modelled

1 Mile C

DrilledNew LocationInjector LocationConvert to Injector

C

C

C

CC

• A Lower CretaceousCarbonate Reservoir in theArabian Peninsula

• Most wells drilled in 1955-1962

• > 600 MMBO produced byearly 1980s

• -this study 1992

(d) Field structural map with 50' contour interval

Figure 7b

Figure 7c

Figure 7d



How some of this output might be used is illustrated schematically in Figure 8. This is an imaginary case where the reservoir study is to consider the best of four options in Field A: Option 1 - to continue as present with the waterflood; Option 2 - upgrade peripheral injection wells; Option 3 - upgrade injectors and drill six new injectors; Option 4 - drill four new infill wells. Clearly, it is much cheaper to model these four cases than to actually do one of them. The important quantities are the oil recovery profiles for each case compared with the scenario where we simple proceed with the current reservoir development strategy (Option 1). Of course, we do not know whether the forward predictions which we are taking as what would happen anyway, are actually correct. Likewise, we may be unsure of how accurate our forward predictions are for each of the various scenarios. In fact, an important aspect of reservoir simulation is to assess each of the various uncertainties which are associated with our model. This would ideally lead to range of profiles for any forward modeling but we will deal with this in detail later. We discuss the handling of uncertainties in rather more detail in Section 3.8. of this Chapter.

In the schematic case shown in Figures 8(a) - 8(g) we note that:

(i) The areal plan of the reservoir is given showing injector and producer well location in Figure 8(a);

(ii) The corresponding stratification/lithology of the field is shown along the well A-B-C-D transect in Figure 8(b);

(iii) Figures 8(c) and 8(d) show the areal grid and the vertical grid, respectively;

(iv) The forward predictions of cumulative oil for the various options are shownin Figure 8(f). Note that Option 3 produces most oil (but it involves drillingsix additional injection wells);

(v) The economic evolution of each option using the predicted oil recovery profiles in Figure 8(f) is shown in Figure 8(g) (where NPV = Net Present Value; IRR = Interval Rate of Return: these are economic measures explained in the economics module of the Heriot-Watt distance learning course). Note that option 4 emerges in the most economic case although it produces rather less oil than option 3.

A

B

C

D

InjectorProducer

(a) Field A areal plan showing injector and producer well locations; lithology is given from wells A, B, C and D

Figure 8Schematic example of how reservoir simulation might be used in a study of four field development options (see text).



A

BC

D

Sand 1

Sand 2Sand 3

Sand 4

(b) Schematic vertical cross-section showing the lithology across the field through 4 wells A, B, C and D

A

B

C

D

A

B

C

D

A B CD

NZ = 8

(c) Reservoir simulation (areal) grid showing current well locations.

A

B

C

D

A

B

C

D

A B CD

NZ = 8

(d) Reservoir simulation vertical cross-sectional grid showing current well locations.

Figure 8 (c)

Figure 8 (d)

Figure 8 (b)



The grid has 8 blocks in the z- direction representing the thickness of theformation as shown below; NZ = 8. Note that the vertical grid is not uniform.

Time

Infill Wells

Periferal Injectors

Option 3Option 4

Option 2

21

34

Continue as at present (do nothing) Option 1Cum

ulat

ive O

il

Option

NPV

or I

RR

A

B

C

D

(e) Option 1- continue as at present; Option 2 - upgrade peripheral injection wells; Option 3- upgrade injectors + add 6 new injectors; Option 4 - drill four new infill wells.

Time

Infill Wells

Periferal Injectors

Option 3Option 4

Option 2

21

34


ulat

ive O

il

Option

NPV

or I

RR

A

B

C

D

(f) Simulated oil recovery results for various options

Time

Infill Wells

Periferal Injectors

Option 3Option 4

Option 2

21

34


ulat

ive O

il

Option

NPV

or I

RR

A

B

C

D

Figure 8 (e)

Figure 8 (f)

Figure 8 (g)



(g) Economic evaluation of options - NPV or IRR

Now consider what we are actually trying to do in a typical full field reservoir simulation study. There is a short answer to this is often said in one form or another: it is that the central objective of reservoir simulation is to produce future predictions (the output quantities listed above) that will allow us to optimise reservoir performance. At the grander scale, what is meant by “optimise reservoir performance” is to develop the reservoir in the manner that brings the maximum economic benefit to the company. Reservoir simulation may be used in many smaller ways to decide on various technical matters although even these - for example the issue illustrated in Figure 8 - are usually reduced to economic calculations and decisions in the final analysis as indicated in Figure 8(g).

3 FIELD APPLICATION OF RESERVOIR SIMULATION

3.1 Reservoir Simulation at Appraisal and in Mature FieldsUp to this point, we have considered what a numerical reservoir simulation model is and we have touched on some of the sorts of things that can be calculated. Rather than continue with a discussion of the various technical aspects of reservoir simulation one by one, we will proceed to three field applications of reservoir simulation. These studies will raise virtually all of the technical terms and concepts and many of the issues that will be studied in more detail later in this course. The important terms and concepts will be italicised and will appear in the Glossary at the front of this chapter.

Reservoir simulation may be applied either at the appraisal stage of a field development or at any stage in the early, middle or late field lifetime. There are clearly differences in what we might want to get out of a study carried out at the appraisal stage of a reservoir and a study carried out on a mature field.

Appraisal stage: at this stage, reservoir simulation will be a tool that can be used to design the overall field development plan in terms of the following issues:

• The nature of the reservoir recovery plan e.g. natural depletion, waterflooding,gas injection etc.

• The nature of the facility required to develop the field e.g. a platform, a subsea development tied back to an existing platform or a Floating Production System (for an offshore fileld).

• The nature and capacities of plant sub-facilities such as compressors forinjection, oil/water/gas separation capability.

• The number, locations and types of well (vertical, slanted or horizontal) to bedrilled in the field.

• The sequencing of the well drilling program and the topside facilites.



It is during the initial appraisal stage that many of the biggest - i.e. most expensive - investment decisions are made e.g. the type of platform and facilities etc. Therefore, it is the most helpful time to have accurate forward predictions of the reservoir performance. But, it is at this time when we have the least amount of data and, of course, very little or no field performance history (there may be some extended production well tests). Therefore, it seem that reservoir simulation has a built-in weakness in its usefulness; just when it can be at its most useful during appraisal is precisely when it has the least data to work on and hence it will usually make the poorest forward predictions. So, does reservoir simulation let us down just when we need it most? Perhaps. However, even during appraisal, reservoir simulation can take us forward with the best current view of the reservoir that we have at that time, although this view may be highly uncertain. As we have already noted, if major features of the reservoir model (e.g. the stock tank oil initially in place, STOIIP) are uncertain, then the forward predictions may be very inaccurate. In such cases, we may still be able to build a range of possible reservoir models, or reservoir scenarios, that incorporate the major uncertainties in terms of reservoir size (STOIIP), main fault blocks, strength of aquifer, reservoir connectivity, etc. By running forward predictions on this range of cases, we can generate a spread of predicted future field performance cases as shown schematically in Figure 9. How to estimate which of these predictions is the most likely and what the magnitude of the “true” uncertainties are is very difficult and will be discussed later in the course.

Time (Year)

Most Probable Case

"Pessimistic" Case

"Optimistic" Case

Cum

ulat

ive

Oil

Rec

over

y (S

TB)

2005 2010 2015

For example, scenarios for various cases may involve:

• Different assumptions about the original oil in place (STOIIP; Stock Tank Oil Originally In Place).

• Different values of the reservoir parameters such as permeability, porosity,net-to-gross ratio, the effect of an aquifer, etc..

• Major changes in the structural geology or sedimentology of the reservoir

e.g. sealing vs. “leaky” faults in the system, the presence/absence of majorfluvial channels, the distribution of shales in the reservoir etc..

Figure 9Spread of future predicted field performances from a range of scenarios of the reservoir at appraisal.



Mature field development: we define this stage of field development for our purposes as when the field is in “mid-life”; i.e. it has been in production for some time (2 - 20+ years) but there is still a reasonably long lifespan ahead for the field, say 3 - 10+years. At this stage, reservoir simulation is a tool for reservoir management which allows the reservoir engineer to plan and evaluate future development options for the reservoir. This is a process that can be done on a continually updated basis. The main difference between this stage and appraisal is that the engineer now has some field production history, such as pressures, cumulative oil, watercuts and GORs (both field-wide and for individual wells), in addition to having some idea of which wells are in communication and possibly some production logs. The initial reservoir simulation model for the field has probably been found to be “wrong”, in that it fails in some aspects of its predictions of reservoir performance e.g. it failed to predict water breakthough in our waterflood (usually, although not always, injected water arrives at oil producers before it is expected). By the way, if the original model does turn out to be wrong, this does not invalidate doing reservoir simulation in the first place. (Why do you think this is so?)

At this development stage, typical reservoir simulation activities are as follows:

• Carrying out a history match of the (now available) field production historyin order to obtain a better tuned reservoir model to use for future field performanceprediction

• Using the history match to re-visit the field development strategy in termsof changing the development plan e.g. infill drilling, adding extra injectionwater capability, changing to gas injection or some other IOR scheme etc.

• Deciding between smaller project options such as drilling an attic horizontalwell vs. working over 2 or 3 existing vertical/slanted wells

• It may be necessary to review the equity stake of various partner companiesin the field after some period of production although this typically involvesa complete review of the engineering, geological and petrophysical data priorto a new simulation study

• The reservoir recovery mechanisms can be reviewed using a carefully historymatched simulation model e.g. if we find that, to match the history, we must reduce the vertical flows (by lowering the vertical transmissibility), we maywish to determine the importance of gravity in the reservoir recovery mechanism.(Coats (1972) refers to this as the “educational value of simulation models”and it is a part of good reservoir management that the engineer has a goodgrasp of the important reservoir physics of their asset.)

There are many reported studies in the SPE literature where the simulation model is re-built in early-/mid-life of the reservoir and different future development options are assessed (e.g. see SPE10022 attached to this chapter).

Late field development: we define this stage of field development as the closing few years of field production before abandonment. A question arises here as to whether the field is of sufficient economic importance to merit a simulation study at this stage.



A company may make the call that it is simply not worth studying any further since the payback would be too low. However there are two reasons why we may want to launch a simulation study late in a fieldʼs lifetime. Firstly, we may think that, although it is in far decline, we can develop a new development strategy that will give the field “a new lease of life” and keep it going economically for a few more years. For example, we may apply a novel cheap drilling technology, or a program of successful well stimulation (to remove a production impairment such as mineral scale) or we may wish to try an economic Improved Oil Recovery (IOR) technique. Secondly, the cost of field abandonment may be so high - e.g. we may have to remove an offshore structure - that almost anything we do to extend field life and avoid this expense will be “economic”. This may justify a late life simulation study. However, there are no general rules here since it depends on the local technical and economic factors which course of action a company will follow. In some countries there may be legislation (or regulations) that require that an oil company produces reservoir simulation calcualtions as part of their ongoing reservoir management.

3.2 Introduction to the Field CasesThree field cases are now presented. We reproduce the full SPE papers describing each of these reported cases. In the text of each of these papers there are margin numbers which refer to the Study Notes following the paper. We use these to explain the concepts of reservoir simulation as they arise naturally in the description of a field application. In fact, you may very well understand many of the term immediately from the context of their description in the SPE paper.

The three field examples are as follows:

Case 1: “The Role of Numerical Simulation in Reservoir Management of a West Texas Carbonate Reservoir”, SPE10022, presented at the International Petroleum Exhibition and Technical Symposium of the SPE, Beijing, China, 18 - 26 March 1982, by K J Harpole and C L Hearn.

Case 2: “Anguille Marine, a Deepsea-Fan Reservoir Offshore Gabon: From Geology Toward History Matching Through Stochastic Modelling”, SPE25006, presented at the SPE European Petroleum Conference (Europec92), Cannes, France, 16-18 November 1992, by C.S. Giudicelli, G.J. Massonat and F.G. Alabert (Elf Aquitaine)

Case 3: “The Ubit Field Rejuvenation: A Case History of Reservoir Management of a Giant Oilfield Offshore Nigeria”, SPE49165, presented at the SPE Annual Technical Conference and Exhibition, New Orleans, LA, 27-30 September 1998, by C.A. Clayton et al (Mobil and Department of Petroleum Resources, Nigeria)

These cases were chosen for the following main reasons:

• They are all good technical studies that illustrate “typical” uses of reservoirsimulation as a tool in reservoir management (we have deliberately taken all cases at the middle and the mature stages of field development since muchmore data is available at that time);

• They introduce virtually all of the main ideas and concepts of reservoirsimulation in the context of a worked field application. As these concepts



and specialised terms arise, they are explained briefly in the study notes althoughmore detailed discussion will appear later in the course. Compact definitions of the various terms are given in the Glossary at the front of this module;

• They are all well-written and use little or no mathematics;

• By choosing an example from the early 1980s, the early/mid 1990s and the late 1990s, we can illustrate some of the advances in applied reservoir simulation that have taken place over that period (this is due to the availability of greater computer processing power and also the adoption of new ideas in areas such as geostatistics and reservoir description).

How you should read the next part of the module is as follows:

• Read right through the SPE paper and just pay particular attention when there is a Study Note number in the margin;

• Go back through the paper but stop at each of the Study Notes and read through the actual point being made in that note.

As noted above, all the main concepts that are introduced can also be found in the Glossary which should be used for quick reference throughout the course or until you are quite familiar with the various terms and concepts in reservoir simulation.

See SPE 10022 paper in Appendix

3.3 Case 1: The West Seminole Field Simulation Study (SPE10022, 1982)Case 1: “The Role of Numerical Simulation in Reservoir Management of a West Texas Carbonate Reservoir”, SPE10022, presented at the International Petroleum Exhibition and Technical Symposium of the SPE, Beijing, China, 18 - 26 March 1982. by K J Harpole and C L Hearn.

Summary: This paper presents a study from the early 1980s where a range of re-appraisal strategies for a mature carbonate field are being evaluated using reservoir simulation. For example, possible development strategies include the blowdown of the gas cap or infill drilling. They explicitly state in their opening remarks that their central objective is to “optimise reservoir performance” by choosing a future development strategy from a range of defined options. The structure of the study is very typical of the work flow of a field simulation study, viz Introduction; Reservoir Description; Simulation Model; History Matching; Future Performance; Conclusions and recommendations. Although this paper is almost 20 years old, it introduces the reader in a very clear way to virtually all the concepts of conventional reservoir simulation.

Location maps and general reservoir structure shown in Figures 1 and 2 of SPE 10022.



Study Notes Case 1: 1. States explicitly that the objective of the study is to “optimise reservoir performance” as discussed in the introductory part of this module.

2. Raises the issue of an accurate reservoir description being required and this is emphasised throughout this paper.

3. An interesting point is raised comparing the carbonate reservoir of this study broadly to sandstone reservoirs. It notes that the post-depositional diagenetic effects are of major importance in the West Seminole field in that they affect the reservoir continuity and quality i.e. the local porosity and permeability. In contrast, it is noted that sandstone reservoir are mainly controlled by their depositional environment and tend to show less diagenetic overprint. However, a point to note is that the broad outline and work flow of a numerical reservoir simulation study are quite similar for both carbonate and sandstone reservoirs.

4. Carbonate diagenetic processes include dolomitisation (dolomite = CaMg(CO3)2), recrystallisation, cementations and leaching. This geochemical information is not directly used in the simulation model but it is important since it leads to identification of reservoir layer to layer flow barriers (see below).

5. Strategy: Previous gas re-injection into the cap + peripheral water injection => not very successful. They want to implement a 40 acre, 5-spot water flood; see Fig. 3. A “5-spot” is a particular example of a “pattern flood” which is appropriate mainly for onshore reservoirs where many wells can be drilled with relatively close spacing (see Waterflood Patterns in the Glossary).

6a. They raise the issue of vertical communication between the oil and gas zones. This is an excellent example of an uncertain reservoir feature that can be modelling using a range of scenarios from free flow between layers to zero interlayer flow + all cases in between. Therefore, we can run simulations of all these cases and see which one agrees best with the field observations (which is what they do, in fact).

6b. The vertical communication - or lack of it - will affect flow between the oil and gas zones which may lead to loss of oil to the gas cap; see Figure 4.

7. States the structure of the simulation study work flow: Accurate reservoir description - Develop the simulation model (perform the history match - see below - use model for future predictions - evaluate alternative operating plans). A history match is when we adjust the parameters in the simulation model to make the simulated production history agree with the actual field performance (expanded on below).

8a. A lengthy geological description of the reservoir is given where the depositional environment is described - reference is made to extensive core data (~7500 ft. of core).

8b. The impact of the geology/diagenesis in the simulation model is discussed here. There is evidence of field wide barriers due to cementation with anhydrite which may reduce vertical flows. This is important since it gives a sound geological interpretation to the existence of the vertical flow barriers. Therefore, if we need to include this to



match the field performance, we have some justification or explanation for it rather than it simply being a “fiddle factor” in the model.

9. Figure 5 shows the 6 major reservoir layers where the interfaces between the layers are low φ, low k anhydrite cement zones. Again, these may be explained from the depositional environment and the subsequent diagenetic history of the reservoir.

10. 7500 ft. of whole core analysis for the W. Seminole field was available which was digitised for computer analysis (not common at that time, late 1970s). This is very valuable information and is often not available.

11. Permeability distributions in the reservoir are shown in Fig. 6 and these data are vital for reservoir simulation. Dake (1994; p.19) comments on this type of data: “What matters in viewing core data is the all-important permeability distribution across the producing formations; it is this, more than anything else, that dictates the efficiency of the displacement process.”

12. They note that no consistent k/φ correlation is found in this system (which is quite common in carbonates). Often some approximate k/φ correlation can be found for sandstones (e.g. see k/φ Correlations in the Glossary).

13. The W. Seminole field does “exhibit a distinctly layered structure” and the corresponding permeability stratification in the model is shown in Fig. 7.

14a. Pressure transient work - again gives important ancillary information on the reservoir. The objectives of this work were to determine whether there was (i) directional permeability effects, directional fracturing or channelling; (ii) the degree of stratification in the reservoir; (iii) evaluation of the pay continuity between the injectors and producers

14b. No evidence of “channelling or obvious fracture flow system”

14c. Distinct evidence was seen for: (a) the presence of a layered system; (b) restricted communication between layers (ΔP ≈ 200 - 250 psi between layers). This is vital information since it gives an immediate clue that there is probably not completely free flow between layers i.e. there are barriers to flow as suspected from the geology.

14d. Finally on this issue, there is pressure evidence of “arithmetically averaged permeabilities”. This is again typical of layered systems.

15. Native state core tests are referred to from which they obtained steady-state relative permeabilities. These could be very valuable results but no details given here. NB it appears that only one native state core relative permeability was actually measured. This is probably too little data but reflects the reality in many practical reservoir studies that often the engineer does not have important information; however, we just have to “get on with it”.

16. In this study the reservoir simulator which they used was a commercial Black Oil Model (3D, 3 phase - oil/water/gas). Modelling carried out on the main dome portion of the reservoir. This is done quite often in order to simplify the model and to focus



on the region of the field of interest (and importance in terms of oil production). A no flow boundary is assumed in the model on the saddle with the east dome (justified by different pressure history). Again, this is supported by field evidence but it may also be a simplifying judgement to avoid unnecessary complication in the model.

17a. The grid structure used in the simulations is shown in Fig 8. The particular grid that is chosen is very important in reservoir simulation. An areal grid of 288 blocks ( 16 x 18 blocks) - about 10 acre each is taken along with six layers in the vertical direction; i.e. a total of 1728 blocks. This would be a very small model by todayʼs standards and could easily be run on a PC - this was not the case in late 1970s.

17b. They refer to changing the transmissibilities between grid blocks in order to reduce flows. (See Glossary for exact definition of transmissibility.)

18. The following three concepts are closely related (see Pseudo-isation and Upscaling in the Glossary):

18a. Grid size sensitivity: Refers to the introduction of errors due to the coarsness of the grid known as numerical dispersion.

18b. The very important concept of pseudo--relative permeability is introduced here (Kyte and Berry, 1975). “Pseudos” are introduced in order to control numerical dispersion and account for layering. In essence, the use of pseudos can be seen as a fix up for using a coarse grid structure.

18c. Corresponding coarse and fine grid reservoir models are shown in Fig. 9. They note that the fine grid model uses rock relative permeabilities while the coarse grid model uses pseudo relative permeabilities.

19. History Matching: The basic idea of history matching is that the model input is adjusted to match the field pressures and production history. This procedure is intended as being a way of systematically adjusting the model to agree with field observations. Hopefully we can change the “correct” variables in the model to get a match e.g. we may examine the sensitivity to changes in vertical flow barriers in order to find which level of vertical flow agrees best with the field (indeed, this is done in this study). See History Matching in the Glossary.

20a. “Early” mechanism identified as solution gas drive and assistance from expansion. Some initial discussion of field experience and numerical simulation conclusions is presented and developed in these points.

20b. They note some problems with data from early field life. (i) Complicated by free gas production; (ii) channelling due to poor well completions; (ii) no accurate records on gas production for the first 6 years. 20c. The actual field history match indicates that approx. 8 - 10 BCF of gas must have been produced over this early period in order to match the field pressures. This is a use of a material balance approach in order to find the actual early STOIIP (STOIIP = Stock Tank Oil Initially In Place).



21a. They present a description of some adjustments to the history match - but overall it is very good (which they attribute to extensive core data).

21b. Some highlighting of problems with earlier water injection .

21c. The actual history match of reservoir pressure and production is shown in Fig. 10. This is a good history match but think of which field observable - gas production, water production or average field pressure - is the easiest/most difficult to match?

22. A good description of their study of the sensitivity to vertical communication is given at this point. This is examined by adjusting the vertical transmissibilities. They look at the following cases: (i) no barriers; (ii) moderate barrier; (iii) strong barriers and (iv) no-flow barriers. Most of the sensitivties are for the moderate and strong barrier cases.

23a. Results showed that => strong barrier case is best but some problem high GOR wells are encountered randomly spaced through the field. They diagnosed and simulated this as “behind the pipe” gas flow in these wells to explain the anomalies in the field observations. This is quite a common explanation that appears in many places.

23b. Layer differential pressures up to 200 - 250 psi can only be reproduced for the strong barrier case. In simulation terms, this is probably the strongest evidence that this is the best case match.

24. The strong barrier case was chosen as the base case and this was used for the predictive runs. The base case predictions refer to the cases which essentially continue the current operations and these are shown in Fig. 11.

25. The strategies looked at for the future sensitivities are listed as follows: (i) change rate of water injection; (ii) management of gas cap voidage i.e. increase of gas and blowdown at different times; (iii) infill drilling.

26a. Outlines the problems/issues for various strategies as follows: (i) shows vertical communication is very importance - it has a major impact on predicted reservoir performance; (ii) shows that can avoid high future ΔP between gas cap and oil zone by high water injection or early blowdown; (iii) shows better development strategy is to keep low ΔP e.g. increase gas injection or infill drill. Finally, shows infill drilling is the most attractive option and the forward prediction for this case is shown in Figure 12.

26b. Table showing some alternatives in text.

27a. A brief summary of the best future development option is given which is: (i) infill drilling as the best option; (ii) water injection increased concurrently with the drilling program to maintain voidage replacement (but prevent the over-injection of water).

27b. For completeness, it is explained why other plans are not as attractive; i.e. blowdown of gas cap before peak in waterflood production rate would significantly reduce oil recovery.



28. A reasonably good initial forward prediction from 1978 - 1981 is shown in Figs. 13 and 14.

29. Conclusions are given which, in summary, are as follows:

1. Detailed reservoir description essential for numerical modelling.

2. Carbonate - both primary and post-depositional diagenetic factors are important.

3. Waterflood in W. Seminole very sensitive to vertical permeability.

4. Vertical permeability is broadly characterised using 3D numerical simulation.

5. Understanding of reservoir response (mechanism) essential to good management.

6. Management of W. Seminole field best if minimum _P between oil zone and gas cap (lower losses of oil --> gas cap) by: (i) infill drilling; (ii) controlling water injection rates to maintain voidage replacement - donʼt over-inject; (iii) careful management of voidage replacement into gas cap.

Important terms and concepts introduced in SPE10022:

Specific to Reservoir Simulation: history match, permeability distribution, black oil model, grid structure, transmissibility, numerical dispersion, pseudo--relative permeabilites.

General terms: 5-spot water flood, permeability distribution, k/φ correlation, steady-state relative permeability, rock relative permeabilities, solution gas drive, material balance, infill drilling, voidage replacement.

3.4 Ten Years Later - 1992An interesting snapshot of where reservoir simulation technology had reached 10 years after the West Seminole study can be seen in the following papers:From the proceedings of the SPE 67th Annual Technical Conference, Washington, DC, 4-7 October 1992:

SPE24890: “From Stochastic Geological Description to Production Forecasting in Heterogeneous Layered Systems”, K. Hove, G. Olsen, S. Nilsson, M. Tonnesen and A. Hatloy (Norsk Hydro and Geomatic)Summary: This paper describes the transfer of data from a detailed gridded stochastic geological model to a more coarsely gridded reservoir simulation model. It is essentially a field application of a methodology described in a previous paper from the same company (Damsleth et al, 1992; Damsleth, E., Tjolsen, C.B., Omre, H. and Haldonsen, H.H., “A Two Stage Stochastic Model Applied to a North Sea Reservoir”, J. Pet. Tech., pp. 402-408, April 1992). The two step procedure involves a first step of constructing the geological architecture of the reservoir followed by a



second stage where the petrophysical values are assigned to each building block in the geological model. The consequences of making various assumptions in the gridding are evaluated for water, gas and WAG (water-alternating-gas) injection. They note that is it very important to represent the main geological features in the gridded model. It was also noted that, when a regular coarse grid was used, the contrast in properties of this heterogeneous reservoir were “smoothed out” by the averaging process and in most cases led to a more optimistic predicted production performance. That is, the more stochastic models led to a reduction in predicted recovery compared with conventional coarse gridded models.

In the proceedings of the SPE European Petroleum Conference, Cannes, France, 16-18 November 1992. A session at this conference produced the following selection of reservoir simulation papers:

SPE25008: “Reservoir Management of the Oseberg Field After Four Years”, S. Fantoft (Norsk Hydro)

Summary: The Oseberg Field (500x106 Sm3 oil; 60x106 Sm3 gas) comprises of seven partly communicating reservoirs. Both water and gas are being injected and modelled in this study and results indicate over 60% recovery in the main reservoir units. The modelling results indicate that the plateau production will be extended by the use of horizontal wells. The objective of the simulation study was exactly this - i.e. to maximise the plateau and improve ultimate oil recovery. This is a very competent simulation study and - although details are not given - it is stated that the geological model is updated annually based on information from new wells. It establishes several aspects of the reservoir mechanics and makes a number of recommendations for operating practice in the future. In other respects, this is quite a “conventional” study.

SPE25057: “The Construction and Validation of a Numerical Model of a Reservoir Consisting of Meandering Channels”, W. van Vark, A.H.M. Paardekam, J.F. Brint J.B. van Lieshout and P.M. George (Shell)Summary: This study focuses on a reservoir which has low sandbody connectivity and which is interpreted as a meandering channel fluvial system. Two years of depletion data is available and one of the aims of the study was to evaluate the possibility of performing a waterflood in this field. They identified a problem in that the sandbody connectivity was lower than might be expected from the sedimentology alone and it was conjectured that this might be due to minor faulting with throws of a few meters. This study again emphasises the importance of the reservoir geology and tries to relate the performance back to this. The geological model is also an early practical example of using a “voxel” representation of the system - approx. 128,000 voxels were used in the model. They noted that the original (sedimentological) models gave over optimistic connectivity. An acceptable match to observed field pressures by including some level of smaller scale faulting.

SPE25059: “Development Planning in a Complex Reservoir: Magnus Field UKCS Lower Kimmeridge Clay Formation (LKCF)”, A.J. Leonard, A.E. Duncan, D.A. Johnson and R.B. Murray (BP Exploration Operating Co.)Summary: This simulation study was carried out on the geologically complex, low net to gross LKCF (rather than on higher net to gross Magnus sands studied



previously). The objective was to formulate a development plan for the LKCF which would accelerate production from these sands. Stochastic modelling techniques were integrated into more conventional “deterministic” models and various options were screened for inherent uncertainty and risks. The study concluded that a phased water injection scheme was the best way forward with the phasing being used to manage and offset the considerable geological risks. Ranges of expected recovery were generated and an incremental recovery of 60 MMstb was predicted increasing the total reserve of the LKCF by a factor of x2.4. This study also demonstrated the importance of inter-disciplinary team work to overcome the previously inhibiting high risks involved.

The proceedings of Europec92 also included the following paper:

SPE25006: “Anguille Marine, a Deepsea-Fan Reservoir Offshore Gabon: From geology Toward History Matching Through Stochastic Modelling”, C.S. Giudicelli, G.J. Massonat and F.G. Alabert (Elf Aquitaine)This paper is such a good example of contemporary studies at that time, that this is chosen as our Case 2 example and is presented in some detail in the next section.

3.5 Case 2: The Anguille Marine Study (SPE25006,1992)Case 2: “ Anguille Marine, a Deepsea-Fan Reservoir Offshore Gabon: From geology Toward History Matching Through Stochastic Modelling”, SPE25006, presented at the SPE European Petroleum Conference (Europec92), Cannes, France, 16-18 November 1992, by C.S. Giudicelli, G.J. Massonat and F.G. Alabert (Elf Aquitaine)


Summary: The Anguille Marine Field in Gabon has 25 years of production history. The waterflood performance indicated severe sedimentary heterogeneity as the field is known to have been deposited in a deep water fan sedimentary environment. This paper is one of the first to refer to the multi-scale nature of the heterogeneity (5 scales were studied) and to refer this back to the sequence stratigraphy of the depositional environment. The sequence stratigraphic approach allowed the field to be divided into the main types of turbiditic geometries (channels, lobes, slumps, laminated facies). Fine scale models (> 2 million grid blocks) were generated using geostatistical techniques and several issues were raised concerning both the geological model and the upscaling process itself. This is a very good example of an early integrated geology(sedimentology)/engineering study in reservoir simulation. The multi-scale nature of the heterogeneity is well related back to the geology.

Study Notes Case 2: 1. Depositional environment: the Anguille Marine field is a deep sea fan environment (i.e a turbidite) with a low sand/shale ratio. This geological description opens the discussion (unusual for previous simulation studies) and the geology features heavily in the flow properties and hence in the geological and reservoir models of this field.

2. Sequence stratigraphy: A more modern feature of reservoir simulation is that the five identified scales of heterogeneity are recognised and some attempt is made to



incorporate them into the 3D simulation model. These scales are also firmly linked to the geology (sedimentology) through the principles of sequence stratigraphy.

3. Geostatistics: Reference is made to how the geological features constrain the fine scale 3D models (of > 2 million blocks - which was large for the time) using geostatistical techniques. By the early 90s, the use of geostatistical methods was becoming more widespread and how it has been applied in this case is covered better by Refs. 1 - 5 in this paper.

Location and structure maps of Anguille Marine are given in Figures 1 - 4.

4. Brief field facts: Discovery 1962; primary depletion commenced in 1966 but reservoir pressure fell rapidly over the next 2 - 3 years and GOR increased; waterflooding from 1971 restored pressure support but channelling led to early water breakthrough; infill drilling not very successful due to lack of current understanding of complex reservoir geology; new approach in 1990 focused more strongly on the reservoir geology of this heterogeneous low sand/shale ratio system recognising the characteristic geometries of a tubditic fan - lobes channels, levees, slumps, laminated facies etc.

5. The approach: It is important in all reservoir simulation studies to have a clear logic to how we approach the simulation of a large complex reservoir system. Here they describe their general methodology although details are in Refs. 1 - 4 at the end of the paper. Basically they: describe and model upper reservoir/ extend to the whole reservoir/ try to translate the geological model to a practical simulation model. On the latter issue they describe the use of “partial models” where just a smaller sector of the reservoir is studied but lessons are taken back into the full model.

6. Reservoir description: Section 2 of the paper gives a sedimentological description of the reservoir as a “slope-apron fan” of complex lithology (depositional model Figure 3) in which 14 (simplified) facies were retained; criteria of composite log recognition of various facies shown in Figure 5. Some contradictory water breakthrough observations were noted. Table 1 gives sedimentary body dimensions (lengths and widths) for channels, lobes. levees/crevasse-splay, slumps, channels (Upper Anguille); Table 2 gives mean petrophysical characteristics. A very important final result for reservoir simulation is the identification of five scales of heterogeneity - Figures 6 and 7; this makes the geological analysis and information numerically useable.

7. Sedimentary history: In earlier reservoir simulation studies, and indeed up to the present time, it is rare to see sedimentary history discussed in terms of a sequence stratigraphic analysis (even mentioning the pioneering work on sea level changes of P. R. Vail et al, “Seismic stratigraphy and global changes of sea level”, in Seismic Stratigraphy, Applications to Hydrocarbon Exploration, AAPG Memoir 26, pp. 49-212, 1977). Chronostratigraphic correlations refer to the “timelines” of simultaneous deposition. This analysis underpins much of the reservoir description but we will not elaborate on it here.

8. Geostatistical modelling: Mainly discussed in Refs. 1 and 2 of this paper. Firstly, focus on geostatistical modelling of the 3D distributions of the major flow units (channels and lobes) and barriers (laminated facies or slumps) for the entire reservoir. This is done as a “conditional” simulation where the distribution is constrained



(or conditioned) to the observed facies and reservoir quality observed at the wells. Secondly, the smaller scale heterogeneities are “unconditionally” simulated (synthetically) to yield average properties within the major flow units (see Ref. 2 in paper). The geostatistical simulation method used was “indicator simulation” (Refs. 1 and 8) which require the average frequency and variogram information.

9. Reservoir zonation: Six unit vertical reservoir zonation shown in Figure 10. Simulations of lateral continuity within each of the units (five - not Middle Anguille, Figure 10) performed independently since they correspond to separate sedimentary phases. For horizontal zonation, Figure11 shows lateral zonation on LA2 and UA2 units showing directional trends and thus variograms with spatially variable anisotropy direction used in final model. Figures 12 - 15 show resulting correlation structures of the various units. Ends up with >2 million grid blocks in the full field 3D model.

10. Flow simulations: Discusses details of upscaling from fine grid stochastic model (>2 million blocks) to coarse grid simulation model (11,000 grid blocks). 11 vertical layers are retained to represent the reservoir layering with more blocks being used in the best reservoir units. Upscaling of absolute permeability at some “aggregation rate” (e.g. 4x4) is applied leading to areal block sizes of 200m x 200m - see Figure14. Relative permeabilitiees were upscaled “on a typical block configuration” (details in Refs. 2 and 4). Additionally: Three major zero-transmissibility faults included in model; some WOC variation across field; depth varying bubble points assigned; 25 years of injection/production for history matching.

11. Simulation results: Initial pressure depletion results shown in Figure 16 - where 14 out of 17 wells show satisfactory pressure behaviour. Pressure behaviour and water breakthrough are poorly predicted during injection stage - Figure 17; water saturations around injectors shown in Figure18 - upscaling has “washed out” the finer scale strong anisotropy.

12. Model changes: Table 8 lists a number of sometimes quite radical changes to the model in order to achieve a better fit to observed field performance - Figure 15 shows differences in upscaled permeability maps. Continuing problems with injection predictions => - is geological model correct? - what is the real effect of upscaling?

13. Partial models: “Thin” model - Figure 19 shows the “thin” partial field model to verify reservoir geology; well AGM18 good water breakthrough match (Figure 20) - early breakthrough for well AGM29 (Figure 21). When thin model upscaled as in full field model (abs. k upscale + rel perm as before) - results in Figures 21 and 22 - breakthrough delayed in both wells but shape of BSW is satisfactory. Conclusions: Thin model partly validates geological model; Some problems with upscaling not supressing breakthrough, making reservoir too connected and eliminating strong anisotropy. Test model - (50 x 20 x 56) model extracted from full field model. Figure 23 shows that an optimum upscaling aggregation rate (2 x 2 x 7) is found - they warn caution on this point. We note that if very reliable and general upscaling techniques were available, then this should be eliminated (more work has been done on this issue since 1992 - much of it at Heriot-Watt!).



14. Conclusions:• Sedimentology controls heterogeneity analysis when very wide variation in

sandbody geometries is found (as in this case)

• Link understanding of reservoir history to sequene stratigraphy

• Litho-interpretation of seismic canʼt give paleo-direction when there is techtonic activity during sedimentation

• Multi-scale heterogeneity analysis essential to quantify sub-grid petrophysical properties

• Geostatistical indicator simulation is a good tool for modelling this multi- scale heterogeneity - trends can also be included

• Stochastic model for Anguille Marine constrained by geology gives hopeful first results

• If aggregation rate in upscaling is optimised, history matching is possible with the use of strictly controlled geological parameters

3.6 Case 3: Ubit Field Rejuvenation (SPE49165,1998)Case 2: “ The Ubit Field Rejuvenation: A Case History of Reservoir Management of a Giant Oilfield Offshore Nigeria”, SPE49165, presented at the SPE Annual Technical Conference and Exhibition, New Orleans, LA, 27-30 September 1998, by C.A. Clayton et al (Mobil and Department of Petroleum Resources, Nigeria)


Summary: This is another good example of where integrated reservoir management has greatly contributed to the success of a field redevelopment plan. In particular, a clearer understanding of the reservoir structural geology has been central to this process. The reinterpretation of the structural geology of the field (the fault blocks, compartments and slump blocks) was achieved using seismic data in a range of complementary ways. The Ubit reservoir is a prograding shallow marine system which has been tectonically disturbed. The downslope movements of the youngest sand sequences resulted in large scale slumping and block sliding although reservoir quality in these sediments is good to excellent. Important facts on the Ubit reservoir and this study are: STOIIP = 2.1 billion bbl oil; 37ºAPI black oil, Bo = 1.38, GOR 612 scf/stb, μo = 0.64 cp and μg = 0.16 cp; production from a relatively thin oil column (160 ft.) and a fairly thick gas cap (50 - 550 ft.). Previous average production = 30 MBD; after implementation of study recommendations (many horizontal wells etc.), expected production ≈ 140 MBD. The notes on this SPE paper will not be very extensive and only a few of the main novel points will be discussed below.

Study Notes Case 3: 1. New data and techniques: The study is a very good example of the close integration of (especially) 3D seismic data used in several ways, computer mapping



and reconstruction of the slump blocks, advanced reservoir simulation procedures, visualisation etc.

2. Recommendations: These have been quite clearly established and stated. The study shows that the key strategies are

• Implement horizontal well drilling (approx. 57 wells).• Full field simulation defining drilling placement and timing.• Balancing a non-uniform gas cap.• Maintaining a stable gas cap (gravity stable displacement) and pressure.• Establish field plateau rate.• Minimising free gas production.

3. Uncertainties: there was an initially erroneous view of certain aspects of the reservoir geology and the key uncertainties at the start of the study were

• Geological complexities in reservoir architecture, particularly structural deformation.

• Sandbody geometries.• Petrophysical rock and fluid properties.• Distribution of flow units.

4. Structural reinterpretation: Figures 2a and 2b show both the original and current interpretations of the structure. The original “rubble beds” are reinterpreted as being techtonically disturbed downslope movements of the youngest sand sequences resulting in large scale slumping and block sliding. The older interpretation saw these facies as being essentially “chaotic” whereas they are now thought to be more ordered and predictable. 3D seismic data is of central importance in the definition of the structural geometries where the “bedded” and “disturbed” strata are shown on a seismic section in Figure 4. Several seismic techniques were applied including attribute analysis, rock physics and amplitude analysis, seismic facies analysis of time slices and conventional reflector mapping. The resulting 70 internal slump and fault blocks are shown in Figure 7.

5. Petrophysics-based facies: Seven flow controlling depositional facies were identified as shown in Figure 8 with rock properties related to grain size (lithology, typical log response, net/gross, k vs. φ, Pc and kro-krw). Depositional facies types present are - marine turbidites and debris flow sands; lower delta plain tidal channels and lagoonal sands; shallow marine upper shoreface and lower shoreface sands and shelf shales (best are turbidites, shoreface and channel sands - comprise 80% pore volume in oil column).

6. Layering and Reservoir Simulation Model: Vertical layering is shown schematically in Figure 9 and the areal grid is given in Figure 12, with a set of rock property maps for a single simulation layer given in Figure10. Grid is Nx x Ny x Nz = 93 x 40 x 18 (67,000 blocks) with most oil leg cells being Δz = 10ft. to resolve the gravity stable gas front. Rock property slices were loaded into the 3D modelling software to “connect up” the stratigraphic layers (using new but unclear developments by authors) as shown in Figure 11.



7. Relative permeability: An interesting point is made concerning the oil relative permeability (kro) close to its end point - see Figure 14. Although kro is low in this region (kro ~10-6 - 10-7), it significantly affects the tail of the reservoir oil production profile - see Figure 15. The adjusted curve in Figure 14 is used to get more realistic longer time recoveries (Figure 15). This is important because gravity stable gas cap expansion and downward displacement is the principal oil recovery mechanism - see schematic view in Figure 9.

8. History match and forward prediction: Average field pressure and GOR are history matched - see Figures 15 and 16. Matching field pressure is not so difficult since Ubit shows good pressure communication (high k - lack of sealing faults). Some discussion of field management is presented. Figure 22 shows the initial part of the improved productivity (up to 140 MBD from 37 horizontal wells) and future predictions. See Recommendations (point 2 ) above.

3.7 Discussion of Changes in Reservoir Simulation; 1970s - 2000From the above field examples (Cases 1 -3), there is clearly a progression in the engineering approach, the degree of reservoir description and the computational capabilities as we go from reservoir simulation in the late 1970s to the present time.

The main changes are as follows:

Computer power: There has been a vast increase in computer processing power over this period because of :

(a) CPU: The growth of powerful CPU (central procesing units - i.e. chips) especially as implemented in Unix machines (workstations) and RISC technology and more recently by the development of modern PCs. The corresponding cost of computing has fallen dramatically. A graph of processing power (Mflops/s) vs. time and a corresponding graph of maximum practical model size vs. time is shown in Figure 10:

Mflo

ps/s

Grid

bloc

ks

1000

100

10

1

0.11970 1975 1980 1985 1990 1995 2000

1000

10000

100000

1000000

1001960 1970 1980 1990 2000

Year Year

(a) State of art CPU performance (b) Maximum practical simulation model size

(b) Parallel Processing: Part of the increase in computing power referred to above is the growth of parallel processing in reservoir simulation. The central idea here is to distribute the simulation calculation around a number of processors ( or “nodes”) which perform different parts of the computational problem simultaneously. A bank of such processors is shown in Figure 11 (from the

Figure 10(a) CPU performance (Mflops/s) vs. time and (b) maximum practical model size vs. time; Mflop/s = mega-flops per second = million floating point operations per second; from J.W. Watts, “Reservoir Simulation: Past, Present and Future”, SPE Reservoir Simulation Symposium, Dallas, TX, 5-7 June 1997.



work of Dogru, SPE57907, 2000). The general impact of parallel simulation is shown according to Dogru (2000) in Figure 12. If the problem gets linearly faster with the number of parallel processors, then it is said to be “scalable” and the closeness to an ideal line is a measure of how well the process “parallelises” (reaches the ideal scaling line); an example is shown in Figure 13. Finally, the type of fine scale calculation that can now be performed using megacell simulation is shown in Figure 14 where it is shown that there is a lengthscale of remaining oil that is missed in the coarser (but still quite fine) simulation. A table of what types of calculation can be performed and some timings for these is also included (although these numbers will probably be out of date very quickly!). For further details, see Megacell Reservoir Simulation - A.H. Dogru - SPE Distinguished Author Series, SPE57907, 2000 and the references therein.

Conventional Simulators

Parallel Simulators

1.2

1

0.8

0.6

0.6

0.2

01988 1990 1992 1994 1996 1997 1998

00

4

8

12

16

4 8 12 16Number of Nodes

Cluster SP2 Ideal

Speedup

Figure 12The impact of parallel reservoir simulation; from “Megacell Reservoir Simulation” - A.H. Dogru - SPE Distinguished Author Series, SPE57907, 2000.

Figure 11A cluster of parallel processors; from “Megacell Reservoir Simulation” - A.H. Dogru - SPE Distinguished Author Series, SPE57907, 2000.



Conventional Simulators

Parallel Simulators

1.2

1

0.8

0.6

0.6

0.2

01988 1990 1992 1994 1996 1997 1998

00

4

8

12

16

4 8 12 16Number of Nodes

Cluster SP2 Ideal

Speedup

Fig. 7—Comparison of conventional simulation (40,500cells, 5 layers) and megacell simulation (2.45 millioncells, 67 layers).

Reservoir Model Size History CPU Hours (Millions of Gridblocks) Length (Years) On 64 Nodes On 128 Nodes

Carbonate 1.2 27 3 1.7

Sandstone 1.3 49 4.5 2.5

Carbonate 3.9 10 - 2.0(With gas cap)

Carbonate 2.5 2.5 - 4.0

(c) Visulisation: Huge improvements in visualisation capabilities have taken which allow us to evaluate vast quantities of numerical data in a more convenient manner. This, in turn allows us to apply our intuitive engineering judgement to reservoir development problems. In addition, the visual representation of output allows a more fruitful communication to take place between geologists and engineers;

(d) Integrated software: The availability of more integrated software suites that handle all the initial data from seismic processing, to petrophysical analysis and data generation, geocellular modelling and upscaling through to the actual simulators themselves.

Figure 14The type of reservoir simulation that becomes more possible with parallel processing. Comparison with fine grid and megacell simulation which identifies the scale of remaining oil in a reservoir displacement process; from “Megacell Reservoir Simulation” - A.H. Dogru - SPE Distinguished Author Series, SPE57907, 2000.

Figure 13A cluster of parallel processors; from “Megacell Reservoir Simulation” - A.H. Dogru - SPE Distinguished Author Series, SPE57907, 2000.



(e) Linked to this increased power, is the ability to handle huge geocellular models and somewhat smaller but still very large reservoir simulation models.

Geostatistics: there have been significant advances in the application of geostatistical techniques in reservoir modelling. Such approaches were quite well known in the mining, mineral processing and prospecting industries but only in the last 10 to 15 years have they been specifically adapted for application in petroleum reservoir modelling. Introductory texts are now available such as:

• “An Introduction to Applied Geostatistics” by E.H. Isaaks and R.M. Srivastava, Oxford University Press, 1989

• “Introduction to Geostatistics: Applications in Hydrogeology” by P.K. Kitanidis, Cambridge University Press, 1997

Both pixed-based point geostatistical techniques and object based modelling have been developed and applied in various reservoirs.

Upscaling: There have been a number of advances in approaches to upscaling (or pseudo-isation) from fine geocellular model → the reservoir simulation model. This still an area of active research and the debate is still in progress on the question:

• Will increasing computing power remove the need for upscaling?

The basic idea of upscaling has been introduced in the SPE examples. Upscaling is dealt with in much more detail in Chapter 7.

Organisational changes in the oil industry: A number of major organisational changes have occurred in the oil industry since the 1970s which have affected the practice of reservoir simulation. The main ones are as follows:

(a) Many companies have taken a more integrated geophysics/geology/engineering view of reservoir development and many studies have made a central virtue of this by organising reservoir studies within more multi-disciplinary asset teams (e.g. SPE25006 clearly shows a strong integration of geology and engineering);

(b) There have been significant organisational changes in the structure of the industry given the sucessive rounds of “downsizing” and “outsourcing” that have occurred. For example, see the short article by Galas, “The future of Reservoir Simulation”, JCPT, p.23, Vol. 36 (1), January 1997, which is reproduced in Appendix B. This article has an interesting slant from the point of view of the smaller consultant and it makes a number of interesting observations;

(c) There have been a number of major mergers and take-overs recently which have formed some very large companies e.g. BP (BP - Amoco - ARCO), Exxon-Mobil, Total-Fina-Elf. Likewise, a number of very “low cost” operations have grown up which may specialise in the successful (i.e. profitable) exploitation of mature assets e.g. Talisman, Kerr-McGee etc. How these changes will affect the future of reservoir simulation remains to be seen.



(d) Up until the late 1970s, almost every major and medium sized oil company had a “research centre” where programmes of applied R&D were carried out by oil company personnel. The view was essentially that this in-house technology development would give the company a competitive and commercial edge in reservoir exploration and development. Most companies have greatly reduced the amount of in-house “R” that takes place and have focused much more heavily on shorter term asset related “D”. Many companies do support research in universities and other independent outside organisations - they also ally themselves with service companies in order to have their R&D needs met in certain areas. Again, the situation is in flux and the longer term effects of this change is yet to be seen.

(e) More specific to reservoir simulation is the fact that, in the 1970s, most companies would have had their own numerical reservoir simulator which was built (programmed up) and maintained in-house. To this day, a few companies still do. However, most oil companies use specialised software service companies to supply their reservoir simulation (and visualisation, gridding etc.) software. Again, the relative merits and demerits of this will emerge in the coming years.

Detailed technical advances: In addition to the changes discussed above, many advances have been made over the past 50 years on how we perform the simulations i.e. on the formulation and numerical methods etc. Our practical capabilities have also expanded greatly as discussed above. Table 1 presents a list of capabilities and major technical advances in reservoir simulation over the last 50 years; this table was adapted from two tables in J.W. Watts “Reservoir Simulation: Past, Present and Future”, SPE38441, SPE Reservoir Simulation Symposium, Dallas, TX, 5-7 June 1997. This article is well worth reading. Most of the technical details in the advances listed in Table 1 are beyond the scope of this course and the introductory student does not need to have any in-depth knowledge on these.



1950’s Two dimensions Aronofsky and Jenkins radial gas model Aronofsky, and Jenkins (1954) Two incompressible phases Alternating-direction implicit (ADI) procedure Peaceman and Rachford (1955) Simple Geometry

1960’s Three dimensions IMPES computational method Sheldon et al (1960) Three phases Upstream weighting Stone and Garder (1961) Black-oil fluid model Understanding of numerical dispersion Lantz (1971) Multiple wells Strongly-implicit procedure (SIP) Stone (1968) Realistic geometry Implicit computational method Well coning Additive correction to line-successive MacDonald and Coats (1970) overrelaxation Watts (1971)

1970’s Compositional Stone relative permeability models Stone (1970, 1973) Miscible Vertical equilibrium concept Coats et al (1971) Chemical Todd-Longstaff miscible displacement Todd and Longstaff (1972) Thermal computation Two-point upstream weighting Todd et al (1972) D4 direct solution method Price and Coats (1974) Total velocity sequential implicit method Spillette et al (1973) Pseudofunctions Kyte and Berry (1975) Variable bubblepoint black-oil treatment Thomas et al (1976) Conjugate gradients and ORTHOMIN Meijerink and Van der Vorst (1977) Vinsome (1976) Iterative methods based on approximate Peaceman (1978) factorizations Peaceman well correction Yanosik and McCracken (1979) Nine-point method for grid orientation effect

1980’s Complex well management Code vectorization Appleyard and Cheshire (1983) Fractured reservoirs Nested factorization Acs et al (1985); Watts (1986) Special gridding at faults Volume balance formulation Young and Stephenson (1983) Graphical user interfaces Young-Stephenson formulation Thomas and Thurnau (1983) Adaptive implicit method Wallis et al (1985) Constrained residuals Ponting (1992) Local grid refinement Cornerpoint geometry Geostatistics Domain decomposition

1990’s Improved ease of use Code parallelization Heinemann et al (1991) Geologic models and upscaling Upscaling Palagi and Aziz (1994) Local grid refinement Voronoi grid Complex geometry Integration with non-reservoir computations

Decade Capabilities Technical Advances References

3.8 The Treatment of Uncertainty in Reservoir SimulationIt is well recognised in modern reservoir development that when we calculate the future oil recovery profile of a reservoir, it is not “accurate”. Suppose a particular reservoir simulation model for Field X - a new development which is currently under appraisal - gives the forecast in Figure 15.

2000 2005 2010Time (Year)

Cum

ulat

ive

Oil

Rec

over

y

Table 1Capabilities and major technical advances in reservoir simulation over the last 50 years (adapted from two tables in J.W. Watts “Reservoir Simulation: Past, Present and Future”, SPE38441, SPE Reservoir Simulation Symposium, Dallas, TX, 5-7 June 1997; (all references are given in Appendix A)

Figure 15A single forward prediction of the oil recovery production profile for a given reservoir.



Clearly, we cannot trust this single curve since there is a considerable amount of uncertainty associated with it for various easily appreciated reasons. The main contributors to this uncertainty are to do with lack of knowledge about the input data although the modelling process itself is not error free. A list of possible sources of error is as follows:

• Lack of knowledge or wide inaccuracies in the size of the reservoir; its areal extent, thickness and net-to-gross ratios

• Lack of knowledge about the reservoir architecture i.e. its geological structure in terms of sandbodies, shales, faults, etc.

• Uncertainties in the actual numerical values of the porosities (φ) and permeabilities (k) in the inter-well regions (which make up the vast majority of the reservoir volume)

• Inaccuracy in the fluid properties such as viscosity of the oil (μo), formation volume factors (Bo, Bw, Bg), phase behaviour etc., or doubts about the representativity of these properties

• Lack of data - or very uncertain data- on the multiphase fluid/rock properties, particularly relative permeability and capillary pressure, and on knowledge as to how these curves vary from rock type within the reservoir volume away from the wells

• Because the representational reservoir simulations model may be poor, e.g. the numerical errors due to the coarse grid block model may significantly affect the answer in either an optimistic or pessimistic manner.

The above list of uncertainties for a given reservoir, especially at the appraisal stage, is really quite realistic and is by no means complete. As we have noted elsewhere, it is at the appraisal stage when, although the future reservoir performance is at its most uncertain, we must make the biggest decisions about the development and hence speed most of our investment money.

At a first glance, the task of doing something useful with reservoir simulation may seem quite hopeless in the face of such a long list of uncertainties. No matter how bleak things look, the only two options are to give up or do something, and reservoir engineers never give up! We must produce an answer - even if it is an educated guess (or even just a guess) - and some estimate of the sort of error sound that we might expect.

Before considering what we can do in practice, let us first consider what the answer might look like for the case above in Figure 15. Figure 16 gives some idea of what is required:



2000 2005 2010Time (Year)

Cum

ulat

ive

Oil

Rec

over

y

Most probable case

Range of cases witha 50% probability

Range of cases witha 90% probability

The results in Figure 16 can be understood qualitatively without worrying about how we actually obtain them right now. Our single curve in Figure 15 may becomes a “most probable” (or “base case”) future oil recovery forecast. The closer set of outer curves is the range of future outcomes that can be expected with a 50% probability. That is, there is a probability, p = 0.5, that the true curve lives within this envelope of curves shown in Figure 16. Such results allow economic forecasts to be made with the appropriate weights being given to the likelihood of that particular outcome. A company can then estimate its risk when it is considering various field development options.

In fact, here we will just discuss doing some simple sensitivities to various factors in the simulation model. We can think of a given calculation as a scenario. Therefore, we can set up various scenarios based on our beliefs about the various input values in our model and we simply compare the recovery curves for each of the cases. For example, suppose we have a layered reservoir as shown in Figure 17 which we think has a field-wide high permeability streak set in background of 100 mD rock.

1000ft

1000ft

100ft

PRODUCER

INJECTOR

High Permeability Streak,khigh, φhi

∆Zhi

φ = 0.18

klow = 100mD

The reservoir is being developed by a five-spot waterflood as shown in Figure 17 However, we are uncertain as to the actual thickness (ΔZhi), the permeability (khi) and the porosity (φhi) of the high permeability streak. From various sources, we derive mean, high and low estimates of each of these quantities as follows:

Figure 17This shows a layered reservoir where we have some uncertainties in the various parameters such as the permeability (khi ), the thickness (ΔZhi ) and the porosity (φhi ) in the high permeability layer.

Figure 16Outcome of reservoir simulation calculations showing a range of recoveries for various reservoir development scenarios.



Low Value Mean Value High Value

Permeability, khi 400 mD 800 mD 1600 mDThickness, ∆Zhi 20 ft 30 ft 40 ftPorosity, φhi 0.18 0.22 0.26

Even with just the three uncertainties in this single model, we can see that there are 3x3x3 = 27 possible scenarios or combinations of input data for which we could run a reservoir simulation model. Alternatively, we could conclude that some input combinations are unlikely (e.g. lower permeability with higher value of porosity) and we could reduce the number. We could simply keep the mean value of two of the factors while varying only the third factor, leading to 7 scenarios to simulate. Taking this view, we can take some measure of the oil recovery e.g. cumulative oil produced (predicted) at year 2010. The notional results could be entered in Table 2

Changed Input Oil Initially in Cumulative % Change in Change in Recovery

Value Place (OIIP) Recovery at Input Value Relative from Base Case

(res. bbl) Year 2010 (stb) to Base Case

Base Casekhi = 400 mDkhi = 1600 mD∆Zhi = 20 ftZhi = 40 ftφhi = 0.18φhi = 0.25

Note: The OIIP will vary somewhat from case to case since the thickness of the high permeability layer and its porosity both change.

In Table 2, we have noted the % change in the varied parameters relative to its base case value. Not that different physical quantities such as k and φ, vary by different percentages for realistic min./max. values. A useful way to plot the variation in recovery is against this % change in input value since all three factors can be represented on the same scale in a so-called “spider diagram”. Such a plot is shown in Figure 18.

X

X Layer Thickness

PorosityPermeability

% Change in Parameter

Change in Recovery From Base Case (STB)

Figure 18“Spider diagram” showing the sensitivity of the cumulative oil to various uncertainties in the reservoir model parameters (khi; ΔZhi; φhi) in Figure 17.

Table 2Results of sensitivity simulations described in the text.



This type of spider plot is very useful since it displays the effect of the different uncertainties on the outcome. It clearly highlights which is the most important input quantity (of those considered) and has the most impact on the result. Thus, if we were going to spend time and effort on reducing the uncertainty in our predictions, then this tells us which quantity to focus on first. Indeed it ranks the effects of the various uncertainties.

There are more sophisticated ways to deal with uncertainty in reservoir performance but these are beyond the scope of the current course. The basic ideas presented above give you enough to go on with in this course.


The examples presented in the above SPE papers should give you a good idea of what reservoir simulation is all about. By this point you should also be familiar with many of the basic concepts that are involved in reservoir simulation from the study notes and the Glossary. However, the reservoir simulation of say a waterflood or a gas displacement is a dynamic process. That is, as time progresses, the water or gas front should be moving through the reservoir interacting with the underlying geological structure in quite a complicated manner. The pressure distribution through the system should also be evolving with time. For example, the water may possibly be advancing preferentially through a high permeability channel between an injector and producer pair. The sequence of the saturation fronts in the series of “snapshots in time” shown in Figure 19 give some idea of the progression of the flood with time. However, this can better be illustrated by looking at an animated sequence of saturation distributions as the flood front moves through the reservoir. An example of such a sequence is shown in file Res_Sim_D1.ppt This can be run on your PC from the CD supplied with this course and double clicking on the PowerPoint presentation.



Figure 19The sequence of saturation distributions as the flood front moves through the reservoir. From Res_Sim_D1.ppt Down arrow injector, up arrow producer.



5 TYPES OF RESERVOIR SIMULATION MODEL

Until now, we have confined our discussion to relative simple reservoir recovery processes such as natural depletion (blowdown) and waterflooding. However, there are many more complex reservoir recovery processes which can also be carried out. Dry gas (methane, CH4) injection, for example, would generally result in the flow of three phases (gas, oil and water) in the reservoir which is more complicated than two phase flow. Another process is where we alternately inject water and gas in repeating sequence - this is water-alternating-gas or WAG flooding. If the injected gas was carbon dioxide (CO2), then quite complex phase behaviour may occur and this requires some particular steps to be taken in order to model this. More exotic Improved Oil Recovery (IOR) processes can also be carried out where we inject chemicals (polymers, surfactants, alkali or foams) into the reservoir to recover oil that is left behind by primary and secondary oil recovery processes.

5.1 The Black Oil ModelDifferent types of simulator are available to model these different types of reservoir recovery process. Throughout the chapters of this course we will focus on the simplest of these (which is quite complex enough!) known as the "Black Oil Model". However, for completeness, we will also list the others and present a table comparing experience of these various models.

The Black Oil Model: This model was used in the three SPE field case studies above and is the most commonly used formulation of the reservoir simulation equations which is used for single, two and three phase reservoir processes. It treats the three phases - oil, gas and water - as if they were mass components where only the gas is allowed to dissolve in the oil and water. This gas solubility is described in oil and water by the gas solubility factors (or solution gas-oil ratios), Rso and Rsw, respectively; typical field units of Rso and Rsw are SCF/STB. These quantities are pressure dependent and this is incorporated into the black oil model.

A simple schematic of a grid block in a black oil simulator is presented in Figure 20 showing the amounts of mass of oil, water and gas present. Note that, because the gas is present in the oil and water there are extra terms in the expression for the mass of gas. These mathematical expressions for the mass of the various phases are important when we come to deriving the flow equations (Chapter 5).

Reservoir processes that can be modelled using the black oil model include:

• Recovery by fluid expansion - solution gas drive (primary depletion).

• Waterflooding including viscous, capillary and gravity forces (secondary recovery).

• Immiscible gas injection.

• Some three phase recovery processes such as immiscible water-alternating- gas (WAG).

• Capillary imbibition processes.



Rock

Gas, Sg

Oil + Gas, So

Water + Gas, Sw

Mass oil = So;Vp.ρosc

BoMass water = Sw

Vp.ρwscBw

Mass gas = (Sg + So.Rso + Sw.Rsw)Vp.ρgsc

Bg

free gas

gas in oil

gas in water

5.2 More Complex Reservoir Simulation ModelsThe Compositional Model: A compositional reservoir simulation model is required when significant inter-phase mass transfer effects occur in the fluid displacement process. It can be considered as a generalisation of the black oil model. This model usually defines three phases (again gas, oil and water) but the actual compositions of the oil and gas phases are explicitly acknowledged due to their more complicated PVT behaviour. That is, the separate components (C1, C2, C3, etc.) in the oil and gas phases are explicitly tracked as is indicated in Figure 21 (which should be compared with Figure 20). The mass conservation is applied to each component rather than just to “oil”, “gas” and “water” as in the black oil model. For example, in a near-critical fluid where small changes in say pressure can result in large compositional changes of the “oil” and “gas” phases which, in turn, strongly affects their physical properties (viscosity, density, interfacial tensions etc.).

Examples of reservoir processes that can be modelled using a compositional model include:

• Gas injection with oil mobilisation by first contact or developed (multi- contact) miscibility (e.g. in CO2 flooding).

• The modelling of gas injection into near critical reservoirs.

• Gas recycling processes in condensate reservoirs.

Figure 20Schematic of a grid block in a black oil simulator showing the amounts of mass of oil, water and gas present. Note that, because the gas is present in the oil and water there are extra terms for the mass of gas; pore volume = Vp = block vol. x φ; ρosc, ρwsc. and ρgsc are densities at standard conditions (60°F and 14.7 psi); Bo, Bw and Bg are the formation volume factors; Rso and Rsw are the gas solubilities (or solution gas/oil ratios).



ROCK

GASSg

OIL + GASSo

WATER + GASSw

Gas: C11, C21, C31....j = 1 = Gas

Oil: C12, C22, C32....

Water: C13, C23, C33....

j = 2 = Oilj = 3 = Water

Component concentrationsin each phase:

Phase Labels:

Mass of component in block = Vp. Σ Sj.Cij

3

j=1

The Chemical Flood Model: This model has been developed primarily to model polymer and surfactant (or combined) displacement processes. Polymer flooding can be considered mainly as extended waterflooding with some additional effects in the aqueous phase which must be modelling e.g. polymer component transport, the viscosification of the aqueous phase, polymer adsorption, permeability reduction etc. Surfactant, flooding however, involves strong phase behaviour effects where third phases may appear which contain oil/water/surfactant emulsions. Specialised phase packages have been developed to model such processes. For economic reasons, activity on field polymer flooding has continued at a fairly low level world wide and surfactant flooding has virtually ceased in recent years. However, if economic factors were favourable (a very high oil price), then interest in these processes may revive. Extended chemical flood models are also used to model foam flooding.

Examples of reservoir processes that can be modelled using a chemical flood model include:

• Polymer flooding which can be thought of as an “enhanced waterflood” to improve the mobility ratio and hence improve the microscopic sweep efficiency and also to reduce streaking in highly heterogeneous layered systems;

• Polymer/surfactant flooding where the main purpose of the surfactant is to lower interfacial tension (IFT) between the oil and water phases and hence to “release” or “mobilise” trapped residual oil; the polymer is for mobility control behind the surfactant slug;

• Low-tension polymer flooding (LTPF) where a more viscous polymer containing injected solution also contains some surfactant to reduce IFT; the combined effect of the lower IFT and viscous drive fluid improves the sweep and also helps to mobilise some of the residual oil;

• Alkali flooding where a solution of sodium hydroxide is injected into the formation. The sodium hydroxide may react with certain conponents in the oil to produce natural "soaps" which lower IFT and which may help to mobilise some of the residual oil;

Figure 21The view of phases and components taken in compositional simulation. Cij - is the mass concentration of component i in phase j (j = gas, oil or water) - dimensions of mass/unit volume of phase; pore volume = Vp = block vol. x φ



• Foam flooding where a surfactant is added during gas injection to form a foam which has a high effective viscosity (lower mobility) in the formation than the gas alone which may then displace oil more efficiently.

Another near-wellbore process that can be modelled using such simulators in water shut-off using either polymer-crosslinked gels or so-called “relative permeabilty modifiers”.

Thermal Models: In all thermal models heat is added to the reservoir either by injecting steam or by actually combusting the oil (by air injection, for example). The purpose of this is generally to reduce the viscosity of a heavy oil which may have μo of order 100s or 1000s of cP. The heat may be supplied to the reservoir by injected steam produced using a steam generator on the surface or downhole. Alternatively, an actual combustion process may be initiated in the reservoir - in-situ combustion - where part of the oil is burned to produce heat and combustion gases that help to drive the (unburned) oil from the system.

Examples of reservoir processes that can be modelled using thermal models include:

• Steam “soaks” where steam in injected into the formation, the well is shut in for a time to allow heat dissipation into the oil and then the well is back produced to obtain the mobilised oil (because of lower viscosity). This is known as a “Huff n Puff” process.

• Steam “drive” where the steam is injected continuously into the formation from an injector to the producer. Again, the objective is to lower oil viscosity by the penetration of the heat front deep into the reservoir.

• In situ combustion where - as noted above - an actual combustion process is initiated in the reservoir by injecting oxygen or air. Part of the oil is burned (oxidised) to produce heat and combustion gases that help to drive the (unburned) oil from the system. This is not a common improved oil recovery method but a number of field cases showing at least technical success have been reported in the SPE literature.

The above more complex reservoir simulation models are really based on the fluid flow process. However, there are also other types of simulator that are more closely defined by their treatment of the rock structure or the rock response. These include:

Dual-Porosity Models of Fractured Systems: These models have been designed explicitly to simulate multiphase flow in fractured systems where the oil mainly flows in fractures but is stored mainly in the rock matrix. Such models attempt to model the fracture flows (and sometimes the matrix flows) and the exchange of fluids between the fractures and the rock matrix. They have been applied to model recovery processes in massively fractured carbonate reservoir such as those found in many parts of the Middle East and elsewhere in the world. There is quite considerable field experience of modelling such systems in certain companies but there are also doubts over the validity of such models to model flow in fractured systems.



Coupled Hydraulic, Thermal Fracturing and Fluid Flow Models: These simulators are still essentially at the research stage although there have been published examples of specific field applications. The main function of these is to model the mechanical stresses and resulting deformations and the effects of these on fluid flow. This is beyond the scope of this course although, in the future, these will be important in many systems.

5.3 Comparison of Field Experience with Various Simulation ModelsWe now consider what field experience exists in the oil industry with the various models from the black oil model through to more complex fracture models and in situ combustion models etc. The vast majority of simulation studies which are carried out involve the black oil model. However, there are pockets of expertise with the various other types of simulation model, depending on the asset base of the particular oil company or regional expertise within regional consultancy groups. For example, there is (or until recently, was) a concentration of expertise in both California and parts of Canada on steam flooding since this process is applied in these regions to recover heavy oil; in the Middle East (and within the companies that operate there) there is great competence in the dual-porosity simulation of fractured carbonate reservoirs.

Black Oil Model

Compositional Model

Compositional Model- Near Crit.

Chemical Model- Polymer

Chemical Model- Surfactant

Thermal Model- Steam

Thermal Model - In SituCombustion

• Primary depletion • Waterflooding• Immiscible gas injection• Imbibition

• Huge• But there are still challenges with upscaling of large models• >90% of cases

Any of the books on reservoir simulation listed inSection 7 (Chapter 1)

• Gas injection • Gas recycling• CO2 injection• WAG

Coats, (1980a),Acs et al (1985),Nolen (1973),Watts (1986), Young and Stephenson(1983).

• Gas injection near crit.• Condensate development• MWAG

Difficult Very expensive(x5 - x30)

as above

• Polymer flooding• Near-well water shut-off

Not too difficult

Moderate (x2 - x5)

• Micellar flooding• Low tension polymer flooding

• Steam soak (Huff n’ Puff)• Steam flooding

• In situ combustion processes

Cheap = 1Routine

DifficultSpecialisd

DifficultSpecialisd

Not too difficult

Very difficultVery specialised

Expensive (x3 - x20)

Expensive(x3 - x10)

Expensive (x5 - x20)

Expensive(x10 - x40)

• Moderate• High in certain companies

• Low to moderate

• Moderate to large

• Low • Mainly “research type” pilot floods

• Moderate• High in limited geographical areas

• Very low

Bondor et al (1972),Vela et al (1976),Sorbie (1991)

Todd and Chase (1979), Todd et al (1978),Van Quy and Labrid (1983);Pope and Nelson (1978)

Coats (1978), Prats (1982), Mathews (1983)

Crookston et al (1979),Youngren (1980),Coats (1980b)

Simulator Type Processes Modelled Degree of Difficulty Relative Computing Amount of Industrial Example References2

Costs Experience

Table 3This is an adapted version of a table in Chapter 11 of Mattax and Dalton (1990). This gives some idea of the problems and issues encountered in applying advanced simulation models relative to applying a black oil simulator. The view about the difficulties and computer time consuming these are is somewhat subjective.




A full alphabetic list of References which are cited in the course is presented in Appendix A. Here, we briefly review some good texts which cover Reservoir Simulation from various viewpoints. The authors have learned something from each of these and we would recommend anyone who wishes to specialise in Reservoir Simulation to consult these.

Archer, J S and Wall, C: Petroleum Engineering: Principles and Practice, Graham and Trotman Inc., London, 1986.

This book is not a specialised reservoir simulation text. However, it offers a good overview of petroleum engineering and it contexts reservoir simulation very well within the overall picture of reservoir development. This book is also one of the earliest proponents of the importance of integrating the reservoir geology within the simulation model.

Aziz, K. and Settari, A.: Petroleum Reservoir Simulation, Elsevier Applied Science Publishers, Amsterdam, 1979. This is a classic text on the discretisation and numerical solution of the reservoir simulation flow equations. It is quite mathematical with a focus on the actual difference equations that arise from the flow equations and how to solve these.

Crichlow, H B: Modern Reservoir Engineering: A Simulation Approach, Prentice-Hall Inc., Englewood Cliffs, NJ, 1977. This book gives a fairly good introduction to reservoir simulation from the viewpoint of it being a central part of current reservoir engineering.

Dake, L P: The Practice of Reservoir Engineering, Developments in Petroleum Science 36, Elsevier, 1994. Again, this book is not about reservoir simulation but it makes a number of interesting and controversial observations on reservoir simulation (not all of which the authors agree with!). An interesting lengthy quote from this book on the relationship between material balance and reservoir simulation is reproduced in Chapter 2.

Fanchi, J R: Principles of Applied Reservoir Simulation, Gulf Publishing Co., Houston, TX, 1997. This recent book provides a good elementary text on reservoir simulation. It is a based around the BOAST4D black oil simulation model which is supplied on disk and can be run on your PC. The software makes this a very attractive way to familiarise yourself with reservoir simulation if you donʼt have ready access to a simulator.

Mattax, C C and Dalton, R L: Reservoir Simulation, SPE Monograph, Vol. 13, 1990. This is an excellent SPE monograph which covers virtually every aspect of traditional reservoir simulation. It is has been put together by a team of Exxon reservoir engineers between them have vast experience of all areas of reservoir simulation.

Peaceman, D W: Fundamentals of Numerical Reservoir Simulation, Developments in Petroleum Science No. 6, Elsevier, 1977.



This book presents an excellent treatment of the mathematical and numerical aspects of reservoir simulation. It discusses the discretisation of the flow equations and the subsequent numerical methods of solution in great detail. SPE Reprint No. 11, Numerical Simulation I (1973) and SPE Reprint No. 20, Numerical Simulation II (19**).

These two collections present some of the classic SPE papers on reservoir simulation. All aspects of reservoir simulation are covered including numerical methods, solution of linear equations, the modelling of wells and field applications. Most of this material is too advanced or detailed for a newcomer to this field but the volumes contain excellent reference material. They are also relatively cheap!

Thomas, G W: Principles of Hydrocarbon Reservoir Simulation, IHRDC, Boston, 1982. This short volume is written - according to Thomas - from a developers viewpoint; i.e. someone who is involved with writing and supplying the simulators themselves. The treatment is quite mathematical with quite a lot of coverage of numerical methods. The treatment of some areas is rather brief; for example, there are only 7 pages on wells.

APPENDIX A: REFERENCES

NOTE: SPEJ = Society of Petroleum Engineers Journal - there was an early version of this and it stopped for a while. Currently, there are SPE Journals in various subjects but reservoir simulation R&D appears in SPE (Reservoir Engineering and Evaluation).

Acs, G., Doleschall, S. and Farkas, E., “General Purpose Compositional Model”, SPEJ, pp. 543 - 553, August 1985.

Allen, M.B., Behie, G.A. and Trangenstein, J.A.: Multiphase Flow in Porous Media: Mechanics, Mathematics and Numerics, Lecture Notes in Engineering No. 34, Springer-Verlag, 1988.

Amyx, J W, Bass, D M and Whiting, R L: Petroleum Reservoir Engineering, McGraw-Hill, 1960.

Appleyard, J.R. and Cheshire, I.M.: “Nested Factorization,” paper SPE 12264 presented at the Seventh SPE Symposium on Reservoir Simulation, San Francisco, CA, November 16-18, 1983.

Archer, J S and Wall, C: Petroleum Engineering: Principles and Practice, Graham and Trotman Inc., London, 1986.

Aronofsky, J.S. and Jenkins, R.: “A Simplified Analysis of Unsteady Radial Gas Flow,” Trans., AIME 201 (1954) 149-154

Aziz, K. and Settari, A.: Petroleum Reservoir Simulation, Elsevier Applied Science Publishers, Amsterdam, 1979.



Bondor, P.L., Hirasaki, G.J and Tham, M.J., “Mathematical Simulation of Polymer Flooding in Complex Reservoirs”, SPEJ, pp. 369-382, October 1972.

Clayton, C.A., et al, “The Ubit Field Rejuvenation: A Case History of Reservoir Management of a Giant Oilfield Offshore Nigeria”, SPE49165, presented at the SPE Annual Technical Conference and Exhibition, New Orleans, LA, 27-30 September 1998.

Coats, K.H., .......... 1969 - tools of res sim

Coats, K.H., “A Highly Implicit Steamflood Model”, SPEJ, pp. 369-383, October 1978.

Coats, K.H., “An Equation of State Compositional Model”, SPEJ, pp. 363-376, October 1980a; Trans. AIME, 269.

Coats, K.H., “ In-Situ Combustion Model”, SPEJ, pp. 533-554, December 1980b; Trans. AIME 269.

Coats, K.H., Dempsey, J.R., and Henderson, J.H.: “The Use of Vertical Equilibrium in Two-Dimensional Simulation of Three-Dimensional Reservoir Performance,” Soc. Pet. Eng. J. 11 (March 1971) 63-71; Trans., AIME 251

Craft, B C, Hawkins, M F and Terry, R E: Applied Petroleum Reservoir Engineering, Prentice Hall, NJ, 1991.

Craig, F F: The Reservoir Engineering Aspects of Waterflooding, SPE monograph, Dallas, TX, 1979.

Crichlow, H B: Modern Reservoir Engineering: A Simulation Approach, Prentice-Hall Inc., Englewood Cliffs, NJ, 1977.

Crookston, R.B., Culham, W.E. and Chen, W.H., “A Numerical Simulation Model for Thermal Recovery Processes”, SPEJ, pp. 35-57, February 1979; Trans. AIME 267.

Dake, L P: The Fundamentals of Reservoir Engineering, Developments in Petroleum Science 8, Elsevier, 1978.

Dake, L P: The Practice of Reservoir Engineering, Developments in Petroleum Science 36, Elsevier, 1994.

Fanchi, J R: Principles of Applied Reservoir Simulation, Gulf Publishing Co., Houston, TX, 1997.

Fantoft, S., “Reservoir Management of the Oseberg Field After Four Years”, SPE25008, proceedings of the SPE European Petroleum Conference, Cannes, France, 16-18 November 1992.

Giudicelli, C.S., Massonat, G.J. and Alabert, F.G., “Anguille Marine, a Deepse-Fan Reservoir Offshore Gabon: From Geology Toward History Matching Through



Stochastic Modelling”, SPE25006, proceedings of the SPE European Petroleum Conference, Cannes, France, 16-18 November 1992.

Harpole, K.J. and Hearn, C.L., “The Role of Numerical Simulation in Reservoir Management of a West Texas Carbonate Reservoir”, SPE10022, presented at the International Petroleum Exhibition and Technical Symposium of the SPE, Beijing, China, 18 - 26 March 1982.

Heinemann, Z.E., Brand, C.W., Munka, M., and Chen, Y.M.: “Modeling Reservoir Geometry with Irregular Grids,” SPERE 6 (1991) 225-232.

Hove, K., Olsen, G., Nilsson, S., Tonnesen, M. and Hatloy, A., “From Stochastic Geological Description to Production Forecasting in Heterogeneous Layered Systems”, SPE24890, the proceedings of the SPE 67th Annual Technical Conference, Washington, DC, 4-7 October 1992.

Katz, D.L., “Methods of Estimating Oil and Gas Reserves”, Trans. AIME, Vol. 118, p.18, 1936 (classic early ref. on Material Balance)Kyte, J.R. and Berry, D.W.: “New Pseudo Functions to Control Numerical Dispersion,” Soc .Pet. Eng. J. 15 (August 1975) 269-276.

Lantz, R.B.: “Quantitative Evaluation of Numerical Diffusion (Truncation Error),” Soc .Pet. Eng. J. 11 (September 1971) 315-320; Trans., AIME 251.

Leonard, A.J., Duncan, A.E., Johnson, D.A. and Murray, R.B., SPE25059: “Development Planning in a Complex Reservoir: Magnus Field UKCS Lower Kimmeridge Clay Formation (LKCF)”, SPE25059, proceedings of the SPE European Petroleum Conference, Cannes, France, 16-18 November 1992.

Mathews, C.W., “Steamflooding”, J. Pet. Tech., pp. 465-471, March 1983; Trans. AIME 275.

MacDonald, R.C. and Coats, K.H.: “Methods for Numerical Simulation of Water and Gas Coning,” Soc. Pet. Eng. J. 10 (December 1970) 425-436; Trans., AIME 249.

Mattax, C C and Dalton, R L: Reservoir Simulation, SPE Monograph, Vol. 13, 1990.

Meijerink, J.A. and Van der Vorst, H.A.: “An Iterative Solution Method for Linear Systems of Which the Coefficient Matrix is a Symmetric M-Matrix,” Mathematics of Computation 31 (January 1977) 148.

Nolen, J.S., “Numerical Simulation of Compositional Phenomena in Petroleum Reservoirs”, SPE4274, proceedings of the SPE Symposium on Numerical Simulation of Reservoir Performance, Houston, TX, 11-12 January 1973.

Palagi, C.L. and Aziz, K.: “Use of Voronoi Grid in Reservoir Simulation,” SPE Advanced Technology Series 2 (April 1994) 69-77.

Peaceman, D.W.: “Interpretation of Well-Block Pressures in Numerical Reservoir



Simulation,” Soc. Pet. Eng. J. 18 (June 1978) 183-194; Trans., AIME 253.

Peaceman, D.W. and Rachford, H.H.: “The Numerical Solution of Parabolic and Elliptic Differential Equations,” Soc Ind. Appl. Math. J. 3 (1955) 28-41

Peaceman, D W: Fundamentals of Numerical Reservoir Simulation, Developments in Petroleum Science No. 6, Elsevier, 1977.

Ponting, D.K.: “Corner point geometry in reservoir simulation,” in The Mathematics of Oil Recovery - Edited proceedings of an IMA/SPE Conference, Robinson College, Cambridge, July 1989; Edited by P.R. King, Clarendon Press, Oxford, 1992.

Pope, G.A. and Nelson, R.C., “A Chemical Flooding Compositional Simulator”, SPEF, pp.339-354, October 1978.

Prats, M., “Thermal Recovery” SPE Monograph Series No. 7, SPE Richardson, TX, 1982.Price, H.S. and Coats, K.H.: “Direct Methods in Reservoir Simulation,” Soc. Pet. Eng. J. 14 (June 1974) 295-308; Trans., AIME 257

Robertson, G., in Cores from the Northwest European Hydrocarbon Provence, C D Oakman, J H Martin and P W M Corbett (eds.), Geological Society, London. 1997.

Schilthuis, R.J., “Active Oil and Reservoir Energy”, Trans. AIME, Vol. 118, p.3, 1936; (original ref. on Material Balance)

Sheldon, J.W., Harris, C.D., and Bavly, D.: “A Method for Generalized Reservoir Behavior Simulation on Digital Computers,” SPE 1521-G presented at the 35th Annual SPE Fall Meeting, Denver, Colorado, October 1960.

Sibley, M.J., Bent, J.V. and Davis, D.W., “Reservoir Modelling and Simulation of a Middle Eastern Carbonate Reservoir”, SPE36540, proceedings of the SPE 71st Annual Conference and Exhibition, Denver, CO, 6-9 October 1996.

Sorbie, K.S., “Polymer Improved Oil Recovery”, Blakie and SOns & CRC Press, 1991.

SPE Reprint No. 11, Numerical Simulation I (1973) and SPE Reprint No. 20, Numerical Simulation II (19**).

Spillette, A.G., Hillestad, J.H., and Stone, H.L.: “A High-Stability Sequential Solution Approach to Reservoir Simulation,” SPE 4542 presented at the 48th Annual Fall Meeting of the Society of Petroleum Engineers of AIME, Les Vegas, Nevada, September 30-October 3, 1973.

Stone, H.L.: “Iterative Solution of Implicit Approximations of Multidimensional Partial Differential Equations,” SIAM J. Numer.Anal. 5 (September 1968) 530-558

Stone, H.L.: “Probability Model for Estimating Three-Phase Relative Permeability,” J. Pet. Tech. 24 (February 1970) 214-218; Trans., AIME 249.



Stone, H.L.: “Estimation of Three-Phase Relative Permeability and Residual Oil Data,” J. Can. Pet. Tech. 12 (October-December 1973) 53-61.

Stone, H.L. and Garder, Jr., A.O.: “Analysis of Gas-Cap or Dissolved-Gas Drive Reservoirs,” Soc .Pet. Eng. J. 1 (June 1961) 92-104; Trans., AIME 222.

Thomas G.W. and Thurnau, D.H.: “Reservoir Simulation Using an Adaptive Implicit Method,” Soc. Pet. Eng.. J. 23 (October 1983) 759-768.

Thomas L.K., Lumpkin, W.B., and Reheis, G.M.: “Reservoir Simulation of Variable Bubble-point Problems,” Soc. Pet. Eng. J. 16 (February 1976) 10-16; Trans., AIME 261.

Todd, M.R. and Longstaff, W.J.: “The Development, Testing, and Application of a Numerical Simulator for Predicting Miscible Flood Performance, “ J. Pet. Tech. 24 (July 1972) 874-882; Trans., AIME 253.

Todd, M.R., OʼDell, P.M., and Hirasaki, G.J.: “Methods for Increased Accuracy in Numerical Reservoir Simulators,” Soc. Pet. Eng. J. 12 (December 1972) 515-530.

Thomas, G W: Principles of Hydrocarbon Reservoir Simulation, IHRDC, Boston, 1982.

Todd, M.R. and Chase, C.A., “A Numerical Simulator for Predicting Chemical Flood Performance”, SPE7689, proceedings of the SPE Symposium on Reservoir Simulation, Denver, CO, 1-2 February 1979.

Todd, M.R. et al , “Numerical Simulation of Competing Chemical Flood Designs”, SPE7077, proceedings of the SPE Symposium on Improved Methods for Oil Recovery, Tulsa, OK, 16-19 April 1978.

Uren, L.C., Petroleum Production Engineering, Oil Field Exploitation, 3rd edn., McGraw-Hill Book Company Inc., New York, 1953.

Van Quy, N. and Labrid, J., “A Numerical Study of Chemical Flooding - Comparison with Experiments”, SPEJ, pp.461-474, June 1983; Trans. AIME 275.

van Vark, W., Paardekam, A.H.M., Brint, J.F., van Lieshout, J.B. and George, P.M., “The Construction and Validation of a Numerical Model of a Reservoir Consisting of Meandering Channels”, SPE25057, proceedings of the SPE European Petroleum Conference, Cannes, France, 16-18 November 1992.

Vela, S., Peaceman, D.W. and Sandvik, E.I., “Evaluation of Polymer Flooding in a Layered Reservoir with Crossflow, Retention and Degradation”, SPEJ, pp. 82-96, April 1976.

Vinsome, P.K.W.: “Orthomin, an Iterative Method for Solving Sparse Banded Sets of Simultaneous Linear Equations,” paper SPE 5729 presented at the Fourth SPE Symposium on Reservoir Simulation, Los Angeles, February 19-20, 1976.



Wallis, J.R., Kendall, R.P., and Little, T.E.: “Constrained Residual Acceleration of Conjugate Residual Methods,” SPE 13536 presented at the Eighth SPE Reservoir Simulation Symposium, Dallas, Texas, February 10-13, 1985.

Watts, J.W.: “An Iterative Matrix Solution Method Suitable for Anisotropic Problems,” Soc Pet. Eng .J. 11 (March 1971) 47-51; Trans., AIME 251.

Watts, J.W., “A Compositional Formulation of the Pressure and Saturation Equations”, SPE (Reservoir Engineering), pp. 243 - 252, March 1986.

Watts, J.W., “Reservoir Simulation: Past, Present and Future”, SPE Reservoir Simulation Symposium, Dallas, TX, 5-7 June 1997.

Yanosik, J.L. and McCracken, T.A.: “A Nine-Point Finite Difference Reservoir Simulator for Realistic Prediction of Unfavorable Mobility Ratio Displacements,” Soc. Pet. Eng. J. 19 (August 1979) 253-262; Trans., AIME 267.

Young, L.C. and Stephenson, R.E., “A Generalised Compositional Approach for Reservoir Simulation”, SPEJ, pp. 727-742, October 1983; Trans. AIME 275.

Youngren, G.K., “Development and Application of an In-Situ Combustion Reservoir Simulator”, SPEJ, pp. 39-51, February 1980; Trans. AIME 269.



This one pager summarises a panel discussion that was held in London in 1990. Given the brevity of the article, it is packed with some genuine wisdom - and some things to disagree with - from a really excellent group of front line “users” of the technology. Briggs comes closest to capturing the principal authors particular prejudices!


This short viewpoint from a Canadian independent consultant is interesting since it contexts reservoir simulation in the current “outsourced” and “downsized” oil industry. He notes that virtually everyone can have PC based powerful simulation technology on their desk tops. However, he concludes that the overall demand for simulation will rise and that, for the sake of efficiency, this will be performed by specialists. At the same time, he promotes a teamwork environment for the simulation engineer where he or she will be involved in the preceding reservoir characterisation process and the subsequent decision making process. This is a well argued position but not all of his conclusion would be generally accepted.




This very interesting pair of articles gives a very good broad brush commentary on a range of technical issues in reservoir simulation e.g. gridding, handling wells, pseudo-relative permeability, error analysis and “consistency checking”. The views are clearly those of someone who has been deeply involved in applied reservoir simulation. They are well presented and quite individual although again there are issues that would provoke disagreement. Read this and decide for yourself what you accept and what you donʼt.

22Basic Concepts in Reservoir Engineering

1. INTRODUCTION

2. MATERIAL BALANCE 2.1 Introduction to Material Balance (MB) 2.2 Derivation of Simplified Material Balance Equations 2.3 Conditions for the Validity of Material Balance

3. SINGLE PHASE DARCY LAW 3.1 The Basic Darcy Experiment 3.2 Mathematical Note: on the Operators “gradient”

∇

∇

.

and “divergence” ∇

∇

. 3.3 Darcy’s Law in 3D - Using Vector and Tensor Notation 3.4 Simple Darcy Law with Gravity 3.5 The Radial Darcy Law

4. TWO-PHASE FLOW 4.1 The Two-Phase Darcy Law

5. CLOSING REMARKS

6. SOME FURTHER READING ON RESERVOIR ENGINEERING





• be familiar with the meaning and use of all the usual terms which appear in reservoir engineering such as, Sw, So, Bo, Bw, Bg, Rso, Rsw, cw, co, cf, kro, krw, Pc etc.

• be able to explain the differences between material balance and reservoir simulation.

• be aware of and be able to describe where it is more appropriate to use material balance and where it is more appropriate to use reservoir simulation.

• be able to use a simple given material balance equation for an undersaturated oil reservoir (with no influx or production of water) in order to find the STOOIP.

• know the conditions under which the material balance equations are valid.

• be able to write down the single and two-phase Darcy Law in one dimension (1D) and be able to explain all the terms which occur (no units conversion factors need to be remembered).

• be aware of the gradient (∇) and divergence (∇.) operators as they apply to the generalised (2D and 3D) Darcy Law (but these should not be committed to memory).

• know that pressure is a scalar and that the pressure distribution, P(x, y, z) is a scalar field; but that ∇P is a vector.

• know that permeability is really a tensor quantity with some notion of what this means physically (more in Chapter 7).

• be able to write out the 2D and 3D Darcy Law with permeability as a full tensor and know how this gives the more familiar Darcy Law in x, y and z directions when the tensor is diagonal (but where we may have kx ≠ ky ≠ kz).

• be able to write down and explain the radial Darcy Law and know that the pressure profile near the well, ΔP(r), varies logarithmically.



REVIEW OF BASIC CONCEPTS IN RESERVOIR ENGINEERING

Brief Description of Chapter 2This module reviews some basic concepts of reservoir engineering that must be familiar to the simulation engineer and which s/he should have covered already. We start with Material Balance and the definition of the quantities which are necessary to carry out such calculations: φ, co , cf , Bo , Swi etc. This is illustrated by a simple calculator exercise which is to be carried out by the student. The same exercise is then repeated on the reservoir simulator. Alternative approaches to material balance are discussed briefly. The respective roles of Material Balance and Reservoir Simulation are compared.

The unit then goes on to consider basic reservoir engineering associated with fluid flow: the single phase Darcy law (k), tensor permeabilities, k , two phase Darcy Law - relative permeabilitites (kro , krw) and capillary pressures (Pc).

Note that many of the terms and concepts reviewed in this section are summarised in the Glossary at the front of this chapter.

1. INTRODUCTIONIt is likely that you will have completed the introductory Reservoir Engineering part of this Course. You should therefore be fairly familiar with the concepts reviewed in this section. The purpose of doing any review of basic reservoir engineering is as follows:

(i) Between them, the review in this section and the Glossary make this course more self-contained, with all the main concepts we need close at hand;

(ii) This allows us to emphasise the complementary nature of “conventional” reservoir engineering and reservoir simulation;

(iii) We would like to review some of the flow concepts (Darcy’s law etc.), in a manner of particular use for the derivation of the flow equations later in this course (in Chapter 5).

An example of point (ii) above concerns the complementary nature of Material Balance (MB) and numerical reservoir simulation. At times, these have been presented as almost opposing approaches to reservoir engineering. Nothing could be further from the truth and this will be discussed in detail below. Indeed, a MB calculation will be done by the student and the same calculation will be performed using the reservoir simulator.

In addition to an introductory review of simple material balance calculations, we will also go over some of the basic concepts of flow through porous media. These flow concepts will be of direct use in deriving the reservoir simulation flow equations in Chapter 5. Again, most of the concepts are summarised in the Glossary.

Exercises are provided at the end of this module which the student must carry out.



The following concepts are defined in the Glossary and should be familiar to you: viscosity (μo, μw, μg), density (ρo, ρw, ρg), phase saturations (So, Sw and Sg), initial or connate water saturation (Swi or Swc), residual oil saturation (Sor). In addition, you should also be familiar with the basic reservoir engineering quantities in Table 1 below:

Bg Bo

Bw

Pb

FVF

P

Pb

Rso

Rso

Rso

Bo

P

STC = Stock Tank Conditions (60°F; 14.7 psi).Likewise for water (usually const.) and gas; Pb= bubble point pressure below.

Symbol Name Field Units Meaning / Formulae

Bo, Bw, BgFormation volume bbl/STBfactors (FVF) for oil, or RB/STBwater and gas

=

=

= = -

Rso, Rsw Gas solubility factors SCF/STB or solution gas oil ratios

co, cw, cg Isothermal fluid psi-1

compressibilities of oil water and gas

Vol. oil + dissolved gas in reservoirVol. oil at STC

Vol. dissolved gas in reservoirVol. gas at STC

ρk and Vk - density and volume of phase k;k = o, w, g

ck1ρk

∂ρk∂P

∂Vk∂P

1Vk

2. MATERIAL BALANCE

2.1 Introduction to Material Balance (MB)The concept of Material Balance (MB) has a central position in the early history of reservoir engineering. MB equations were originally derived by Schilthuis in 1936. There are several excellent accounts of the MB equations and their application to different reservoir situations in various textbooks (Amyx, Bass and Whiting, 1960; Craft, Hawkins and Terry, 1991; Dake, 1978, 1994). For this reason, and because this subject is covered in detail in the Reservoir Engineering course in this series, we only present a very simple case of the material balance equation in a saturated reservoir case. The full MB equation is presented in the Glossary for completeness. Our objectives in this context are as follows:

• To introduce the central idea of MB and apply it to a simple case which we will then set up as an exercise for simulation;

Table 1: Basic reservoir engineering quantities to revise



• To demonstrate the complementary nature of MB and reservoir simulation calculations.

Material balance has been used in the industry for the following main purposes:

1. Determining the initial hydrocarbon in place (e.g. STOIIP) by analysing mean reservoir pressure vs. production data;

2. Calculating water influx i.e. the degree to which a natural aquifer is supporting the production (and hence slowing down the pressure decline);

3. Predicting mean reservoir pressure in the future, if a good match of the early pressure decline is achieved and the correct reservoir recovery mechanism has been identified.

Thus, MB is principally a tool which, if it can be applied successfully, defines the input for a reservoir simulation model (from 1 and 2 above). Subsequently, the mean field pressure decline as calculated in 3 above can be compared with the predictions of the numerical reservoir simulation model.

Before deriving the restricted example of the MB equations, we quote the introduction of Dake’s (1994) chapter on material balance.

Material Balance Applied to Oilfields (from Chapter 3; L. P. Dake, The Practice of Reservoir Engineering, Developments in Petroleum Science 36, Elsevier, 1994.) Dake says:

It seems no longer fashionable to apply the concept of material balance to oilfields, the belief being that it has now been superseded by the application of the more modern technique of numerical reservoir simulation modelling. Acceptance of this idea has been a tragedy and has robbed engineers of their most powerful tool for investigating reservoirs and understanding their performance rather than imposing their wills upon them, as is often the case when applying numerical simulation directly in history matching.

As demonstrated in this chapter, by defining an average pressure decline trend for a reservoir, which is always possible, irrespective of any lack of pressure equilibrium, then material balance can be applied using simply the production and pressure histories together with the fluid PVT properties. No geometrical considerations (geological models) are involved, hence the material balance can be used to calculate the hydrocarbons in place and define the drive mechanisms. In this respect, it is the safest technique in the business since it is the minimum assumption route through reservoir engineering. Conversely, the mere act of construction of a simulation model, using the geological maps and petrophysically determined formation properties implies that the STOIIP is “known”. Therefore, history matching by simulation can hardly be regarded as an investigative technique but one that merely reflects the input assumptions of the engineer performing the study.



There should be no competition between material balance and simulation, instead they must be supportive of one another: the former defining the system which is then used as input to the model. Material balance is excellent at history matching production performance but has considerable disadvantages when it comes to prediction, which is the domain of numerical simulation modelling.

Because engineers have drifted away from oilfield material balance in recent years, the unfamiliarity breeds a lack of confidence in its meaningfulness and, indeed, how to use it properly. To counter this, the chapter provides a comprehensive description of various methods of application of the technique and included six fully worked exercises illustrating the history matching of oilfields. It is perhaps worth commenting that in none of these fields had the operators attempted to apply material balance, which denied them vital information concerning the basic understanding of the physics of reservoir performance.

Notes on Dake’s comments1. The authors of this Reservoir Simulation course would very much like to echo Dake’s sentiments. Performing large scale reservoir simulation studies does not replace doing good conventional reservoir engineering analysis - especially MB calculations. MB should always be carried out since, if you have enough data to build a reservoir simulation model, you certainly have enough to perform a MB calculation.

2. Note Dake’s comments on the complementary nature of MB in defining the input for reservoir simulation, as we discussed above.

3. Take careful note of Dake’s comment on where a reservoir simulation model is used for history matching. The very act of setting up the model means that you actually input the STOIIP, whereas, this should be one of the history matching parameters. The reservoir engineer can get around this to some extent by building a number of alternative models of the reservoir and this is sometimes, but not frequently, done.

2.2 Derivation of Simplified Material Balance EquationsMaterial balance (MB) is simply a volume balance on the changes that occur in the reservoir. The volume of the original reservoir is assumed to be fixed. If this is so, then the algebraic sum of all the volume changes in the reservoir of oil, free gas, water and rock, must be zero. Physically, if oil is produced, then the remaining oil, the other fluids and the rock must expand to fill the void space left by the produced oil. As a consequence, the reservoir pressure will drop although this can be balances if there is a water influx into the reservoir. The reservoir is assumed to be a “tank” - as shown in Figure 5 Chapter 1. The pressure is taken to be constant throughout this tank model and in all phases. Clearly, the system response depends on the compressibilities of the various fluids (co, cw and cg) and on the reservoir rock formation (crock). If there is a gas cap or production goes below the bubble point (Pb), then the highly compressible gas dominates the system response. Typical ranges of fluid and rock compressibilities are given in Table 2:



Fluid or formation Compressibility (10-6 psi-1)

Formation rock, crock 3 - 10Water, cw 2 - 4Undersaturated oil, co 5 - 100Gas at 1000psi, cg 900 - 1300Gas at 5000psi. cg 50 - 200

The simple example which we will take in order to demonstrate the main idea of material balance is shown in Figure 1 where the system is simply an undersaturated oil, with possible water influx.

Water influx Water influx We

Oil

N

NBoi = Vf.(1-Swi)

Water, Swi

W = Vf.Swi

Oil

(N - Np)Bo

NBoi = Vf.(1-Swi)

Water, Swi

W + We - Wp

Water, Wp

Oil, Np

Initial conditionspressure = po

After production (Np)pressure = p

Definitions: N = initial reservoir volume (STB) Boi = initial oil formation volume factor (bbl/STB or RB/STB) Np = cumulative produced oil at time t, pressure p (STB) Bo = oil formation volume factor at current t and p (bbl/STB) W = initial reservoir water (bbl) Wp = cumulative produced water (STB) Bw = water formation volume factor (bbl/STB) We = water influx into reservoir (bbl) cw = water isothermal compressibility (psi-1) Δ P = change in reservoir pressure, p - po Vf = initial void space (bbl); Vf = N.Boi/(1- Swi); W = Vf.Swi Swi = initial water saturation (of whole system)

c f = void space isothermal compressibility (psi-1); cV

V

pff

f=∂∂

1

(NB: (i) bbl = reservoir barrels, sometimes denoted RB; and (ii) in the figures above, the oil and water are effectively assumed to be uniformly distributed throughout the system)

Table 2: Typical rock and fluid compressibilities (from Craft, Hawkins and Terry, 1991)

Figure 1.Simplified system for material balance (MB) in a system with an undersaturated oil above the bubble point and possible water influx.



Definitions of the various quantities we need for our simplified MB equation for the depletion of an undersaturated oil reservoir above the bubble point (Pb) are given in Figure 1. (NB a more extensive list of quantities required for a full material balance equation in any type of oil or gas reservoir is given in the Glossary for completeness).

In going from initial reservoir conditions shown in Figure 1 at pressure, po, to pressure, p, volume changes in the oil, water and void space (rock) occur, ΔVo, ΔVw, ΔVvoid (ΔVvoid = - ΔVrock). The pressure drop is denoted, Δ P = p - po. The volume balance simply says that:

∆ ∆ ∆ ∆ ∆ ∆

∆ ∆

∆ ∆

∆

∆

∆ ∆

V V V V V V

W c p

W c p

V c p

V c p

o w rock o w void

w

w

f f

f f

+ + = + − = 0

V = W - (W - W B + W +

V = W B - W -

= V - (V - )

=

V = -

w p w e

w p w e

f f

rock

. .

. .

. .

. .

__

__

__

__

VV = - void V c pf f. .__

∆

(1)

Each of these volume changes can be calculated quite straightforwardly as follows:

Oil volume change, ΔVo

Initial oil volume in reservoir = N.Boi (bbl = RB)

Oil volume t time t, pressure p = (N - Np). Bo (bbl)

Change in oil volume, ΔVo = N.Boi - (N - Np). Bo (bbl) (1)

Water volume change, ΔVw

Initial reservoir water volume = W (bbl)

Cumulative water production at time = Wp (STB) Reservoir volume of cumulative water production at time = Wp.Bw (bbl)

Volume of water influx into reservoir = We (bbl)

Water volume change due to compressibility = W.cw. Δ P (bbl)

Change in water volume, ΔVw = W - (W - Wp Bw + We + W.cw. Δ P ) (bbl) ΔVw = Wp Bw - We - W.cw. Δ P (2)

Change in the void space volume, ΔVvoid

Initial void space volume = Vf (bbl)



Change in void space volume, ΔVvoid = Vf - (Vf - Vf.cf. Δ P )

= Vf.cf. Δ P

Change in rock volume, ΔVrock = - ΔVvoid = - Vf.cf. Δ P (3)

Now adding the volume changes as follows:

∆ ∆ ∆ ∆ ∆

∆

V V V N B N N B W B W W c P V c P

N B N B N B W W B N BS c cf

SP

N B N

o w rock oi p o p w e w f f

oi o p o e p w oiwi w

wi

oi

+ + = + − + − − − =

− + − + − +−

=

−

. ( ). . . . . .

. . . . ..

.

.

__ __

__

0

10

.. . ..

.

. . . ..

.

. . . ..

__

__

B N B N BS c c

SP

N B N B N B N BS c c

SP


o p o oiwi w f

wi

oi o p o oiwi w rock

wi


+ −+

−

=

− + + +−

=

− + + +

10

10

1

∆

∆

−−

=S

Pwi

.__

∆ 0

(4)

Rearranging equation 4 and noting that W = Vf.Swi and Vf = N.Boi/(1-Swi), we obtain:∆ ∆ ∆ ∆ ∆

∆



SP

N B N



wi

oi

+ + = + − + − − − =

− + − + − +−

=

−

. ( ). . . . . .

. . . . ..

.

.

__ __

__

0

10

.. . ..

.

. . . ..

.

. . . ..

__

__

B N B N BS c c

SP


SP


o p o oiwi w f

wi


wi


+ −+

−

=

− + + +−

=

− + + +

10

10

1

∆

∆

−−

=S

Pwi

.__

∆ 0

(5)

Equation 5 is the (simplified) material balance expression for the undersaturated system given in Figure 1 (as long as it remains above its bubble point).

To illustrate the use of material balance in an even simpler example, let us assume that there is no water influx (We =0) or production (Wp = 0). Therefore, the MB equation simplifies even further to:

∆ ∆ ∆ ∆ ∆

∆



SP

N B N



wi

oi

+ + = + − + − − − =

− + − + − +−

=

−

. ( ). . . . . .

. . . . ..

.

.

__ __

__

0

10

.. . ..

.

. . . ..

.

. . . ..

__

__

B N B N BS c c

SP


SP


o p o oiwi w f

wi


wi


+ −+

−

=

− + + +−

=

− + + +

10

10

1

∆

∆

−−

=S

Pwi

.__

∆ 0

(6)Note that we can divide through equation 6 by N (the initial reservoir oil volume, bbl = RB) to obtain:


SP

B BN

NB B

S c c

SP

N

NBB

BB

S c c


wi

oi op

o oiwi w f

wi

p oi

o

oi

o

wi w f

. . . ..

.

..

.

.

__

__

− + + +−

=

− + −+

−

=

= − ++

10

10

11

∆

∆

−−

=

=

=

= − = − ∂

∂

SP

p

uQA

k PL

k Px

wi

.

. .

__

__

∆

∆

∆

0

0

µ µ

(7)

which rearranges easily to:


SP

B BN

NB B

S c c

SP

N

NBB

BB

S c c


wi

oi op

o oiwi w f

wi

p oi

o

oi

o

wi w f

. . . ..

.

..

.

.

__

__

− + + +−

=

− + −+

−

=

= − ++

10

10

11

∆

∆

−−

=

=

=

= − = − ∂

∂

SP

p

uQA

k PL

k Px

wi

.

. .

__

__

∆

∆

∆

0

0

µ µ

(8)

where the quantity (Np/N) is the Recovery Factor (RF) as a fraction of the STOIIP. It is seen from equation 8 that, at t = 0, Bo = Boi and and therefore (Np/N)= 0, as expected. Note also in equation 8 that ∆P is negative in depletion ( ∆P = p-po, where po. is the higher initial pressure for depletion).



It is convenient to rearrange equation 8 above as follows:

1

1−

= −+

−

N

NBB

BB

S c c

SPp oi

o

oi

o

wi w f

wi

∆

(9)

We then identify 1-(Np/N) as the fraction of the initial oil still in place. We can then plot this quantity vs. - ∆P shown in Figure 2 (we take - ∆P since it plots along the positive axis, since ∆P is negative).

-∆P

1-

1

0

Np

N

"almost" straight line for w/o systems

As noted in Figure 2, this decline plot is not necessarily a straight line but for oil water systems, it is very close in practice. Figure 2 suggests a way of applying a simple material balance equation to the case of an undersaturated oil above the bubble point (with no water influx or production). This is a pure depletion problem driven by the oil (mainly), water and formation compressibilities. Suppose we know the pressure behaviour of B0 (i.e. B0(P)) as shown in Figure 3.

1.4

1.3

4000 P (psi) 5500

Oil FVFBo

Bo(P) = m.P + c

If we draw the reservoir pressure down by an amount ∆P (known or measured) and we know that to do this we had to produce a volume Np (STB) of oil. This point of depletion is shown in Figure 4.

-∆P

1-

1

0

Y

XNp

N

Figure 3B0 as a function of pressure for a black oil.

Figure 4Reservoir depletion on a plot following equation 9.

Figure 2Plot of remaining oil,

1 −

−N

Nvs Pp . ∆



We know Y ( it is ∆P ), we can calculate X (the RHS of equation 9). X is equal to 1-(Np/N) and we know Np (the amount of oil we had to produce to get drawdown ∆P ). Hence, we can find N the initial oil in place. An exercise to do this is given below.

2.3 Conditions for the Validity of Material BalanceThe basic premise for the material balance assumptions to be correct is that the reservoir be “tank like” i.e. the whole system is at the same pressure and, as the pressure falls, then the system equilibrates immediately. For this to be correct, the pressure communication through the system must at least be very fast in practice (rather than instantaneous which is strictly impossible). For a pressure disturbance to travel very quickly through a system, we know that the permeability should be very high and the fluid compressibility should be low (pressure changes a re communicated instantaneously through and incompressible fluid). Indeed, we will show later (Chapter 5) that pressure equilibrates faster - or “diffuses” through the system faster - for larger values of the “hydraulic diffusivity”, which is given by k/(φµc) (Dake, 1994, p.78).

Dake (1994, p.78), also points out two “necessary” conditions to apply material balance in practice as follows:

(i) We must have adequate data collection (production/pressures/PVT); and

(ii) we must have the ability to define an average pressure decline trend i.e. the more “tank like”, the better and this is equivalent to having a large k/(φμc) as discussed above.

EXERCISE 1.

Material Balance problem for an undersaturated reservoir using equation 8 above. This describes a case where production is by oil, water and formation expansion above the bubble point (Pb) with no water influx or production.

Exercise:

Suppose you have a tank - like reservoir with the fluid properties given below (and

in Figure 4). Plot a figure of 1 −

N

Np

vs. - ∆P over the first 250 psi of depletion

of this reservoir. Suppose you find that after 200 psi of depletion, you have produced 320 MSTB of oil. What was the original oil in place in this reservoir?

Input data: The initial water saturation, Swi = 0.1. The rock and water compressibilities are, as follows:

cf = 5 x 10-6 psi-1; cw = 4 x 10-6 psi-1.

The initial reservoir pressure is 5500 psi at which Boi = 1.3 and the bubble point is at Pb = 4000 psi where Bo = 1.4. That is, the oil swells as the pressure drops as shown in Figure 4.



3. SINGLE PHASE DARCY LAW

We review the single phase Darcy Law in this section in order to put our own particular “slant” or viewpoint to the student. This will prove to be very useful when we derive the flow equations of reservoir simulation in Chapter 5. We also wish to extend the idea of permeability (k) somewhat further than is covered in basic reservoir engineering texts. In particular, we wish to introduce the idea of permeability as a tensor property, denoted by k . Some useful mathematical concepts will also be introduced in this section associated with vector calculus; in particular, the idea of gradient

∇

∇

.

and divergence

∇

∇

.

• will be discussed in the context of the generalised formulation of the single phase Darcy law. Note that for reference, many of the terms discussed here are also summarised in the Glossary.

3.1 The Basic Darcy ExperimentDarcy in 1856 conducted a series of flow tests through packs of sands which he took as approximate experimental models of an aquifer for the ground water supply at Dijon. A schematic of the essential Darcy experiment is shown in Figure 5 where we imagine a single phase fluid (e.g. water) being pumped through a homogeneous sand pack or rock core. (Darcy used a gravitational head of water as his driving force whereas, in modern core laboratories, we would normally use a pump.)

The Darcy law given in Figure 5, is in its “experimental” form where a conversion factor, β, is indicated that allows us to work in various units as may be convenient to the problem at hand. In differential form, a more useful way to express the Darcy Law and introducing the Darcy velocity, u, is as follows:


SP

B BN

NB B

S c c

SP

N

NBB

BB

S c c


wi

oi op

o oiwi w f

wi

p oi

o

oi

o

wi w f

. . . ..

.

..

.

.

__

__

− + + +−

=

− + −+

−

=

= − ++

10

10

11

∆

∆

−−

=

=

=

= − = − ∂

∂

SP

p

uQA

k PL

k Px

wi

.

. .

__

__

∆

∆

∆

0

0

µ µ (9)

where the minus sign in equation 9 indicates that the direction of fluid flow is down the pressure gradient from high pressure to low pressure i.e. in the opposite direction to the positive pressure gradient.

∆P

Definitions:

Symbol Dimensions Meaning Consistent Units c.g.s lab. field SI - field

Q L3/T Volumetric cm3/s cm3/s bbl/day m3/day flow rate

L L Length of cm cm ft. m system

A L2 Cross - sectional cm2 cm2 ft.2 m2

area

µ Viscosity cP cP cP Pa.s

∆P M.L.T.2 Pressure drop atm dyne/cm2 psi Pa (Force/Area)

k L2 Permeability# darcy darcy mD mD

β dimensionless Conversion 1.00 9.869x10-6 1.127x10-3 8.527 factor x10-3

# permeability - dimensions L2; e.g. units m2, Darcies (D), milliDarcies (mD); 1 Darcy= 9.869 x 10-9 cm2 = 0.98696 x 10-12 m2 ≈ 1 µm2.

L

Q = β. k.Aµ

∆PL

Q Q

.



∆P

Definitions:

Symbol Dimensions Meaning Consistent Units c.g.s lab. field SI - field

Q L3/T Volumetric cm3/s cm3/s bbl/day m3/day flow rate

L L Length of cm cm ft. m system

A L2 Cross - sectional cm2 cm2 ft.2 m2

area

µ Viscosity cP cP cP Pa.s

∆P M.L.T.2 Pressure drop atm dyne/cm2 psi Pa (Force/Area)

k L2 Permeability# darcy darcy mD mD

β dimensionless Conversion 1.00 9.869x10-6 1.127x10-3 8.527 factor x10-3

# permeability - dimensions L2; e.g. units m2, Darcies (D), milliDarcies (mD); 1 Darcy= 9.869 x 10-9 cm2 = 0.98696 x 10-12 m2 ≈ 1 µm2.

L

Q = β. k.Aµ

∆PL

Q Q

.

Note on Units Conversion for Darcy’s Law: the various units that are commonly used for Darcy’s Law are listed in Figure 2 above. Sometimes, the conversion between various systems of units causes confusion for some students. Here, we briefly explain how to do this using the examples in the previous figure; that is, we go from c.g.s. (centimetre - gram - second) units where β = 1, indeed, the Darcy was defined such that β = 1. Starting from the Darcy Law in c.g.s. units:

Qk A P

L

Q cm sk Darcy A cm

cpP atmL cm

Q bbl dayk Darcy A ft

cpP psiL ft

Qbblday

=

=

=

βµ

µ

µ

..

.

( / )( ) . ( )

( ).

( )( )

( / )( ) . ( )

( ).

( )( .)

.

∆

∆

∆

1.00

??

1.58999

32

2

xx5

x4

x4

x5

10 10

10

1000 . 1.58999 10

8 641000

30 4814 7

30 48

8 64 30 4814 7 30 48

2 2

2

.

( ) . ( . ). .

( ). .

( .). .

. . .. . . .

=( )

=

kmD A ft

cp

Ppsi

L ft

Qbblday

µ

∆

( )

= ( )

1.126722 10

x

-3

k Darcy A ftcp

P psi

L ft

Qbblday

k Darcy A ftcp

P psi

L ft

( ) . ( . )( )

.( .)

( ) . ( . )( )

.( .)

2

2

µ

µ

∆

∆

Suppose we now wish to convert to field units as follows:

Qk A P

L

Q cm sk Darcy A cm

cpP atmL cm


cpP psiL ft

Qbblday

=

=

=

βµ

µ

µ

..

.

( / )( ) . ( )

( ).

( )( )

( / )( ) . ( )

( ).

( )( .)

.

∆

∆

∆

1.00

??

1.58999

32

2

xx5

x4

x4

x5

10 10

10

1000 . 1.58999 10

8 641000

30 4814 7

30 48

8 64 30 4814 7 30 48

2 2

2

.

( ) . ( . ). .

( ). .

( .). .

. . .. . . .

=( )

=

kmD A ft

cp

Ppsi

L ft

Qbblday

µ

∆

( )

= ( )

1.126722 10

x

-3

k Darcy A ftcp

P psi

L ft

Qbblday

k Darcy A ftcp

P psi

L ft

( ) . ( . )( )

.( .)

( ) . ( . )( )

.( .)

2

2

µ

µ

∆

∆

How do we find the correct conversion factor for these new units? Essentially, we convert it unit by unit starting from the c.g.s. expression where we know that β = 1. We do need to know a few conversion factors as follows: 1 ft. = 30.48 cm (exact), 14.7 psi = 1 atm., 1 bbl = 5.615 ft3 = 5.615 x 30.483 cm3 = 1.58999 x 105 cm3, 1 day = 24 x 3600 s = 8.64 x 104 s. Thus, we now convert everything in the field units to c.g.s. units as follows (except for cp. which are the same):

Figure 5.The single phase Darcy Law



Qk A P

L

Q cm sk Darcy A cm

cpP atmL cm


cpP psiL ft

Qbblday

=

=

=

βµ

µ

µ

..

.

( / )( ) . ( )

( ).

( )( )

( / )( ) . ( )

( ).

( )( .)

.

∆

∆

∆

1.00

??

1.58999

32

2

xx5

x4

x4

x5

10 10

10

1000 . 1.58999 10

8 641000

30 4814 7

30 48

8 64 30 4814 7 30 48

2 2

2

.

( ) . ( . ). .

( ). .

( .). .

. . .. . . .

=( )

=

kmD A ft

cp

Ppsi

L ft

Qbblday

µ

∆

( )

= ( )

1.126722 10

x

-3

k Darcy A ftcp

P psi

L ft

Qbblday

k Darcy A ftcp

P psi

L ft

( ) . ( . )( )

.( .)

( ) . ( . )( )

.( .)

2

2

µ

µ

∆

∆

Thus, collecting the numerical factors together we obtain:

Qk A P

L

Q cm sk Darcy A cm

cpP atmL cm


cpP psiL ft

Qbblday

=

=

=

βµ

µ

µ

..

.

( / )( ) . ( )

( ).

( )( )

( / )( ) . ( )

( ).

( )( .)

.

∆

∆

∆

1.00

??

1.58999

32

2

xx5

x4

x4

x5

10 10

10

1000 . 1.58999 10

8 641000

30 4814 7

30 48

8 64 30 4814 7 30 48

2 2

2

.

( ) . ( . ). .

( ). .

( .). .

. . .. . . .

=( )

=

kmD A ft

cp

Ppsi

L ft

Qbblday

µ

∆

( )

= ( )

1.126722 10

x

-3

k Darcy A ftcp

P psi

L ft

Qbblday

k Darcy A ftcp

P psi

L ft

( ) . ( . )( )

.( .)

( ) . ( . )( )

.( .)

2

2

µ

µ

∆

∆which simplifies to

Qk A P

L

Q cm sk Darcy A cm

cpP atmL cm


cpP psiL ft

Qbblday

=

=

=

βµ

µ

µ

..

.

( / )( ) . ( )

( ).

( )( )

( / )( ) . ( )

( ).

( )( .)

.

∆

∆

∆

1.00

??

1.58999

32

2

xx5

x4

x4

x5

10 10

10

1000 . 1.58999 10

8 641000

30 4814 7

30 48

8 64 30 4814 7 30 48

2 2

2

.

( ) . ( . ). .

( ). .

( .). .

. . .. . . .

=( )

=

kmD A ft

cp

Ppsi

L ft

Qbblday

µ

∆

( )

= ( )

1.126722 10

x

-3

k Darcy A ftcp

P psi

L ft

Qbblday

k Darcy A ftcp

P psi

L ft

( ) . ( . )( )

.( .)

( ) . ( . )( )

.( .)

2

2

µ

µ

∆

∆

and hence β = 1.127 x 10-3 for these units (as given in Figure 5).

3.2 Mathematical Note: on the Operators “gradient”

∇

∇

.

and “divergence”

∇

∇

.

• Before generalising the Darcy Law to 3D, we first make a short mathematical digression to introduce the concepts of gradient and divergence operators. These will be used to write the generalised flow equation of single and two phase flow in Chapter 5.

Gradient (or grad) is a vector operation as follows:

∇ = ∂∂

+ ∂∂

+ ∂∂

∇ = ∂∂

+ ∂∂

+ ∂∂

∇

∇ =

∂∂∂∂∂∂

∂∂

∂∂

, and

xi

yj

zk

i j k

PPx

iPy

jPz

k

P

P

Px

i

Py

j

Pz

k

Px

iPy

j and, , ∂∂

Pz

k

where i, j and k are the unit vectors which point in the x, y and z directions, respectively. The gradient operation can be carried out on a scalar field such as pressure, P, as follows:

∇ = ∂∂

+ ∂∂

+ ∂∂

∇ = ∂∂

+ ∂∂

+ ∂∂

∇

∇ =

∂∂∂∂∂∂

∂∂

∂∂

, and

xi

yj

zk

i j k

PPx

iPy

jPz

k

P

P

Px

i

Py

j

Pz

k

Px

iPy

j and, , ∂∂

Pz

k

where

∇

∇

.

P is sometimes written as grad P. The quantity

∇

∇

.

P is actually a vector of the pressure gradients in the three directions, x, y and z as follows:

∇ = ∂∂

+ ∂∂

+ ∂∂

∇ = ∂∂

+ ∂∂

+ ∂∂

∇

∇ =

∂∂∂∂∂∂

∂∂

∂∂

, and

xi

yj

zk

i j k

PPx

iPy

jPz

k

P

P

Px

i

Py

j

Pz

k

Px

iPy

j and, , ∂∂

Pz

k



This is shown schematically in Figure 6 where the three components of the vector ∇

∇

.

P, i.e.

∇ = ∂∂

+ ∂∂

+ ∂∂

∇ = ∂∂

+ ∂∂

+ ∂∂

∇

∇ =

∂∂∂∂∂∂

∂∂

∂∂

, and

xi

yj

zk

i j k

PPx

iPy

jPz

k

P

P

Px

i

Py

j

Pz

k

Px

iPy

j and, , ∂∂

Pz

k , and are shown by the dashed lines.

Figure 3: The definition of grad P or

Unit vectorsz

y

x

k

j

i

P

∆

Divergence (or div) is the dot product of the gradient operator and acts on a vector to produce a scalar. The operator is denoted as follows:

∇ = ∂∂

∂∂

∂∂

∇ = ∂∂

∂∂

∂∂

∇ = ∂∂

∂∂

∂∂

. . . .

. . . .

. . . .

xi

yj

zk

ux

iy

jz

k

u i

u j

u k

ux

iy

jz

k

u i

u j

u k

x

y

z

x

y

z

= ∂∂

+∂∂

+ ∂∂

= = =

∇ = ∂∂

+∂∂

+ ∂∂

∇ ∇ ∇

∇ ∇ = ∂∂

ux

i iu

yj j

uz

k k

i i j j k k

uux

u

yuz

P P

Px

i

x y z

x y z

. . .

. . .

.

. .

.

1

∂∂∂

∂∂

∂∂

∂∂

∂∂

= ∂∂

+ ∂∂

+ ∂∂

y

jz

k

Px

i

Py

j

Pz

k

Px

i iP

yj j

Pz

. . . . 2

2

2

2

2

2

= = =

∇ ∇ = ∂∂

+ ∂∂

+ ∂∂

= ∇

k k

i i j j k k

PP

xP

yP

zP

.

. . .

.

1

2

2

2

2

2

22

For example, taking the divergence of the Darcy velocity vector, u, gives the following:

∇ = ∂∂

∂∂

∂∂

∇ = ∂∂

∂∂

∂∂

∇ = ∂∂

∂∂

∂∂

. . . .

. . . .

. . . .

xi

yj

zk

ux

iy

jz

k

u i

u j

u k

ux

iy

jz

k

u i

u j

u k

x

y

z

x

y

z

= ∂∂

+∂∂

+ ∂∂

= = =

∇ = ∂∂

+∂∂

+ ∂∂

∇ ∇ ∇

∇ ∇ = ∂∂

ux

i iu

yj j

uz

k k

i i j j k k

uux

u

yuz

P P

Px

i

x y z

x y z

. . .

. . .

.

. .

.

1

∂∂∂

∂∂

∂∂

∂∂

∂∂

= ∂∂

+ ∂∂

+ ∂∂

y

jz

k

Px

i

Py

j

Pz

k

Px

i iP

yj j

Pz

. . . . 2

2

2

2

2

2

= = =

∇ ∇ = ∂∂

+ ∂∂

+ ∂∂

= ∇

k k

i i j j k k

PP

xP

yP

zP

.

. . .

.

1

2

2

2

2

2

22

where we can expand the RHS of the above equation by multiplying out the first (1x3) matrix by the second (3x1) matrix to obtain a “1x1 matrix” which is a scalar as follows:

∇ = ∂∂

∂∂

∂∂

∇ = ∂∂

∂∂

∂∂

∇ = ∂∂

∂∂

∂∂

. . . .

. . . .

. . . .

xi

yj

zk

ux

iy

jz

k

u i

u j

u k

ux

iy

jz

k

u i

u j

u k

x

y

z

x

y

z

= ∂∂

+∂∂

+ ∂∂

= = =

∇ = ∂∂

+∂∂

+ ∂∂

∇ ∇ ∇

∇ ∇ = ∂∂

ux

i iu

yj j

uz

k k

i i j j k k

uux

u

yuz

P P

Px

i

x y z

x y z

. . .

. . .

.

. .

.

1

∂∂∂

∂∂

∂∂

∂∂

∂∂

= ∂∂

+ ∂∂

+ ∂∂

y

jz

k

Px

i

Py

j

Pz

k

Px

i iP

yj j

Pz

. . . . 2

2

2

2

2

2

= = =

∇ ∇ = ∂∂

+ ∂∂

+ ∂∂

= ∇

k k

i i j j k k

PP

xP

yP

zP

.

. . .

.

1

2

2

2

2

2

22

where we use the relationships,

∇ = ∂∂

∂∂

∂∂

∇ = ∂∂

∂∂

∂∂

∇ = ∂∂

∂∂

∂∂

. . . .

. . . .

. . . .

xi

yj

zk

ux

iy

jz

k

u i

u j

u k

ux

iy

jz

k

u i

u j

u k

x

y

z

x

y

z

= ∂∂

+∂∂

+ ∂∂

= = =

∇ = ∂∂

+∂∂

+ ∂∂

∇ ∇ ∇

∇ ∇ = ∂∂

ux

i iu

yj j

uz

k k

i i j j k k

uux

u

yuz

P P

Px

i

x y z

x y z

. . .

. . .

.

. .

.

1

∂∂∂

∂∂

∂∂

∂∂

∂∂

= ∂∂

+ ∂∂

+ ∂∂

y

jz

k

Px

i

Py

j

Pz

k

Px

i iP

yj j

Pz

. . . . 2

2

2

2

2

2

= = =

∇ ∇ = ∂∂

+ ∂∂

+ ∂∂

= ∇

k k

i i j j k k

PP

xP

yP

zP

.

. . .

.

1

2

2

2

2

2

22

, to obtain:

∇ = ∂∂

∂∂

∂∂

∇ = ∂∂

∂∂

∂∂

∇ = ∂∂

∂∂

∂∂

. . . .

. . . .

. . . .

xi

yj

zk

ux

iy

jz

k

u i

u j

u k

ux

iy

jz

k

u i

u j

u k

x

y

z

x

y

z

= ∂∂

+∂∂

+ ∂∂

= = =

∇ = ∂∂

+∂∂

+ ∂∂

∇ ∇ ∇

∇ ∇ = ∂∂

ux

i iu

yj j

uz

k k

i i j j k k

uux

u

yuz

P P

Px

i

x y z

x y z

. . .

. . .

.

. .

.

1

∂∂∂

∂∂

∂∂

∂∂

∂∂

= ∂∂

+ ∂∂

+ ∂∂

y

jz

k

Px

i

Py

j

Pz

k

Px

i iP

yj j

Pz

. . . . 2

2

2

2

2

2

= = =

∇ ∇ = ∂∂

+ ∂∂

+ ∂∂

= ∇

k k

i i j j k k

PP

xP

yP

zP

.

. . .

.

1

2

2

2

2

2

22

Figure 6The definition of grad P or

∇

∇

.

P



Likewise, we can take the divergence of the grad P vector,

∇

∇

.

P, to obtain the quantity,

∇

∇

.

•

∇

∇

.

P, (sometimes denoted div grad P), as follows:

∇ = ∂∂

∂∂

∂∂

∇ = ∂∂

∂∂

∂∂

∇ = ∂∂

∂∂

∂∂

. . . .

. . . .

. . . .

xi

yj

zk

ux

iy

jz

k

u i

u j

u k

ux

iy

jz

k

u i

u j

u k

x

y

z

x

y

z

= ∂∂

+∂∂

+ ∂∂

= = =

∇ = ∂∂

+∂∂

+ ∂∂

∇ ∇ ∇

∇ ∇ = ∂∂

ux

i iu

yj j

uz

k k

i i j j k k

uux

u

yuz

P P

Px

i

x y z

x y z

. . .

. . .

.

. .

.

1

∂∂∂

∂∂

∂∂

∂∂

∂∂

= ∂∂

+ ∂∂

+ ∂∂

y

jz

k

Px

i

Py

j

Pz

k

Px

i iP

yj j

Pz

. . . . 2

2

2

2

2

2

= = =

∇ ∇ = ∂∂

+ ∂∂

+ ∂∂

= ∇

k k

i i j j k k

PP

xP

yP

zP

.

. . .

.

1

2

2

2

2

2

22

Again using the relationships,

∇ = ∂∂

∂∂

∂∂

∇ = ∂∂

∂∂

∂∂

∇ = ∂∂

∂∂

∂∂

. . . .

. . . .

. . . .

xi

yj

zk

ux

iy

jz

k

u i

u j

u k

ux

iy

jz

k

u i

u j

u k

x

y

z

x

y

z

= ∂∂

+∂∂

+ ∂∂

= = =

∇ = ∂∂

+∂∂

+ ∂∂

∇ ∇ ∇

∇ ∇ = ∂∂

ux

i iu

yj j

uz

k k

i i j j k k

uux

u

yuz

P P

Px

i

x y z

x y z

. . .

. . .

.

. .

.

1

∂∂∂

∂∂

∂∂

∂∂

∂∂

= ∂∂

+ ∂∂

+ ∂∂

y

jz

k

Px

i

Py

j

Pz

k

Px

i iP

yj j

Pz

. . . . 2

2

2

2

2

2

= = =

∇ ∇ = ∂∂

+ ∂∂

+ ∂∂

= ∇

k k

i i j j k k

PP

xP

yP

zP

.

. . .

.

1

2

2

2

2

2

22

, we obtain the familiar expression:

∇ = ∂∂

∂∂

∂∂

∇ = ∂∂

∂∂

∂∂

∇ = ∂∂

∂∂

∂∂

. . . .

. . . .

. . . .

xi

yj

zk

ux

iy

jz

k

u i

u j

u k

ux

iy

jz

k

u i

u j

u k

x

y

z

x

y

z

= ∂∂

+∂∂

+ ∂∂

= = =

∇ = ∂∂

+∂∂

+ ∂∂

∇ ∇ ∇

∇ ∇ = ∂∂

ux

i iu

yj j

uz

k k

i i j j k k

uux

u

yuz

P P

Px

i

x y z

x y z

. . .

. . .

.

. .

.

1

∂∂∂

∂∂

∂∂

∂∂

∂∂

= ∂∂

+ ∂∂

+ ∂∂

y

jz

k

Px

i

Py

j

Pz

k

Px

i iP

yj j

Pz

. . . . 2

2

2

2

2

2

= = =

∇ ∇ = ∂∂

+ ∂∂

+ ∂∂

= ∇

k k

i i j j k k

PP

xP

yP

zP

.

. . .

.

1

2

2

2

2

2

22

where, in summary,

∇

∇

.

2 is the Laplacian operator:

∇ = ∇ ∇ = = ∂∂

+ ∂∂

+ ∂∂

=

∇ ∇

∇ =

22

2

2

2

2

2

.

.

div gradx y y

k

k k k

k k k

k k k

P k P

k P

k k k

k k k

k k k

xx xy xz

yx yy yz

zx zy zz

xx xy xz

yx yy yz

zx zy zz

.

.

∂∂

∂∂

∂∂

∇ =

∂∂

∂∂

Px

Py

Pz

k P

k k k

k k k

k k k

Px

Py

xx xy xz

yx yy yz

zx zy zz

.

∂∂

=

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

Pz

kPx

kPy

kPz

kPx

kPy

kPz

xx xy xz

yx yy yz

+ +

+ +

∂∂

∂∂

∂∂

∇ =

∂∂

∂∂

kPx

kPy

kPz

k P

kPx

kPy

zx zy zz

xx xy

+ +

+

.

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

+

+ +

+ +

kPz

kPx

kPy

kPz

kPx

kPy

kPz

xz

yx yy yz

zx zy zz

The final issue we wish to discuss in this mathematical note is the rule for taking the dot product of a tensor and a vector. N.B. We now omit the explicit inclusion of the unit vectors, i, j and k in the following developments.

3.3 Darcy’s Law in 3D - Using Vector and Tensor NotationSuppose we have a tensor k which in 3D can be represented by a 3 x 3 matrix as follows:

∇ = ∇ ∇ = = ∂∂

+ ∂∂

+ ∂∂

=

∇ ∇

∇ =

22

2

2

2

2

2

.

.

div gradx y y

k

k k k

k k k

k k k

P k P

k P

k k k

k k k

k k k

xx xy xz

yx yy yz

zx zy zz

xx xy xz

yx yy yz

zx zy zz

.

.

∂∂

∂∂

∂∂

∇ =

∂∂

∂∂

Px

Py

Pz

k P

k k k

k k k

k k k

Px

Py

xx xy xz

yx yy yz

zx zy zz

.

∂∂

=

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

Pz

kPx

kPy

kPz

kPx

kPy

kPz

xx xy xz

yx yy yz

+ +

+ +

∂∂

∂∂

∂∂

∇ =

∂∂

∂∂

kPx

kPy

kPz

k P

kPx

kPy

zx zy zz

xx xy

+ +

+

.

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

+

+ +

+ +

kPz

kPx

kPy

kPz

kPx

kPy

kPz

xz

yx yy yz

zx zy zz

Suppose we now wish to take a dot product of this tensor, k , with the vector

∇

∇

.

P; that is k •

∇

∇

.

P. The dot product of a tensor and a vector is a vector and the operation is carried out like a matrix multiplication as follows:

∇ = ∇ ∇ = = ∂∂

+ ∂∂

+ ∂∂

=

∇ ∇

∇ =

22

2

2

2

2

2

.

.

div gradx y y

k

k k k

k k k

k k k

P k P

k P

k k k

k k k

k k k

xx xy xz

yx yy yz

zx zy zz

xx xy xz

yx yy yz

zx zy zz

.

.

∂∂

∂∂

∂∂

∇ =

∂∂

∂∂

Px

Py

Pz

k P

k k k

k k k

k k k

Px

Py

xx xy xz

yx yy yz

zx zy zz

.

∂∂

=

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

Pz

kPx

kPy

kPz

kPx

kPy

kPz

xx xy xz

yx yy yz

+ +

+ +

∂∂

∂∂

∂∂

∇ =

∂∂

∂∂

kPx

kPy

kPz

k P

kPx

kPy

zx zy zz

xx xy

+ +

+

.

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

+

+ +

+ +

kPz

kPx

kPy

kPz

kPx

kPy

kPz

xz

yx yy yz

zx zy zz



which multiplies out as follows:

∇ = ∇ ∇ = = ∂∂

+ ∂∂

+ ∂∂

=

∇ ∇

∇ =

22

2

2

2

2

2

.

.

div gradx y y

k

k k k

k k k

k k k

P k P

k P

k k k

k k k

k k k

xx xy xz

yx yy yz

zx zy zz

xx xy xz

yx yy yz

zx zy zz

.

.

∂∂

∂∂

∂∂

∇ =

∂∂

∂∂

Px

Py

Pz

k P

k k k

k k k

k k k

Px

Py

xx xy xz

yx yy yz

zx zy zz

.

∂∂

=

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

Pz

kPx

kPy

kPz

kPx

kPy

kPz

xx xy xz

yx yy yz

+ +

+ +

∂∂

∂∂

∂∂

∇ =

∂∂

∂∂

kPx

kPy

kPz

k P

kPx

kPy

zx zy zz

xx xy

+ +

+

.

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

+

+ +

+ +

kPz

kPx

kPy

kPz

kPx

kPy

kPz

xz

yx yy yz

zx zy zz

giving the final result:

∇ = ∇ ∇ = = ∂∂

+ ∂∂

+ ∂∂

=

∇ ∇

∇ =

22

2

2

2

2

2

.

.

div gradx y y

k

k k k

k k k

k k k

P k P

k P

k k k

k k k

k k k

xx xy xz

yx yy yz

zx zy zz

xx xy xz

yx yy yz

zx zy zz

.

.

∂∂

∂∂

∂∂

∇ =

∂∂

∂∂

Px

Py

Pz

k P

k k k

k k k

k k k

Px

Py

xx xy xz

yx yy yz

zx zy zz

.

∂∂

=

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

Pz

kPx

kPy

kPz

kPx

kPy

kPz

xx xy xz

yx yy yz

+ +

+ +

∂∂

∂∂

∂∂

∇ =

∂∂

∂∂

kPx

kPy

kPz

k P

kPx

kPy

zx zy zz

xx xy

+ +

+

.

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

+

+ +

+ +

kPz

kPx

kPy

kPz

kPx

kPy

kPz

xz

yx yy yz

zx zy zz

Using the above concepts from vector calculus (div. and grad), we can extend the Darcy Law (in the absence of gravity) to 3D as follows by introducing the tensor permeability, k :

u k P

k k k

k k k

k k k

Px

Py

Pz

kPx

xx xy xz

yx yy yz

zx zy zz

xx

= ∇ =

∂∂

∂∂

∂∂

=

∂∂

-1

-1

-1

. µ µ µ

++ +

+ +

+ +

kPy

kPz

kPx

kPy

kPz

kPx

kPy

k

xy xz

yx yy yz

zx zy zz

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂∂

=

=

∂∂

∂∂

∂∂

Pz

u

u

u

u

kPx

kPy

kPz

k

x

y

z

xx xy xz

-1

+ +

µ yxyx yy yz

zx zy zz

Px

kPy

kPz

kPx

kPy

kPz

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

+ +

+ +

= ∂∂

∂∂

∂∂

= ∂∂

∂∂

∂∂

=

u kPx

kPy

kPz

u kPx

kPy

kPz

u

x xx xy xz

y yx yy yz

z

-1

+ +

-1

+ +

-1

µ

µ

µµk

Px

kPy

kPz

k

k

k

k

zx zy zz

xx

yy

zz

∂∂

∂∂

∂∂

=

+ +

0 0

0 0

0 0

which we may write as:

u k P

k k k

k k k

k k k

Px

Py

Pz

kPx

xx xy xz

yx yy yz

zx zy zz

xx

= ∇ =

∂∂

∂∂

∂∂

=

∂∂

-1

-1

-1

. µ µ µ

++ +

+ +

+ +

kPy

kPz

kPx

kPy

kPz

kPx

kPy

k

xy xz

yx yy yz

zx zy zz

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂∂

=

=

∂∂

∂∂

∂∂

Pz

u

u

u

u

kPx

kPy

kPz

k

x

y

z

xx xy xz

-1

+ +

µ yxyx yy yz

zx zy zz

Px

kPy

kPz

kPx

kPy

kPz

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

+ +

+ +

= ∂∂

∂∂

∂∂

= ∂∂

∂∂

∂∂

=

u kPx

kPy

kPz

u kPx

kPy

kPz

u

x xx xy xz

y yx yy yz

z

-1

+ +

-1

+ +

-1

µ

µ

µµk

Px

kPy

kPz

k

k

k

k

zx zy zz

xx

yy

zz

∂∂

∂∂

∂∂

=

+ +

0 0

0 0

0 0



and we can identify the three components of the velocity as follows:

u k P

k k k

k k k

k k k

Px

Py

Pz

kPx

xx xy xz

yx yy yz

zx zy zz

xx

= ∇ =

∂∂

∂∂

∂∂

=

∂∂

-1

-1

-1

. µ µ µ

++ +

+ +

+ +

kPy

kPz

kPx

kPy

kPz

kPx

kPy

k

xy xz

yx yy yz

zx zy zz

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂∂

=

=

∂∂

∂∂

∂∂

Pz

u

u

u

u

kPx

kPy

kPz

k

x

y

z

xx xy xz

-1

+ +

µ yxyx yy yz

zx zy zz

Px

kPy

kPz

kPx

kPy

kPz

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

+ +

+ +

= ∂∂

∂∂

∂∂

= ∂∂

∂∂

∂∂

=

u kPx

kPy

kPz

u kPx

kPy

kPz

u

x xx xy xz

y yx yy yz

z

-1

+ +

-1

+ +

-1

µ

µ

µµk

Px

kPy

kPz

k

k

k

k

zx zy zz

xx

yy

zz

∂∂

∂∂

∂∂

=

+ +

0 0

0 0

0 0

If the permeability tensor is diagonal i.e. the cross-terms are zero as follows:

u k P

k k k

k k k

k k k

Px

Py

Pz

kPx

xx xy xz

yx yy yz

zx zy zz

xx

= ∇ =

∂∂

∂∂

∂∂

=

∂∂

-1

-1

-1

. µ µ µ

++ +

+ +

+ +

kPy

kPz

kPx

kPy

kPz

kPx

kPy

k

xy xz

yx yy yz

zx zy zz

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂∂

=

=

∂∂

∂∂

∂∂

Pz

u

u

u

u

kPx

kPy

kPz

k

x

y

z

xx xy xz

-1

+ +

µ yxyx yy yz

zx zy zz

Px

kPy

kPz

kPx

kPy

kPz

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

+ +

+ +

= ∂∂

∂∂

∂∂

= ∂∂

∂∂

∂∂

=

u kPx

kPy

kPz

u kPx

kPy

kPz

u

x xx xy xz

y yx yy yz

z

-1

+ +

-1

+ +

-1

µ

µ

µµk

Px

kPy

kPz

k

k

k

k

zx zy zz

xx

yy

zz

∂∂

∂∂

∂∂

=

+ +

0 0

0 0

0 0

then the various components of the Darcy law revert to their normal form and :

u kPx

u kPy

u kPz

u kPx

gzx

zx

u kPx

g

x xx

y yy

z zz

x xx

x xx

= ∂∂

= ∂∂

= ∂∂

= ∂∂

∂∂

∂∂

=

= ∂∂

-1

-1

-1

-1

-1

µ

µ

µ

µρ

θ

µρ

-

cos

- . ..cosθ

πµ

µπ

=

=

Qkhr dP

dr

dPQkh

drr

2

2

3.4 Simple Darcy Law with GravityIn the presence of gravity the 1D Darcy Law becomes:

u kPx

u kPy

u kPz

u kPx

gzx

zx

u kPx

g

x xx

y yy

z zz

x xx

x xx

= ∂∂

= ∂∂

= ∂∂

= ∂∂

∂∂

∂∂

=

= ∂∂

-1

-1

-1

-1

-1

µ

µ

µ

µρ

θ

µρ

-

cos

- . ..cosθ

πµ

µπ

=

=

Qkhr dP

dr

dPQkh

drr

2

2

where, in the case of a simple inclines system at a slope of θ, as shown in Figure 7,

u kPx

u kPy

u kPz

u kPx

gzx

zx

u kPx

g

x xx

y yy

z zz

x xx

x xx

= ∂∂

= ∂∂

= ∂∂

= ∂∂

∂∂

∂∂

=

= ∂∂

-1

-1

-1

-1

-1

µ

µ

µ

µρ

θ

µρ

-

cos

- . ..cosθ

πµ

µπ

=

=

Qkhr dP

dr

dPQkh

drr

2

2

, as shown in the figure above and:

u kPx

u kPy

u kPz

u kPx

gzx

zx

u kPx

g

x xx

y yy

z zz

x xx

x xx

= ∂∂

= ∂∂

= ∂∂

= ∂∂

∂∂

∂∂

=

= ∂∂

-1

-1

-1

-1

-1

µ

µ

µ

µρ

θ

µρ

-

cos

- . ..cosθ

πµ

µπ

=

=

Qkhr dP

dr

dPQkh

drr

2

2



x

θ

Note that:

= cos θ∂z∂x

3.5 The Radial Darcy LawIn the above discussion, in both 1D and 3D we considered the Darcy Law in normal Cartesian coordinates (x, y and z). In Chapter 6, we will explain how wells are treated in reservoir simulation. Because a radial (r/z) geometry is appropriate for the near-well region, it is useful to consider the Darcy Law in radial coordinates. In 1D, this simply involves the radial coordinate, r. In fact, the radial form of the Darcy law can be derived from the linear form as shown in Figure 8.

rh

Qdr

Area, A = 2π.r.h

Radial Darcy Law is:

Q = k.Aµ

dPdr

2πkhrµ

dPdr=

Notation: Q = volumetric flow rate of fluid into well r = radial distance from well h = height of formation dP = incremental pressure drop from r→ (r + dr) i.e. over dr A = area of surface at r = 2π.r.h μ = fluid viscosity k = formation permeability rw = wellbore radius

dr = incremental radius

Starting from the radial form of the Darcy Law, as follows:

u kPx

u kPy

u kPz

u kPx

gzx

zx

u kPx

g

x xx

y yy

z zz

x xx

x xx

= ∂∂

= ∂∂

= ∂∂

= ∂∂

∂∂

∂∂

=

= ∂∂

-1

-1

-1

-1

-1

µ

µ

µ

µρ

θ

µρ

-

cos

- . ..cosθ

πµ

µπ

=

=

Qkhr dP

dr

dPQkh

drr

2

2

we can rearrange this to obtain:

u kPx

u kPy

u kPz

u kPx

gzx

zx

u kPx

g

x xx

y yy

z zz

x xx

x xx

= ∂∂

= ∂∂

= ∂∂

= ∂∂

∂∂

∂∂

=

= ∂∂

-1

-1

-1

-1

-1

µ

µ

µ

µρ

θ

µρ

-

cos

- . ..cosθ

πµ

µπ

=

=

Qkhr dP

dr

dPQkh

drr

2

2

Figure 8Single phase Darcy Law in an inclines system - effect of gravity

Figure 7Radial form of the single-phase Darcy Law



Taking rw as the wellbore radius and r some appropriate radial distance, we can easily integrate the above equation to obtain the radial pressure profile in a radial system as follows:

dPQkh

drr

P rQkh

rr

uk k P

xg

zx

uk k P

xg

zx

Px

and

r

r

r

r

w

wrw w

w

oro o

o

w

w w

∫ ∫=

∆ =

= ∂∂

∂∂

= ∂∂

∂∂

∂∂

∂

µπ

µπ

µρ

µρ

2

2( ) ln

.-

.-

-

-

PPx

P S P P

P S P P

k k k

k k k

k and k

o

c w o w

c w non wett wett

w rw

o ro

w o

∂

( ) = −

( ) = −

=

=

−

. .

which gives:

dPQkh

drr

P rQkh

rr

uk k P

xg

zx

uk k P

xg

zx

Px

and

r

r

r

r

w

wrw w

w

oro o

o

w

w w

∫ ∫=

∆ =

= ∂∂

∂∂

= ∂∂

∂∂

∂∂

∂

µπ

µπ

µρ

µρ

2

2( ) ln

.-

.-

-

-

PPx

P S P P

P S P P

k k k

k k k

k and k

o

c w o w

c w non wett wett

w rw

o ro

w o

∂

( ) = −

( ) = −

=

=

−

. .

where we have denoted the radial pressure drop (or increase for a producer) from rw to r as, ΔP(r). Note that, unlike the linear Darcy Law, the pressure profile is logarithmic in the radial case. This means that pressure drops are much higher closer to the well. This is exactly what we expect physically since the area is decreasing with r as we approach the well and Q is the same; therefore, the pressure drop, dP, over a given dr is higher. This is shown schematically for an injector and a producer in Figure 9. The formulae and the ideas developed here will be used later in Chapter 4 on well modelling in reservoir simulation and we will not discuss this further here.

QQProducer

Pwf

Pwf

rw rrw r

∆P(r)∆P(r)

∆P(r) = P(r) - Pwf

∆P(r) = Pwf - P(r)

Injector

4. TWO-PHASE FLOW

4.1 The Two-Phase Darcy LawDarcy’s Law was originally applied to single phase flow only. However, in reservoir engineering, it has been convenient to extend it to describe the flows of multiple phases such as oil, water and gas. To do this, the Darcy Law has been modified empirically to include a term - the relative permeability - which is intended to describe the impairment of the flow of one phase due to the presence of another. A schematic representation of a steady-state two phase Darcy type (relative permeability) experiment is shown in Figure 10, where all of the quantities are defined. Examples of the relative permeability curves which can be measured in this way are also shown schematically in Figure 10 and actual experimental examples are given for rock curves of different wettability states in the Glossary.

Figure 9 Pressure profiles, ΔP(r), in radial single-phase flow; Pwf is the well flowing pressure (at rw)



0

0 1

kro

krw

Qw

Sw

Qo

Qw

QoL

∆Pw

∆Po

1

Schematic of relativepermeabilities, krw and kro

Rel.Perm.

Qw = k.krw.Aµw

Qo = k.kro.Aµo

∆PwL.

∆PoL.

The two - phase Darcy Law is as follows:

At steady - state flow conditions, the oil and water flow rates in and out,Qo and Qw, are the same:

Where: Qw and Qo = volumetric flow rates of water and oil; A = cross-sectional area; L = system length; µw and µo = water and oil viscosities; k = absolute permeabilities; ΔPw and ΔPo = the pressure drops across the water and oil phases at steady-state flow conditions krw and kro = the water and oil relative permeabilities

NB the Units for the two-phase Darcy Law are exactly the same as those in Figure 5.

The differential form of the two phase Darcy Law in 1D, again including gravity which is taken to act in the z-direction, is as follows:

dPQkh

drr

P rQkh

rr

uk k P

xg

zx

uk k P

xg

zx

Px

and

r

r

r

r

w

wrw w

w

oro o

o

w

w w

∫ ∫=

∆ =

= ∂∂

∂∂

= ∂∂

∂∂

∂∂

∂

µπ

µπ

µρ

µρ

2

2( ) ln

.-

.-

-

-

PPx

P S P P

P S P P

k k k

k k k

k and k

o

c w o w

c w non wett wett

w rw

o ro

w o

∂

( ) = −

( ) = −

=

=

−

. .

where we note that the flow of the two phases (water and oil, in this case) depends

on the pressure gradient in that phase; i.e. on

dPQkh

drr

P rQkh

rr

uk k P

xg

zx

uk k P

xg

zx

Px

and

r

r

r

r

w

wrw w

w

oro o

o

w

w w

∫ ∫=

∆ =

= ∂∂

∂∂

= ∂∂

∂∂

∂∂

∂

µπ

µπ

µρ

µρ

2

2( ) ln

.-

.-

-

-

PPx

P S P P

P S P P

k k k

k k k

k and k

o

c w o w

c w non wett wett

w rw

o ro

w o

∂

( ) = −

( ) = −

=

=

−

. .

.

Figure 10 The two-phase Darcy Law and relative permeability



The phase pressures, Po and Pw, at a given saturation, Sw (So = 1 - Sw), are generally not equal. However, they are related through the capillary pressure, as follows:

dPQkh

drr

P rQkh

rr

uk k P

xg

zx

uk k P

xg

zx

Px

and

r

r

r

r

w

wrw w

w

oro o

o

w

w w

∫ ∫=

∆ =

= ∂∂

∂∂

= ∂∂

∂∂

∂∂

∂

µπ

µπ

µρ

µρ

2

2( ) ln

.-

.-

-

-

PPx

P S P P

P S P P

k k k

k k k

k and k

o

c w o w

c w non wett wett

w rw

o ro

w o

∂

( ) = −

( ) = −

=

=

−

. .

More strictly, the capillary pressure is the difference between the non-wetting phase pressure and the wetting-phase pressure;

dPQkh

drr

P rQkh

rr

uk k P

xg

zx

uk k P

xg

zx

Px

and

r

r

r

r

w

wrw w

w

oro o

o

w

w w

∫ ∫=

∆ =

= ∂∂

∂∂

= ∂∂

∂∂

∂∂

∂

µπ

µπ

µρ

µρ

2

2( ) ln

.-

.-

-

-

PPx

P S P P

P S P P

k k k

k k k

k and k

o

c w o w

c w non wett wett

w rw

o ro

w o

∂

( ) = −

( ) = −

=

=

−

. .. We can think of the capillary pressure as a constraint on the phase pressures. That is, if we know the capillary pressure function - from experiment , say - then, if we have Po at a given saturation, we can calculate Pw. Examples of capillary pressure curves are also shown in the Glossary.

Note that, as in the single-phase Darcy Law, we may generalise the two-phase Darcy expressions to 3D. Defining the combination of absolute permeability in its full tensor form, k , with the phase relative permeabilities gives:

dPQkh

drr

P rQkh

rr

uk k P

xg

zx

uk k P

xg

zx

Px

and

r

r

r

r

w

wrw w

w

oro o

o

w

w w

∫ ∫=

∆ =

= ∂∂

∂∂

= ∂∂

∂∂

∂∂

∂

µπ

µπ

µρ

µρ

2

2( ) ln

.-

.-

-

-

PPx

P S P P

P S P P

k k k

k k k

k and k

o

c w o w

c w non wett wett

w rw

o ro

w o

∂

( ) = −

( ) = −

=

=

−

. .

where

dPQkh

drr

P rQkh

rr

uk k P

xg

zx

uk k P

xg

zx

Px

and

r

r

r

r

w

wrw w

w

oro o

o

w

w w

∫ ∫=

∆ =

= ∂∂

∂∂

= ∂∂

∂∂

∂∂

∂

µπ

µπ

µρ

µρ

2

2( ) ln

.-

.-

-

-

PPx

P S P P

P S P P

k k k

k k k

k and k

o

c w o w

c w non wett wett

w rw

o ro

w o

∂

( ) = −

( ) = −

=

=

−

. .

are the effective phase permeability tensors of water and oil, respectively. Using this notation, the Darcy velocity vectors for the water and oil, uw and uo, may be written in 3D as follows:

u k P g z

u k P g z

ww

w w w

oo

o o o

= − ∇ − ∇( )

= − ∇ − ∇( )

1

1

µρ

µρ

.

.

This form of these equations is particularly useful in deriving the two-phase flow equations in their most general form (this will done in Chapter 5).

5. CLOSING REMARKS

The purpose of Chapter 2 is to review some key concepts in reservoir engineering which impact directly on the subject matter of reservoir simulation. The topics reviewed specifically involved:

- Material balance and its particular relationship with reservoir simulation;

- The single-phase Darcy law and its extension using vector calculus terminology to a 3D version of the Darcy Law including tensor permeabilities;



- The two-phase Darcy Law and the related concepts that arise in two-phase flow e.g. relative permeabilities (kro and krw), phase pressures (Po and Pw), capillary pressure (Pc(Sw) = Po - Pw), etc.

Ideas and concepts developed here will be used in other parts of this course.

6. SOME FURTHER READING ON RESERVOIR ENGINEERING

A full alphabetic list of References which are cited in the course is presented in Appendix A. Many excellent texts have appeared over the years covering the basics of Reservoir Engineering. Some of these are listed below, although this list is far from comprehensive.

Amyx, J W, Bass, D M and Whiting, R L: Petroleum Reservoir Engineering, McGraw-Hill, 1960. This is still an excellent petroleum engineering text although the coverage in some areas a little old fashioned. It has a very good chapter on material balance.

Archer, J S and Wall, C: Petroleum Engineering: Principles and Practice, Graham and Trotman Inc., London, 1986. This book offers a good overview of petroleum engineering and covers many of the basics of reservoir engineering. This book is also one of the earliest proponents of the importance of integrating the geology within the reservoir model.

Craft, B C, Hawkins, M F and Terry, R E: Applied Petroleum Reservoir Engineering, Prentice Hall, NJ, 1991. The original text by Craft and Hawkins was already an early classic. This was revised and updated by Terry and reissued in 1991. This has very good clear coverage of material balance and its application in various reservoir systems.

Craig, F F: The Reservoir Engineering Aspects of Waterflooding, SPE monograph, Dallas, TX, 1979. This text is confined to the underlying principles and reservoir engineering applications of waterflooding. It is an excellent monograph on the subject and an essential reference text for the reservoir engineer who is interested in the traditional analytical methods for assessing waterflooding.

Dake, L P: The Fundamentals of Reservoir Engineering, Developments in Petroleum Science 8, Elsevier, 1978. This has become a modern classic on the basics of reservoir engineering. It is very widely referenced and draws on Dake’s vast experience of teaching reservoir engineering basics. It has particularly good coverage of material balance and Buckley-Leverett theory.

Dake, L P: The Practice of Reservoir Engineering, Developments in Petroleum Science 36, Elsevier, 1994. This book is a modern plea for the continued application traditional reservoir engineering principles and techniques in performance analysis and prediction. It gives central place to the interpretation of well testing, the application of material balance and the use of Buckley Leverett theory. It has many examples from the hundreds of reservoirs that Dake himself worked on. This book also makes a number of interesting and controversial observations on reservoir simulation (not all of which the authors agree with!).



Solution To Exercises

EXERCISE 1:

Material Balance problem for an undersaturated reservoir using equation 8 above. This describes a case where production is by oil, water and formation expansion above the bubble point (Pb) with no water influx or production.

Exercise: For the input data below, do the following:

(i) Plot the function (1 - N/Np) as calculated by equation 8 vs. -ΔP.

As a reminder equation 8 is 1 11

−

= − ++

−

N

NBB

BB

S c

SPp oi

o

oi

o

wi f

wi

∆

This is shown below

Series 1

(1-Np/N) vs. -DP

0.999

0.997

0.995

0.993

0.991

0.989

0.987

0.985

0 50 100 200 300150 250

(1-N

p/N

)

-∆p (psi)

(ii) Note from the graph (or from your numerical calculation when plotting the graph) that, at - ΔP = 200 psi, then (1 - Np/N) = 0.991. However, we know by field observation that this 200 psi drawdown was caused by the production of 320 MSTB. That is, we know that Np = 320 MSTB. Hence,

(1 - 320/N) = 0.991 => N = 35555.5 MSTB ≈ 35.6 MMSTB

Answer: the STOOIP = 35.6 MMSTB.

Input data: The initial water saturation, Swi = 0.1. The rock and water compressibilities are, as follows:

crock = 5 x 10-6 psi-1; cw = 4 x 10-6 psi-1.

The initial reservoir pressure is 5500 psi at which Boi = 1.3 and the bubble point is at Pb = 4000 where Bo = 1.4. That is, the oil swells as the pressure drops as shown below:



1.4

1.3

4000 P (psi) 5500

Oil FVFBo

Bo(P) = m.P + c

26

CONTENTS

1. INTRODUCTION TO CHAPTER 3

2. SETTING UP A RESERVOIR SIMULATION MODEL 2.1. Defining The Objectives Of A Simulation Study 3. DATA INPUT AND OUTPUT 4. EXAMPLE INPUT DATA FILE 4.1. Reservoir System to be Modelled 4.2. ECLIPSE Syntax 4.3. Model Dimensions 4.4. Grid and Rock Properties 4.5. Fluid Properties 4.6. Initial Conditions 4.7. Output Requirements 4.8. Production Schedule 5. RUNNING ECLIPSE AND FILE NAME CONVENTIONS 5.1. Running ECLIPSE on a PC 5.2. File Name Conventions

6. CLOSING REMARKS

33Reservoir Simulation Model Set-Up



LEARNING OBJECTIVES

Having worked through this chapter and the associated tutorials the student should be able to:

Simulation Input• Identify what questions the simulation is expected to address.• Identify what data is required as input to perform the desired calculations.• Format data correctly, taking account of keyword syntax and required units.

Simulation Output• Select required output of calculations.• Quality check output data to check for errors in input.• Identify purpose of each output file and use post-processors to analyse data.

Analysis of Results• Identify impact of reservoir engineering principles in calculation performed.• Identify numerical effects and impact of grid block size and orientation on results.• Perform simple upscaling calculation to address numerical diffusion.




In this module, a step-by-step approach is given to setting up a 3D reservoir simulation model. This is done by working through an actual case which is complex enough to demonstrate most of the basic ideas, can demonstrate various sensitivities and can also show the effects of well controls. The simulator used in this course is ECLIPSE which is a software product of Schlumberger GeoQuest. However, the general approach and methodology for setting up a field simulation calculation is very similar for other commercial simulators.

This example will be used to illustrate the power of reservoir simulation in understanding reservoir recovery mechanisms.

1. INTRODUCTION TO CHAPTER 3

In this section of the course, we set up a practical 3D, two phase (oil/water) reservoir simulation model using the ECLIPSE reservoir simulator. This is proprietary software of Schlumberger GeoQuest. The central objective of this exercise is to get you actually applying reservoir simulation to a realistic (but quite simple) case. However, there are also some tasks in the study itself - you can think of these as the “objectives” if this were a real field case. One of the tasks in the exercise is as follows: in your calculation, you should observe initial short-term rise in BHP (bottom hole pressure) in the injection well and drop in BHP in the production well. You are asked to explain these trends. This is good example of where you may have run a calculation without necessarily thinking about what was going to happen to the BHP (or it could be watercut at the producer, or field average pressure etc.). However, when you study the simulator output - usually as graphs and figures - you would notice the BHP trends. This would catch the attention of a good engineer who would not be happy just to note it and move on. She or he would immediately stop and think and ask a few questions, “What’s going on here?”, “Is this something physical that I should expect or is there something wrong with the calculation?”. The engineer would stop and work it out from their basic reservoir engineering knowledge ... just as you are going to! The engineer would conclude that - although I possibly didn’t expect it - this behaviour is perfectly understandable and predictable.

From the above discussion, you can see that it is not just the mechanics of running a numerical simulator and getting the results out that we want you to achieve in this course. We want you to be able to formulate the right questions for a given reservoir application, carry out the appropriate simulations and then interpret the results correctly. The mechanics of running a simulation - if this was all you did - is really a technician’s job, the important job of correctly formulating the simulation problem, understanding the results and predicting reservoir performance is an engineer’s job and this course is intended for the latter (or for the former strongly intent on becoming the latter in the future!).



2. SETTING UP A RESERVOIR SIMULATION MODEL

This section will not be very long since there are many excellent examples of reservoir simulation studies elsewhere in this course e.g. Cases 1 - 3 in Chaper 1, the SPE field cases. Some generalities on how to set up a reservoir simulation study were also discussed in Chaper 1. Here, we will lay out more formally, the general procedues broadly following the workflow of a typical simulation study.

2.1 Defining The Objectives Of A Simulation StudyDefining the objectives is a vitally important stage of any field simulation study. The general spirit which is suggested for approaching this (in Chaper 1) is to correctly formulate the question you are trying to ask in order to make a particular decision. For example, the decision may be: “do I need to infill drill in this field in order to significantly (i.e. economically) improve reservoir performance?”. This is like the schematic example in Chaper 1, Figure 8 where the question was not “will I get more oil by ...”, since you could get more oil but at too great a cost - the decision must be economically based. Having said this, some reservoir decisions are made that may not in themselves be economic; however, they may be strategic or may lead to some knowledge or experience which will be economic in the future. The important matter in that you know what sort of decision you are trying to make.

3. DATA INPUT AND OUTPUT

When run, every reservoir simulator will require input data that defines the system to be modelled, and should generate output data that represents the results of the calculations which have been performed. Although different reservoir simulation codes have different formats for entering data, they all have some basic components in common. Most will read data from an input file. The input file must therefore be set up before starting the simulation run. The data file may be set up by manual editing (if it is in ASCII format), or by using a Graphical User Interface (GUI). Whichever method is used, most data files will be divided into certain key sections that define:

• Model dimensions• Grid and rock properties• Fluid properties• Initial conditions• Output requirements• Production schedule

Additional optional sections may allow for manipulation of an imported grid structure and for subdivision of the grid into regions. We will find that there are a huge number of possible refinements in all of the above general sections representing special models for particular applications but we will focus on the simpler common features of most black oil simulations.

Individual parts of the input data may be set up by other programs that may be supplied by the same supplier as the simulation code, or by other companies. These are referred to as pre-processors: They are used to perform calculations that set up the model in



advance of the actual reservoir fluid flow calculation. Typically, a pre-processor is used to set up the grid and to define the rock properties (permeability, porosity, etc.) of each cell. Setting up a model with more than a few hundred cells would be very laborious if performed by hand. These gridding packages are usually designed to generate output that conforms to the input format of several of the more widely used simulators. Thus, the reservoir simulation input data file need only refer to these grid files by means of a simple “include” statement, and no further manipulation of the grid is required by the flow simulator.

Pre-processors may be used to:• Define grid and rock properties• Define fluid properties• Convert the results of special core analysis data to a form that can be used in the simulation• Upscale rock data so that it is appropriate for the size of grid cells being used• Define vertical flow performance tables• Set up the production schedule

Indeed, any software that is used for setting up a part or the whole of an input data file is termed a pre-processor.

The output of the reservoir flow calculations usually comes in two forms, which in both cases results in the creation of files that can be stored and read at a later date. The first category of output data is typically referred to as “summary” data and the second type of output consists of grid data. The two types are as follows:

(a) Summary data: this consists of calculated parameters such as oil, water and gas production rates, well bottom hole or tubing head pressures, etc. These data may be plotted as line charts, usually as a function of time, either by using specialised post-processing software, or by using standard graphing software such as Microsoft Excel or Lotus 1,2,3.

(b) Grid data: in this type of data, values such as pressure or saturation to be plotted for each cell at a given time step. These files are typically in binary format, which means that they may only be read by appropriate post-processing software. The reason that ASCII format is not generally used is one of disk space usage. For example, a 100,000 cell model, with output data for 20 time steps, would generate 2 million values of pressure (usually to eight significant figures) during the course of the simulation, and similarly for every other property such as phase saturations, etc.

Two major, and usually understated, elements of good reservoir simulation practice are thus:

• Keeping a record of what each calculation represents (by choosing sensible file names and inserting comments)• Minimising disk usage (by outputting only data that is actually required).

Current software developments are addressing automatic report generation and minimising the time taken to obtain a good history match by automatically varying specified parameters.



4. EXAMPLE INPUT DATA FILE

4.1 Reservoir System to be ModelledIn the first tutorial exercise, Tutorial 1A, the reservoir system to be modelled consists of a five-spot pattern of four production wells surrounding each injection well. However, the symmetry of the system allows us to model a single injection production pair, which will be located at opposite corners of a grid, as shown in Figure 1.

The system is initially at connate water saturation (Swc), and a waterflood calculation is to be performed to evaluate oil production and water breakthrough time. The reservoir will be maintained above the bubble point pressure (Pb) at all times, and thus there is no need to perform calculations for a free gas phase (in the reservoir).

Reservoir and fluid properties, such as layer permeabilities, porosity, oil and water PVT and relative permeability data, are provided, as are the initial reservoir conditions and production schedule (proposed injection and production rates).

The input data file, whether generated using a text editor or by GUI, consists of various sections that incorporate all of these components just described. Here we will go through the input file, TUT1A.DATA, used for this calculation. TUT1A.DATA will also be used as a base case for other tutorial sessions associated with this course. The data format is that required by the Schlumberger GeoQuest Reservoir Technologies model, ECLIPSE 100, but other than syntactical differences, the style of data entry is similar for most other simulators.

While the data file may be set up using a GUI, it is useful in the first instance to set up a simple model using a text editor, thus ensuring by the end of the exercise that every line of data is familiar and relatively well understood. This file can then be used as a starting point for other models, which may be set up by modifying the appropriate parts of this data file.

Production well

Injection well

Two well quarterfive-spot grid

Figure 1. A five spot pattern consists of alternating rows of production and injection wells. The symmetry of the system means that the flow between any two wells can be modelled by placing the wells at opposite corners of a Cartesian grid, and is referred to as a quarter five-spot calculation.



4.2 ECLIPSE SyntaxEach item of data, such as porosities or relative permeabilities, are identified by use of set keywords. The individual sections are also designated by keywords. The syntax, format and function of each keyword may be found in the online manuals. These also give examples of how the keywords may be used.

There are certain rules governing data entry for any simulator, and effort must be made at the outset to get these right; otherwise setting up the input data file can be very frustrating and as time consuming as performing the calculations themselves.

ECLIPSE uses free format. This means that, with a few exceptions, as many or as few spaces, tabs and new lines may be used as desired. However, arranging the file appropriately, such as by lining data up in columns, etc., can improve readability, reducing unnecessary typographical mistakes, and saving time in the long run.

The following additional rules should be noted• Each section starts with a keyword• There must be no other characters (or spaces) on the same line as a keyword (i.e. each keyword must start in column 1, and be immediately followed by a new line keystroke)• All data associated with a keyword must appear on the subsequent lines• Data entry is terminated by a forward slash symbol (/)• Lines beginning with two dashes (--) are ignored, and treated as comment lines• Blank lines are ignored

To illustrate the use of keywords, data and comments, the following style conventions, illustrated in Figure 2, will be used here.

FEATURE EXAMPLE FROM DATA FILE

Comments Number of cells NX NY NZ

KEYWORDS DIMENS

Data (followed by /) 5 5 3 /

4.3 Model DimensionsThe first step in setting up a model is to define:

• Title of run• Type of geometry to be used (Cartesian or radial, though Cartesian is often the default)• Number of cells in each direction (x, y, z, or r, θ, z)• Phases to be modelled (oil, water, gas, vapourised oil in the gas, dissolved gas in the oil)

Figure 2. Example of conventions used to identify components of input data file. It should be noted that the actual input file should be in ASCII text format only (as produced by Notepad, WordPad or other basic text editor), and should not contain italic, bold or coloured letters.



• Units to be used (field, metric or lab)• Number of wells• Start date for simulation (usually corresponding to the date of first oil production)

The model set up in Tutorial 1A consists of a Cartesian grid of 5 x 5 x 3 cells, each cell having dimensions of 500 ft x 500 ft x 50 ft, as shown in Figure 3.

123

505050

1

1

2

2

3

3

4

4

5

5

500

500

500

500

500

500

500

500

500

500

Cartesian is the default geometry used by ECLIPSE, so it does need to be specified explicitly. A two-phase oil and water calculation is to be performed in this model, with the injection and production wells to be located at opposite corners, and completed in all three layers. ECLIPSE allows wells to be grouped together so that the cumulative production or injection rates may be specified or calculated. Here, we will assume that the injection and production wells are in two separate groups. Field units are to be used throughout the input data file. (Note that once a choice of units has been made, it must be used consistently for all data entry. This precludes, for example, using feet (field units) for depths and bars (metric units) for pressures in the same run.) The title for this calculation will be “3D 2-Phase”, and it is assumed that first oil was on 1st January 2001. Generated output should be written to a single unified output file. (The ECLIPSE default is to create a separate output file for every time step, which has the advantage that not all the data output data is lost if one file is in some way corrupted, but this may result in an unmanageable number of files being generated.)

The above information is all that is required for defining the dimensioning data that goes in the first section of an ECLIPSE data file, referred to as the RUNSPEC section. The form in which this data should be entered is shown in Figure 4. The following keywords are used:

RUNSPEC Section headerDIMENS Number of cells in X, Y and Z directionsOIL Calculate oil flowsWATER Calculate water flows FIELD Use field units throughout (i.e. feet, psi, lb, bbl, etc.)WELLDIMS Number of wells, connections per well, groups, wells per group

Figure 3. Cartesian grid of 5 x 5 x 3 cells used to represent reservoir system to be modelled in Tutorial 1A.



UNIFOUT Unified output fileSTART Start date of simulation (1st day of production)

Every time an unfamiliar keyword is encountered, it is well worth looking it up in the online manual, and this is probably as good a point to start as any! Particular attention should be paid to units. For example, when using field units gas rates are entered in MSCF/day. Entering a value in SCF/day would be allowed by the simulator, but would lead to completely wrong results.

TUT1A. DATA

Base case for tutorials

RUNSPEC

TITLE 3D 2-Phase

Number of cells NX NY NZ

DIMENS 5 5 3 /

PhasesOIL

WATER

UnitsFIELD

Maximum well / connection / group values #wells #cons/w #grps #wells/grp

WELLDIMS 2 3 2 1 /

Unified output filesUNIFOUT

Simulation start dateSTART 1 JAN 2001 /

Figure 4. RUNSPEC Section of input data file.



4.4 Grid and Rock PropertiesHaving identified the number of grid cells in the X, Y and Z directions required to model the reservoir or part of the reservoir being studied, the following grid properties must be defined:

• Dimensions of each cell• Depth of each cell (or at least the top layer)• Cell permeabilities in each direction (x, y, z, or r, θ, z)• Cell porosities

If a Cartesian grid is being used, as here, then the size of each cell may be specified by providing data on the length, width and height of each cell. The current grid has 75 cells (5 x 5 x 3), and thus 75 values must be specified for each property.

Each cell in the model is to be 500 ft long, by 500 ft wide, by 50 ft thick. There are three layers, the top layer being at a depth of 8,000 ft. All three layers are assumed to be continuous in the vertical direction, so there is no need to specify the depths of the second and third layers - the simulator can calculate these implicitly from the depth of the top layer and the thickness of the top and middle layers. The formation has a uniform porosity of 0.25, and the layer permeabilities in each direction are given below.

Permeability (mD) Layer Horizontal Vertical X direction Y direction Z direction1 200 150 202 1000 800 1003 200 150 20

This represents the minimum information that is required for defining the grid and rock properties for the second section of an ECLIPSE data file, referred to as the GRID Section. It is useful to output a file that allows these values to be viewed graphically by one of the post-processors. This enables a quick visual check that the grid data has been entered correctly. The following keywords are used:

GRID Section headerDX Size of cells in the X directionDY Size of cells in the Y directionDZ Size of cells in the Z directionTOPS Depth of cellsPERMX Cell permeabilities in the X directionPERMY Cell permeabilities in the Y directionPERMZ Cell permeabilities in the Z directionPORO Cell porositiesINIT Output grid values to .INIT file

ECLIPSE normally assumes that grid values, such as DY, DZ, PERMX, PORO, etc., are being entered for the whole grid. If values are only being entered for a subsection of the grid, then the BOX and ENDBOX keywords may be used to identify this subsection (an example is given later). If no BOX is defined, or after an ENDBOX



keyword, ECLIPSE assumes that cell values are being defined for the entire grid.

The values of each grid property, such as cell length, DX, are read in a certain order. If the co-ordinates of each cell are specified by indices (i, j, k), where i is in the X direction, j is in the Y direction, and k is in the Z direction, then the values are read in with i varying fastest, and k slowest. The first value that is read in is for cell (1, 1, 1), and the last one is for cell (NX, NY, NZ), where NX, NY and NZ are the number of cells in the X, Y and Z directions respectively.

Thus, in this (NX=5, NY=5, NZ=3) model, the values of DX (and every other grid value such as DY, DZ, PERMX, PORO, etc.) will be read in in the order shown in Figure 5.

123

505050

1

1

2

2

3

3

4

4

5

5

500

500

500

500

500

500

500

500

500

500No i j k1 1 1 12 2 1 13 3 1 14 4 1 15 5 1 16 1 2 17 2 2 18 3 2 1. . . .. . . .24 4 5 125 5 5 126 1 1 227 2 1 2. . . .. . . .75 5 5 3

Y

Z

X

The length (DX) of each of the 75 cells in Tutorial 1A is the same: 500 ft. Thus the data may be entered as:

DX500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 /

Most simulators will allow the definition of multiple cells, each with the same size, to be lumped together. In ECLIPSE this is done by prefixing the value (cell size) by the number of cells to be assigned that value, and separating these two numbers by a “*”. Thus, since all 75 cells in the model have a length of 500 ft, this may be entered as:

DX75*500 /

Note that the multiplier comes first, then the “*” operator, then the value. There should be no spaces on either side of the “*”.

Figure 5. Order in which cell property values are read in by ECLIPSE, starting at (1, 1, 1), and finishing at (5, 5, 3), with the i index varying the fastest, and the k index the slowest.



This convention may be used for all other grid parameters.

If cell depths are only to be defined for the top layer of cells using the TOPS keyword, then a box must be used to identify this top layer as the only section of the grid for which depths are being defined. The box that encompasses the top layer is defined as from 1 to 5 in the X direction, 1 to 5 in the Y direction, but only 1 in the Z direction. Instead of 75 cells for the whole model, there are only 25 cells in this section of the model, and thus only 25 values of TOPS need be defined:

BOX 1 5 1 5 1 1 /

TOPS25*8000 /

ENDBOX

The GRID Section of the Tutorial 1A input data file should be as shown in Figure 6.



GRID

Size of each cell in X, Y and Z directionsDX75*500 /

DY75*500 /

DZ75*50

TVDSS of top layer only X1 X2 Y1 Y2 Z1 Z2

BOX 1 5 1 5 1 1 /

TOPS25*8000 /

ENDBOX

Permeability in X, Y and Z directions for each cellPERMX25*200 25*1000 25*200 /

PERMY25*150 25*800 25*150 /

PERMZ25*20 25*100 25*20 /

Porosity of each cellPORO75*0.2 /

Output file with geometry and rock properties (.INIT)INIT

Figure 6. GRID Section of input data file.



4.5 Fluid PropertiesHaving defined the grid and rock properties such as permeability and porosity, the following Pressure/Volume/Temperature (PVT), viscosity, relative permeability and capillary pressure data must be defined:

• Densities of oil, water and gas at surface conditions• Formation factor and viscosity of oil vs. pressure• Pressure, formation factor, compressibility and viscosity of water• Rock compressibility• Water and oil relative permeabilities, and oil-water capillary pressure vs. water saturation

The densities of the three phases ρoil, ρwater and ρgas are given below. (Note that all three densities must be supplied, even though free gas is not modelled in the system.)

Oil Water Gas(lb/ft3) (lb/ft3) (lb/ft3)49 63 0.01

The oil formation volume factor (Bo) and viscosity (μo) is provided as a function of pressure (P).

Pressure Oil FVF Oil Viscosity(psia) (rb/stb) (cP)300 1.25 1.0800 1.20 1.16000 1.15 2.0

At a pressure of 4,500 psia, the water formation volume factor (Bw) is 1.02 rb/stb, the compressibility (cw) is 3 x 10-6 PSI-1 and the viscosity (μw) is 0.8 cP. Water compressibility does not change with pressure within the pressure ranges encountered in the reservoir, and thus viscosibility (∂μw/∂P) is 0. The rock compressibility at a pressure of 4,500 psia is 4 x 10-6 PSI-1. Water and oil relative permeability data and capillary pressures are given as functions of water saturation below.

Sw krwater kroil capillary pres. (psi)0.25 0.00 0.90 4.00.50 0.20 0.30 0.80.70 0.40 0.10 0.20.80 0.55 0.00 0.1

This data should be inserted in the third section of the ECLIPSE data file, the PROPS section. The form in which this data should be entered is shown in Figure 7. The following keywords are used:

PROPS Section headerDENSITY Surface density of oil, water and gas phasesPVDO PVT data for dead oil relating FVF and viscosity to pressurePVTW PVT data for water relating FVF, compressibility and viscosity to pressure ROCK Compressibility of the rock



SWOF Table relating oil and water relative permeabilities and oil-water capillary pressure to water saturations

PROPS Densities in lb/ft3 Oil Water Gas

DENSITIES 49 63 0.01 /

PVT data for dead oil P Bo Vis

PVDO 300 1.25 1.0 800 1.20 1.1 6000 1.15 2.0 /

PVT data for water P Bw Cw Vis Viscosibility

PVTW 4500 1.02 3e-06 0.8 0.0 /

Rock compressibility P Cr

ROCK 4500 4e-06 /

Water and oil rel perms and capillary pressure Sw Krw Kro Pc

SWOF 0.25 0.0 0.9 4.0 0.5 0.2 0.3 0.8 0.7 0.4 0.1 0.2 0.8 0.55 0.0 0.1 /Figure 7.

PROPS Section of input data file.



4.6 Initial ConditionsOnce the rock and fluid properties have all been defined, the initial pressure and saturation conditions in the reservoir must be specified. This may be done in one of three ways:

1) Enumeration2) Equilibration3) Restart from a previous run

1) Enumeration. In this method of initialising the model, the pressure, oil and water saturations in each cell at time = 0 are set in much the same way as permeabilities and porosities are set. This method is the most complicated and least commonly used. A failure to correctly account for densities when setting the pressures in cells at different depths will result in a system that is not initially in equilibrium.

2) Equilibration. This is the simplest and most commonly used method for initialising a model. A pressure at a reference depth is defined in the input data, and the model then calculates the pressures at all other depths using the previously entered density data to account for hydrostatic head. The depths of the water-oil and gas-oil contacts are also specified if they are within the model, and the initial saturations can then be set depending on position relative to the contacts. (In a water-oil system, above the oil-water contact the system is at connate water saturation, below the contact Sw = 1.)

3) Restart from a previous run. If a model has already been run, then one of the output time steps can be used to provide the starting fluid pressures and saturations for a subsequent calculation. This option will typically be used where a model has been history matched against field data to the current point in time, and various future development scenarios are to be compared. A restart run will use the last time step of the history-matched model as the starting point for a predictive calculation, which may then be used to assess future performance. Time is saved by not repeating the entire calculation.

In this example the model is being set up to predict field performance from first oil, and thus there is no previous run to use as a starting point. The equilibration model is to be used, with an initial pressure of 4,500 psia at 8,000 ft. The model should initially be at connate water saturation throughout. To achieve this, the water-oil contact should be set at 8,200 ft, 50 ft below the bottom of the model. The water saturation in each cell will be set to the first value in the relative permeability (SWOF) table, which is 0.25. (If any cells were located below the water-oil contact, they would be set to the last value in the relative permeability table, which would thus have to include relative permeability and capillary pressure values for Sw = 1.)

An output file containing initial cell pressures and saturations for display should be requested so that a visual check can be made that the correct initial values of these properties have been calculated.

This initialisation data should be inserted in the fourth section of the ECLIPSE data file, the SOLUTION section. The form in which this data should be entered is shown in Figure 8. The following keywords are used:

SOLUTION Section header



EQUIL Equilibration data (pressure at datum depth and contact depths)RPTRST Request output of cell pressures and saturations at t = 0

SOLUTION

Initial equilibration conditions Datum Pi@datum WOC Pc@WOC

EQUIL 8075 4500 8200 0 /

Output to restart file for t=0 (.UNRST) Restart file Graphics for init cond only

RPTRST BASIC=2 NORST=1 /

4.7 Output RequirementsClearly there is no point in performing a reservoir simulation if no results are output. The parameters that should be calculated are specified in the SUMMARY section by the use of appropriate keywords, but for this section only the keywords are not found in the main section of the manual, but in the Summary Section Overview.

Most of the summary keywords consist of four letters that follow a basic convention.

1st letter: F - field R - region W - well C - connection B - block

2nd letter: O - oil (stb in FIELD units) W - water (stb in FIELD units) G - gas (Mscf in FIELD units) L - liquid (oil + water) (stb in FIELD units) V - reservoir volume flows (rb in FIELD units) T - tracer concentration S - salt concentration C - polymer concentration N - solvent concentration

Figure 8. SOLUTION Section of input data file.



3rd letter: P - production I - injection

4th letter: R - rate T - total

Thus, use of the keyword FOPR requests that the Field Oil Production Rate be output, and WWIT represents Well Water Injection Total, etc.

Keywords beginning with an F refer to the values calculated for the field as a whole, and require no further identification. However, keywords beginning with another letter must specify which region, well, connection or block they refer to. Thus, for example, a keyword such as FOPR requires no accompanying data, but WWIT must be followed by a list of well names, terminated with a /. If no well names are supplied, and the keyword is followed only by a /, the value is calculated for all wells in the model. An example would be

FOPR

WWITInj /

WBHP/

Here, the following will be calculated:• Oil production rate for the entire field• Cumulative water injection for well “Inj”• Well bottomhole pressure for all wells in the model

In Tutorial 1A the following parameters should be calculated and output:• Field average pressure• Bottomhole pressure of all wells• Field oil production rate• Field water production rate• Field oil production total• Field water production total• Water cut in well PROD• CPU usage

In addition, the output Run Summary file (.RSM) should be defined such that it can easily be read into MS Excel.

The form in which this data should be entered is shown in Figure 9. The following keywords are used:

SUMMARY Section headerFPR Field average pressure



WBHP Well Bottomhole pressureFOPR Field Oil Production RateFWPR Field Water Production RateFOPT Field Oil Production TotalFWPT Field Water Production TotalWWCT Well Water CutCPU CPU usage EXCEL Create summary output as Excel readable Run Summary file

SUMMARY

Field average pressureFPR

Bottomhole pressure of all wellsWBHP/

Field oil production rateFOPR

Field water production rateFWPR

Field oil production totalFOPT

Field water production totalFWPT

Water cut in PRODWWCTPROD /

CPU usageTCPU

Create Excel readable run summary file (.RSM)EXCEL

4.8 Production ScheduleHaving defined the initial conditions (t = 0) in the SOLUTION Section, the final part of the input data file defines the well controls and time steps (t > 0) in the SCHEDULE Section. The main functions that are performed here are:

• Specify grid data to be output for display or restart purposes• Define well names, locations and types• Specify completion intervals for each well• Specify injection and production controls for each well for each given period

Figure 9. SUMMARY Section of input data file.



(time step)Basic pressure and saturation data should be generated at every time step to enable a 3-D display of the model to be viewed at each time step.

A production well, labeled PROD and belonging to group G1, is to be drilled in cell position (1,1), and a water injection well, INJ belonging to group G2, is to be drilled in cell (5,5). Both wells will have 8 inch diameters, and should be completed in all three layers. They both have pressure gauges at their top perforation (8,000 ft). Both will be open from the start of the simulation, enabling production of 10,000 stb/day of liquid (oil + water) from the system, and pressure support provided by injection of 11,000 stb/day of water. The simulation should run for 2,000 days, outputting data every 200 days.

A number of keywords in the SCHEDULE section will be able to read in data that the user may wish to default or not supply all. This can be done by using the “*” character, with the number of values to be defaulted or ignored on the left, and the space to the right left blank. Thus “1* “ ignores one value, “2* “ ignores the next two values, etc. For example, in the COMPDAT keyword, we may wish to specify values for items 1 to 6, and item 9 (which is the wellbore diameter), but items 7 and 8 should remain unspecified. This may be achieved as follows:

Completion interval Well Location Interval Status Well name I J K1 K2 O or S ID

Item number 1 2 3 4 5 6 7 8 9 PROD 1 1 1 3 OPEN 2* 0.6667 //

COMPDAT

The keywords to be used are:

SCHEDULE Section headerRPTRST Request output of cell pressures and saturations at all time steps (t > 0)WELSPECS Define location of wellhead and pressure gaugeCOMPDAT Define completion intervals and wellbore diameterWCONPROD Production controlWCONINJ Injection controlTSTEP Time step sizes (for output of calculated data)END End of input data file

These keywords should appear as the last section of the data file as shown in Figure 10. Although not the case in this simple example, this section will typically be the longest, containing flow rates for each well on a monthly basis for the history of the field. It should be noted that the time steps input here refer to time intervals at which



data are output. The simulator will try to use these time step sizes as numerical time step also, but if the calculations do not converge, it will automatically cut the numerical time step sizes.

SCHEDULE

Output to Restart file for t > 0 (.UNRST) Restart file Graphics every step only

RPTRST BASIC=2 NORST=1 /

Location of wellhead and pressure gauge Well Well Location BHP Pref. name group I J datum phase

WELSPECS PROD G1 1 1 8000 OIL / INJ G2 5 5 8000 WATER //

Completion interval Well Location Interval Status Well name I J K1 K2 0 or S ID

COMPDAT PROD 1 1 1 3 OPEN 2* 0.6667 / INJ 5 5 1 3 OPEN 2* 0.6667 //

Production control Well Status Control Oil Wat Gas Liq. Resv BHP name mode rate rate rate rate rate lim

WCONPROD PROD OPEN LRAT 3* 10000 1* 2000 //

Injection control Well Fluid Status Control Surf Resv Voidage BHP Name TYPE mode rate rate frac flag lim

WCONINJ INJ WATER OPEN RATE 11000 3* 20000 //

Number and size (days) of timestepsTSTEP10*200 /

END

Figure 10. SCHEDULE Section of input data file.



5. RUNNING ECLIPSE AND FILE NAME CONVENTIONS

5.1 Running ECLIPSE on a PCOnce the input data file has been edited, it should be saved with the file extension “.DATA”. For Tutorial 1A choose a file name such as “TUT1A.DATA”. Care should be taken that the text editor does not append the suffix “.txt” onto the file name, as this will render the file unreadable to ECLIPSE. This can be avoided by using Menu->File->Save As and selecting “All Files” instead of “Text Documents” as the “Save as type”.

Having saved the input data file, the GeoQuest Launcher may be used to run ECLIPSE, and the user will be prompted to locate the input file. The simulation will then start, and will run by reading every keyword in the order in which they appear in the input file.

5.2 File Name ConventionsIf any of the keywords or data are incorrectly entered, the run will stop without performing the required flow calculations. If the simulator is satisfied that all data has been entered correctly, then it will perform the requested flow calculations, and various output files will be generated during the run, as follow:

TUT1A.PRT The .PRT file is an ASCII file that is generated for every successful and unsuccessful run. It contains a list of the keywords, and will indicate if any keywords have been incorrectly entered. If the simulation fails, this file should be checked for the cause of the failure. A search for an ERROR in this file will usually reveal which keyword was the culprit. If the run was successful, this file will contain summary data such as field average pressure and water cut for each time step.

TUT1A.GRID The .GRID file is a binary file that contains the geometry of the model, and is used by post processors for displaying the grid outline.

TUT1A.INIT The .INIT file is a binary file that contains initial grid property data such as permeabilities and porosities. These may be displayed using a post-processor to check that the data have been entered correctly, and to display a map of field permeabilities, etc.

TUT1A.UNRST The .UNRST file is a unified binary file that contains pressure and saturation data for each time step. These may be displayed using a post-processor, or may be used as the starting point for an ECLIPSE restart run.

TUT1A.RSM The .RSM file is an ASCII file that can be read into MS Excel to display summary data in line chart format. This file is only created once the run has completed. During the run the summary data is stored in file TUT1A.USMRY, which is a binary file readable only by the GeoQuest post-processors.

The GeoQuest post-processors are Graf and FloViz. During this course FloViz is used for 3D displays of the model, showing, for example, progression of the water flood



by displaying saturations varying with time (Figure 11). Excel is used to display line charts such as water cut vs. time, etc. Both grid and line charts may be displayed in Graf, which is a powerful though more complicated post-processor than FloViz. The functionality of Graf is being replaced by ECLIPSE Office, which is a GUI that may be used for setting up data files and viewing results, and may optionally be used for subsequent tutorial sessions. However, students are encouraged to use a basic text editor for pre-processing, and Excel and FloViz for post-processing for Tutorial 1A, since this will give a better understanding of the calculations being performed.

6. CLOSING REMARKS

In this section of the course, we have presented the working details of how to set up a a practical reservoir simulation model. We have used the Schlumberger GeoQuest ECLIPSE software for the specific case presented here. However, the general procedures are very similar for most other commercially available simulators. The various input data that are required should be quite familiar to you from the discussion in the introductory chapter of the course (Chapter 1). However, how these are systematically organised as input for the simulator should now be clear. The vast possibilities for simulation output have also been discussed in this section and you should know be aware of how to choose this output, organise it is files and then visualise it later. The issue of visualisation was also discussed previously but its value should be better appreciated by the student.

Figure 11 FloViz visualisation of water saturation for four time steps, showing progression of water flood in Tutorial 1A. The injection well is on the left and the production well on the right of the 5 x 5 x 3 model.



ECLIPSE TUTORIAL 1

(A 3D 2-Phase Reservoir Simulation Problem)

A. Prepare an input data file for simulating the performance of a two-phase (water/oil) three dimensional reservoir of size 2500’ x 2500’ x 150’, dividing it into three layers of equal thickness. The number of cells in the x and y directions are 5 and 5 respectively. Other relevant data are given below, using field units throughout:

Depth of reservoir top: 8000 ftInitial pressure at 8075’: 4500 psiaPorosity: 0.20

Permeability in x direction: 200 mD for 1st and 3rd layers and 1000 mD for 2nd layer.Permeability in y direction: 150 mD for 1st and 3rd layers and 800 mD for 2nd layer.Permeability in z direction: 20 mD for 1st and 3rd layers and 100 mD for 2nd layer.

123

505050

1

1

2

2

3

3

4

4

5

5

500

500

500

500

500

500

500

500

500

500

Water and Oil Relative Permeability and Capillary Pressure Functions. Water Saturation krw kro Pcow (psi) 0.25* 0.0 0.9 4.0 0.5 0.2 0.3 0.8 0.7 0.4 0.1 0.2 0.8 0.55 0.0 0.1 1.0 1.00 0.0 0.0 * Initial saturation throughout.

Figure 1 Schematic of model.



Water PVT Data at Reservoir Pressure and Temperature.

Pressure Bw cw μw Viscosibility (psia) (rb/stb) (psi-1) (cp) (psi-1)

4500 1.02 3.0E-06 0.8 0.0

Oil PVT Data, Bubble Point Pressure (Pb) = 300 psia.

Pressure Bo Viscosity (psia) (rb/stb) (cp)

300 1.25 1.0 800 1.20 1.1 6000 1.15 2.0

Rock compressibility at 4500 psia: 4E-06 psi-1

Oil density at surface conditions: 49 lbs/cfWater density at surface conditions: 63 lbs/cfGas density at surface conditions: 0.01 lbs/cf

The oil-water contact is below the reservoir (8,200 ft), with zero capillary pressure at the contact.

Drill a producer PROD, belonging to group G1, in Block No. (1, 1) and an injector INJ, belonging to group G2, in Block No. (5, 5). The inside diameter of the wells is 8”. Perforate both the producer and the injector in all three layers. Produce at the gross rate of 10,000 stb liquid/day and inject 11,000 stb water/day. The producer has a minimum bottom hole pressure limit of 2,000 psia, while the bottom hole pressure in the injector cannot exceed 20,000 psia. Start the simulation on 1st January 2000, and use 10 time steps of 200 days each.

Ask the program to output the following data:

• Initial permeability, porosity and depth data (keyword INIT in GRID section)

• Initial grid block pressures and water saturations into a RESTART file (keyword RPTRST in SOLUTION section)

• Field Average Pressure (FPR) Bottom Hole Pressure for both wells (WBHP) Field Oil Production Rate (FOPR) Field Water Production Rate (FWPR) Total Field Oil Production (FOPT) Total Field Water Production (FWPT) Well Water Cut for PROD (WWCT) CPU usage (TCPU)

to a separate Excel readable file (using keyword EXCEL) in the SUMMARY section.



• Grid block pressures and water saturations into RESTART files at each report step of the simulation (keyword RPTRST in SCHEDULE section)

Procedure:

1 Edit file TUT1A.DATA in folder \eclipse\tut1 by dragging it onto the Notepad icon, fill in the necessary data, and save the file.

2 Activate the ECLIPSE Launcher from the Desktop or the Start menu.

3 Run ECLIPSE and use the TUT1A dataset.

4 When the simulation has finished, use Excel to open the output file TUT1A.RSM, which will be in the \eclipse\tut1 folder. You will need to view. “Files of type: All files (*.*)” and import the data as “Fixed width” columns.

5 Plot the BHP of both wells (WBHP) vs. time and the field average pressure (FPR) vs. time on Figure 1.

6 Plot the water cut (WWCT) of the well PROD and the field oil production rate (FOPR) vs. time on Figure 2.

7 Plot on Figure 3 the BHP values for the first 10 days in the range 3,500 psia to 5,500 psia.

Explain the initial short-term rise in BHP in the injection well and drop in BHP in the production well. Account for the subsequent trends of these two pressures and of the field average pressure, relating these to the reservoir production and injection rates, water cut and the PVT data of the reservoir fluids.

B. Make a copy of the file TUT1A.DATA called TUT1B.DATA in the same folder tut1.

By modifying the keyword TSTEP change the time steps to the following:

15*200

Modify the WCONINJ keyword to operate the injection well at a constant flowing bottom hole pressure (BHP) of 5000 psia, instead of injecting at a constant 11,000 stb water/day (RATE).

Add field volume production rate (FVPR) to the items already listed in the SUMMARY section.

Run Eclipse using the TUT1B.DATA file, and then plot the two following pictures in Excel:



Figure 4: Both well bottom hole pressures and field average pressure vs. time, showing pressures in the range 3,700 psia to 5,100 psia.Figure 5: Field water cut and field volume production rate vs. time.

Account for the differences between the pressure profiles in this problem and Tutorial 1A. To assist with the interpretation, calculate total mobility as a function of water saturation for 4 or 5 saturation points, using:

MTOT(Sw ) =

Kro(Sw )

µo

+Krw (Sw )

µw

and show how this would change the differential pressure across the reservoir as the water saturation throughout the reservoir increases. From Figure 5, explain the impact of the WWCT profile (fraction) on the FVPR (rb/day).

C.Copy file TUT1B.DATA to TUT1C.DATA in the same folder.

This time, instead of injecting at a constant flowing bottom hole pressure of 5000 psi, let the simulator calculate the injection rate such that the reservoir voidage created by oil and water production is replaced by injected water. To do this, modify the control mode for the injection well (keyword WCONINJ) from BHP to reservoir rate (RESV), and use the voidage replacement flag (FVDG) in item 8. Set the upper limit on the bottom hole pressure for the injection well to 20,000 psia again.

Note the definitions given in the manual for item 8 of the WCONINJ keyword. Based on the definition for voidage replacement, reservoir volume injection rate = item 6 + (item 7 * field voidage rate)

Therefore, to inject the same volume of liquid as has been produced, set item 6 to 0, and item 7 to 1.

Run Eclipse using the TUT1C.DATA file, and then run Floviz, to display the grid cell oil saturations (these displays need NOT be printed).

Discuss the profile of the saturation front in each layer, and explain how it is affected by gravity and the distribution of flow speeds between the wells.



TUT1A.DATA

RUNSPEC

TITLE

NDIVIX NDIVIY NDIVIZ

DIMENS

OILWATERFIELD

NWMAXZ NCWMAX NGMAXZ NWGMAX

WELLDIMS

START

GRIDDX

DZ

PORO

X1 X2 Y1 Y2 Z1 Z2

BOXTOPS

?DBOX

PERMX

PERMY

PERMZ

INIT



PROPS

OIL WAT GAS

DENSITY

P Bo Vis

PVDO

P Bw Cw Vis Viscosibility

PVTW

P Cr

ROCK

Sw Krw Kro Pc

SWOF

SOLUTION

DATUM Pi@DATUM WOC Pc@WOC GOC Pc@GOC

EQUIL Block Block Create initial P Sw restart file

RPTSOL

SUMMARY

RPTSMRY

30

SCHEDULE Block Block Create restart file P Sw at each time step

RPTSCHED

WELL WELL LOCATION BHP PREF. NAME GROUP I J DATUM PHASE

WELSPECS

WELL LOCATION INTERVAL STATUS WELL NAME I J K1 K2 O or S ID

COMPDAT

WELL STATUS CONTROL OIL WAT GAS LIQ RESV BHP NAME MODE RATE RATE RATE RATE RATE LIMIT

WCONPROD

WELL FLUID STATUS CONTROL SURF RESV VOIDAGE BHP NAME TYPE MODE RATE RATE FRAC FLAG LIMIT

WCONINJ

DAYSTSTEP

END

CONTENTS

1. INTRODUCTION

2. GRIDDING IN RESERVOIR SIMULATION 2.1. Introduction 2.2. Accuracy of Simulations and Numerical Dispersion 2.3. Grid Orientation Effects 2.4 Local Grid Refinement (LGR) 2.5 Distorted Grids and Corner Point Geometry 2.6 Issues in Choosing a Reservoir Simulation Grid 2.7 Streamline Simulation

3. THE CALCULATION OF BLOCK TO BLOCK FLOWS IN RESERVOIR SIMULATORS 3.1. Introduction to Averaging of Block to Block Flows 3.2. Averaging of the Two-Phase Mobility Term,

λ λ

λ λ

λ

T o i o o i

pp

p

p

p

i i

i i

p

p

i i

S S

Q kAB

Px

kAB

P Px x

kA

B

x

x x

( )( ) ( )( )

= −

∆∆

= −

−

∆ + ∆

∆

= ∆ + ∆

− +

−

−

−

1 2 1 2

1

1

1

2

2

/ /

. . . .

or

kkrp

pµ

4. WELLS IN RESERVOIR SIMULATION 4.1. Basic Idea of a Well Model 4.2. Well Models for Single and Two-Phase Flow 4.3 Well Modelling in a Multi-Layer System 4.4. Modelling Horizontal Wells 4.5 Hierarchies of Wells and Well Controls

5. CLOSING REMARKS - GRIDDING AND WELL MODELLING

44Gridding And Well Modelling





• understand and be able to describe the basic idea of gridding and of spatial and temporal discretisation.

• be aware of all the main types of grid in 1D, 2D and 3D used in reservoir simulation and be able to describe examples of where it is most appropriate to use the different grid types.

• be able to give a short description with simple diagrams of the phenomena of numerical dispersion and grid orientation and to explain how these numerical problems can be overcome.

• be familiar with more sophisticated issues in gridding such as the use of local grid refinement (LGR), distorted, PEBI and corner point grids

• given a specific task for reservoir simulation, the student should be able to select the most appropriate grid dimension (1D, 2D, 3D) and geometry/structure (Cartesian, r/z, corner point etc.).

• be able to discuss the issues of grid fineness/coarseness (i.e. how many grid blocks do we need to use) in terms of some examples of what can happen if an inappropriate number of grid blocks are used in a reservoir simulation calculation.

• be able to describe the basic ideas behind streamline simulation and to compare it with conventional reservoir simulation in terms of its advantages and disadvantages.

• be familiar with the different types of average used for single phase kA, two phase relative permeabilities (krp) and for μp and Bp (p = o, w, g phase) when calculating the block to block flows (Qp) in a reservoir simulator.

• be able to describe the physical justification for using the upstream value of two phase relative permeabilities when calculating the block to block flows (Qp) in a reservoir simulator.

• understand the origin of all the pressure drops that are experienced by the reservoir fluids from deep in the reservoir, through to the wellbore and then to the surface facilities and beyond

• know what a well model is and what productivity index (PI) is, including knowing the radial Darcy Law and how this gives a mathematical expression for PI for single phase flow (know the expression from memory).

• be able to describe the main issues in relating the pressure in the reservoir, Pe, at some drainage radius, re, to an average grid block pressure and how this leads to the Peaceman formula (Δr = 0.2 Δx) which is then used to calculate PI.

• be able to extend PI to the concept of multi-phase flow to calculate PIw and PIo.



• describe a well model for a multi layer system where there is two phase flow into the wellbore.

• understand and be able to describe the various types of well constraint that can be applied e.g. injection volume constrained wells, well flowing pressure constraints and voidage replacement constraints.

GRIDDING AND WELL MODELLING IN RESERVOIR SIMULATION

This chapter deals with the related issues of grid selection and well modelling in reservoir simulation.

Gridding: Examples of various grid geometries are presented including 1D linear, 2D areal, 2D cross-sectional , 2D radial (r/z) and 3D Cartesian cases. Selection of grid geometry, how fine a grid to take and potential problems are discussed. Non-mathematical introductions to the concepts of numerical dispersion and grid orientation are given along with some examples of the consequences of these effects. More sophisticated gridding such as PEBI grids, distorted grids and local grid refinement (LGR) are illustrated with examples. A brief discussion of streamline simulation is also presented. Treatment of the block to block flows in reservoir simulation is presented and it shown how the various inter-block quantities such as the permeability and relative permeabilities are averaged.

Well Modelling: All interactions between the surface facilities and the reservoir takes place through the injector and producer wells. It is therefore very important to model wells accurately in the reservoir simulation model. The central issue with a “well model” in a simulator is that it must represent a near singular line source within (usually) a very large grid block. The basic ideas of well modelling are explained and how simple well controls are applied is introduced. Modelling of horizontal wells and the control of well hierarchies are also briefly discussed.

2 GRIDDING IN RESERVOIR SIMULATION

2.1 IntroductionBy this point in the course, you will be familiar with the idea of gridding since it has been discussed in Chapter 1 both in general terms and with reference to the SPE examples (Cases 1 - 3; Section 1.3). You will also have seen how to set up a 3D grid in Chapter 3. Basically, the gridding process is simply one of chopping the reservoir into a (large) number of smaller spatial blocks which then comprise the units on which the numerical block to block flow calculations are performed. More formally, this process of dividing up the reservoir into such blocks is known as spatial discretisation. Recall that we also divide up time into discrete steps (denoted Δt) and this related process is known as temporal discretisation. The numerical details of how the discretisation process is carried out using finite difference approximations of the governing flow equations is presented in Chapter 6. However, we would point out that the grid used for a given application is a user choice - it is certainly not a



reservoir “given” - although, as we will see, there are some practical guidelines that help us to make sensible choices in grid definition.

In a Cartesian grid, which to date has been the mot common type of grid used, we denote the block size as Δx, Δy and Δz and these may or may not be equal. This grid can also be in one dimension (1D), two dimensions (2D) or three dimensions (3D). Some typical examples of 1D, 2D and 3D Cartesian grids are shown in Figure 1. This is the most straightforward type of grid to set up and typical application of such grids are as follows:

- 1D linear grids may be used to simulate 1D Buckley-Leverett type water displacement calculations (x-direction) or for single column vertical displacements (z-direction) such as gravity stable gas displacement of oil (Figures 1(a) and 1(b));

- 2D Cartesian grids: 2D cross-sectional (x/z) grids may be used to; (a) study vertical sweep efficiency in a heterogeneous layered system; (b) calculate water/oil displacements in a geostatistically generated cross-section; (c) generate pseudo-relative permeabilites (can be used to collapse a 3D calculation down to a 2D system); (d) to study the mechanism of a gas displacement process - e.g. to determine the importance of gravity etc. (Figure 1(e));

2D areal (x/y) grids may be used to; (a) calculate areal sweep efficiencies in a waterflood or a gas flood; (b) to examine the stability of a near-miscible gas injection within a heterogeneous reservoir layer; (c) examine the benefits of infill drilling in an areal pattern flood etc. (Figures 1(c) and 1(d));

- 3D (x/y/z) Cartesian grids are used to model a very wide range of field wide reservoir production processes and would often be the default type of calculation for a typical full field simulation of waterflooding, gas flooding, etc. (Figure 1(f)).

Cartesian grids are clearly quite versatile but they are not appropriate for all flow geometries that can occur in a reservoir. For example, close to the wellbore (of a vertical well), the flows are more radial in their geometry. For such systems, an r/z - geometry may be more appropriate as shown in Figure 2. An r/z grid is frequently used when modelling coning either of water or gas into a producer. The pressure gradients near the well are very steep and, indeed, we know from the discussion in Chapter 2, section 3.5 that the pressure varies as ln(r/rw), where rw is the well radius. For this reason, in coning studies, a logarithmically spaced grid is often used for the grid block size, Δr; a logarithmic spacing divides up the grid such that (ri/ ri-1) is constant where Δri = (ri- ri-1) as shown in Figure 2. For example, if we take rw = 0.5 ft and the first grid block size is, Δr1 = 1ft, then this sets (ri/ ri-1) = 3 since r0 = rw = 0.5ft and r1 = 1.5ft.; hence r2 = 3x1.5ft. = 4.5ft. and Δr2 = 3ft., r3 = 13.5ft and Δr3 = 9ft., and so on.



(a) 1D horizontal grid

(b) 1D vertical grid

(d) 2D areal grid

∆x

∆x

∆y

∆z

∆z

1D vertical displacemente.g. 1D vertical gravitydrainage calculations

(c) 2D Areal grid - top view showinginjectors ( ) and producers ( )

Just 1 x z - block in 2D areal grid

Perspective view ofa 2D areal (x/y)reservoir simulationgrid: W = well

W1

W2

W3

y

x

(e) 2D (x/z) cross-sectional model showing a waterflood

∆y

∆x

∆x

∆z

∆z

Water Injector Producer



(f) a 3D Cartesian grid with variable vertical grid (Δz varies from layer to layer)

∆y

∆x

∆x

∆z

∆z

Water Injector Producer

r = radial distance from well∆r = radial grid size (can vary ∆r1, ∆r2, ...)z = vertical coordinate∆zi = vertical grid size (can vary ∆z1, ∆z2, ...)h = height of formationrw = wellbore radius

Notation:

∆zi

top view

z

h

r

Q∆r

ri

∆ri

rw

ri-1

2.2 Accuracy of Simulations and Numerical DispersionThe issue of numerical dispersion was touched upon briefly in Chapter 1 in connection with Case 1 (SPE10022). Here we expand on the concept in a non-mathematical manner. Numerical dispersion is essentially an error due to the fact that we use a grid block approximation for solving the flow equations. A more mathematical description of numerical dispersion is presented in Chapter 6, Section 8 but here we focus on explaining physically how it arises and what its consequences are. We also

Figure 1 Examples of 1D, 2D and 3D Cartesian grids

Figure 2r/z grid geometry - more appropriate for modelling flows in the near well region even in a heterogeneous layered system as shown.



discuss how to reduce this source of error in our simulations or to make it a minor effect. The balance, as we will see, is between accuracy (usually by taking more grid blocks) and computational cost. Ideally, we would like to capture all the main reservoir processes (e.g. frontal displacement, crossflow, gravity segregation etc.) and accurately forecast recovery to some acceptable percentage error, for the minimum number of grid blocks.

A simple schematic of the way that the numerical dispersion error arises is shown step by step in Figure 3. This figure illustrates a simple linear waterflood and we imagine that each of the sub-figures shown from (a) to (e) represents a time step, Δt, of the water injection process. Block i = 1 contains the injector well which is injecting water at a constant volumetric rate of Qw, and block i = 5 contains the producer. The system is initially at water saturation, Swc, and this water is immobile i.e. the relative permeability of water is zero, krw(Swc) = 0. Each block has constant pore volume, Vp = Δx.A.φ where φ is the porosity. In Figure 3(a), we see that after time, t = Δt, some quantity of fluid has been injected into block i = 1; the volume of water injected is Qw. Δt and this would cause a water saturation change in grid block i = 1, ΔSw1 = (Qw. Δt)/Vp i.e. the new water saturation in this block is now, Swc+ (Qw. Δt)/Vp. Over the first time step, no fluid flowed from block to block since the relative permeability of all blocks was zero (krw(Swc) = 0). However, the relative permeability in block i = 1 is now krw(Sw1) > 0. The second time period of water injection is shown in Figure 3(b). Another increment of water, Qw. Δt, is injected into block i = 1 causing a further increase in water saturation Sw1. However, over the second time period, krw(Sw1) > 0 and therefore water can flow from block i = 1 to i = 2, increasing the water saturation such that Sw2 > Swc making the relative permeability in this block, krw(Sw2) > 0. In the third time step, shown in Figure 3 (c), the same sequence occurs except that fluid can now from block 1 → 2 and also 2 → 3, where for the same reasons as explained, krw(Sw3) > 0. In the fourth and fifth time steps (Figures 3(d) and 3(e)), flow can now go from block 3 → 4 and from 4 → 5 where, since krw(Sw5) > 0, then it can be produced from this block, although the relative permeability in block 5 will be very small. Hence, after only five time steps to time , t = 5Δt, the water has reached the producer in block 5 from whence it can be produced (although not a very high rate because the relative permeability is very small) and this is an unsatisfactory situation. If we had taken 10 grid blocks, then clearly a similar argument would apply and water would be produced after 10 time steps - with an even lower relative permeability in block i = 10 - and this is more satisfactory. Indeed, this underlies why we take more grid blocks. This simple illustration explains in a quite physical way the basic idea of numerical dispersion.



1.0

00 20 40 60 80 100

krw

Swc = 20% Sor = 30%

Sw %

Relative Permeability to Water

(a)

(b)

(c)

(d)

(e)

WaterInjection

Qw

Sw = Swc

Mixing due to numerical dispersion& relative permeability effects

i = 1 i = 2 i = 3 i = 4 i = 5

∆x

OilProduction

Time Step

1t = ∆t

2t = 2.∆t

3t = 3.∆t

4t = 4.∆t

5t = 5.∆t

∆Sw1

The frontal spreading effect of numerical dispersion can be seen when we try to simulate the actual saturation profile, Sw(x,t), in a 1D waterflood. Under certain conditions, this may have an analytical solution, e.g. the well known Buckley-Leverett solution described in Chapter 2, Section 4.2. This often has a shock front solution for the advancing water saturation profile, Sw(x,t) as shown in Figure 4. Clearly, this sharp front may be “lost” in a grid calculation since, at time t, the front will have a definite position, x(t). However, in a grid system with block size Δx, any saturation front can only be located within Δx as shown in Figure 4. If we take more grid blocks (Δx decreases), then we will locate the front more accurately. Indeed, taking more and more blocks we will gradually get closer to the analytical (correct) solution. Hence, one method of reducing numerical dispersion is to increase the number of grid blocks. An alternative is to use a numerical method which has inherently less dispersion in it but we will not pursue this here. Another approach is to use pseudo functions to control numerical dispersion - as we briefly introduced in Chapter 1 - and this is discussed further below.

Figure 3Effect of grid on water breakthrough time - numer-ical dispersion.



1.0

0

Sw

∆x x

Numerical Grid block size = ∆x

Water saturation = Sw(x,t1) time = t1

Front moving in this directionwith velocity Vw

xf(t1)

Analyticalsolution

2.3 Grid Orientation EffectsAnother numerical problem arising in 2D and 3D grids is the grid orientation effect. This is illustrated in Figure 5 where the distance between wells I - P1 and I - P2 are the same. However I - P1 are joined by a row of cells oriented to the flow as shown. The flow between I - P2 is rather more tortuous as also shown in Figure 5. The Grid Orientation effect arises when we have fluid flow both oriented with the principal grid direction and diagonally across this grid. Numerical results are different for each of the fluid “paths” through the grid structure. This problem arises mainly due to the use of 5-point difference schemes (in 2D) in the Spatial Discretisation. It may be alleviated by using more sophisticated numerical schemes such as 9-point schemes (in 2D).

P1

P2

I I = InjectorP = Producer

Flow arrows showthe fluid paths inoriented grid anddiagonal flowleading to gridorientation errors

The effects on the breakthrough time and in recoveries of these two flow orientations are shown in Figure 6. The I-P1 orientation tends to lead to somewhat earlier breakthrough and a less optimistic recovery that the I-P2 orientation. The reasons for this are intuitively fairly obvious.

Figure 4The frontal spreading of a Buckley-Leverett shock front when calculated using a 1D grid block model

Figure 5Flow between an injector (I) and 2 producers (P1 and P2) where the injector-producer separations are identical but flow is either oriented with the grid or diagonally across it illustrating the grid orientation effect.



Oilrecovery

Time

gridalignedresult

diagonal flow result

"true"recovery

The grid orientation error can be emphasised for certain types of displacement. For example, in gas injection, gas viscosity is much less than the oil viscosity (μg << μo) leading to viscous fingering instability. This is exaggerated when flow is along the grid as in the I-P1 orientations.

Again, as for numerical dispersion, some grid refinement can help to reduce the grid orientation effect. Alternative numerical schemes can also be devised to reduce this source of error. In particular, in a 2D grid the flows are usually worked out using the neighbouring grid blocks shown in Figure 7. The 5-point numerical scheme uses the neighbours shown in Figure 7(a). If information from the blocks in Figure 7(b) are used, the 9-point scheme that emerges helps greatly to reduce the grid orientation error. We will not go into further technical details here.

(a) (b)

2.4 Local Grid Refinement (LGR)In a reservoir, the changes in pressure, saturations and flows tend to be quite different in different parts of the system. For example, close to a well which is changing production rate every day or so, there will be large pressure and saturation changes. On the contrary, on a flank of the field which is connected to, but is remote from, the active wells, the pressure may be quite slowly changing and the saturations may hardly be changing at all. To represent regions with rapidly changing waterfronts will require a finer grid than will be required for relatively stagnant regions of the system. Thus, a single uniform grid with fixed Δx, Δy and Δz will often not be suitable to represent all regions of an active reservoir. Instead, the application of some local grid refinement (LGR) may be much more appropriate.

LGR options are supported by most major simulation models and the simplest version is shown in Figure 8 (indeed, this simpler version is sometimes not referred to as LGR). “True” LGR is shown in Figure 9 where the refined grid is clearly seen.

Figure 6Oil recoveries for “true”, aligned and diagonal flows in 2D grid

Figure 7 5 - point and 9 - point schemes for discretising the grid - the latter helps to reduce grid orientation effects



In addition to conventional LGR (Figure 9), we can also define Hybrid Grid LGR as shown in Figure 10. Hybrid Grids are mixed geometry combinations of grids which are used to improve the modelling of flows in different regions. The most common use of hybrid grids are Cartesian/Radial combinations where the radial grid is used near a well. Hybrid Grid LGR can be used in a similar way to other LGR scheme.

Hybrid Grid

injector

producer Coarse gridin aquifer

Schematic of Local Grid Refinement (LGR)

2.5 Distorted Grids and Corner Point GeometryIn recent years, many studies have used grids that have tried to distort either to reservoir geometry or to the particular flow field and an examples of a distorted grid is shown in Figure 11.

Figure 8 A simple version of local grid refinement where the grid is finer in the (central) area of interest in the reservoir

Figure 9 “True” local grid refine-ment (LGR) where the refined grid is embedded in the coarser grid.

Figure 10 A simple example of LGR and Hybrid Grid structure



An interesting class of non-Cartesian grids are PEBI grids; PEBI = Perpendicular Bisector. In the PEBI grid the chosen grid points are blocked off into volumes using a geometrical construction which is not shown here. PEBI grids have been developed very extensively by Aziz and co-workers at Stanford University and by Heinemann in Austria (Heinemann, et al 1991; Palagi and Aziz, 1994).

An example of a study using PEBI grids is shown in Figure 12 where the particularly flexible form of this grid is used to model faults in this particular case.

PEBI grids can be orientated to follow major reservoirfaults

This example from:R.E Phelps, T. Pham and A.M. Shahri, “Rigorous Inclusion of Faults and Fracturesin 3D Simulation”, SPE59417, 2000 SPE Asia Pacific Conference, Yokohama, Japan, 25-26 April 2000

Another way of building distorted grids where the individual blocks retain some broad relationship with an underlying Cartesian form is using corner point geometry (Ponting, 1992). This is shown in Figure 13. This scheme is implemented in the reservoir simulator Eclipse (GeoQuest, Schlumberger) where it has been applied quite widely. In corner point geometry it appears rather tedious to build up a grid by specifying all 8 corners of every block (although some are shared with neighbours). However, if this approach is used, the engineer would virtually always have access to grid building software although building complex grids can still be time consuming. The engineer may be reluctant to use corner point geometry if there is a high likelihood

Figure 11 An example of a distorted grid

Figure 12An example of a study using a PEBI grid



that the reservoir model will change radically. In future, this may be overcome by auto-generating the corner point mesh directly from the geo-model (although some upscaling may also be necessary in this process).

At present, corner point geometry is probably more common in fairly fixed base case reservoir models that the engineer has fairly high confidence in. The model shown in Figure 14 is constructed using corner point geometry since it has a major fault in it between the aquifer and the main reservoir.

Highly distortedgrid blocks

Coordinates ofvertices ( )specified.

Block centres ( )

Block <-> BlockTransmissibility

Corner Point Geometry

2.6 Issues in Choosing a Reservoir Simulation GridThe main issues in choosing a grid for a given reservoir simulation calculation are as follows:

(i) Grid Dimension: Refers to whether we should use a 1D, 2D or 3D grid structure;

Figure 13Corner point geometry

Figure 14Complex reservoir model constructed using corner point geometry



(ii) Grid Geometry/Structure: The next issue is whether we should use a simple Cartesian grid (x, y, z) or some other grid structure such as r/z. This choice also includes where local grid refinement, a distorted grid or corner point geometry is appropriate;

(iii) Grid Fineness/Coarseness: How many grid blocks do we need to use? This asks whether a few hundred or thousand is adequate or whether we need 10s or 100s of thousands for an adequate simulation calculation.

We remind the student of the advice in Chapter 1. This was to carry out the reservoir simulation calculation keeping firmly in mind the question which had to be answered or the decision which had to be taken. Therefore, the issues of grid dimension, type and fineness are directly related to the appropriate question/decision. However, as we will see, there are several technical considerations that can guide us in these choices.

Essentially, all 3 choices (grid dimension, type and fineness) depend strongly on the problem we are trying to solve. Consider the issues of grid dimension and type together. A 2D x/z cross-sectional model (with dip if necessary) may be used to study the effects of vertical heterogeneity - layering for example - on the sweep efficiency or water breakthrough time. For a near-well coning study, an r/z grid is usually more appropriate since it more closely resembles the geometry of the near well radial flow. 2D x/z grids are also used to generate pseudo relative permeabilities for possible use in 2D areal models. For full field simulations, 3D grids are generally used which in most models are still probably Cartesian with varying grid spacing in all three dimensions. In recent years, other types of grid such as distorted or corner point grids are being applied - especially if a geocellular model has been generated as part of the reservoir description process. Such guides are also applied in some studies to model major faults in reservoirs. Flow through major faults can lead to communication between non neighbour blocks and this can be modelled in some simulators by defining non-neighbour grid block connections as shown schematically in Figure 15.

Fault

L1L2

L3

L4

L1L2

L3

L4

The issue of grid fineness/coarseness, or how many grid blocks to use in a given simulation, can sometimes be quite subtle as we will show below. However, in many practical calculations, some “reasonable” and practical number of grid block is chosen by the engineer. Then, this can be checked by refining the grid and seeing if the answers are close enough to the coarser calculation. If, as we carry out this grid

Figure 15 Grid system at a fault which may have non-neighbour connections



refinement, the answer - e.g. recovery profiles and water etc. - no longer changes, the calculation is said to be “converged” and is probably quite reliable. By “reliable”, we mean that the grid errors are probably a small contribution of to the overall uncertainties of the whole calculation. If the grid is far from being converged, then comparisons between different sensitivity calculations may be masked by numerical errors. These various points are illustrated by the two examples below.

The two examples used to show the importance of the number of grid blocks are:

(i) Example 1: the effect of vertical grid fineness (i.e. NZ blocks) on a miscible water-alternating-gas (MWAG) process.

(ii) Example 2: resolving the vertical equilibrium (VE) limit of a gas displacement calculation.

Example 1: Figures 16(a) and 16(b) shows the recovery results (at a given time or pore volume throughput) for both a waterflood and an MWAG flood in the same system each as a function of 1/NZ. The difference between the two calculations is the incremental oil recovered by the MWAG process. The economics of performing MWAG depends on how large this difference is. The purpose of plotting this vs. (1/NZ) is that we can extrapolate this to zero i.e., effectively to NZ → ∞. Taking the results at NZ = 2 (1/NZ = 0.5) shows an incremental recovery of (72% - 36%) = 36% of STOIIP which is a huge increase and would make such a project very attractive. However, as we refine the vertical grid, the waterflood recovery increases while the MWAG recovery decreases, i.e. the calculations move closer together and the incremental oil is greatly reduced. Indeed, as we extrapolate to (1/NZ) = 0, we see that the incremental oil is only (47.5% - 47%) = 0.5% which is well within the error band of the calculation. So, rather than having a very attractive project, we appear to have a completely marginal or non-existent improved oil recovery scheme. Certainly, performing just one coarse grid calculation and taking the results at face value would be very misleading in this case.

Example 2: The vertical equilibrium (VE) condition in a gas flood is where the gas if fully segregated by gravity from the oil. This limit has a simple analytical form (not discussed here) which can be written down without doing a grid block calculation. However, we can test the numerical simulation by seeing how many blocks (NZ again) we need to correctly reproduce the VE limit. The answer is rather surprising as shown by the results in Figure 17. These results show that 200 layers are needed to fully resolve the gas “tongue” at the top of the reservoir. Clearly, if we just guessed that 5 vertical blocks would be enough and did not check, then our calculation would be significantly in error.

The two examples above illustrate how the number of blocks chosen for a simulation can strongly affect the results. It shows the need to check that a calculation has converged or that changing the number of grid blocks does not significantly change the answers.



Figure 16: (a) Extrapolation of Predicted Waterflood Recovery Efficiency for 2D Stratified Model C Sand Base Case

100

90

80

70

60

50

40

30

20

10

00.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60

Water

Oil

Gas

Process ==> Waterflooding

1/nz grid blocks

Rec

over

y E

ffici

ency

, %oo

IP

Vertical Grid RefinementNX

As vertical grid is refined --> 0

Homogeneous Model, kv/kh = 0.1Stratified Model, 1.2 HCPVI, kv/kh = 0.02Homogeneous Model, kv/kh = 0.01

100

90

80

70

60

50

40

30

20

10

00.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60

Water

Oil

Gas

1/nz grid blocks

Rec

over

y E

ffici

ency

, %oo

IP

NXVertical Grid Refinement


Homogeneous Model, kv/kh = 0.1Variable Width Homogeneous Model, side solverStratified Model, 25% slug, 1.2 HCPVI, kv/kh = 0.02Homogeneous Model, kv/kh = 0.01

Process ==> Wiscible Water - Alternating - Gas (MWAG)

1NZ

1NZ

123

NZ

123

NZ

Extrapolated RE = 27.6%


(b) Extrapolation of Predicted MWAG Recovery Efficiency for 2D Stratified Model C Sand Base Case

100

90

80

70

60

50

40

30

20

10

00.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60

Water

Oil

Gas

Process ==> Waterflooding

1/nz grid blocks

Rec

over

y E

ffici

ency

, %oo

IP

Vertical Grid RefinementNX


Homogeneous Model, kv/kh = 0.1Stratified Model, 1.2 HCPVI, kv/kh = 0.02Homogeneous Model, kv/kh = 0.01

100

90

80

70

60

50

40

30

20

10

00.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60

Water

Oil

Gas

1/nz grid blocks

Rec

over

y E

ffici

ency

, %oo

IP

NXVertical Grid Refinement


Homogeneous Model, kv/kh = 0.1Variable Width Homogeneous Model, side solverStratified Model, 25% slug, 1.2 HCPVI, kv/kh = 0.02Homogeneous Model, kv/kh = 0.01

Process ==> Wiscible Water - Alternating - Gas (MWAG)

1NZ

1NZ

123

NZ

123

NZ



Figure 16The effect of vertical grid refinement on recovery in (a) a waterflood and (b) a MWAG displacement in a 2D cross-sectional model



0 0.2 0.4 0.6 0.8 1

PVI

Rec

over

y fa

ctor

0.50.450.40.350.30.250.20.150.10.050

fine grid : 5 layersfine grid : 25 layersfine grid : 50 layersfine grid : 200 layerscoarse grid

VE Limit

Oil

x

x

Gravity Dominated

x

x

x

x

xxx

xx

xx

x

xx

xx

x

x

x

xx

x

x

xx x

Gas

2.7 Streamline SimulationThe problem of numerical dispersion was discussed above. One approach to have a more accurate transport calculation is to use streamlines. We will describe this qualitatively in a non-mathematical manner with reference to Figure 18 from the work of Gautier et al (1999). The basic procedure in streamline simulation for a given permeability field (Figure 18(a)) is to calculate the pressure distribution by solving a conventional pressure equation (see Chapter 5 and 6). From this the iso-potentials (pressure contours) can be calculated as shown in Figure 18(b); the gradient of the pressures locally perpendicular to the iso-potentials are the streamlines as shown in Figure 18(c). These streamlines are essentially the “paths” of the injected fluid from the injectors (sources) to the producers (sinks). Since the velocity along these paths is known (from Darcy’s Law using the calculated ∇P), we can work out how far the saturation front moves along the streamline, Δl = v.Δt, where v is the (local) velocity at that point on the streamline. Since v is known quite accurately, the advance of the front along the streamline can be calculated accurately without the problem of block to block numerical dispersion. After we propagate the front along the streamlines, the saturations will change over the reservoir domain. These saturation changes are then projected back onto the Cartesian grid as shown in Figure 18(d), hence changing fluid mobilities. These updated mobilities can be used to recalculate the pressures which, in turn, can be used to update the streamline pattern. This process can be continued throughout the calculation. However, the calculation of the pressure equation is what takes most computational time in most reservoir simulation.

Figure 17Resolving the gas “tongue” in the Vertical Equilibrium (VE) limit in a gas - oil displacement by increasing the number of vertical grid blocks (from Darman et al, 1999)



From Gautier et al (1999)

Outline of Streamline Method

a) Permeability map with an injector and a producer wells

b) Solve the pressure field

c) Compute the velocity field and trace streamlines

d) Move saturation along streamlines and compute the values of the saturation on the grid.

Prod

Inj

(a) (b)

(c) (d)

Injected water concentration

I1

P1

2D Areal Displacements with Streamlines

• Pair Injector / Producer

• IW flows faster in a direct line between the wells and slower in the corners

• Arrival of different streamlines at producer at the same time.

An example of a streamline calculation in a five-spot pattern is shown in Figure 19. In streamline simulation, it may be possible to recalculate the pressures after many transport (saturation update) time steps. This relies on the assumption that the streamlines do not vary too rapidly as the flood progresses. This is a good assumption for many applications. Clearly, if the wells change very significantly, or we switch off some wells and add new ones, it will almost certainly be necessary to recalculate the pattern of streamlines in the reservoir domain.

Streamline simulation has gained some popularity in recent years since 3D streamline codes have been developed e.g. by (Blunt and coworkers at Imperial College in London) and are available commercially.

Streamline simulation is fastest compared with conventional simulation for viscous dominated flow where the assumption of slowly changing streamlines is probably best. For flow where gravity effects are very prominent, there tends to be “side flow” between streamlines and hence it is necessary to recalculate the pressure field quite often. This slows the streamline simulation down quite significantly in many cases although it can sometimes remain competitive with conventional simulation. At the present time, streamline simulation has a place in our simulation “toolbox” but it

Figure 19 An example of a streamline calculation in a five-spot pattern

Figure 18Schematic of streamline simulation from the work of Gautier et al (1999)



is not suitable for all types of reservoir calculation. It is strictly for incompressible flow and compressibility effects can cause some errors. Also, streamline codes tend not to be as developed in terms of “bells and whistles” as conventional simulators e.g. complex well models, “difficult” PVT oil behaviour etc.

3 THE CALCULATION OF BLOCK TO BLOCK FLOWS IN RESERVOIR SIMULATORS

3.1 Introduction to Averaging of Block to Block FlowsIn reservoir simulation, block to block flow terms arise between blocks which we often denoted by mobility terms such as (λT(S0))i-1/2 or (λ0(S0))i+1/2 etc, where the (i+1/2) and (i-1/2) denote the block boundary as the location where that term is evaluated. However, we do not specify properties directly on the boundaries, instead we define them within the grid blocks themselves e.g. the saturations (Sw and So), the permeability, porosity, relative permeability etc. A typical block-to-block flow is shown in Figure 20 where the appropriate terms in Darcy’s Law are also shown:

Applying Darcy’s law to the inter block flow shown in Figure 20 and using the notation in that figure, we obtain:

λ λ

λ λ

λ

T o i o o i

pp

p

p

p

i i

i i

p

p

i i

S S

Q kAB

Px

kAB

P Px x

kA

B

x

x x

( )( ) ( )( )

= −

∆∆

= −

−

∆ + ∆

∆

= ∆ + ∆

− +

−

−

−

1 2 1 2

1

1

1

2

2

/ /

. . . .

or

kkrp

pµ

(1)

where the overbar denotes an average of that quantity. The issue here is: which average should we take? The two main specific questions are:

What is the correct average for the permeability-area product,

λ λ

λ λ

λ

T o i o o i

pp

p

p

p

i i

i i

p

p

i i

S S

Q kAB

Px

kAB

P Px x

kA

B

x

x x

( )( ) ( )( )

= −

∆∆

= −

−

∆ + ∆

∆

= ∆ + ∆

− +

−

−

−

1 2 1 2

1

1

1

2

2

/ /

. . . .

or

kkrp

pµ

?

What is the correct average for the phase mobility,

λ λ

λ λ

λ

T o i o o i

pp

p

p

p

i i

i i

p

p

i i

S S

Q kAB

Px

kAB

P Px x

kA

B

x

x x

( )( ) ( )( )

= −

∆∆

= −

−

∆ + ∆

∆

= ∆ + ∆

− +

−

−

−

1 2 1 2

1

1

1

2

2

/ /

. . . .

or

kkrp

pµ

?

Further issues involve (i) whether we should average the separate terms - krp and μp - within the phase mobility,

λ λ

λ λ

λ

T o i o o i

pp

p

p

p

i i

i i

p

p

i i

S S

Q kAB

Px

kAB

P Px x

kA

B

x

x x

( )( ) ( )( )

= −

∆∆

= −

−

∆ + ∆

∆

= ∆ + ∆

− +

−

−

−

1 2 1 2

1

1

1

2

2

/ /

. . . .

or

kkrp

pµ

; and (ii) which average we should use for

λ λ

λ λ

λ

T o i o o i

pp

p

p

p

i i

i i

p

p

i i

S S

Q kAB

Px

kAB

P Px x

kA

B

x

x x

( )( ) ( )( )

= −

∆∆

= −

−

∆ + ∆

∆

= ∆ + ∆

− +

−

−

−

1 2 1 2

1

1

1

2

2

/ /

. . . .

or

kkrp

pµ

although the formation volume factor does not usually vary very rapidly from block to block.

3.2 Averaging of the k-A Product,

λ λ

λ λ

λ

T o i o o i

pp

p

p

p

i i

i i

p

p

i i

S S

Q kAB

Px

kAB

P Px x

kA

B

x

x x

( )( ) ( )( )

= −

∆∆

= −

−

∆ + ∆

∆

= ∆ + ∆

− +

−

−

−

1 2 1 2

1

1

1

2

2

/ /

. . . .

or

kkrp

pµ

We first examine the kA averaging by considering what this should be for single-phase flow. Consider the volumetric flow, Q, of a single phase and the related pressure drops as shown in Figure 21:



∆x

Permeability = ki-1 Permeability = ki

∆x i-1 ∆x i

Area, A i-1 Area, A i

Qp

Notation: Qp = volumetric flow of phase p (p = o, w, g)

Δxi-1, Δxi = sizes of the (i-1) and i blocks (may not be equal)

∆x = distance between grid block centres

λ λ

λ λ

λ

T o i o o i

pp

p

p

p

i i

i i

p

p

i i

S S

Q kAB

Px

kAB

P Px x

kA

B

x

x x

( )( ) ( )( )

= −

∆∆

= −

−

∆ + ∆

∆

= ∆ + ∆

− +

−

−

−

1 2 1 2

1

1

1

2

2

/ /

. . . .

or

kkrp

pµ

Ai-1, Ai = areas of the (i-1) and i blocks (may not be equal)

ki-1, ki = permeabilites of the (i-1) and i blocks (usually not equal)

λp = mobility of phase p =

λ λ

λ λ

λ

T o i o o i

pp

p

p

p

i i

i i

p

p

i i

S S

Q kAB

Px

kAB

P Px x

kA

B

x

x x

( )( ) ( )( )

= −

∆∆

= −

−

∆ + ∆

∆

= ∆ + ∆

− +

−

−

−

1 2 1 2

1

1

1

2

2

/ /

. . . .

or

kkrp

pµ krp = relative permeability of phase p

ΔP = pressure drop between grid block centres

μp, Bp = viscosity and formation volume factors of phase p

PressureP

Pf

Pi

Pi-1

x

∆xi-1

Ai-1

ki-1

Pi-1 Pi

ki

Ai

∆xi

Q

Pf = face pressure between blocks

We now consider flows from Block (i-1) to the interface of the two blocks, where we denote the interface pressure as Pf (see Figure 21).

Figure 21 Single-phase flow between blocks to determine the correct kA average to use in flow simulation.

Figure 20Block to block flow in a simulator



In block (i-1): Qk A P P

x

P P Qx

k A

Qk A P P

x

P P Qx

k A

P

i i f i

i

f ii

i

i i i f

i

i fi

i

i

= − −∆

−( ) = − ∆( )

= − −∆

−( ) = − ∆( )

−

− − −

−

−−

−

1 1 1

1

11

1

2

21

2

21

..

..

..

..

µ

µ

µ

µ

PPQ x

k Ax

k A

Q xk A

xk A

P P

ii

i

i

i

i

i

i

i

i i

−−

−

−

−

−

( ) = − ∆( )

+ ∆( )

= − ∆( )

+ ∆( )

−( )

11

1

1

1

1

2

1 2

µ

µ

.. .

.

. .

.

(2)

from which we can find the pressure difference (Pf - Pi-1):

Qk A P P

x

P P Qx

k A

Qk A P P

x

P P Qx

k A

P

i i f i

i

f ii

i

i i i f

i

i fi

i

i

= − −∆

−( ) = − ∆( )

= − −∆

−( ) = − ∆( )

−

− − −

−

−−

−

1 1 1

1

11

1

2

21

2

21

..

..

..

..

µ

µ

µ

µ

PPQ x

k Ax

k A

Q xk A

xk A

P P

ii

i

i

i

i

i

i

i

i i

−−

−

−

−

−

( ) = − ∆( )

+ ∆( )

= − ∆( )

+ ∆( )

−( )

11

1

1

1

1

2

1 2

µ

µ

.. .

.

. .

.

(3)

In block (i):

Qk A P P

x

P P Qx

k A

Qk A P P

x

P P Qx

k A

P

i i f i

i

f ii

i

i i i f

i

i fi

i

i

= − −∆

−( ) = − ∆( )

= − −∆

−( ) = − ∆( )

−

− − −

−

−−

−

1 1 1

1

11

1

2

21

2

21

..

..

..

..

µ

µ

µ

µ

PPQ x

k Ax

k A

Q xk A

xk A

P P

ii

i

i

i

i

i

i

i

i i

−−

−

−

−

−

( ) = − ∆( )

+ ∆( )

= − ∆( )

+ ∆( )

−( )

11

1

1

1

1

2

1 2

µ

µ

.. .

.

. .

.

(4)

from which we can find the pressure difference (Pi - Pf):

Qk A P P

x

P P Qx

k A

Qk A P P

x

P P Qx

k A

P

i i f i

i

f ii

i

i i i f

i

i fi

i

i

= − −∆

−( ) = − ∆( )

= − −∆

−( ) = − ∆( )

−

− − −

−

−−

−

1 1 1

1

11

1

2

21

2

21

..

..

..

..

µ

µ

µ

µ

PPQ x

k Ax

k A

Q xk A

xk A

P P

ii

i

i

i

i

i

i

i

i i

−−

−

−

−

−

( ) = − ∆( )

+ ∆( )

= − ∆( )

+ ∆( )

−( )

11

1

1

1

1

2

1 2

µ

µ

.. .

.

. .

.

(5)

Adding equations 3 and 5 above to eliminate the Pf term gives that:

Qk A P P

x

P P Qx

k A

Qk A P P

x

P P Qx

k A

P

i i f i

i

f ii

i

i i i f

i

i fi

i

i

= − −∆

−( ) = − ∆( )

= − −∆

−( ) = − ∆( )

−

− − −

−

−−

−

1 1 1

1

11

1

2

21

2

21

..

..

..

..

µ

µ

µ

µ

PPQ x

k Ax

k A

Q xk A

xk A

P P

ii

i

i

i

i

i

i

i

i i

−−

−

−

−

−

( ) = − ∆( )

+ ∆( )

= − ∆( )

+ ∆( )

−( )

11

1

1

1

1

2

1 2

µ

µ

.. .

.

. .

.

(6)

which rearranges to the following form of the single-phase Darcy Law:

Qk A P P

x

P P Qx

k A

Qk A P P

x

P P Qx

k A

P

i i f i

i

f ii

i

i i i f

i

i fi

i

i

= − −∆

−( ) = − ∆( )

= − −∆

−( ) = − ∆( )

−

− − −

−

−−

−

1 1 1

1

11

1

2

21

2

21

..

..

..

..

µ

µ

µ

µ

PPQ x

k Ax

k A

Q xk A

xk A

P P

ii

i

i

i

i

i

i

i

i i

−−

−

−

−

−

( ) = − ∆( )

+ ∆( )

= − ∆( )

+ ∆( )

−( )

11

1

1

1

1

2

1 2

µ

µ

.. .

.

. .

.

(7)

But, taking kA as the correct average, then, by definition, the single-phase Darcy Law is of the form:

QkA P P

x x

kAx x x

k Ax

k A

kAx x

xk A

xk A

i i

i i

i i i

i

i

i

i i

i

i

i

i

= −−( )

∆ + ∆

∆ + ∆ = ∆( )

+ ∆( )

= ∆ + ∆∆

( )+ ∆

( )

−

−

− −

−

−

−

−

µ.

. .

. .

1

1

1 1

1

1

1

1

2

2

2

= +

=

= + = +

−

− −

1

12

and

k k k

k

BB B

H i i

prp

p

pi i

pi i

1 1

2 2

1

1 1

λµ

µ µ µ

(8)

Comparing identical terms in equations 7 and 8 above gives that:

QkA P P

x x

kAx x x

k Ax

k A

kAx x

xk A

xk A

i i

i i

i i i

i

i

i

i i

i

i

i

i

= −−( )

∆ + ∆

∆ + ∆ = ∆( )

+ ∆( )

= ∆ + ∆∆

( )+ ∆

( )

−

−

− −

−

−

−

−

µ.

. .

. .

1

1

1 1

1

1

1

1

2

2

2

= +

=

= + = +

−

− −

1

12

and

k k k

k

BB B

H i i

prp

p

pi i

pi i

1 1

2 2

1

1 1

λµ

µ µ µ

(9)which easily rearranges to:



kA =

∆xi−1 + ∆xi

∆xi −1

k.A( )i−1

+ ∆xi

k.A( )i

(10)

Note: This is an important result and in particular, we observe the following:

(i) The appropriate average

λ λ

λ λ

λ

T o i o o i

pp

p

p

p

i i

i i

p

p

i i

S S

Q kAB

Px

kAB

P Px x

kA

B

x

x x

( )( ) ( )( )

= −

∆∆

= −

−

∆ + ∆

∆

= ∆ + ∆

− +

−

−

−

1 2 1 2

1

1

1

2

2

/ /

. . . .

or

kkrp

pµ

is not the arithmetic average, it is the harmonic average weighted by the grid block sizes.

(ii) This harmonic average gives much more weighting to the lower permeability value. If the grid sizes are equal, this reduced to the exact harmonic average, kH, of the permeabilities as follows:

QkA P P

x x

kAx x x

k Ax

k A

kAx x

xk A

xk A

i i

i i

i i i

i

i

i

i i

i

i

i

i

= −−( )

∆ + ∆

∆ + ∆ = ∆( )

+ ∆( )

= ∆ + ∆∆

( )+ ∆

( )

−

−

− −

−

−

−

−

µ.

. .

. .

1

1

1 1

1

1

1

1

2

2

2

= +

=

= + = +

−

− −

1

12

and

k k k

k

BB B

H i i

prp

p

pi i

pi i

1 1

2 2

1

1 1

λµ

µ µ µ

(11)

This can be seen for the following example: k1 = 200 mD, k2 = 2 mD ⇒ kH = 3.9 mD. We would expect the flows to be much more strongly affected by the lower permeability since, if one of the permeabilities were zero, then the flow would be zero, no matter how large the other permeability was.

(iii) The above arguments carry over into the averaging of

λ λ

λ λ

λ

T o i o o i

pp

p

p

p

i i

i i

p

p

i i

S S

Q kAB

Px

kAB

P Px x

kA

B

x

x x

( )( ) ( )( )

= −

∆∆

= −

−

∆ + ∆

∆

= ∆ + ∆

− +

−

−

−

1 2 1 2

1

1

1

2

2

/ /

. . . .

or

kkrp

pµ

for multi-phase flow.

EXERCISE 1.

1. For the two grid blocks below, calculate

λ λ

λ λ

λ

T o i o o i

pp

p

p

p

i i

i i

p

p

i i

S S

Q kAB

Px

kAB

P Px x

kA

B

x

x x

( )( ) ( )( )

= −

∆∆

= −

−

∆ + ∆

∆

= ∆ + ∆

− +

−

−

−

1 2 1 2

1

1

1

2

2

/ /

. . . .

or

kkrp

pµ

(in mD.ft.2)

15 ft. k2

100 ft.

20 ft.

120 ft.

25 ft.

15 ft.

k1

(i) For k1 = 200 mD and k1 = 185 mD. Compare the answer with

λ λ

λ λ

λ

T o i o o i

pp

p

p

p

i i

i i

p

p

i i

S S

Q kAB

Px

kAB

P Px x

kA

B

x

x x

( )( ) ( )( )

= −

∆∆

= −

−

∆ + ∆

∆

= ∆ + ∆

− +

−

−

−

1 2 1 2

1

1

1

2

2

/ /

. . . .

or

kkrp

pµ

calculated as the arithmetic average.

(ii) For k1 = 200 mD and k1 = 5 mD. Compare the answer with

λ λ

λ λ

λ

T o i o o i

pp

p

p

p

i i

i i

p

p

i i

S S

Q kAB

Px

kAB

P Px x

kA

B

x

x x

( )( ) ( )( )

= −

∆∆

= −

−

∆ + ∆

∆

= ∆ + ∆

− +

−

−

−

1 2 1 2

1

1

1

2

2

/ /

. . . .

or

kkrp

pµ

calculated as the arithmetic average.

2. If k1 ≈ k2, and A1 ≈ A2, show that

λ λ

λ λ

λ

T o i o o i

pp

p

p

p

i i

i i

p

p

i i

S S

Q kAB

Px

kAB

P Px x

kA

B

x

x x

( )( ) ( )( )

= −

∆∆

= −

−

∆ + ∆

∆

= ∆ + ∆

− +

−

−

−

1 2 1 2

1

1

1

2

2

/ /

. . . .

or

kkrp

pµ

is approximately equal to the arithmetic average.



3.3 Averaging of the Two-Phase Mobility Term,

λ λ

λ λ

λ

T o i o o i

pp

p

p

p

i i

i i

p

p

i i

S S

Q kAB

Px

kAB

P Px x

kA

B

x

x x

( )( ) ( )( )

= −

∆∆

= −

−

∆ + ∆

∆

= ∆ + ∆

− +

−

−

−

1 2 1 2

1

1

1

2

2

/ /

. . . .

or

kkrp

pµ

To recap, the single-phase

λ λ

λ λ

λ

T o i o o i

pp

p

p

p

i i

i i

p

p

i i

S S

Q kAB

Px

kAB

P Px x

kA

B

x

x x

( )( ) ( )( )

= −

∆∆

= −

−

∆ + ∆

∆

= ∆ + ∆

− +

−

−

−

1 2 1 2

1

1

1

2

2

/ /

. . . .

or

kkrp

pµ

term is taken as a weighted harmonic average and this leaves us with the issue of what averages to take for the two-phase flow term,

λ λ

λ λ

λ

T o i o o i

pp

p

p

p

i i

i i

p

p

i i

S S

Q kAB

Px

kAB

P Px x

kA

B

x

x x

( )( ) ( )( )

= −

∆∆

= −

−

∆ + ∆

∆

= ∆ + ∆

− +

−

−

−

1 2 1 2

1

1

1

2

2

/ /

. . . .

or

kkrp

pµ

. In fact, it is convenient to separate the relative permeability term from the viscosity and consider these separately as follows:

QkA P P

x x

kAx x x

k Ax

k A

kAx x

xk A

xk A

i i

i i

i i i

i

i

i

i i

i

i

i

i

= −−( )

∆ + ∆

∆ + ∆ = ∆( )

+ ∆( )

= ∆ + ∆∆

( )+ ∆

( )

−

−

− −

−

−

−

−

µ.

. .

. .

1

1

1 1

1

1

1

1

2

2

2

= +

=

= + = +

−

− −

1

12

and

k k k

k

BB B

H i i

prp

p

pi i

pi i

1 1

2 2

1

1 1

λµ

µ µ µ

(12)

We do this since the relative permeability of phase p is far more variable than the phase viscosity. If fact, without further discussion, we will simply note that the viscosity and formation volume factors are very accurately calculated as the arithmetic averages, since they usually do not vary very much from block to block i.e. the averages between grid blocks (i-1) and i are:

QkA P P

x x

kAx x x

k Ax

k A

kAx x

xk A

xk A

i i

i i

i i i

i

i

i

i i

i

i

i

i

= −−( )

∆ + ∆

∆ + ∆ = ∆( )

+ ∆( )

= ∆ + ∆∆

( )+ ∆

( )

−

−

− −

−

−

−

−

µ.

. .

. .

1

1

1 1

1

1

1

1

2

2

2

= +

=

= + = +

−

− −

1

12

and

k k k

k

BB B

H i i

prp

p

pi i

pi i

1 1

2 2

1

1 1

λµ

µ µ µ (13)

Therefore the situation is summarised in Figure 22:

Qp

Qp = - kA.krp

µp.Bp

∆P

∆x.

Harmonic averageArithmetic averages

Whichaverage??

We will determine which relative permeability average to take by considering the physical situation of two-phase oil/water flow from block i → (i+1) as shown in Figure 23. Consider the situation in Figure 23 where:

Block i is at Sw = (1-Sor) i.e. only water can flow; krw > 0 and kro = 0;

Block (i+1) is at Sw = Swc i.e. only oil can flow; krw = 0 and kro > 0.

Figure 22Which average should be taken for the relative permeability of phase p in the averaging of block to block flows?



Block i Block i + 1

1 - Sor

Swc

krw > 0

kro = 0

Sw

kro > 0

krw = 0

Sor

Physically, it is clear in Figure 23 that water can flow from i → (i+1) but oil cannot. Therefore, for flow in this direction, the average water relative permeability must be non-zero but the average oil relative permeability must be zero. Let us consider the different averages that are possible in turn:

Harmonic Average: First consider if the harmonic average can be used since this was appropriate for the single-phase permeability averaging.

Harmonic average of water relative permeabilities = Harm. Av. {krw i >0; krw i+1 = 0} = 0

Likewise, for oil, Harm. Av. {kro i = 0; kro i+1 > 0} = 0

Thus, the harmanic average gives zero flow for both water (incorrect) and oil (correct). Therefore, it cannot be the harmonic average which is correct for the relative permeability.

Arithmetic Average: Now consider if the arithmetic average can be used since this is a natural thing to try and it is certainly the simplest.

Arithmetic average of water relative permeabilities

= ( ) ( )k k

k

rw i rw i

rw

> + = >

>

+0 02

0

0

1

So, the arithmetic average could be physically correct for the water phase since it gives the average

( ) ( )k k

k

rw i rw i

rw

> + = >

>

+0 02

0

0

1

and thus allows water to flow.

But, for oil the oil phase, we find that : Arith. Av. {kro i = 0; kro i+1 > 0} > 0 and this is physically incorrect, since oil cannot flow from i → (i+1).

Therefore, it cannot be the arithmetic average which is correct for the relative permeability.

What average does this leave? We simply state the answer and then give some physical justification.

Upstream Value: In fact, it turns out that the physically correct value of the relative permeability is simply the upstream value. The upstream value refers to the block from

Figure 23Flow of water from block i → (i+1) and the corresponding relative permeability values for water and oil.



which the flow is coming i.e. in the flow from left to right in Figure 23. This would be block i. This can be seen to be consistent with the physical situation since:

Flowing from i → (i+1), the “average” water relative permeabilities = Upstream {krw i} > 0, as required.

Likewise, flowing from i → (i+1), the “average” oil relative permeabilities = Upstream {kro i } = 0, as required.

Now reversing the flows from (i+1) → i, we find that only oil should flow and this is again consistent since:

Flowing from (i+1) → i, the “average” water relative permeabilities = Upstream {krw i+1} = 0, as required since water cannot flow in this direction.

Likewise, flowing from (i+1) → i, the “average” oil relative permeabilities = Upstream {kro i+1} > 0, as required since only oil can flow in this direction.

The situation is summarised in Figure 24.

Qp

Qp = - kA.krp

µp.Bp

∆P

∆x.

Harmonic averageArithmetic averages

Upstreamvalue of krp

4 WELLS IN RESERVOIR SIMULATION

4.1 Basic Idea of a Well ModelThe only way fluids can be produced from or injected into a reservoir is through the wells and we must therefore include them in our reservoir simulation model. As you may know, the area of Well Technology is vast and in addition to the long wellbore between the reservoir and the surface, there are many other technical features of wells that can have a major impact on the flows into and out of the reservoir. For example, there will be safety valves at the surface and many different types of completion in the well construction itself. Here, we will simplify things as much as possible in order to extract the central functions of the well that we will have to model in the simulator. A schematic of the total well is shown in Figure 25 where the details of the near well formation are shown inset. The near wellbore flows are thought to be radial in an ideal vertical well and this will have some relevance in modelling the near-well pressure behaviour, as discussed in Chapter 2 and elaborated upon below. In addition to these near-well pressure drops, there are several other identifiable

Figure 24The correct inter-block averages for all terms in the two-phase block-to-block flows in a reservoir simulator.



pressure drops between the fluids in the reservoir and the surface oil storage facilities and we may have to model at least some of these. Indeed, it is this topside pressure behaviour that “links” or “couples” the surface with the pressure and flows that we are trying to model in the reservoir using reservoir simulation. The main decision is to determine how much of the formation to surface well assembly we will actually have to model. The main pressure drops are shown in Figure 26 (based on Figure 25 of whole well + ΔPs) and are associated with:

(i) Formation → wellbore flow, ΔPf→w: where fluids flow from a “drainage radius”, re, at pressure, Pe, to the wellbore. Figure 26 shows the near-well pressure profile, in the near-sandface region with bottom hole flowing well pressure (BHFP), Pwf. Thus the formation → wellbore pressure drop, ΔPf→w , is:

∆

∆

∆

∆

P P P

P P P

Q PI P P

Q PI P

P rQ

k hrr

P r P PQ

k hrr

Qk h

f w e wf

wf atm r s

o e wf

o e

w

e e wfe

w

→

→

= −( )

= +

= −( )

=

( ) =( )

( ) = −( ) =( )

= ( )

.

.

.ln

.ln

.

.

max .

µπ

µπ

π

µ

2

2

2

lnln

.

.

.ln

rr

P P

PIk h

rr

e

w

e wf

e

w

−( )

= ( )

2π

µ

(14)

(ii) The pressure drop, ΔPwell, that may occur along the completed region of the wellbore from the bottom of the well (or the “toe” of a horizontal well) to the wellbore just at the top of the completed interval. In very long wells, this pressure drop along the wellbore due to friction may be quite significant although there will be cases where it can be ignored;

Well tubulars Well

Reservoir showingtwo geological layers

Well completed in reservoir

Fluid flow fromreservoir layers to wellbore

Well head

To storage / export

Surface facilities(separator etc...)

Figure 25Schematic of the fluid flows into a well in a grid block model of the reservoir through to their storage or export from the field.



Well tubulars Well

Reservoir showingtwo geological layers

Well completed in reservoir

Fluid flow fromreservoir layers to wellbore

Well head

To storage / export

Surface facilities(separator etc...)

∆Pwh -> export

∆Pr -> spressure dropfrom reservoirtop to surface

Pwf

rw rer

Pe

∆Pf -> w

∆Pwell

Near wellbore formationto wellbore ∆Pf -> w

pressure drop along the wellwithin reservoir section

(iii) Reservoir → surface pressure drop, ΔPr→s : the pressure drop from the well at the top of the completed formation just above the reservoir to the wellhead which is at pressure, Pwh . This ΔP is quite significant and, locally in any sector of the well, there will be a local pressure drop vs. flow rate/fluid composition relationship. This may be calculated from models (often correlations) of multi-phase flow in pipes. As the fluids move up the wellbore, the pressure drops in oil/water production and free gas may also appear; thus, we can have three phase flow in the well tubular to the surface and we may have to incorporate this flow rate/pressure drop behaviour in our modelling.

(iv) There will frequently be further pressure drops as the fluids flow from the wellhead through the surface facilities such as the separators, various chokes, etc. We will not consider this in detail here although it can be an important consideration in some field cases e.g. if the well is feeding into a network gathering system which other wells are also feeding into. This could be a complex surface gathering system network or a multiple-well manifold of a subsea production system.

In this section, we will mainly focus on the formation to wellbore pressure drops. Thus, our main task is to either calculate or set the well flowing pressure (Pwf) although we will return briefly to the issue of calculating the pressure drops between the reservoir and the surface in the discussion below.

To set the scene in modelling wells in a simulator, we will first consider a very simple model well producing only oil into the wellbore. How do we decide what

Figure 26Schematic of the fluid flows from the well through to storage or export showing the associated pressure drops that occur in the system.



the volumetric flow rate, Qo, of this well is? Indeed, do we decide or is it set for us by the reservoir and well properties? We will start with the simplest case where we basically “take what we can get” by drawing the wellhead pressure, Pwh, down as low as possible. Suppose that we simple open it up such that the oil pressure drops essentially to atmospheric. There is then the additional reservoir to surface pressure drop, ΔPr→s, to consider. Thus, the well flowing pressure, Pwf, would be given by:

∆

∆

∆

∆

P P P

P P P

Q PI P P

Q PI P

P rQ

k hrr

P r P PQ

k hrr

Qk h

f w e wf

wf atm r s

o e wf

o e

w

e e wfe

w

→

→

= −( )

= +

= −( )

=

( ) =( )

( ) = −( ) =( )

= ( )

.

.

.ln

.ln

.

.

max .

µπ

µπ

π

µ

2

2

2

lnln

.

.

.ln

rr

P P

PIk h

rr

e

w

e wf

e

w

−( )

= ( )

2π

µ

(15)

So, what happens ? Clearly, if this value of Pwf > Pe (the reservoir pressure), then no oil can be produced. However, if Pwf < Pe, then some oil will flow into the well and we can now calculate how much. As we will see, this will depend on the physical properties of the system such as the permeability of the rock, the viscosity of the oil, the precise geometry of the well etc. However, in our simple conceptual well, we will take all of these quantities as “givens” for the moment. Suppose the well does flow at a volumetric flow rate, Qo, for reservoir pressure, Pe, and well flowing pressure, Pwf . We can then define a productivity index, PI, of the well as follows:

∆

∆

∆

∆

P P P

P P P

Q PI P P

Q PI P

P rQ

k hrr

P r P PQ

k hrr

Qk h

f w e wf

wf atm r s

o e wf

o e

w

e e wfe

w

→

→

= −( )

= +

= −( )

=

( ) =( )

( ) = −( ) =( )

= ( )

.

.

.ln

.ln

.

.

max .

µπ

µπ

π

µ

2

2

2

lnln

.

.

.ln

rr

P P

PIk h

rr

e

w

e wf

e

w

−( )

= ( )

2π

µ

(16)

where possible units of PI could be bbl/day/psi, for example. The above equation basically states how much oil is produced per psi of drawdown. This simple equation takes us back to our original question on “what/who decides on Qo?”. In our simple case, the answer is now clear; i.e. some things are “givens” - e.g. PI and Pe in a virgin oil producing system - and some we can set within limits - e.g. Pwf by setting the wellhead pressure. However, you may be able to set a flowrate, Qo, by installing a downhole pump (an ESP - electrical submersible pump). In that case, you would set Qo and then calculate Pwf where we are still considering the reservoir pressure (Pe) as a given. But clearly we cannot set Qo to any arbitrarily large value since the lowest possible value of Pwf = 0 (a vacuum), which would then set a maximum value of Qo given by:

∆

∆

∆

∆

P P P

P P P

Q PI P P

Q PI P

P rQ

k hrr

P r P PQ

k hrr

Qk h

f w e wf

wf atm r s

o e wf

o e

w

e e wfe

w

→

→

= −( )

= +

= −( )

=

( ) =( )

( ) = −( ) =( )

= ( )

.

.

.ln

.ln

.

.

max .

µπ

µπ

π

µ

2

2

2

lnln

.

.

.ln

rr

P P

PIk h

rr

e

w

e wf

e

w

−( )

= ( )

2π

µ

(17)

So, in summary, we can set either a pressure or a flowrate but (a) not both and (b) in either case, within limits.

But, can’t we affect the well PI or the reservoir pressure, Pe? We can actually affect the PI of a well by “stimulating” it possibly by locally hydraulically fracturing the well or by acidising it to increase the effective permeability of the near well region. In addition, we can increase (or more commonly maintain) the reservoir pressure (which relates to the “reservoir energy”) to some extent by injecting a fluid - usually water or gas in another injector well. However, the basic well controls are either setting pressure or flow rate and this must be kept in mind when we model wells in reservoir simulation. We elaborate on these ideas in the following section where we introduce the central idea of a well model.



4.2 Well Models for Single and Two-Phase FlowWe now consider how a well model can be developed, firstly in our simple conceptual reservoir producing only oil. Figure 27 shows the local pressure profiles in a simple homogeneous well system in single phase flow (see Chapter 2, section 3.5). The pressure profile close to the wellbore, assuming radial flow, was derived in Chapter 2 and is given by (section 3.5):

∆

∆

∆

∆

P P P

P P P

Q PI P P

Q PI P

P rQ

k hrr

P r P PQ

k hrr

Qk h

f w e wf

wf atm r s

o e wf

o e

w

e e wfe

w

→

→

= −( )

= +

= −( )

=

( ) =( )

( ) = −( ) =( )

= ( )

.

.

.ln

.ln

.

.

max .

µπ

µπ

π

µ

2

2

2

lnln

.

.

.ln

rr

P P

PIk h

rr

e

w

e wf

e

w

−( )

= ( )

2π

µ

(18)

Taking the pressure at radius re as being the reservoir pressure, Pe, then gives:

∆

∆

∆

∆

P P P

P P P

Q PI P P

Q PI P

P rQ

k hrr

P r P PQ

k hrr

Qk h

f w e wf

wf atm r s

o e wf

o e

w

e e wfe

w

→

→

= −( )

= +

= −( )

=

( ) =( )

( ) = −( ) =( )

= ( )

.

.

.ln

.ln

.

.

max .

µπ

µπ

π

µ

2

2

2

lnln

.

.

.ln

rr

P P

PIk h

rr

e

w

e wf

e

w

−( )

= ( )

2π

µ

(19)

(a) (b)

∆P(r) = P(r) - Pwf

Q

Pe

Pwf

Pwf

Pwf

rw rer

P

Well at BHFP

(assume ∆x = ∆y)

∆x ∆y

h

P

Grid blockpressure

which can easily be arranged to obtain:

∆

∆

∆

∆

P P P

P P P

Q PI P P

Q PI P

P rQ

k hrr

P r P PQ

k hrr

Qk h

f w e wf

wf atm r s

o e wf

o e

w

e e wfe

w

→

→

= −( )

= +

= −( )

=

( ) =( )

( ) = −( ) =( )

= ( )

.

.

.ln

.ln

.

.

max .

µπ

µπ

π

µ

2

2

2

lnln

.

.

.ln

rr

P P

PIk h

rr

e

w

e wf

e

w

−( )

= ( )

2π

µ

(20)

and hence from equation 16 above, we can identify the productivity index (PI) of the well as:

∆

∆

∆

∆

P P P

P P P

Q PI P P

Q PI P

P rQ

k hrr

P r P PQ

k hrr

Qk h

f w e wf

wf atm r s

o e wf

o e

w

e e wfe

w

→

→

= −( )

= +

= −( )

=

( ) =( )

( ) = −( ) =( )

= ( )

.

.

.ln

.ln

.

.

max .

µπ

µπ

π

µ

2

2

2

lnln

.

.

.ln

rr

P P

PIk h

rr

e

w

e wf

e

w

−( )

= ( )

2π

µ (21)

This now demonstrates exactly how the quantities k, h, μ, rw and re affect the well productivity. All of these factors behave as we might expect them to physically e.g. as k↑, PI↑; as μ ↑, PI↓ etc.

Now consider how this relates to the pressures in the simulation block shown in Figures 27 and 28. In a grid block, the pressure is thought of as being constant throughout the block although we know that it should be varying continuously across the block; we will refer to this as the average block pressure, P . The size of the grid block in

Figure 27Schematic showing (a) the near well pressure profiles that occur in a radial system and (b) corresponding quantities in a Cartesian grid block



our example is (Δx, Δy) and, for simplicity, we will assume that, Δx = Δy. Looking at the expression for PI in equation 21 and the quantities we have in the grid block, it is easy to make direct relations for some of them - obviously k, μ and h and also possibly rw and Pwf , although these latter two do not seem to appear in the grid model. The drainage radius, re, and the reservoir pressure, Pe, which appear in the radial model do not appear in the grid model - instead, the block size (Δx, Δy) and average block pressure ( P ) appear. This immediately suggests the following 2 questions:

1. what is the relation between re , and the block size (Δx, Δy)?

2. what is the relation between Pe and the average grid block pressure, P ?


Pe

Pwf

Pwf

rw rer

P

Well at BHFP= Pwf

re <-> ∆x, ∆y ??

re => Pe =P

∆x ∆y

h

P

Grid blockpressure

How do we choosere such that

Pe =P ??

Relation

re

∆x

∆yre is where P(re) =P

and re = re (∆x, ∆y) butwhat is the formula ?

re

rw

The issue is defined quite clearly in Figure 28. From this figure, we would like to choose re such that Pe coincides with the average grid block pressure, P . The latter quantity ( P ) is calculated in the simulation itself. In fact, we need to know how to calculate re from the quantities Δx and Δy as indicated in Figure 29 where we show the re as function of Δx and Δy, i.e. re(Δx, Δy).

If we know the formula for re(Δx, Δy), then we can calculate the PI (equation 21) and use this in the simulation to couple the quantities Qo (oil flow rate) and P (average grid block pressure); i.e.

P

Q PI P P

r

rk h P r P

Q

r x

r x y

Q PI P P Q PI P P

PIkh k S

rr

o wf

w

wf

e

e

o o wf w w wf

oro o

oe

w

= −( )

= ( ) −( )

≈

≈ +

= −( ) = −( )

= ( )

.

ln.

.( )

.

.

. ; .

.

. ln

2

0 2

0 14

2

2 2

πµ

π

µ

∆

∆ ∆

= ( )

PIkh k S

rr

wrw w

we

w

2π

µ

.

. ln

(22)

Figure 29The relation between re and the block dimensions, Δx and Δy.

Figure 28Schematic indicating how the near well pressures relate to the corresponding quantities in a Cartesian grid block



Here, we can set either Qo or Pwf and then calculate the other one from P (and the known PI ). This was achieved in a very simple but ingenious way in a classic paper by Donald Peaceman (1978), another pioneer of numerical reservoir simulation. Peaceman did this by carrying out a 2D numerical solution of the pressure equation on a Cartesian (x, y) grid for a quarter five-spot configuration as shown in Figure 30. But, from equation 19, we know that:

P

Q PI P P

r

rk h P r P

Q

r x

r x y

Q PI P P Q PI P P

PIkh k S

rr

o wf

w

wf

e

e

o o wf w w wf

oro o

oe

w

= −( )

= ( ) −( )

≈

≈ +

= −( ) = −( )

= ( )

.

ln.

.( )

.

.

. ; .

.

. ln

2

0 2

0 14

2

2 2

πµ

π

µ

∆

∆ ∆

= ( )

PIkh k S

rr

wrw w

we

w

2π

µ

.

. ln

(23) Hence, if we plot the pressure at grid blocks away from the well block vs. the well block spacing on a logarithmic scale as shown in Figure 31, then we can extrapolate back to find the equivalent radius where P = Pe in terms of the well block dimension (Δx). It turns out that the simple relation is (for Δx = Δy):

P

Q PI P P

r

rk h P r P

Q

r x

r x y

Q PI P P Q PI P P

PIkh k S

rr

o wf

w

wf

e

e

o o wf w w wf

oro o

oe

w

= −( )

= ( ) −( )

≈

≈ +

= −( ) = −( )

= ( )

.

ln.

.( )

.

.

. ; .

.

. ln

2

0 2

0 14

2

2 2

πµ

π

µ

∆

∆ ∆

= ( )

PIkh k S

rr

wrw w

we

w

2π

µ

.

. ln

(24)

Therefore, we have a very simple way of calculating the PI or “well connection factor” as it is sometimes called of a well in a simulation grid block.

The simple Peaceman formula applies to a well in a radial environment (the five-spot configuration is as close as we can get to radial using a common 2D Cartesian grid) and for Δx = Δy. In fact, some modification to the simple formula is required for wells in “corner” locations or set close to a boundary and these are shown in Figure 32. Also, if Δx ≠ Δy, but the well is isolated (radial flow), then:

P

Q PI P P

r

rk h P r P

Q

r x

r x y

Q PI P P Q PI P P

PIkh k S

rr

o wf

w

wf

e

e

o o wf w w wf

oro o

oe

w

= −( )

= ( ) −( )

≈

≈ +

= −( ) = −( )

= ( )

.

ln.

.( )

.

.

. ; .

.

. ln

2

0 2

0 14

2

2 2

πµ

π

µ

∆

∆ ∆

= ( )

PIkh k S

rr

wrw w

we

w

2π

µ

.

. ln

(25)

11109876543210

-1

-1 0 1 2 3 4 5 6 7 8 9 10 11

Figure 30The 2D areal grid used to compare pressures with the expected radial profiles (from Peaceman, 1978)



Figure 32Well factors for wells in various positions relative to the boundaries; after Kuniansky and Hillstad (1980)

• On Edge: i = 0 or j = 0• On Diagonal: i = j• i ≠ j ≠ 0

0.6

0.5

0.4

0.3

0.2

0.1

00.1 0.2 0.4 0.6 1 2 3 4 5 6

(1.0)

(1.1)

(2.1)(2.0)

(2.2)

From areal average pressure

p ij -

po

qµ /

kh

ro / ∆x r o / ∆xA

Factor in terms of (re / ∆x)

(re / ∆x) = 0.2 (Peaceman's equation)

(re / ∆x) = 0.196

implies no flow boundary

(re / ∆x) = 0.433

(re / ∆x) = 0.193

(re / ∆x) = 0.72

For anisotropicpermeabilitieskx ≠ ky

ky

kx

kx

ky

ky

kx

kx

ky

re = 0.28

+

.∆x2 + .∆y21/2 1/2 1/2

1/4 1/4

Figure 31Log plot of the pressures from the 2D areal grid used to find re (from Peaceman, 1978)



We may find that, in a given simulation of a field case, that we input all the known or estimated data but the well in our simulation does not perform like the real case. It may produce more (higher PI) or it may produce less (lower PI) than expected. The former case may be due to the well being stimulated and the second case may be due to well damage. Within limits, we may adjust the calculated well PI in the simulation model in order to reproduce the observed field behaviour. However, we should think carefully before making such changes since the simulated well productivity may be wrong because some (or several) other aspects of the reservoir simulator input data are wrong.

It is now relatively straightforward to extend our discussion on PI and simple well models in a homogeneous single layer system to the flow of two phases - say, oil and water - as shown in Figure 33. Since two phases are being produced, then the saturations of both oil and water (So and Sw) must be at values where their relative permeabilities are > 0 (i.e. = > So > Sor ; Sw > Swc).


Well radius = rw

∆x ∆y

h

Qw Qo

SaturationSw and So

We now apply the same ideas as were developed above for the single phase case. From the radial two phase Darcy Law, the volumetric production rates of oil and water are given by:

P

Q PI P P

r

rk h P r P

Q

r x

r x y

Q PI P P Q PI P P

PIkh k S

rr

o wf

w

wf

e

e

o o wf w w wf

oro o

oe

w

= −( )

= ( ) −( )

≈

≈ +

= −( ) = −( )

= ( )

.

ln.

.( )

.

.

. ; .

.

. ln

2

0 2

0 14

2

2 2

πµ

π

µ

∆

∆ ∆

= ( )

PIkh k S

rr

wrw w

we

w

2π

µ

.

. ln

(26)

where the PI0 and PIw are the oil and water productivity indices, respectively, and

P

Q PI P P

r

rk h P r P

Q

r x

r x y

Q PI P P Q PI P P

PIkh k S

rr

o wf

w

wf

e

e

o o wf w w wf

oro o

oe

w

= −( )

= ( ) −( )

≈

≈ +

= −( ) = −( )

= ( )

.

ln.

.( )

.

.

. ; .

.

. ln

2

0 2

0 14

2

2 2

πµ

π

µ

∆

∆ ∆

= ( )

PIkh k S

rr

wrw w

we

w

2π

µ

.

. ln

(27)and

P

Q PI P P

r

rk h P r P

Q

r x

r x y

Q PI P P Q PI P P

PIkh k S

rr

o wf

w

wf

e

e

o o wf w w wf

oro o

oe

w

= −( )

= ( ) −( )

≈

≈ +

= −( ) = −( )

= ( )

.

ln.

.( )

.

.

. ; .

.

. ln

2

0 2

0 14

2

2 2

πµ

π

µ

∆

∆ ∆

= ( )

PIkh k S

rr

wrw w

we

w

2π

µ

.

. ln (28)

Figure 33 Near well two-phase flows in a Cartesian grid block.



where, again, re is calculated using Peaceman’s formula. In the above equation, we have not incorporated the separate phase pressure, Po and Pw, in the well block but these may be used in a given calculation.

4.3 Well Modelling in a Multi-Layer SystemThe most common case which is modelled is where we have multi-phase (e.g. two phase oil/water) flow in a layered system where the layers are of different permeability. This situation is shown for a simple four layer system in Figure 34.

Clearly, there are additional issues in this system since all four layers may be producing both oil and water and the proportions of each phase may be changing as the saturations (and hence relative permeabilities) change. In addition, there is also a gravitational potential in each layer which we may have to take into account. Using the notation in Figure 34, we note that the oil flows in layer k (k = 1, 2, 3, 4, in this example) are:

Q PI P Pkhk S

rr

P P

Q Q Q PI PI P P

Q Q Q PI PI P P

Q PI

ok ok k wfk

ro o k

oek

w

k wfk

T ok wkk

ok wkk

k wfk

Tk ok wk ok wk k wf

o i j k

= −( ) =( )( )

−( )

= +( ) = +( ) −( )

= +( ) = +( ) −( )

=

= =∑ ∑

2

1

3

1

3

π

µ . ln

, , oo i j k i j k i j k

w injo o

ww

w injo o

ww

P P

QB Q

BQ

QB Q

BQ

wf

, , , , , ,. −( )

= +

> +

(29)

and a similar expression applies for water. The total oil and water flows in the well are:

Q PI P Pkhk S

rr

P P

Q Q Q PI PI P P

Q Q Q PI PI P P

Q PI

ok ok k wfk

ro o k

oek

w

k wfk

T ok wkk

ok wkk

k wfk

Tk ok wk ok wk k wf

o i j k

= −( ) =( )( )

−( )

= +( ) = +( ) −( )

= +( ) = +( ) −( )

=

= =∑ ∑

2

1

3

1

3

π

µ . ln


w injo o

ww

w injo o

ww

P P

QB Q

BQ

QB Q

BQ

wf

, , , , , ,. −( )

= +

> +

(30)

where we have taken the mean block pressure and also the well flowing pressure in layer k as being the same for both oil and water phases. Again, in the layered case, we can specify the total flow or the flowing bottom hole pressure and then we can calculate the other one using the above relations. (In fact, we can also specify the flow of either phase - oil or water - and then find the flow of the other and the bottom hole flowing pressure). For example, suppose we specify the total flow, QT. We need to decide how this total flow is made up - i.e. what are the separate Qo and Qw (QT = Qo + Qw) and how this flow is allocated from each of the layers in the system. For simplicity, we make the assumption that there is a single well flowing pressure, Pwf (i.e. Pwf1 = Pwf2 = Pwf3 = Pwf4). Hence, for each layer, k :

Q PI P Pkhk S

rr

P P

Q Q Q PI PI P P

Q Q Q PI PI P P

Q PI

ok ok k wfk

ro o k

oek

w

k wfk

T ok wkk

ok wkk

k wfk

Tk ok wk ok wk k wf

o i j k

= −( ) =( )( )

−( )

= +( ) = +( ) −( )

= +( ) = +( ) −( )

=

= =∑ ∑

2

1

3

1

3

π

µ . ln


w injo o

ww

w injo o

ww

P P

QB Q

BQ

QB Q

BQ

wf

, , , , , ,. −( )

= +

> +

(31)

Everything is known in the above equation which allows us to determine the allocation of all fluids from each layer and we can calculate the corresponding bottom hole flowing pressure, Pwf.

In a very similar manner, we could specify the well flowing pressure, Pwf, and then calculate the individual flows, Qok and Qwk etc. in each layer.



Layer1234

Wellcompletion

h

QT = Qo + Qw

Qo1Qw1

Qo2Qw2

Qo3Qw3

Qo4Qw4

k1, h1, P1, Sw1

k2, h2, P2, Sw2

k3, h3, P3, Sw3

k4, h4, P4, Sw4

(a) (b)

4.4 Modelling Horizontal WellsFigure 35 shows the trajectory of a “horizontal” well in a reservoir simulation model. This is not well represented by the purely radial r/z model grid discussed in Section 2.1 above in the context of a vertical well. Hence, it is less likely that the well connection factors calculated as shown in previous sections will apply for a horizontal well. This is broadly true although the basic principle is very similar. That is, each well sector intersects a grid block (i,j,k) even although the well may be going through this block in say the x(i) direction and the flows between the well sector and the grid are given by an expression of the form:

Q PI P Pkhk S

rr

P P

Q Q Q PI PI P P

Q Q Q PI PI P P

Q PI

ok ok k wfk

ro o k

oek

w

k wfk

T ok wkk

ok wkk

k wfk

Tk ok wk ok wk k wf

o i j k

= −( ) =( )( )

−( )

= +( ) = +( ) −( )

= +( ) = +( ) −( )

=

= =∑ ∑

2

1

3

1

3

π

µ . ln


w injo o

ww

w injo o

ww

P P

QB Q

BQ

QB Q

BQ

wf

, , , , , ,. −( )

= +

> +

(32)

where the actual form of the productivity index expression, PIoi,j,k , may be rather more complex since (a) the well may intersect the block in a more complex way and (b) the aspect ratio of the block is rather different when a horizontal well intersects it in that the x-direction well is very close to the z-boundaries since Δz is often smaller than Δx or Δy.

Figure 34Well modelling of two-phase flow in a multi-layered system



Figure 35Cartesian grid cut from a 3D reservoir model showing two horizontal wells going through the system; two vertical wells also shown.

4.5 Hierarchies of Wells and Well Controls Simple well control can be understood in terms of the well models discussed above. For a single well, we can essentially set the well flowing pressure and then calculate the flows or vice versa but not both and with certain constraints (see above). Alternatively, we may set the wellhead pressure and then calculate the Pwf from the - calculated or input - well formation to surface pressure drops etc. We now consider controls on pairs, then groups and then clusters of groups of wells in a field - indeed, we can even couple together the wells from several reservoirs and set more global controls and this will be described briefly.

For a simple injector/producer well pair, for example in a 2D x/z cross-section, we can apply a range of well controls. Suppose this is a simple waterflood in an oil/water system. One of the most common controls is to fix the water injection rate at the injector but with (upper) limits on the well flowing pressure. The corresponding producer is then controlled by setting the bottom hole flowing pressure and then allowing the calculation of the oil and water phase flows (Qo and Qw). The volumetric production will be approximately balanced with the total production volume being of the order of the injected water volume - but not quite the same. Do you know why this is? Clearly the formation volume factors (Bo and Bw) will affect the exact production volumes; when we are injecting water and producing 100% oil, the reservoir volume of injected water per day is Qw.Bw and this will displace (virtually) the same reservoir volume of oil. The volume of oil produced per day is Qo stb which is actually Qo.Bo reservoir bbls, equating these reservoir volumes shows that if we inject water at a rate of Qw (stb/day), we produce oil at a rate of Qw.Bw/Bo stb/day. Since Bo > Bw, then the volumetric production rate of oil (in stb/day) is lower than the injected water injection rate (in stb/day). This must be taken into account in considering well control by voidage replacement as discussed below.

If we wish to set injected Qw to precisely voidage replace whatever is produced, then we can do so to a good approximation by noting that if the production rate of oil



and water in our simulation is currently, Qo and Qw. What volume of injected water must we inject to exactly replace the reservoir volume of these two phases? This is now quite straightforward since and, from the above discussion, it is evident that the quantity of water that must be injected, Qw inj (in stb/day) is given by:

Q PI P Pkhk S

rr

P P

Q Q Q PI PI P P

Q Q Q PI PI P P

Q PI

ok ok k wfk

ro o k

oek

w

k wfk

T ok wkk

ok wkk

k wfk

Tk ok wk ok wk k wf

o i j k

= −( ) =( )( )

−( )

= +( ) = +( ) −( )

= +( ) = +( ) −( )

=

= =∑ ∑

2

1

3

1

3

π

µ . ln


w injo o

ww

w injo o

ww

P P

QB Q

BQ

QB Q

BQ

wf

, , , , , ,. −( )

= +

> +

(33)

Hence, we would gradually adjust the volume of water injected in the simulation model based on what we have just produced (at the last time step say) to the above quantity in order to voidage replace. At the producer, the most common option would be to constrain by bottom hole flowing pressure as described above.

Note that it is possible but less common to constrain all wells by volumetric injection/production rate. We can see why if we consider an incompressible fluid where it is clearly impossible for the injection and production rates to be different since the pressure would go to + ∞ or - ∞, depending on whether we over- or under-injected, respectively. Although it is possible to specify different volumetric flow rates at injector and producer for a compressible fluid, this can only be done within very tight limits and the pressures tend to go to unrealistic limits e.g. if we over-inject

(i.e.

Q PI P Pkhk S

rr

P P

Q Q Q PI PI P P

Q Q Q PI PI P P

Q PI

ok ok k wfk

ro o k

oek

w

k wfk

T ok wkk

ok wkk

k wfk

Tk ok wk ok wk k wf

o i j k

= −( ) =( )( )

−( )

= +( ) = +( ) −( )

= +( ) = +( ) −( )

=

= =∑ ∑

2

1

3

1

3

π

µ . ln


w injo o

ww

w injo o

ww

P P

QB Q

BQ

QB Q

BQ

wf

, , , , , ,. −( )

= +

> + ), the pressure tends to rise to unphysically high levels well above fracture pressure of the reservoir rock.

4.5 CLOSING REMARKS - GRIDDING AND WELL MODELLING

In this section, the student has been presented with a largely non-mathematical description of gridding and well modelling in reservoir simulation. A more mathematical treatment of these issues will be given as we develop the flow equations and consider their numerical solution in Chapter 5 and 6, respectively.

CONTENTS

1 INTRODUCTION

2 THE SINGLE PHASE PRESSURE EQUATION 2.1 The Physics of Single Phase Compressible Systems 2.2. The Single Phase Pressure Equation 2.3 The Simplified Compressible Pressure Equation 2.4 Extension of the Single Phase Pressure Equation to 2D and 3D 2.5 Mathematical Shorthand for the 3D Single- Phase Pressure Equation

3 THE TWO-PHASE FLOW EQUATIONS 3.1 Review of Two-Phase Flow Concepts 3.2 Derivation of the Two-Phase Conservation Equations 3.3 The Two-Phase Pressure Equation 3.4 Schematic Strategy for Solving the Two- Phase Pressure and Saturation Equation 3.5 The Simplified Pressure and Saturation Equations

4 CLOSING REMARKS

55The Flow Equations




Note that in this chapter, there are a number of lengthy derivations of the flow equations. The student should work through all of these in detail and understand each step in the process. In the learning objectives below, we clarify what parts of this the student should know and be able to reproduce and/or apply in an exam question and what parts need only be appreciated or understood at a general level.


• know that the central principles in deriving the flow equations in porous media are (a) to apply material balance to the flow and accumulation terms; and (b) to then apply Darcy’s Law. • know how to apply material balance to a control volume with flow and accumulation terms, recognising the difference between mass flow rates and volumetric flow rates.

• understand be able to describe the basic physics of single phase compressible flow through porous media.

• be able to derive the (pressure) equation for single phase compressible flow.

• know physically, in the context of the single phase compressible flow equation, what a non-linear partial differential equation (PDE) is and why it is difficult to solve.

• be able to work through all of the simplifying assumptions on the single phase compressible flow equation to arrive at the much simpler pressure equation (linear PDE) for slightly compressible flow involving the hydraulic diffusivity, Dh =k/(μφcf)

• understand the extension of the single phase pressure equation to 2D.

• appreciate (but not memorise) the use of the gradient (∇) and divergence (∇.) operators from vector calculus in writing the multi-phase flow equations.

• be able to apply conservation + Darcy’s law in the two phase case to arrive at the two phase flow equations for compressible fluids and rock. • follow the argument on how the full two phase pressure equation is derived.

• be able to identify the relation of each of the terms in the two phase pressure equation to viscous, gravity and capillary forces.

• be able to reproduce (in words and diagrams) the outline solution scheme for the two phase pressure equations for both the full equations and for the simplified pressure and saturation equations.




The central activity of reservoir simulation is the numerical solution of the multi-phase flow equations in real reservoir systems. This chapter introduces the underlying equations of flow through porous media which are solved in a reservoir simulation code. We start with the single phase flow equation for a compressible fluid and go on to develop the two-phase flow equations. The same principle is used in the derivation of these equations viz. the application of mass (material) balance between flow and accumulation and then the application of Darcy’s Law.

We show that the flow equations are non-linear partial differential equations (PDEs) which can usually not be solved analytically. However, some simplified cases of the equations are considered in order to establish some important physical insights into these equations.

1. INTRODUCTION

In the course so far, we have used many of the concepts of day to day numerical simulation as it is applied in reservoir engineering. However, we have not yet studied the mathematical equations which underpin the whole subject of reservoir simulation. This section will introduce you to these equations in a step-by step manner. You should be able to follow all the details of the derivations of these equations since only elementary undergraduate mathematics is involved.

The general approach to the flow equations is basically the same for both single and two-phase flow. We first derive a mass balance between the flows and the accumulation (local mass build up or decline) in a local control volume. A control volume can be thought of as a typical isolated grid block in the system, as will be evident below. We note that the mass balance equation that we will set up must be correct. That is, it is simply a mathematical way of expressing something that is a fact. Mass balance simply states that over a given time period (say, Δt), then the sum of all the mass that flows into (+) the system and out of it (-) is the change of mass in that block. Once we have set up the mass balance between the flows and the accumulations, we then need some expressions to describe these mass flows. We do not usually think of fluid flow laws as being mass flow laws. Within a porous medium, the principal flow law is Darcy’s law (either for one or two phases) which is a volumetric flow law. That is, in Darcy’s law, the volumetric flow rate, usually denoted Q, is proportional to the pressure gradient; Q ~ -(dP/dx) for a single one dimensional system oriented along the x- direction. (Note the minus sign here since fluids flow down the pressure gradient, from high P to low P). So, to derive the flow equations we simply use mass balance + Darcy’s law. This is shown schematically for single phase fluid flow in Figure 1, where the operator, Δt, implies the difference in the quality (pφΔxΔyΔz) between times t and t + Δt (of course, only the p and φ quantities can change).



i,j+l

i+1,ji-1,j

i,j-l

i,j

Qj+l/2

Qj-l/2

Qi+l/2Qi-l/2

Mass in block (i,j) = ρ.φ.∆x.∆y.∆z

Change in mass over ∆t

where m counts over the 4 neighbours. And Darcy's law is:

Therefore:

where we then substitute Darcy's law

∆m = ρ.Q( )m

m =1

4

∑

.∆t

Qi −1/ 2 = −kA

µ

i−1/ 2

.Pi , j − Pi −1, j( )

∆x

∆t ρφ∆x∆y∆z( ) = ρ.Q( )m

m=1

4

∑

.∆t

in x direction

In Section 2, we start by deriving the single-phase flow equation for a compressible system. This is essentially a pressure equation since this is the only quantity we need to find. The pressure distribution in space is the main unknown in the system and we need to find this as a function of time as the system evolves; pressure is denoted, P(x,y,z;t), for a three-dimensional (3D) system. If we have a way of finding P(x,y,z;t), we can then find the flow rates from the gradient of this quantity i.e. we use Darcy’s law. This sort of “completes the circle” since we use Darcy’s law to derive the pressure equation and, once we solve for the pressure, we find the flows from Darcy’s law etc.

Although we can obtain the single-phase pressure equation for a compressible fluid/rock system in 1D, it turns out to be a non-linear partial differential equation (PDE). There is no known analytical solution to this equation for the general case. That is, even the apparently simple single phase pressure equation cannot be solved using “well known” mathematical functions. You may remember that we discussed the idea of an analytical model in the opening section of this course (Chapter 1 Section 1). If we are faced with this difficult non-linear PDE for the pressure there are two things we might do to solve it as follows: (i) we could possibly use a numerical method to solve it; or (ii) we may be able to simplify the equation by making various assumptions that then allow us to solve the equation analytically. In fact, we will take the latter course of action at this point. (We return to the topic of numerical solution of the flow equation in Chapter 6). Section 2 describes how we simplify the full equation for a compressible fluid in order to obtain an equation that we can solve. Indeed, the simplified equation that arises is none other than the main equation used in single phase well testing.

In Section 3, we go on to derive the equations of two-phase flow in their complete form i.e. for a compressible fluid/rock system. Clearly, if the single-phase pressure equation is not analytically soluble in general, then the two-phase equivalent will certainly not be soluble since it is even more complicated (again, except for some special simple cases).

In the two-phase equations, we also have a pressure equation but, in addition, there is also a saturation equation. Other features are also different from single phase flow,

Figure 1The basic principles of Mass Balance + Darcy’s Law as the basis for all equations for flow through porous media; the Qs are volumetric fluid flows.



such as the capillary pressure, Pc (Sw), leading to two different phase pressures, Po and Pw (where Pc (Sw) = Po-Pw). The two-phase Darcy law also introduces the concept of relative permeability into the flow equations. These concepts have been fully reviewed in Chapter 2 and they are also explained in the Glossary.

2. THE SINGLE PHASE PRESSURE EQUATION

2.1 The Physics of Single Phase Compressible SystemsBefore deriving the single phase pressure equation, consider first the “physics” of what is happening in a compressible single-phase system. Figure 2 shows a long thin “reservoir” containing compressible fluid and rock, which we can consider as being essentially one-dimensional (1D). The fluid and the rock have compressibilities, cf and cr (where cf >> cr; see Glossary). Imagine the “reservoir” is horizontal and has two wells in it at either end as shown in Figure 2a. We take the location of Well 1 as being at x = 0, and Well 2 is located at x = L. If the wells are both shut-in and the reservoir is left to achieve steady-state, then clearly the pressure in the system will reach a constant value, Po (ignoring the gravitational potential since the reservoir is also “thin” in the z-direction). The bottom hole pressure also equalises in both wells and will also be Po. This is shown as the dashed line (at t = 0) in Figure 2b where the “pressure profile” through the reservoir, P(x)=Po, is also shown.

t=t1 t=t2 t=t3

t=0

x L

Pout = PoPo

P

Pin

0

Injector Well

Long thin (1D) reservoir

Producer Well(a)

(b)

Now consider what happens to the pressure profile if we cause the pressure to rise suddenly at Well 1, say by injecting fluid (identical to that already in the reservoir).

Is this pressure disturbance immediately “felt” at Well 2 at x = L? The answer is of course “no” since the pressure wave will take some time to transmit from Well 1

Figure 2(a) Schematic of a long thin “reservoir” containing compressible fluid and rock, which we can consider as being essentially one-dimensional (1D); (b) pressure profile along the system at various times from t = 0, t1 , t2 etc. - this shows the pressure disturbance travelling (“diffusing”) along the system.



to Well 2. At a short time after this disturbance, at Well 1 say at t = t1, the pressure profile, P(x,t1), is shown schematically in Figure 2b. Likewise, at a later time, t2, it is shown as, P(x,t2).

But what happens at Well 2? This depends on what we do at Well 2: Suppose, we leave it shut-in, what will eventually happen? If we were injecting a volumetric flow rate, q1(t), into Well 1 such that it maintained a constant pressure, P1(x = 0,t); i.e. we would be varying (reducing) q1 with time, the outcome is quite predictable. We would keep pumping in at (decreasing) rate q1(t) until we had “blown-up” the pressure in the reservoir – like pumping up a car tyre – to pressure P1. Hence, the q1 value would reduce to q1 = 0 and the reservoir pressure – and the pressure in both wells – would settle to P(x,t) = P1.

On the other hand, suppose we simply pumped into Well 1 at constant rate q1(now fixed) and we held the bottom hole pressure of Well 2 to be constant at P(x =L) = Po (the original reservoir pressure). What would happen for this case? With a little thought, you can probably visualise that the sequence of events would be as follows:

• Firstly, at early times up to t2 and a little later, nothing would happen at Well 2. It would still have pressure Po and would not be flowing;

• The pressure “wave” would propagate or “diffuse” through the reservoir as shown in Figure 2b; the “speed” of this wave is governed by the “diffusivity” which is quantified by k/(cfφμ) as we will show later in this chapter. Note that the larger the fluid compressibility (cf) or porosity (φ), the “slower” the wave propagates.

• At a certain time, denoted, t3, the pressure disturbance just reaches the produced Well 2 as shown in Figure 2b;

• At t = t3, Well 2 must start to flow in order to maintain the pressure at P = Po (this works like a back-pressure regulator);

• As time proceeds (t → ∞), the pressure wave starts to be felt all across the reservoir such that the flowrate at Well 1 is (still fixed) at q1 and the pressure settles to a constant value P1;

• Likewise, as t → ∞, Well 2 (still at fixed pressure Po) will go to steady-state flow exactly at flowrate, q1;

• At t → ∞, the pressure profile P(x,t → ∞) will tend to a straight line as shown by the inclined dashed line in Figure 2b.

This “thought experiment” is extremely useful to appreciate the physics of flow in a compressible system. It also introduces the ideas of boundary conditions. That is, not only do we need a way of modelling the development of pressure in our reservoir (i.e. finding an equation for P(x,t)), we also need to appreciate what boundary conditions to apply in a given situation. We have been quite explicit here in identifying the boundary conditions with different well constraints. For example, for the main case considered above, Well 1 is a rate-constrained injector and Well 2 is a pressure–constrained producer.



EXERCISE 1.

Can you explain the sequence of events that would occur in the simple 1D system of Figure 2 if both wells were pressure constrained to Pin (Well 1) and Po (Well 2) with Pin > Po?

2.2 The Single Phase Pressure EquationWe now return to the task of deriving the single-phase flow equations for a compressible system. Throughout this derivation we refer to the control volume shown as block i in Figure 3 where certain terminology is also explained. In Figure 3, we have divided up the x-axis into increments of (constant) size, Δx, and constant cross-sectional area, A. Flow is considered to be in the positive x-direction (i increasing). The fluid has density, ρ, which may depend on pressure i.e. ρ(P). The porosity is denoted as φ and it too may depend on pressure, φ(P); in block i, the porosity is, φi. The volumetric flows across the boundaries of block i are given by: qi-1/2 and qi+1/2 as shown in Figure 3; the dimensions of these quantities are volume/time and typical units might be STB/day, m3/sec etc.

Let us now apply the mass conservation conditions as were applied in the introductory section of this chapter. Mass conservation states that:

The mass accumulation(increase or decrease)

in block i over a time step, ∆t

The mass thatflows IN over

time ∆t

The mass thatflows OUT

over time ∆t = - (1)

We can easily write mathematical expressions for each of the terms in equation 1 as follows:

The mass of fluid

flowing IN to block i

over time t

Volumetric rate

of in flow to

block i

density of

the fluidx t = q. t

i-12

∆=

( )x ∆ ∆ρ . (2a)

Llikewise:

The mass of fluid flowing OUT of = (qρ)

i+1/2.Δt (2b)

block i over time Δt



Boundary

Boundaryi-1/2

qi-1/2

qi+1/2

x

∆x

∆x

∆x

i-1

i+1

Block i

i+1/2

Notation:

1 The boundaries of block i are denoted (i-1/2) between blocks (i-1)and i (i+1/2) between blocks i and i+1

2 q(i-1/2) denotes the volumetric flow rates across the (i-1/2) boundary q(i+1/2) denotes the volumetric flow rates across the (i+1/2) boundary Note – units bbl/day, m3/s etc…

3 The porosity of block i = φi, the permeability is ki etc..

Therefore, the change in mass of fluid in block i over time Δt is given by:

Change in mass over Δt in block i = ( ) ( )

= ( ) − ( )[ ]= − ( ) − ( )[ ]

+

−

+

q t q t

q q t

q q t

i 12 i

i i+

i i

ρ ρ

ρ ρ

ρ ρ

-

-

. - .

.

∆ ∆

∆

∆

12

12

12

12

12

(3)

Thus, equation 3 expresses the change in mass due to flow that occurs in block i using quantities – volumetric flow rates and densities – defined on the boundaries. Note that we have changed the signs consistently in the final step of equation 3 for convenience below. Which quantity do you think we mean when we refer to ρi-1/2 : the density “on the boundary”, i-1/2? This seems a bit strange. However, you can think of ρi-1/2 being some sort of “average” density between ρi-1 and ρi, the densities in blocks (i-1) and i, respectively. In fact, we have discussed this matter of grid block to grid block average in Chapter 4.

We now turn to the alternative way of expressing the mass of fluid in a grid block i at the different times, t and t+Δt. This is shown in Figure 4 where we denote the mass of fluid in grid block i at times t and (t + Δt) by (ρφΔxA)t and (ρφΔxA)t+Δt , respectively. Therefore:

Figure 3Control volume grid block for application of material balance for the single phase flow equation; notation is given below.



The change of mass of fluid in block i over the time from = The mass at - The masst to t+Δt (t + Δt) at t = [(ρφ)

t+Δt - (ρφ)

t].ΔxA (4)

where we have used the fact that Δx and A are constants in equation 4.

∆xArea = A

RockPoreSpace

The mass of fluid in block i

= (Pore volume of block i) x density

= (Volume of block x porosity) x density

= (∆x.A.φ.)ρ

= ρφ.∆x.A

We now apply the material balance condition (equation 1) by equating the expressions in equations 3 and 4 as follows:

ρφ ρφ ρ ρ( ) − ( )[ ] = − ( ) − ( )[ ]t+ t t i+12 i-1

2xA q q t∆ ∆ ∆. . (5)

Now divide through equation 5 by the (constant) Δx.AΔt to obtain:

ρφ ρφ

ρ ρ( ) ( )[ ] = −

−

−t+ t t i+1

2 i-

t

qA

qA

x∆

∆ ∆1

2 (6)

where we have taken the (constant) area, A, to the inner parenthesis on the RHS of equation 6.

As it stands, equation 6 is exactly true since it is simply a statement of mass balance. There are no assumptions in this equation. Indeed, we can simplify equation 6 a little more by noting that q/A is the Darcy velocity, u. This becomes:

ρφ ρφ ρ ρ( ) ( )[ ] = −( ) ( )[ ]t+ t t i+1

2 i-12

-

t

u. - u.

x∆

∆ ∆ (7)

Figure 4Expressions for the mass of single-phase fluid in a grid block



which is still an exact statement of fact. We can take the limits of the difference equation 7 at Δt → 0 and Δx → 0 to obtain the equivalent differential equation.

Clearly:

Lim

t.

∆ ∆∆

t

-

t t+ t t

→( ) ( )[ ] =

∂( )∂0

ρφ ρφ ρφ (8)

and

Lim.

∆ ∆x

u. - u.

x

u.

x

i+12 i

→−

( ) ( )[ ]= −

∂( )∂

−

0

12

ρ ρ ρ (9)

and therefore:

∂( )

∂= −

∂( )∂

ρφ ρt

u.

x (10)

Equation 10 is the differential form of the conservation equation and again, it is “exact” in the sense discussed above. Before going on to use Darcy’s law (for u), look at the structure of equation 10; in particular look at the symmetry between ρ and φ. These two quantities appear in an identical manner in the LHS of this equation. Hence, if the mass in a grid block stayed the same with time, i.e. ∂(ρφ)/∂t = 0, then this could be because the fluid density went down (the fluid expands) and the rock porosity went up (the rock contracts or compresses).

We now assume Darcy’s law for u, that is:

u - k

.Px

= ∂∂

µ

(11)

(as given in the Glossary etc.). We may substitute this form of the Darcy law directly into equation 10 (taking care with the signs) to obtain:

∂( )

∂= ∂

∂∂∂

ρφ ρµt

x

k Px

(12)

This equation is now inexact, in that its validity depends on whether Darcy’s law is or is not a good assumption. However, it was necessary to use a flow law such as Darcy’s law since pressure (i.e. P(x,t)) does not explicitly appear in equation 10. Thus, Darcy’s law is our “link” between fluid velocities and pressure gradients. Equation 12 still does not have pressure (P(x,t)) explicitly shown on the LHS; it simply has a term ∂(φρ)/∂t. However, we know that φ and ρ depend locally only on pressure i.e. they can be written ρ(P) and φ(P). We therefore manipulate the LHS of equation 12 as follows using the chain rule of differentiation:

∂( )

∂=

∂( )∂

∂∂

ρφ ρφt P

Pt

. (13)



Thus, substituting equation 13 back into equation 12 we obtain a “true” single phase pressure equation as follows:

∂( )

∂

∂∂

= ∂

∂∂∂

ρφ ρµP

Pt x

k Px

. . (14)

where we can consider the term ∂(φρ)/∂P as a generalised fluid and rock compressibility, term, C(P). Equation 14 would then become:

C PPt x

k Px

( ) ∂∂

= ∂

∂∂∂

. .

ρµ

(15)

Some points should be noted about equation 15, as follows:

(i) It is a non-linear partial differential equation (PDE). By “non-linear” we mean that the coefficients in the equation – the “input”, if you like- depend on the quantity we are trying to find, the unknown pressure, P(x,t). In other words, quantities such as C(P), ρ(P), μ(P) etc. “depend on the answer”. We will not go into mathematical detail but such problems are notoriously difficult to solve analytically for general cases.

(ii) Following on from point (i) above, such equations can only usually be solved by one or two approaches: • By solving them numerically where we can handle the non-linearities using certain types of iterative methods (see Chapter 6).

• We may simplify the equation to the extent that it becomes soluble.

Clearly, by its nature, a numerical solution will be approximate although it is an approximation to the full equation (equation 15, in this case). With a simplified equation, we may have an exact solution, but the simplifications may have “thrown away” some of the important physics. In fact, in the following section, we will take the second approach.

(iii) Going back to equation 10, we could include gravity effects by taking the Darcy equation with gravity as follows:

u = − ∂∂

− ∂

∂

k Px

g zxµ

ρ (16)

(instead of equation 11). If the 1D system is at constant incline, z(x), then (∂z/∂x) = cos θ where θ is the angle of incline, therefore, we obtain equation 17 below:

u = − ∂∂

−

k Px

g cos µ

ρ θ (17)



This could simply lead to the generalised single phase equation as follows:

∂( )

∂

∂∂

= ∂

∂∂∂

−

ρφ ρ ρµ

ρ ρ θP t x

kx

g cos . (18)

but this is fairly straightforward.

2.3 The Simplified Compressible Pressure EquationIn this section, we will take equation 15 as our starting point and introduce some simplifying assumptions. Our hope is to get to a simpler equation which will have analytical solutions for given boundary conditions. Indeed, for this to be the case, we would normally aim to derive a simplified linear PDE.

In fact, to make clear what is happening to the LHS of the equation we start from equation 14, which is repeated below:

∂( )

∂

∂∂

= ∂

∂∂∂

ρφ ρµP

Pt x

k Px

. . (14)

We first list the assumptions which we will make in the above equation and then we will go on to examining their consequences.

Simplifying Assumptions

1 Viscosity, μ, is constant (with x and P);2 Permeability and porosity, k and φ, are constant, (with x and P), i.e. the system is homogeneous and the rock is incompressible;3 That pressure gradients, (∂P/∂x), are “small” such that:

∂∂

≈P

x

2

0

4 The fluid has a constant compressibility, cf, i.e. cf = 1ρ

ρ∂∂

≈

Pcons ttan .

Assumption 1 is probably quite reasonable since viscosity does not vary greatly for most oils (or water) over small pressure ranges. Assumption 2 is quite drastic since it says that permeability (k) is constant through the reservoir i.e. that the system is homogeneous in k (and φ). For a real system, this is indeed a very simplifying assumption. However, we will indicate below how we can turn this assumption round, in a sense. The second part of Assumption 2 is that the rock is incompressible and this is quite reasonable (usually, cf >> crock). Assumption 3 is rather odd: it is clearly designed to get rid of “difficult” terms with terms like (∂P/∂x)2 in them. Assumption 4 – fluid compressibility, cf, does not vary with pressure, is again quite reasonable for most (non-critical) reservoir oils.

Applying Assumptions 1 and 2 results in the following simplification of equation 14:



φ ρµ

ρ∂∂

∂∂

= ∂

∂∂∂

P t

kx x

.P P

(19)

which rearranges quite simply to:

µφ ρ ρk P

Pt x

Px

∂∂

∂∂

= ∂

∂∂∂

. . (20)

Now expand the RHS of equation 20 as follows using the product rule:

∂

∂∂∂

= ∂∂

∂∂

+ ∂

∂

x

Px x

Px

Px

2

2ρ ρ ρ. (21)

and use the fact that ρ is a function of pressure only, i.e. ρ(P), to further expand the (∂ρ/∂P) in equation 21 as follows:

∂

∂∂∂

= ∂∂

∂∂

∂∂

+ ∂

∂

x

Px P

Px

Px

Px2ρ ρ ρ. .2

(22)

Equation 22 can be then simplified as follows:

∂

∂∂∂

= ∂∂

∂∂

+ ∂

∂

x

Px P

Px

Px2ρ ρ ρ.

2 2

(23)

Now use Assumption 3 to eliminate the first term on the RHS of equation 23 above

(i.e. ∂∂

≈P

x

2

0 ) to obtain:

µφ ρ ρk P

Pt

Px

2

2

∂∂

∂∂

= ∂

∂

(24)

Note that we can divide both sides of 24 by ρ and use the fact that the fluid has constant compressibility, cf (assumption 4), to obtain:

µφc

kPt

Px

f

∂∂

= ∂

∂

2

2 (25)

where the coefficient (μφcf/k) is a constant.

This is more commonly written with the constant on the RHS as follows:

∂∂

= ( )

∂∂

Pt

kc

Pxfµφ

.2

2 (26)

where the constant in this form is normally referred to as the hydraulic diffusivity, Dh=k/(μφcf).



Equation 26 is now a linear PDE and has the form of a linear diffusion equation as follows:

∂∂

= ∂

∂

Pt

DP

xh 2

2

(27)

Note the following about the above equations (26 or 27):

(i) It is the simplified (slightly) compressible 1D flow equation in a linear porous media i.e. in Cartesian form.

(ii) Analytical solutions are available for a range of boundary conditions (see Crank, The Mathematics of Diffusion, 2nd edition, Oxford Clarendon Press, 1975).

(iii) In its radial form (i.e. in radial coordinates, r, rather than x) equation 27 becomes the following:

∂∂

= ∂

∂∂∂

Pt

Dr r

rPr

h (28)

which is a well known equation of well testing (Stanislav and Kabir, 1990). Likewise, this equation has a number of well known analytical solutions for various boundary conditions.

(iv) The reason these equations have many ready-made analytical solutions available is because these diffusion equations are well-known and are identical in form to the equations of heat conduction which have been studied for many years (Carslaw and Jaeger, 1959).

2.4 Extension of the Single Phase Pressure Equation to 2D and 3DFor the 2D case, the control volume is now grid block (i,j), as shown in Figure 5. The flows are shown at the boundaries as before and are labelled as (i- 1/2) for flow from (i-1) → i in the x-direction, (j + 1/2) for the flow from block (i,j) → (i,j + 1) etc. as listed on Figure 5. The flow areas in the x- and y-directions are given by, Ax = (Δy.h) and Ay = (Δx.h), respectively, as shown in Figure 5. In addition, we show a source/sink term due to a (single phase) well. This well again injects or produces exactly the same fluid as is in the reservoir already. The well flow rate into block (i,j) is described by q ij which is the volumetric flow rate per unit volume of block (i,j); possible units for q ij are m3/ s or bbl/day etc. For this definition, the volume rate of flow into (sign +) or out of (sign -) the well in block (i,j) is as follows:

Mass rate of flow into or out of block (i, j) = qdensity of injected or

produced fluid, volume block (i, j)

= q

ijij

ij ij

˜

˜ .

×

×

×

ρ

ρ ∆ ∆x y h (29)



Well

x (i)

y (j)

∆x ∆y

Area, Ay = ∆x.h Area, A

x = ∆y.h

h

i,j+1

i,ji,j-1 i+1,j

i-1,j

Thickness = h

Note for the control volume, block (i,j), the flow areas in the x and y directions,Ax and Ay are given by:

NOTATIONi, j control volumei - 1/2 = (i-1) → ii + 1/2 = i→ (i+1)j - 1/2 = (j-1) → jj + 1/2 = j→(j+1)

Mij

We now apply the material balance equation as follows:

Change in mass in block (i,j) over time Δt due to flows across boundaries and the well

= ( ) − ( )[ ] + ( ) − ( )[ ]− + − +q q t q q t + q x y.h. t

(the x - flows) (the y - flows) (well source / sink term)

i i ij ijρ ρ ρ ρ ρ12

12

12

12

.∆ ∆ ∆ ∆ ∆j j

(30)

The accumulation term is the same as previously (Figure 4) as follows:

Change in mass in block (i,j) between time t and t+Δt

= ( ) −( ) = ( ) − ( )[ ]φρ φρ φρ φρ x y.h x yh x yht+ t t t+ t t

∆ ∆ ∆ ∆ ∆ ∆∆ ∆ (31)

We then equate expressions in equations 30 and 31 to obtain:

φρ φρ

ρ ρ ρ ρ ρ

( ) − ( )[ ]= − ( ) − ( )[ ] − ( ) − ( )[ ]+ − + −

t+ t t

i i j j ij ij

x y.h

q q t q q t + q x y.h t

∆ ∆ ∆

∆ ∆ ∆ ∆ ∆12

12

12

12

. . ˜

(32)

Figure 5The 2D x/y Grid Showing the Control Volume, Block (i,j)



where we note that the signs have been adjusted slightly on the RHS. Dividing through by Δx.Δy.h.Δt and cancelling gives the following:

φρ φρ ρ ρ ρ ρρ

( ) − ( )[ ] = −( ) − ( )[ ]

−( ) − ( )[ ] ( )+ − + −t+ t t i i

x

j j

yijt

q q

x.A

q q

y.A + q∆

∆ ∆ ∆1

21

21

21

2 ˜

(33)

where we have used the fact that ΔxΔyh = ΔxAx and ΔxΔyh = Δy.Ay. The flow areas, Ax and Ay, may then be divided into the q’s to give Darcy velocity terms (e.g.

uqAi

x i

ρ ρ( ) =

+

+1

21

2

) as follows:

φρ φρ ρ ρ ρ ρρ

( ) − ( )[ ] = −( ) − ( )[ ]

−( ) − ( )[ ] ( )+ − + −t+ t t i i j j

ijt

u. u.

x

u. u.

y + q∆

∆ ∆ ∆1

21

21

21

2 ˜

(34)

Again, taking limits as Δt, Δx, Δy and → 0, gives:

∂( )∂

= −∂( )

∂−

∂( )∂

+ ( )φρ ρ ρ ρt

u

x

u

y

qx y ˜ (35)

This is the 2D mass conservation equation is still exact in that no approximations have yet been made. Clearly, we can easily generalise this to 3D by simply adding a z-flow term (for vertical flows, uz) to give:

∂( )

∂= −

∂( )∂

−∂( )

∂−

∂( )∂

+ ( )φρ ρ ρ ρρ

t

u

x

u .

y

u .

z q

y y z ˜ (36)

As before, the pressure P(x,y;t) (in 2D) or P(x,y,z;t) (in 3D) does not yet appear. We must manipulate the LHS of equation 35 (or 36) to yield, exactly as before:

∂( )

∂=

∂( )∂

∂∂

φρ φρt P

Pt

. (37)

And on the RHS of equation 35 (or 36) we must use Darcy’s law.

Darcy’s law in each of the 3 directions, x, y, and z, is given as follows, where we choose z as the vertical direction (of gravity):

uk P

xg

zxx

x= − ∂∂

− ∂

∂

µ

ρ (38a)



uk P

yg

zyy

y= − ∂∂

− ∂∂

µ

ρ (38b)

uk P

zgz

z= − ∂∂

−

µ

ρ (38c)

where we note that the permeability may be anisotropic i.e. kx ≠ ky ≠ kz. Using the above expressions (equations 38a-c) for the Darcy velocity along with equation 37 for the LHS in the 3D equations 36 gives:

∂( )∂

∂∂

= ∂

∂∂∂

− ∂

∂

+ ∂

∂∂∂

− ∂∂

+ ∂∂

φρ ρµ

ρρµ

ρ

ρµ

PPt x

k Px

gzx

y

k Py

gzy

z

k

x y

z

.

∂∂∂

−

+ ( )Py

g qρ ρ˜

(39)

Equation 39 is the 3D generalisation of equation 14 including gravity and also a well (source/sink) term. Again, this is a 3D non-linear partial differential equation (PDE) which cannot generally be solved analytically. Note also that the equation is rather lengthy and it would be useful if there was a more shorthand way of writing this equation. This is dealt with in the following section.

2.5 Mathematical Shorthand for the 3D Single-Phase Pressure EquationIn order to write equation 39 in a more compact form we must use the notation of vector calculus. Before doing this, we review the following mathematical concepts:

Mathematical Concepts Review: Review the meaning of:

(i) The divergence operator, ∇., on a vector V:

∇ ∂∂

+∂∂

+ ∂∂

∇ = ∂∂

∂∂

∂∂

.V

k

=Vx

V

y

Vz

where vector V =

V

V

V

PPx

i +Py

j +Pz

x y x

x

y

z

(ii) The gradient operator, ∇, on a scalar, such as pressure, P

∇ ∂∂

+∂∂

+ ∂∂

∇ = ∂∂

∂∂

∂∂

.V

k

=Vx

V

y

Vz

where vector V =

V

V

V

PPx

i +Py

j +Pz

x y x

x

y

z



where i, j and k are the unit vectors in the x-, y- and z-directions.

(iii) see also operations on a tensor (Chapter 2, Section 3.2).

Using this notation, equation 35 (the exact conservation equation) becomes:

∂( )

∂= −∇ ( ) + ( )φρ

ρ ρt

q. ˜u (40)

or expanding the LHS of equation 40 as before:

∂( )

∂

∂∂

= −∇ ( ) + ( )φρ

ρ ρP

Pt

u q. ˜ (41)

After the Darcy law has been used – as in equation 38 – the vector calculus version of this equation becomes:

∂( )

∂

∂∂

= ∇ ∇ − ∇( )

+ ( )φρ ρ

µρ ρ

PPt

k P g z q. . . ˜ (42)

We will not expand on equation 42 here but it can be seen that it does give a compact way of writing the 3D pressure equation. This approach can also be used to write the 3D multi-phase flow equations, as we will develop below.

Exercise 2

The following equation is the full 2D equation for a compressible fluid (in the absence of gravity and with no well terms):

∂( )∂

∂∂

= ∂

∂∂∂

+ ∂

∂∂∂

φρ ρµ

ρµP

Pt x

k Px y

k Py

x y.

Simplify this equation as far as possible by making all the Assumptions (1) - (4) in Section 2.3 but keep kx ≠ ky (although both are constant).

3. THE TWO-PHASE FLOW EQUATIONS

3.1 Review of Two-Phase Flow ConceptsWe now consider the equations which govern the flow of two phases through a porous medium e.g. oil-water, gas-oil, air-water. In certain respects, the approach is very similar to that used for single-phase flow in Section 2 above. We apply the mass conservation equation to each of the phases separately. However, there is a little more “two-phase physics” that we must be aware of before proceeding. For this reason, we first review some of the key concepts from two-phase flow (see Chapter 2 or the Glossary).



Key Concepts: Ensure you are familiar with the following main ideas on two-phase flow (where we assume oil/water in the examples below):

• Phase saturations, So and Sw; where So + Sw = 1

• Formation volume factors, Bo and Bw (units RB/STB) where, for example, ρo = ρosc/Bo and ρosc is the oil density at standars conditions (likewise for ρw).

• The two-phase Darcy Law (with gravity) and relative permeabilities, krw(Sw) and kro(So)

u = -k k P

xg

zx

u = -k k P

xg

zx

oro

o

oo

wrw

w

ww

µρ

µρ

∂∂

− ∂

∂

∂∂

− ∂

∂

• Phase pressures, Po and Pw, and the concept of capillary pressure, Pc (Sw) = Po–Pw, as a constraint of the phase pressure difference at various saturations, Sw.

Exercise 3

Calculate the mass of oil and water phases for the control volume (grid block) shown using the usual notation.

∆x

∆zRockOil, S

oWater, Sw ∆y

Ans:Mass oil = Volume of oil x density of oil= (∆x∆y∆z φ So) x ρo

Mass oil = ∆x∆y∆z ( φSoρo) = ∆x∆y∆zρosc (φ So/Bo)

Likewise:Mass water = ∆x∆y∆z (φSwρw) = ∆x∆y∆zρwsc (φ Sw/Bw)

3.2 Derivation of the Two-Phase Conservation EquationsAs for the single phase case, we will use the material balance in a control volume as shown in Figures 6 and 7.

Definition: A very useful quantity to define is the oil flux, Jo, and the water flux, Jw. The oil (water) flux is the mass rate of flow of oil (water) per unit cross sectional area. Dimensions are M.L-2.T-1 and possible units would be kg.m-2.s-1 or lbs.ft-2.day-1.

The oil flux, Jo (and Jw) can easily be related to other more familiar quantities as follows: consider the flow area across the block area shown below.



∆y ∆x∆z

Area, A = ∆y.∆

z

qo Volumetric

Oil

Flow Rate

Oil Flux:

Jo= ρo.qo

Aρosc.uo

Bo=

The volumetric flow rate of oil, qo, can be used to obtain an expression for the flux, Jo. Clearly:

∆x

So

Oil

Oil

Water

Water

∆x

∆x

Boundary(i-1/2) Boundary(i+1/2)Permeability = k

Porosity = φ

∆z

∆y

Sw

x

i-1

i+1

i

Fluid(φ)

Area=A=∆z. ∆y

Mass flow rate of oil

Across area Ax

,=

=volumetric flow rate

of oil across A

density of

oil, Q .

oo oρ

ρ (43)

∴

=

Mass flow rate of oil per unit area = q

A= u

uB

o oo o

o osc

o

ρ ρ ρ (44)

But, by definition the above quantity is the flux, Jo (Figure 6). (Where we note that qo/A = uo, the Darcy velocity of oil). Thus, the flux expressions are:

J u J uo o o w w w= ( ) = ( )ρ ρ; (45)

Figure 6Definition of oil (water) flux, Jo .

Figure 7Control Volume (block i) for Two-Phase Flow



However, we wish to work with reference to standard conditions (sc ⇒ standard conditions; 60°F and 14.7 psia; ~15.6°C and 1 bar). Therefore, introducing the densities of oil and water at standard conditions, ρosc and ρwsc, we obtain (by definition above):

Ju

B

Ju

B

oo osc

o

ww wsc

w

=

=

ρ

ρ (46)

Now use the flux expression to perform the mass balance on control block i in Figure 7.

The flow terms are as follows:

Mass of oil flowing into block i over time step ΔtMass of oil flowing out of block i over time step t

Flux of

oilArea. t J .A t

i-12

o i-12

∆

∆ ∆=

= ( ). . (47)

Mass of oil flowing out of block i over time step Dt = (Jo.A.Δt)i-1/2 (48)

Change in mass of oil in block over t due to flow J - J A. to i+12

o i-12

∆ ∆= − ( ) ( )[ ] (49)

(where we note in Figure 7 that A = ΔyΔz)

The accumulation - i.e. change in static mass of oil in block i - is calculated. The expression for this is given in the Exercise above:

Change of mass in block over Δt (accumulation) = Mass in block at t+Δt – Mass in block at t

= ( ) − ( ) = ( ) − ( )[ ]∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆∆ ∆

x y z S x y z S S S x y zo o t+ t o o t o o t+ t o o tφ ρ φ ρ φ ρ φ ρ

(50)

Equate the expressions in equations 49 and 50 to obtain

φ ρ φ ρS S x y z = - J J A. to o t+ t o o t o i o i-( ) − ( )[ ] ( ) − ( )[ ]+∆∆ ∆ ∆ ∆1

21

2 (51)

Divide through the above equation by ΔxΔyΔzΔt to obtain:

φ ρ φ ρS S

t= -

J J

xo o t+ t o o t

o i o i-( ) − ( )[ ] ( ) − ( )[ ]+∆

∆ ∆1

21

2 (52)



Taking limits as Δt, Δx → 0, we obtain the differential form of the mass conservation equation:

∂( )

∂− ∂

∂

φ ρS

t=

Jx

o o o (53)

Using the fact that ρρ

oosc

o

=B

in the LHS of the above equation gives:

∂∂

− ∂∂

t

SB

=Jx

o osc

o

oφ ρ (54)

Substituting for Ju

Boo osc

o

=

ρ gives:

∂∂

− ∂∂

t

SB

= x

uB

o osc

o

o

o

φ ρ ρosc (55)

However, ρosc is a constant reference density (at standard conditions) which cancels to yield:

∂∂

− ∂∂

t

SB

= x

uB

o

o

o

o

φ (56)

Likewise,

∂∂

− ∂∂

t

SB

= x

uB

w

w

w

w

φ (57)

The above equations are the (exact) differential forms of the oil and water mass conservation equations. They involve no assumptions; they simply arise as a result of the definition of the various terms. However, the equations are of little practical use in this form since they do not mention pressure, and it is this that we measure most directly in a reservoir – not local (spatially distributed) velocities, uo and uw. Clearly, it is now time to introduce Darcy’s Law.

The expressions for the two phase Darcy Law are reviewed at the start of this sub-section. Substituting for uo and uw in equations 56 and 57 above gives:

∂∂

∂∂

∂∂

− ∂

∂

t

SB

=x

k kB

Px

gzx

o

o

ro

o o

oo

φµ

ρ (58)

∂∂

∂∂

∂∂

− ∂

∂

t

SB

=x

k kB

Px

gzx

w

w

rw

w w

ww

φµ

ρ (59)



These equations now express mass conservation of oil and water, on the assumption that the 2-phase Darcy Law applies. These are our “working equations” for the two phase flow (in their 1D form).

At a first glance, it appears that equations 58 and 59 present us with two equations in four unknowns. So, Sw, Po and Pw. But, as we have noted above in the review of key concepts, there are two constraints on these quantities; viz So + Sw = 1 and Pc(Sw) = Po – Pw. Thus, only two of these quantities e.g. Po, Sw or So, Pw etc. are truly independent. It is now clear why we described the Pe as a constraint equation earlier in this chapter.

The above equations still require some manipulation since we would like to have a pressure equation which was free of terms of the type (∂So/∂t) and (∂Sw/∂t). This is derived in the next section.

3.3 The Two-Phase Pressure EquationIn order to eliminate the saturation derivatives with respect to time in equations 58 and 59 above, we proceed as follows. Firstly, we decide to retain the oil pressure, Po, (P = Po) as our reference pressure (this is arbitrary).

Next we expand the LHS of the oil equation (equation 58) as follows:

∂∂

= ∂∂

+ ∂∂

+ ∂

∂

= ∂∂

∂∂

+ ∂

∂

+ ∂

∂

tS

BS

t B BSt

SB t

SP B

Pt B

St

SB P

o

oo

o o

o o

o

oo o

o

o

o o

o

φ φ φ φ

φ φ φ

1

1. . .

oo

o

o

oo

o o

o

o o

o

Pt

B

St

SP B

SB P

Pt

∂∂

= ∂∂

+ ∂

∂

+ ∂∂

∂∂

.

φ φ φ1

(60)

where the underlined term is the one we wish to eliminate. Likewise, for the LHS of the water equation (equation 59):

∂∂

= ∂∂

+ ∂

∂

+ ∂∂

∂∂t

SB B

St

SP B

SB P

Pt

w

w w

ww

o w

w

w o

oφ φ φ φ1 (61)

Note that the underlined terms in equations 60 and 61 are to be eliminated and the following terms are essentially of the form = (compressibility terms) x (∂P/∂t).

To eliminate the time derivative terms of the saturations above, then note that

∂∂

+ ∂

∂

St

St

= 0w o (62)

by definition (since Sw + So = ∴ ∂∂

+ ∂∂

1St

St

= 0w o )



Therefore, multiply equation 60 (oil) by Bo/φ on both sides and multiply 61 (water) by Bw/φ on both sides to obtain:

B

tS

B=

St

B SP B

SP

Pt

o o

o

oo o

o o

o

o

o

φφ

φφ∂

∂

∂∂

+ ∂∂

+ ∂∂

∂∂

.

1 (63)

and

B

tS

B=

St

B SP B

SP

Pt

w w

w

ww w

o w

w

o

o

φφ

φφ∂

∂

∂∂

+ ∂∂

+ ∂∂

∂∂

.

1 (64)

Now ADDING the above equations 63 and 64, we see that the ∂S/∂t terms vanish (equation 62) as follows:

Bt

SB

+B

tS

BB S

P BB S

P BS

PS

PPt

o o

o

w w

wo o

o ow w

o w

o

o

w

o

o

φφ

φφ

φφ

φφ∂

∂

∂∂

= ∂∂

+ ∂∂

+ ∂∂

+ ∂∂

∂∂

=

. .1 1

BB SP B

B SP B

1P

Pto o

o ow w

o w o

o. .∂

∂

+ ∂∂

+ ∂∂

∂∂

1 1φ

φ

(65)

where we have used So + Sw = 1 on the RHS in the last simplification in equation 65.

The expression above (equation 65) is now equal to the RHS of equation 58 multiplied by Bo/φ + RHS of equation 61 multiplied by Bw/φ. First, we simplify a little by denoting all of the compressibility terms above by, α(So,Po) - that is:

αφ

φS ;P B S

P BB S

P B1

Po o o oo o

w wo w o

( ) = ∂∂

+ ∂∂

+ ∂∂

. .

1 1 (66)

Therefore:

αφ µ

ρφ µ

ρS ;PPt

Bx

k kB

Px

gzx

Bx

k kB

Px

gzxo o

o o ro

o o

oo

w rw

w w

ww( ) ∂

∂

= ∂

∂∂∂

− ∂

∂

+ ∂

∂∂∂

− ∂

∂

(67)

Now use capillary pressure, Pc(Sw)=Po-Pw, to eliminate the water pressure as follows:

∂ ( )

∂= ∂

∂

− ∂

∂

∂∂

= ∂

∂− ∂

∂

P S

xPx

Px

Px

Px

Px

c w o w w o ci e. . (68)

Substituting this expression from (∂Pw/∂x) into equation 67, we obtain



αφ µ

ρ

φ µρ

S ;PPt

Bx

k kB

Px

gzx

B

xk k

BPx

Px

gzx

o oo o ro

o o

oo

w rw

w w

o cw

( ) ∂∂

= ∂

∂∂∂

− ∂

∂

+ ∂∂

∂∂

− ∂

∂

− ∂

∂

(69)

This is the 1D pressure equation for a two-phase compressible (fluids and rock) system. Each of the terms is physically interpretable and we expand this out to see each of the contributions more clearly.

φαµ µ

ρµ

S ;PPt

Bx

k kB

Px

Bx

k kB

Px

OIL FLOW WATER FLOW

Bx

k k gB

zx

B

o oo

oro

o o

ow

rw

w w

o

oro o

o o

( ) ∂∂

= ∂

∂∂∂

+ ∂

∂∂∂

− ∂∂

∂∂

−

. .

. wwrw w

w w

wrw

w w

c

xk k g

Bzx

GRAVITY TERMS

Bx

k kB

Px

CAPILLARY PRESSURE TERM

.

.

∂∂

∂∂

− ∂∂

∂∂

ρµ

µ (70)

Notes on equation 70

(i) The two-phase pressure equation for the fully compressible system is clearly very complex. Again, it is a non-linear partial differential equation (PDE) which cannot be solved analytically for the general case;

(ii) Despite its complexity, all of the terms in equation 70 have a clear physical interpretation in terms of viscous, gravity and capillary forces;

(iii) Following from (i) above, our two alternatives are either to use numerical methods to solve equation 70 or to simplify it greatly such that an analytical solution may be possible. We will consider the simplified pressure equation in the next section;

(iv) Recall that in two-phase flow, the dependent variables (the unknown we want to find) were chosen to be:

Po(x,t) and So(x,t)

Therefore, once we solve the pressure equation 70, we must then calculate the saturation. In fact, ideally we would like to solve the pressure equation at the same time as solving for the saturation. Where is our saturation equation in any case? In fact, we have already met this – as equation 59 above.



3.4 Schematic Strategy for Solving the Two-Phase Pressure and Saturation Equations

In fact, we can write the two equations for pressure and saturation in the following schematic way:

PRESSURE φαµ

S ;PP

tB

xk k

BPxo o o

ro

o o

o( ) ∂∂

= ∂∂

∂∂

. + gravity terms + cap. terms

etc. (70)

SATURATION ∂∂

∂∂

∂∂

− ∂

∂

t

SB

=x

k kB

Px

gzx

o

o

ro

o o

oo

φµ

ρ (58)

Clearly, these two equations are coupled. That is, the coefficients in the pressure equation e.g. ko, etc. depend on So which is the unknown we are trying to find. Likewise, in the saturation equation, flow terms appear with (∂Po/∂x) in them – we are also trying to find Po. Hence, if we solve these equations “one at a time”, we face the problem that there are unknowns which we don’t yet know. We will go into more detail in Chapter 6 where we discuss the numerical solution of the flow equations. However, we will just think in general terms about how we may go about solving the pair of pressure/saturation equations above.

Think of the unknowns in our grid block i, Poi and Soi. We imagine the time levels n meaning time = t or now where we know Poi and Soi. We denote this as follows:

Pnoi

Snoi

Time level n (time = t) (KNOWN)

Pn+1oi

Sn+1oi

Time level n+1 (time = t+∆t) (UNKNOWN)

We are trying to find Poi and Soi at the next time, t = t +Δt or n + 1 denoted

Pnoi

Snoi

Time level n (time = t) (KNOWN)

Pn+1oi

Sn+1oi

Time level n+1 (time = t+∆t) (UNKNOWN)

The problem is shown schematically if Figure 8 below as follows:

Solve equations 70

and 59 to find ⇒

Time step, ∆t GRID BLOCK iAt time = t+∆t

(time step n+1)

GRID BLOCK iAt time = t

(time step n)

Rock

S Pn+1oi

n+1oi

(S ) (P )n+1wi

n+1wi

Rock

S Pnoi

noi

(S ) (P )nwi

nwi

Figure 8Schematic showing update of the pressure and saturation in a grid block (i) over a time step, Δt.



Now the pressure equation has lots of terms that depend on So, so strictly we need to know Soi

n+1 to solve the pressure. BUT suppose we try the following strategy:

(a) Use Snoi i.e. the known saturation value to calculate all the coefficients in the

pressure equation (equation 70).

(b) Solve the pressure equation to get a first estimate of Poin+1

(c) Now solve the saturation equation (equation 58) using the latest Poin+1

for the pressure dependent flow terms, to get ⇒ Soi

n+1

(d) We now have Poi

n+1 and Soin+1 as we required – but we may be able to do a

bit better than this by going back to step a and using our (from this step) to get a better solution to the pressure equation, and hence an “improved” Poi

n+1

(e) Clearly, we could iterate through steps a to d until our process “converges” i.e. the newly calculated Soi

n+1 and Poi

n+1 don’t change any more (or change

only by a tiny amount). We would then accept these as our “accurate” new time level values and then go on to the next time level.

The description above outlines – in words – an algorithm or numerical strategy for solving the pressure and saturation equations that arise in two-phase flow. We used no mathematics since it is just the basic idea that we want to get across just at this point.

3.5 The Simplified Two-Phase Pressure and Saturation EquationsIn the case of single phase flow, we found it quite useful to simplify the (analytically insoluble) compressible flow equation. This allowed us to see its structure quite clearly and it turned out that the simplified single phase equation was a diffusion equation. This diffusive process underlay how pressure “waves” spread across an oil reservoir.

For the two-phase flow case, we will make some slightly different assumptions when we simplify the equations as follows.

Assumptions:1 The viscosity of the oil and water are constant μo and μw (with x and P);2 Both the rock and the fluids are incompressible (φ constant; Bo = Bw = 1);3 We neglect both capillary pressure (Pc = 0; Po = Pw = P) and gravity (g = 0).

It is easiest to see the consequences of the above assumptions by going back to the conservation equations 58 and 59. In the absence of gravity and capillarity (Assumption 3), these become as follows:

∂∂

= ∂∂

∂∂

t

SB x

k kB

Px

o

o

ro

o o

φµ

(71)

∂∂

= ∂∂

∂∂

t

SB x

k kB

Px

w

w

rw

w w

φµ

(72)



Now using Assumptions 1 and 2, we obtain:

φµ

∂∂

= ∂

∂∂∂

St x

k k Px

o ro

o

(73)

φµ

∂∂

= ∂

∂∂∂

St x

k k Px

w rw

w

(74)

Clearly, it is now very simple to eliminate the (∂S/∂t) terms since we simply need to add the above two equations to obtain the simplified pressure equation:

∂

∂∂∂

+ ∂

∂∂∂

=

xk k P

x xk k P

x0ro

o

rw

wµ µ (75)

which simplifies to:

∂

∂( ) ∂

∂

=x

+ Px

0o wλ λ (76)

where λo and λw are the phase mobilities given by:

λµ

λµo

ro

ow

rw

w

k k k k= =and (77)

These phase mobilities are strong functions of the saturation So (So = 1 – Sw) through the relative permeabilities i.e. λo(So); λw(So). Indeed, we can define the total mobility, λT(S0), as:

λ λ λT o w oS = ( ) + (78)

thus simplifying the pressure equation to:

∂

∂( ) ∂

∂

=x

SPx

0T oλ (79)

The simplified saturation equation is simply equation 73 above which, in the above notation, becomes:

φ λ∂∂

= ∂∂

( ) ∂∂

St x

SPx

oo o (80)

In summary, the simplified (incompressibility no Pc, no gravity) two-phase pressure and saturation equations are as follows:

∂

∂( ) ∂

∂

=x

SPx

0T oλ (81)



φ λ∂∂

= ∂

∂( ) ∂

∂

St x

SPx

oo o (82)

It is probably clearer for these equations how we would apply our schematic strategy (Section 3.4) to these two equations. We illustrate this in the flowchart in Figure 9.

Time level n ⇒ KNOW Snoi

niP

Calculate mobilities at these

To solve SATURATION equation

n+1iP

n+1oiSAre these ;

satisfactory? (i.e. converged)

n+1iP

n+1oiSSet the latest

to "current" values and ITERATE through calculation again

Keep them andset to "current" values.

Take next time step

noS )=λo(S )+λw(S )λT(

no

no

Solve PRESSURE equation with

"current"∂∂x

λT (Son )

∂P∂x

= 0

⇒obtain Pin +1

φ = ∂∂x

λo (Son

n+1

)∂P∂x

⇒obtain Soin +1

∂So∂t

Use andPin +1 λo(S )

no

NO YES

4. CLOSING REMARKS

The purpose of this module is to familiarise the student with the fundamental flow equations of single- and two-phase flow. We have shown that these can be derived in a unified way by applying:

MATERIAL BALANCE + DARCY’S LAW ⇒ FLOW EQUATIONS

In the two-phase case, the mass conservation equation was applied to each of the two phases – oil and water, in the example here. There is also a little more “two-phase physics” to be dealt with due to capillary pressure and relative permeability.

For both single- and two-phase flow, the resulting full equations for the compressible system were non-linear PDE’s (indeed, 2 non-linear, completed PDE’s for two-phase flow). These could not be solved analytically but, in simplified form, they illustrate some interesting features of the processes. The numerical solution of these equations is discussed in some detail in Chapter 6.

Figure 9Schematic Strategy for the Iterative Solution of the (simplified) Pressure and Saturation Equations for Two-Phase Flow



Solution to Exercise 2:

Expand the LHS (φ constant) ⇒∂( )

∂= ∂

∂

⇒ ∂∂

∂∂

+ ∂∂

∂∂

∂∂

∂∂

= ∂∂

+ ∂∂

∂∂

φρφ ρ

µρ

µρ

ρ ρ ρ

P P

kx

Px

k

yPy

xPx

Px

Px x

x y

2

2

== ∂∂

∂∂

∂∂

= ∂

∂

∴ ∂∂

∂∂

= ∂

∂+ ∂

∂

=

∂∂

ρ

ρ ρ

φ ρ ρµ

ρµ

µρ ρ

ρ

2

2

2

1

Px

yPy

Py

PPt

k Px

k Py

.k

where kk k

2 P

2

2

x2

2y

2

x y ==

∂∂

= ∂

∂

+ ∂∂

( )

= ( )

c

ck

Pt

kk

Px

k

kP

y

k = k + k / z

Dkc

k / k

k / k

f

f x2

y2

2

x y

hf

x

y

µφ

µφ

2

The RHS becomes (using assumptions)

⇒∂( )

∂= ∂

∂

⇒ ∂∂

∂∂

+ ∂∂

∂∂

∂∂

∂∂

= ∂∂

+ ∂∂

∂∂

φρφ ρ

µρ

µρ

ρ ρ ρ

P P

kx

Px

k

yPy

xPx

Px

Px x

x y

2

2

== ∂∂

∂∂

∂∂

= ∂

∂

∴ ∂∂

∂∂

= ∂

∂+ ∂

∂

=

∂∂

ρ

ρ ρ

φ ρ ρµ

ρµ

µρ ρ

ρ

2

2

2

1

Px

yPy

Py

PPt

k Px

k Py

.k

where kk k

2 P

2

2

x2

2y

2

x y ==

∂∂

= ∂

∂

+ ∂∂

( )

= ( )

c

ck

Pt

kk

Px

k

kP

y

k = k + k / z

Dkc

k / k

k / k

f

f x2

y2

2

x y

hf

x

y

µφ

µφ

2

The x- and y- RHS terms are identical in form and so we just need to expand up one of these (the other will be very similar).

⇒∂( )

∂= ∂

∂

⇒ ∂∂

∂∂

+ ∂∂

∂∂

∂∂

∂∂

= ∂∂

+ ∂∂

∂∂

φρφ ρ

µρ

µρ

ρ ρ ρ

P P

kx

Px

k

yPy

xPx

Px

Px x

x y

2

2

== ∂∂

∂∂

∂∂

= ∂

∂

∴ ∂∂

∂∂

= ∂

∂+ ∂

∂

=

∂∂

ρ

ρ ρ

φ ρ ρµ

ρµ

µρ ρ

ρ

2

2

2

1

Px

yPy

Py

PPt

k Px

k Py

.k

where kk k

2 P

2

2

x2

2y

2

x y ==

∂∂

= ∂

∂

+ ∂∂

( )

= ( )

c

ck

Pt

kk

Px

k

kP

y

k = k + k / z

Dkc

k / k

k / k

f

f x2

y2

2

x y

hf

x

y

µφ

µφ

2

Likewise

⇒∂( )

∂= ∂

∂

⇒ ∂∂

∂∂

+ ∂∂

∂∂

∂∂

∂∂

= ∂∂

+ ∂∂

∂∂

φρφ ρ

µρ

µρ

ρ ρ ρ

P P

kx

Px

k

yPy

xPx

Px

Px x

x y

2

2

== ∂∂

∂∂

∂∂

= ∂

∂

∴ ∂∂

∂∂

= ∂

∂+ ∂

∂

=

∂∂

ρ

ρ ρ

φ ρ ρµ

ρµ

µρ ρ

ρ

2

2

2

1

Px

yPy

Py

PPt

k Px

k Py

.k

where kk k

2 P

2

2

x2

2y

2

x y ==

∂∂

= ∂

∂

+ ∂∂

( )

= ( )

c

ck

Pt

kk

Px

k

kP

y

k = k + k / z

Dkc

k / k

k / k

f

f x2

y2

2

x y

hf

x

y

µφ

µφ

2

multiply both sides by

⇒∂( )

∂= ∂

∂

⇒ ∂∂

∂∂

+ ∂∂

∂∂

∂∂

∂∂

= ∂∂

+ ∂∂

∂∂

φρφ ρ

µρ

µρ

ρ ρ ρ

P P

kx

Px

k

yPy

xPx

Px

Px x

x y

2

2

== ∂∂

∂∂

∂∂

= ∂

∂

∴ ∂∂

∂∂

= ∂

∂+ ∂

∂

=

∂∂

ρ

ρ ρ

φ ρ ρµ

ρµ

µρ ρ

ρ

2

2

2

1

Px

yPy

Py

PPt

k Px

k Py

.k

where kk k

2 P

2

2

x2

2y

2

x y ==

∂∂

= ∂

∂

+ ∂∂

( )

= ( )

c

ck

Pt

kk

Px

k

kP

y

k = k + k / z

Dkc

k / k

k / k

f

f x2

y2

2

x y

hf

x

y

µφ

µφ

2

and note that

⇒∂( )

∂= ∂

∂

⇒ ∂∂

∂∂

+ ∂∂

∂∂

∂∂

∂∂

= ∂∂

+ ∂∂

∂∂

φρφ ρ

µρ

µρ

ρ ρ ρ

P P

kx

Px

k

yPy

xPx

Px

Px x

x y

2

2

== ∂∂

∂∂

∂∂

= ∂

∂

∴ ∂∂

∂∂

= ∂

∂+ ∂

∂

=

∂∂

ρ

ρ ρ

φ ρ ρµ

ρµ

µρ ρ

ρ

2

2

2

1

Px

yPy

Py

PPt

k Px

k Py

.k

where kk k

2 P

2

2

x2

2y

2

x y ==

∂∂

= ∂

∂

+ ∂∂

( )

= ( )

c

ck

Pt

kk

Px

k

kP

y

k = k + k / z

Dkc

k / k

k / k

f

f x2

y2

2

x y

hf

x

y

µφ

µφ

2

to obtain:

⇒∂( )

∂= ∂

∂

⇒ ∂∂

∂∂

+ ∂∂

∂∂

∂∂

∂∂

= ∂∂

+ ∂∂

∂∂

φρφ ρ

µρ

µρ

ρ ρ ρ

P P

kx

Px

k

yPy

xPx

Px

Px x

x y

2

2

== ∂∂

∂∂

∂∂

= ∂

∂

∴ ∂∂

∂∂

= ∂

∂+ ∂

∂

=

∂∂

ρ

ρ ρ

φ ρ ρµ

ρµ

µρ ρ

ρ

2

2

2

1

Px

yPy

Py

PPt

k Px

k Py

.k

where kk k

2 P

2

2

x2

2y

2

x y ==

∂∂

= ∂

∂

+ ∂∂

( )

= ( )

c

ck

Pt

kk

Px

k

kP

y

k = k + k / z

Dkc

k / k

k / k

f

f x2

y2

2

x y

hf

x

y

µφ

µφ

2

We have simply introduced k = (kx+ky)/2 to retain the general form of the simplified pressure equation. The hydraulic diffusivity in this case is therefore given by:

⇒∂( )

∂= ∂

∂

⇒ ∂∂

∂∂

+ ∂∂

∂∂

∂∂

∂∂

= ∂∂

+ ∂∂

∂∂

φρφ ρ

µρ

µρ

ρ ρ ρ

P P

kx

Px

k

yPy

xPx

Px

Px x

x y

2

2

== ∂∂

∂∂

∂∂

= ∂

∂

∴ ∂∂

∂∂

= ∂

∂+ ∂

∂

=

∂∂

ρ

ρ ρ

φ ρ ρµ

ρµ

µρ ρ

ρ

2

2

2

1

Px

yPy

Py

PPt

k Px

k Py

.k

where kk k

2 P

2

2

x2

2y

2

x y ==

∂∂

= ∂

∂

+ ∂∂

( )

= ( )

c

ck

Pt

kk

Px

k

kP

y

k = k + k / z

Dkc

k / k

k / k

f

f x2

y2

2

x y

hf

x

y

µφ

µφ

2

The coefficients on the RHS of the simplified pressure equation here are kx/ k and ky/ k which reflect the anisotropy of the problem. Obviously, if kx = ky then these terms would be unity.



CONTENTS

1. INTRODUCTION

2. REVIEW OF FINITE DIFFERENCES

3. APPLICATION OF FINITE DIFFERENCES TO PARTIAL DIFFERENTIAL EQUATIONS (PDEs) 3.1. Explicit Finite Difference Approximation of the Linear Pressure Equation 3.2. Implicit Finite Difference Approximation of the Linear Pressure Equation 3.3. Implicit Finite Difference Approximation of the 2D Pressure Equation 3.3.1Discretisation of the 2D Pressure Equation 3.3.2Numbering Schemes in Solving the 2D Pressure Equation 3.4. Implicit Finite Difference Approximation of Non-linear Pressure Equations

4. APPLICATION OF FINITE DIFFERENCES TO TWO-PHASE FLOW 4.1 Discretisation of the Two-Phase Pressure and Saturation Equations 4.2 IMPES Strategy for Solving the Two-Phase Pressure and Saturation Equations

5. THE NUMERICAL SOLUTION OF LINEAR EQUATIONS 5.1. Introduction to Linear Equations 5.2. General Methods for Solving Linear Equations 5.3. Direct Methods for Solving Linear Equations 5.4. Iterative Methods for Solving Linear Equations 5.5. A Comparison of Iterative and Direct Methods for Solving Linear Equations

6. DIRECT SOLUTION OF THE NON-LINEAR EQUATIONS OF MULTI-PHASE FLOW 6.1. Introduction to Sets of Non-linear Equations 6.2. Newton’s Method for Solving Sets of Non- linear Equations 6.3. Newton’s Method Applied to the Non-linear Equations of Two-Phase Flow

66Numerical Methods in Reservoir Simulation

7. NUMERICAL DISPERSION - A MATHEMATICAL APPROACH 7.1. Introduction to the Problem 7.2. Mathematical Derivation of Numerical Dispersion

8. CLOSING REMARKS

APPENDIX A: Some Useful Matrix Theorems.




Having worked through this chapter the student should be able to:

• write down from memory simple finite difference expressions for derivatives, (∂P/∂x), (∂P/∂t) and (∂2P/∂x2) explaining your spatial (space) and temporal (time) notation ( P P P Pi

nin

in

in, , , etc.+

++

−+1

11

11 ); for (∂P/∂x), the student should know the

meaning of the forward difference, the backward difference and the central difference and the order of the error associated with each, O(Δx) or O(Δx2).

• apply finite difference approximations to a simple partial differential equation (PDE) such as the diffusion equation and explain what is meant by an explicit and an implicit numerical scheme.

• write a simple spreadsheet to solve the explicit numerical scheme for a simple PDE for given boundary and initial conditions and be able to describe the effect of time step size, Δt.

• show how the implicit finite difference scheme applied to a simple linear PDE leads to a set of linear equations which are tridiagonal in 1D and pentadiagonal in 2D.

• derive the structure of the pentadiagonal A-matrix in 2D for a given numbering scheme going from (i, j) notation to m-notation where m is an ordered numbering scheme e.g. for the natural numbering scheme, m = (j - 1).NX + i

• describe a solution strategy for the non-linear single phase 2D pressure equation where the fluid and rock compressibility (and density, ρ, and viscosity, μ) are functions of the dependent variable, pressure, P(x,y,t). • write down the discretised form of both the pressure and saturation equation for two-phase flow given the governing equations (in simplified form in 1D), and be able to explain why these lead to sets of non-linear algebraic equations.

• outline with an explanation and a simple flow chart the main idea behind the IMPES solution strategy for the discretised two-phase flow equations.

• write down the expanded expressions for a set of linear equations which, in compact form are written A.x = b, where the matrix A is an nxn matrix of (known) coefficients (aij; i = rows, j = columns), b is a vector of n (known) values and x is the vector of n unknowns which we are solving for.

• explain clearly the main differences between a direct and an iterative solution method for the set of linear equations, A.x = b.

• write down the algorithm for a very simple iterative scheme for solving A.x = b, and be able to describe the significance of the initial guess, x(0) , what is meant by iteration (and iteration counter, ν), the idea of convergence of x(ν) as ν → ∞; and be able to comment on the number of iterations required for convergence, Niter.



• explain how to apply (without derivation) the Newton-Raphson method for solving a single non-linear algebraic equation, f x( ) = 0 ; Newton-Raphson scheme

=> x xf xf x

( ) ( )( )

' ( )

( )( )

ν νν

ν+ = −1 .

• extend the application of the Newton-Raphson to sets of non-linear algebraic equations such as those arising from the discretisation of the two-phase pressure and

saturation equations; F X where XS

P( ) = =

0, and S and P are the (unknown) vectors of saturation and pressure; (given the Newton-Raphson expression, X X J X F X( ) ( ) ( ) ( ).ν ν ν ν+ −

= − ( )[ ] ( )1 1

).

• understand, but not be able to reproduce the detailed derivation of, the more mathematical explanation of numerical dispersion.

NUMERICAL METHODS IN RESERVOIR SIMULATION

As noted in Chapter 5, the multi-phase flow equations for real systems are so complex that it is not remotely possible to solve them analytically. In practice these equations can only be solved numerically. The most commonly applied numerical methods are based on finite difference approximations of the flow equations and this approach will be followed in this chapter.

After a brief review of finite differences, we go on to apply them to very simple systems such as for the simplified 1D pressure equation (derived in Chapter 5). This equation does not need to be solved numerically but it demonstrates how explicit and implicit finite difference solution can be developed. We then show how sets of linear equations arise in solving implicit equations and we consider solution of the linear equations in an elementary manner. We then consider how the 2D pressure equation is solved.

In the numerical solution of the multi-phase flow equations, we need to solve for pressure and flow. An outline of how this is done is presented but we do not go into great detail.

1. INTRODUCTION

At the very start of this course, we considered a very simple “simulation model” for a growing colony of bacteria. The number of bacteria, N, grew with a rate proportional to N itself i.e. (dN/dt) = α.N, where α is a constant. We saw that this simple equation had a well-known analytical solution, N t N eo

t( ) . .= α, where No is the number of

bacteria at time, t =0. This exponential growth law provides the solution to our model. However, we also introduced the idea of a numerical model where, even although we could solve the problem analytically, we formulated it this way, in any case. The numerical version of the model came up with the algorithm (or recipe) N t Nn n+ = +1 1( . ).α ∆ . In the exercise in Chapter 1, you should have compared the results of the analytical and numerical models and found out that, they get closer as



we take successively smaller time steps, Δt. In this case, we say that the numerical model converges to the analytical model. In fact, in many areas of science and engineering, we often apply a numerical technique to a problem we can already solve analytically i.e. where we “know the answer”. Why would we do this? The answer is that we might be testing the numerical method to see how closely it gives the right answer. More commonly, we might test several - maybe 3 or 4 - numerical techniques to determine which one works best. The phrase “works best” in the context of a numerical method usually means gives the most accurate numerical agreement with the analytical answer for the least amount of computational work. Note the importance of this balance between accuracy and work for a numerical method. There may be no point in having a numerical method that is “twice” as accurate (in some sense) for ten times the amount of computational work.

In Chapter 5, we already met the equations for single-phase and two-phase flow of compressible fluids through porous media. These turned out to be non-linear partial differential equations (PDEs). Recall that a non-linear PDE is one where certain coefficients in the equation depend on the answer we are trying to find e.g. Sw(x,t), P(x,y,z,t) etc. For example, for single phase compressible flow, the equation for pressure, P(x,t), in 1D is given by:

c P

Pt x

k Px

( ).∂

∂

= ∂

∂∂∂

ρµ (1)

In this equation, both the generalised compressibility term, c(P) (of both the rock and the fluid), and the density, ρ(P), terms depend on the unknown pressure, P, which we are trying to find. As noted previously, such non-linear PDEs are very difficult to solve analytically and we must usually resort to numerical methods. The main topic of this module is on how we solve the reservoir flow equations numerically. This process involves the following steps:

(i) Firstly, we must take the PDE describing the flow process and “chop it up” into grid blocks in space. This is known as spatial discretisation and, in this course, we will exclusively use finite difference methods for this purpose.

(ii) When we apply finite differences, we usually end up with sets of non-linear algebraic equations which are still quite difficult to solve. In some cases, we do solve these non-linear equations. However, we usually linearise these equations in order to obtain a set of linear equations.

(iii) We then solve the resulting sets of linear equations. Many numerical options are available for solving sets of linear equations and these will be discussed below. This is often done iteratively by repeatedly solving them until the numerical solution converges.

This module will deal successively with each of the parts of the numerical solution process, (i) - (iii) above.



2. REVIEW OF FINITE DIFFERENCES

Definition: A finite difference scheme is simply a way of approximating

derivatives of a function (e.g. dPdx

Pt

Px

etc

∂∂

∂∂

, , . 2

2)

numerically from either point or block values of the function.

The main concept of finite difference approximation is best illustrated by the following simple example where we refer to Figure 1. Study the Notation in this figure, since it is the basis of that used throughout this chapter. The main task of the finite difference approach is to represent the derivatives of the function, P(x), in an approximate manner

i.e. dPdx

Pt

Px

etc

∂∂

∂∂

, , . 2

2

First consider how we might approximate (dP/dx) at xi using the quantities in Figure 1. In fact, it is easily seen that there are three ways we may do this as follows:

Approximation 1 -Forward Differences (fd): we may take the slope between Pi and Pi+1 as being approximately dP

dx i

at xi to obtain:

dPdx

P Pxi fd

i i

= −

∆+

1

(2)

Approximation 2 -Backward Differences (bd): we may take the slope between Pi-1

and Pi as being approximately dPdx i

at xi to obtain:

dPdx

P Pxi bd

i i

= −

∆−

1

(3)

Approximation 3 -Central Differences (cd): thirdly, we could take the average of the forward difference (fd) and backward difference (bd) approximations to give us

dPdx i

at xi. This is known as central differences (cd) and is given by:

dPdx

dPdx

dPdx

P Px

P Px

dPdx

P Px

i cd i fd i bd

i i i i

i cd

i i

=

+

= −

∆+ −

∆

= −

∆

+ −

+ −

12

12

2

1 1

1 1

. (4)



∆x = constant

P(x)

xi-1

Pi-1

Pi+1

Pi

xi-1xix

∆x ∆x

Notation:∆x = the x- grid spacing or distance between grid points(i - 1), i, (i + 1) = the x - label (subscript) for the grid point or block numbers

Pi-1, Pi, Pi+1 = the corresponding function values at grid point i-1, i, i+1 etc.

Each of the above approximations to dPdx i

is shown graphically in Figure 2.

You may wonder why we bother with three different numerical approximations to dPdx i

and ask: which is “best”? There is not an unqualified answer to this question

just yet, but let us take a simple numerical example where we know the right answer and examine each of the forward, backward and central difference approximations. The values given in Figure 3 will illustrate how the methods perform. Firstly, simply calculate the values given by each methods for the data in Figure 3:

dPdx

err

dPdx

err

dPdx

err

i fd

i bd

i cd x

≈ −

= ≈ +

≈ −

= ≈ −

≈ −

= ≈ −

3 0042 2 71830 1

2 859 0 14

2 7183 2 45960 1

2 587 0 13

3 0042 2 71832 0 1

2 723

. ..

. [ . ]

. ..

. [ . ]

. ..

. [ 00 005. ] (5)

The quantity in square brackets after each of the finite difference approximations above is the error i.e. the difference between that method and the right answer which is 2.7183. As expected from the figure for this case, the forward difference answer is a little too high (by ≈ +0.14) and the backward difference answer is a little too low (by ≈ -0.13). The central difference approximation is rather better than either of the previous ones. In fact, we note that the fd and bd methods give an error of order Δx, the grid spacing (where, as engineers, we are saying 0.14 ≈ 0.1 !). The cd approximation, on the other hand, has an error of order Δx2 (where again we are

Figure 1Notation for the application of finite difference methods for approximating derivatives.



saying 0.005 (0.1x0.1 = 0.01). More formally, we say that the error in the fd and bd approximations are "of order Δx" which we denote, O(Δx), and the cd approximation has an error "of order Δx2" which we denote, O(Δx2).

Now consider the finite difference approximation of the second derivative,

∂∂

2

2

Px

Going back to Figure 1, the definition of second derivative at ∂∂

Px xi is the rate of

change of slope (dP/dx) at xi. Therefore, we can evaluate this derivative between xi-1 and xi (i.e. the bd approximation) and then do the same between xi and xi+1 (i.e. the fd approximation) and simply take the rate of change of these two quantities with respect to x, as follows:

∂∂

≈

−

∆

≈

−∆

− −

∆

∆

∂∂

≈+ −( )

∆

+ −

+ −

2

2

1 1

2

21 1

2

2

Px

dPdx

dPdx

x

P Px

P Px

x

Px

P P Px

i fd i bd

i i

i fd

i i

i bd

i i i

(6)

Calculating the numerical value of ∂∂

2

2

Px for the example in Figure 3 (where again

∂∂

2

2

Px

= 2.7183 since the example is the exponential function), gives:

∂∂

≈ + − = ≈2

2 2

3 0042 2 4596 2 2 71830 1

2 7200 0 0017P

xerr

x. . ..

. [ . ]

(7)

Thus, we can see that the error in this case (err ≈ 0.0017 ≈ 0.12) is actually O(Δx2).

In the introductory section of Chapter 1, we already applied the idea of finite differences (although we didn’t call it that at the time) to the simple ordinary differential equation

(ODE); dNdt

N

= α. .

Here, N (number of bacteria in the colony) as a function of time, N(t), is the unknown we want to find. Denoting Nn and Nn+1 as the size of the colony at times t (labelled n) and t+Δt (labelled n+1), we applied finite differences to obtain:



dNdt

N Nt

N

N t N

n nn

n n

≈ −

∆≈

≈ + ∆( )

+

+

1

1 1

α

α

.

. (8)

This gave us our very simple algorithm to explicitly calculate Nn+1. We then took this as the current value of N and applied the algorithm repeatedly.

dP Pi+1 - Pi-1

dx 2.∆x=

=

i cd

dP Pi+1 - Pi

dx ∆xi fd

=dP Pi - Pi-1

dx ∆xi bd

∆x = constantxi-1

Pi+1

Pi

Pi-1

xi xi+1

P(x)

∆x ∆x

x

dP

dx x=1.0 x=1.0

d2P

dx2

0.9

Pi+1

Pi

Pi-1

1.0 1.1

P(x)

0.1 0.1

x

3.0042

2.7183

2.4596

= 2.7183 = 2.7183

Figure 2Graphical illustration of the finite difference derivatives calculated by backward differences (bd), forward differences (cd) and central differences.

Figure 3A numerical example where the function values and derivative values are known (P(x) = ex)



EXERCISE 1.

Apply finite differences to the solution of the equation:

dydt

y

= +2 42.

where, at t = 0, y(t = 0) = 1. Take time steps of Δt = 0.001 (arbitrary time units) and step the solution forward to t = 0.25. Use the notation yn+1 for the (unknown) y at n+1 time level and yn for the (known) y value at the current, n, time level.

Plot the numerically calculated y as a function of t between t = 0 and t = 0.25 and plot it against the analytical value (do the integral to find this).

Answer: is given below where the working is shown in spreadsheet CHAP6Ex1.xls. This gives the finite difference formula, a spreadsheet implementing it and the analytical solution for comparison.

3. APPLICATION OF FINITE DIFFERENCES TO PARTIAL DIFFERENTIAL EQUATIONS (PDEs)

3.1 Explicit Finite Difference Approximation of the Linear Pressure Equation We have seen in Chapter 5 that the flow equations are actually partial differential equations (PDEs) since the unknowns, P(x,t) and Sw(x,t) say, depend on both space and time. As an example of a linear PDE, we will take the simplified pressure equation (equation 26; Chapter 5) as follows:

∂∂

= ∂

∂

Pt

kc

Pxφµ

2

2

(9)

where the constant k

cφµ is the hydraulic diffusivity, which we denoted previously by Dh. As we noted in Chapter 5, this PDE is linear which has known analytical solutions for various boundary conditions. However, we will again neglect these and apply numerical methods as an example of how to use finite differences to solve PDEs numerically. To make things even simpler, we will take Dh = 1, giving the equation:

∂∂

= ∂

∂

Pt

Px

2

2

(10)

This is the pressure equation for a 1D system where 0 ≤ x ≤ L, where L is the length of the system. We can visualise this physically - much as we did in Chapter 5, Section 2.1 - using Figure 4. After the system is held constant at P = Po, the inlet pressure is raised (at x = 0) instantly to P = Pin while the outlet pressure is held at Pout = Po.



x0 Lt = 0

t = t1 t = t2P

Pin

Po Pout = Po

These pressures, Pin and Pout, represent the constant pressure boundary conditions (Mathematically these are sometimes called Dirichlet boundary conditions). In other words, these are set, as if by experiment, and the system between 0 < x < L must “sort itself out” or "respond" by simply obeying equation 10 above. We now approach this problem using finite differences as follows:

• discretise the x-direction by dividing it into a numerical grid of size Δx; • choose a time step, Δt; • use the following notation

P

P

in

in+1

→ time level n = 0, 1, 2 ... → x-grid block label, i = 1, 2, 3 ... NX (at x = L)

P

P

in

in+1

current (known) P at time level

P

P

in

in+1 new (unknown) P at time level

• fix the boundary conditions which, in this case, are as follows (see Figure 4):

P1 = P in and PNX = Pout = Po which are fixed for all t.

• apply finite differences to equation 10 using the above notation to obtain:

∂∂

≈ −

∆

+Pt

P Pti

in

in1

(11)

and

∂∂

≈ + −∆

+ −2

21 1

2

2Px

P P Pxi

i i i?? ?? ??

(12)

However, an issue arises in equation 12 above as shown by the question marks on the spatial derivative time levels. It is simply: which time level should we take for the spatial derivative terms in equation 12? This is important and we will return to this matter soon. However, for the moment, let us take these spatial derivatives at time level n (the “known” level) since this will turn out to be the simplest thing we can do. Thus, we obtain:

Figure 4 Physical picture of pressure propagation in a 1D (com-pressible) system described by. ∂

∂

= ∂

∂

Pt

Px

2

2



∂∂

≈ + −∆

+ −2

21 1

2

2Px

P P Pxi

in

in

in

(13)

Equating the numerical finite difference approximations of each of the above derivatives as required by the original PDE, equation 10 (i.e. equating the expressions in equations 11 and 13) gives:

P Pt

P P Px

in

in

in

in

in+

+ −−∆

≈ + −∆

11 1 2

(14)

which easily rearranges to obtain an explicit expression for, Pin+1, the only

unknown in the above equation:

P P

tx

P P Pin

in

in

in

in+

+ −= + ∆∆

+ −( )12 1 1 2

(15)

In words, we can interpret this above algorithm as saying:

New (n+1 level) value of Pi = Old (n level) value of Pi + a “correction term”

Equation 15 gives the algorithm for propagating the solution of the PDE forward in time from the given set of initial conditions.

We now consider how to set the initial conditions. The initial conditions are the values of all the Pi (i = 1, 2, 3 ... , NX) at t = 0. From Figure 4, these are clearly:

P

P

P

i

NX

10

0

0

= 1 for all time (boundary condition)

P

P

P

i

NX

10

0

0

= 0 at time t = 0 for 2 < i < NX-1

P

P

P

i

NX

10

0

0 = Po for all time (boundary condition)

Let us now apply the above algorithm to the solution of equation 10. For this problem, suppose we take the following data: 0 ≤ x ≤ 1.0

Δx = 0.1 => implies 11 grid points, P1 , P2 , P3, .... , P11 (NX = 11).

Δt = 0.001

The problem is then to calculate the solution, P(x,t), - that is Pin+1 for all i (at all

grid points) and all future times up to some final time (possibly when the equation comes to a steady-state as will happen in this example). This can be done by filling in the “solution chart” Table 1 - see Exercise 2 below.



Grid Blocks

Time x = 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t n (BC↓) (BC↓) i = 1 2 3 4 5 6 7 8 9 10 110 0 P0

→ 1 0 0 0 0 0 0 0 0 0 0 (IC)0.1 100 P100 1 0

0.2 200 P200 1 0

0.3 300 P300 1 0

0.4 400 P400 1 0

0.5 500 P500 1 0

0.6 600 P600 1 0

0.7 700 P700 1 0

0.8 800 P800 1 0

0.9 900 P900 1 0

1 1000 P1000 1 0

i

i

i

i

i

i

i

i

i

i

Note: BC = Boundary Conditions - these points are fixed; IC = Initial Conditions, i.e. values of P(x, t = 0) for all i at t= 0

i

EXERCISE 2.

Fill in the above table using the algorithm (where (Δt/Δx2)=0.1):

P P

tx

P P Pin

in

in

in

in+

+ −= + ∆∆

+ −( )12 1 1 2

Hint: make up a spread sheet as above and set the first unknown block (shown grey shaded in table above) with the above formula. Copy this and paste it into all of the cells in the entire unknown area (surrounded by bold border above).

Answer: If you get stuck, look at spreadsheet CHAP6Ex2.xls on the disk.

Table 1.Solution chart for the solu-tion of the simple pressure equation



The assumption we made in equation 12 was that the spatial derivative was taken at the n (known) time level. This allowed us to develop an explicit formula for Pi

n+1 (for all i). This method is therefore known as an explicit finite difference method. We can learn about some interesting and useful features of the explicit method which is encapsulated in equation 15 by simply experimenting with the spreadsheet CHAP6Ex2.xls. Three features of this method can be illustrated by “numerical experiment” as follows:

(i) the effect of time step size, Δt;

(ii) the effect of refining the spatial grid size, Δx (or number of grid cells, NX);

(iii) the effect of running the calculation to steady-state as t → ∞ (in practice, until the numerically computed solution stops changing).

It is best if you do these yourself by modifying the spreadsheet (CHAP6Ex2.xls).

EXERCISE 3.

Experiment with the spreadsheet in CHAP6Ex2.xls to examine the effects of the three quantities above - Δt, Δx (or NX) and the solution as t → ∞.

HINTS for Exercise 6.3:• Sensitivity to Δt: When you make Δt too big, the predicted numerical solution of the PDE goes badly wrong - indeed negative pressures can occur which is physically impossible. In fact, the solution has become unstable for the larger time steps. This means that this explicit numerical method does have some time step limitations which we must be careful of.

• Sensitivity to Δx: the effects of grid refinement are that, as the grid blocks get smaller (i.e. Δx → 0 or NX → ∞). The answer should get more accurate although, to make this happen, you need to reduce the time step as well.

• Behaviour as t → ∞ : Finally, considering the long-time behavior of the solution of the PDE, you should find that P(x, t → ∞) tends to a straight line. Some other intermediate pressure profiles can also be plotted using CHAP6Ex2.xls. In fact, a little bit of analysis shows us that this is quite expected. As t → ∞, then if steady-state is reached, then:

∂∂

= => ∂

∂

=Pt

which impliesP

x0 0

2

2,

But the “curve” with a zero second derivative (i.e. a first derivative which is constant) is a straight line. Therefore, this result is physically reasonable and our numerical model appears to be behaving correctly.



Another way to represent this explicit finite difference solution to the PDE is shown in Figure 5, where we indicate the time levels for the solution of the equations and we also show the dependency of the unknown pressure ( Pi

n+1).

Time level n

Time level n + 1

Time∆x

∆t

∆x

i - 1 i i + 1

P ni-1 P n

i

P n+1i-1 P n+1

i P n+1i+1

P ni+1

3.2 Implicit Finite Difference Approximation of the Linear Pressure Equation We now return to the original finite difference equation 12, where we had to

make a choice of time level for the spatial derivative, ∂∂

2

2

Px . We now examine

the consequences of taking this derivative at the (n+1) time level - this is the “unknown” time level. The finite difference equation for this case is as follows: The time derivative is the same, i.e.

∂∂

≈ −

∆

+Pt

P Pti

in

in1

(16)

but the spatial derivative now becomes:

∂∂

≈ + −∆

++

−+ +2

21

11

1 1

2

2Px

P P Pxi

in

in

in

(17)

As before, we now equate the numerical finite difference approximations of each of the above derivatives (as required by the original PDE, equation 10) to obtain:

P Pt

P P Px

in

in

in

in

in+

++

−+ +−

∆≈ + −

∆

11

11

1 12

(18)

This equation should be compared with equation 14. In the previous case, this could easily be rearranged into an explicit equation for P

P P P

in

in

in

in

+

−+ +

++

1

11 1

11, and

(equation 15). However, equation 18 above cannot be rearranged to give a simple expression for the pressure at the new time step (n+1),

P

P P P

in

in

in

in

+

−+ +

++

1

11 1

11, and . Indeed, there now appear to be three unknowns at

time level (n+1), viz.

P

P P P

in

in

in

in

+

−+ +

++

1

11 1

11, and . This appears to be a bit of a paradox:

how do we find three unknowns from a single equation (equation 18 above)? The answer is not really too difficult: Basically, we have an equation - like equation 18 - at every grid point. We will show below that this leads to a set of linear equations where we have exactly the same number of unknowns as we have linear equations.

Figure 5.Schematic of the explicit finite difference algorithm for solving the simple pressure equation (a PDE).



Therefore, if these equations are all linearly independent, then we can solve them numerically. This is a bit more trouble than our earlier simple explicit finite difference method. Because, we do not get an explicit expression for our unknowns - instead we get an implicit set of equations - this method is known as an implicit finite difference method.

To see where these linear equations come from, rearrange equation 18 above to obtain:

P

xt

P Pxt

Pin

in

in

in

−+ +

++− + ∆

∆

+ = − ∆∆

1

12

11

12

2 (19)

where all the unknowns (

P

P P P

in

in

in

in

+

−+ +

++

1

11 1

11, and ) are on the LHS of the equation

and the term on the RHS is “known”, since it is at the old time level, n. We can write this equation as follows:

a P a P a P bi in

i in

i in

i− −+ +

+ +++ + =1 1

1 11 1

1

(20)

and where and a axt

a bxt

Pi i i i in

− += = − + ∆∆

= = − ∆∆

1

2

1

2

1 2 1; ;

are all constants. The ai do not change throughout the calculation but the quantity bi is updated at each time step as the newly calculated Pn+1 is set to the Pn for the next time step. We can see how this works for the 5 grid block system in Figure 6 below:

Time level n=1

Time level n

1 2 3 4 5

Boundarycondition(fixed)

Boundarycondition(fixed)

∆x

∆t

P P P P P

P P P P Pn1

n2

n3

n4

n5

n+11

n+12

n+13

n+14

n+15

At each grid point, then (a) we know the value from the boundary condition (i = 1 and i = 5), (b) it uses a boundary condition (i=2 and i=4) or (c) it is specified completely by equation 20 (only i = 3, in this case - but it would be most points for a large number of grid points).

Consider each grid point in turn as follows:

i = 1 a boundary; therefore P1 is fixed, say as P1

Figure 6.Simple example of a 5 grid block system showing how the implicit finite difference scheme is applied



iP a P P b

xt

P

a P Pxt

P P

i P a P P

n n n

n n n

n n n

=+ + = = − ∆

∆

=> + = − ∆∆

−

= + +

+ +

+ +

+ + +

2

3

1 2 21

31

2

2

2

2 21

31

2

2 1

21

3 31

41

== = − ∆∆

=+ + = = − ∆

∆

=> + = − ∆∆

−

+ +

+ +

bxt

P

iP a P P b

xt

P

P a Pxt

P P

n

n n n

n n n

3

2

3

31

4 41

5 4

2

4

31

4 41

2

4 5

4

i = 5 a boundary; therefore P5 is fixed, say as P

P P Pn n n

5

21

31

41( , )+ + + and

Therefore there are only three unknowns in the above set of linear equations

P

P P Pn n n

5

21

31

41( , )+ + + and which can be summarised as follows:

i = 2:

i = 3:

i = 4:

a P Pxt

P P

P a P Pxt

P

P a P

n n n

n n n n

n n

2 21

31

2

2 1

21

3 31

41

2

3

31

4 41

+ +

+ + +

+ +

+ = − ∆∆

−

+ + = − ∆∆

+ = − ∆∆

−xt

P Pn2

4 5

(21)

Note that the set of linear equations above can be represented as a simple matrix equation as follows:

0

1 1

0 1

a

a

a

P

P

P

xt

P P

xt

P

xt

P P

n

n

n

n

n

n

2

3

4

21

31

41

2

2 1

2

3

2

4 5

1

=

− ∆∆

−

− ∆∆

− ∆∆

−

+

+

+

(22)



The structure of this matrix equation is clearer when there are more equations involved. For example, it is quite easy to show that, if we take 12 grid points instead of the 5 above, we obtain 10 equations (using the two fixed boundary conditions, P P

P P P Pn n n n

1 12

21

31

41

111

and

, + + + +, ....

) for the quantities,

P P

P P P Pn n n n

1 12

21

31

41

111

and

, + + + +, .... , of the form:

0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 1 1 0 0 0 0

0 0 0 1 1 0 0 0

0 0 0 0 1 1 0 0

0 0 0 0 0 1 1 0

0 0 0 0 0 0 1 1

0 0 0 0 0 0 0 1

a

a

a

a

a

a

a

a

a

a

2

3

4

5

6

7

8

9

10

11

1

1

0

0

0

0

0

0

=

− ∆ ∆( )+

+

+

+

+

+

+

+

+

+

P

P

P

P

P

P

P

P

P

P

x t Pn

n

n

n

n

n

n

n

n

n

21

31

41

51

61

71

81

91

101

111

222 1

23

24

25

26

27

28

29

210

211 12

n

n

n

n

n

n

n

n

n

n

P

x t P

x t P

x t P

x t P

x t P

x t P

x t P

x t P

x t P P

−

− ∆ ∆( )− ∆ ∆( )− ∆ ∆( )− ∆ ∆( )− ∆ ∆( )− ∆ ∆( )− ∆ ∆( )− ∆ ∆( )− ∆ ∆( ) −

(23)

And 20 grid points in 1D would lead to the following set of 18 equations:

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

2

3

4

5

6

7

8

9

10

11

12

13

14

15

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1616

17

18

19

21

31

41

51

61

71

1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

a

a

a

P

P

P

P

P

P

n

n

n

n

n

n

+

+

+

+

+

+

PP

P

P

P

P

P

P

P

P

P

P

P

n

n

n

n

n

n

n

n

n

n

n

n

81

91

101

111

121

131

141

151

161

171

181

191

+

+

+

+

+

+

+

+

+

+

+

+

=

− ∆ ∆( ) −

− ∆ ∆( )− ∆ ∆( )− ∆ ∆( )− ∆ ∆( )− ∆ ∆( )− ∆ ∆( )− ∆ ∆( )− ∆ ∆( )−

x t P P

x t P

x t P

x t P

x t P

x t P

x t P

x t P

x t P

n

n

n

n

n

n

n

n

n

22 1

23

24

25

26

27

28

29

210

∆∆ ∆( )− ∆ ∆( )− ∆ ∆( )− ∆ ∆( )− ∆ ∆( )− ∆ ∆( )− ∆ ∆( )− ∆ ∆( )− ∆ ∆( ) −

x t P

x t P

x t P

x t P

x t P

x t P

x t P

x t P

x t P P

n

n

n

n

n

n

n

n

n

211

212

213

214

215

216

217

218

219 20

(24)



For this simple 1D PDE, it is clear that the matrix arising from our implicit finite difference method has the following properties:

(i) It is tridiagonal - that is, it has a maximum of three non-zero elements in any row and these are symmetric around the central diagonal;

(ii) It is very sparse - that is, most of the elements are zero. In an MxM matrix, there are only 3M non-zero terms but M2 actual elements. If M = 100, then the matrix is only (300/1002)x100% = 3% filled with non-zero terms.

As it happens, a very simple computational procedure, called the Thomas algorithm, can be used to solve tridiagonal systems very quickly. The FORTRAN code for this is shown for interest in Figure 7. Note that it is very compact and quite simple in structure. You are not expected to know this or to necessarily understand how this algorithm works.



THE THOMAS ALGORITHM

subroutine thomas c Thomas algorithm for tridiagonal systems + (N ! matrix size + ,a ! diagonal + ,b ! super-diagonal + ,c ! sub-diagonal + ,s ! rhs + ,x ! solution + )c----------------------------------------c This program accompanies the book:c C. Pozrikidis; Numerical Computation in Science and Engineeringc Oxford University Press, 1998c------------------------------------------------c Coefficient matrix:c | a1 b1 0 0 ... 0 0 0 |c | c2 a2 b2 0 ... 0 0 0 |c | 0 c3 a3 b3 ... 0 0 0 |c | .............................. |c | 0 0 0 0 ... cn-1 an-1 bn-1 |c | 0 0 0 0 ... 0 cn an |c------------------------------------------ Implicit Double Precision (a-h,o-z) Dimension a(200),b(200),c(200),s(200),x(200) Dimension d(200),y(200) Parameter (tol=0.000000001) this is a measure of how close to coverage we arec prepare Na = N-1c reduction to upper bidiagonal d(1) = b(1)/a(1) y(1) = s(1)/a(1) DO i=1,Na i1 = i+1 Den = a(i1)-c(i1)*d(i) d(i1) = b(i1)/Den y(i1) =(s(i1)-c(i1)*y(i))/Den End DOc Back substitution x(N) = y(N) DO i=Na,1,-1 x(i)= y(i)-d(i)*x(i+1) End DOc Verification and alarm Res = s(1)-a(1)*x(1)-b(1)*x(2) If(abs(Res).gt.tol) write (6,*) “ thomas: alarm” DO i=2,Na Res = s(i)-c(i)*x(i-1)-a(i)*x(i)-b(i)*x(i+1) If(abs(Res).gt.tol) write (6,*) “ thomas: alarm” End DO Res = s(N)-c(N)*x(N-1)-a(N)*x(N) If(abs(Res).gt.tol) write (6,*) “ thomas: alarm”c Done 100 Format (1x,f15.10) Return End

Figure 7.The Thomas algorithm for the solution of tridiagonal matrix systems.



Finally, we note that the implicit finite difference method can be viewed as shown in Figure 8 below. The choice of the (n+1) time level for the spatial discretisation leads to a set of linear equations which are somewhat more difficult to solve than the simple algebraic equation that arises in the explicit method (equation 15). However, there are exactly the same number of linear equations as there are unknowns and so it is possible to solve these.

Time level n

Time level n + 1

Time∆x

∆t

∆x

i - 1 i i + 1

P ni-1 P n

i

P n+1i-1 P n+1

i P n+1i+1

P ni+1

3.3 Implicit Finite Difference Approximation of the 2D Pressure EquationBefore we go on to discuss how to solve the sets of linear equations that arise in the finite difference approximation of PDEs arising in reservoir simulation, we will first consider the discretisation of the 2D single-phase pressure equation. This raises some additional important issues which occur when we try to solve more complicated systems, as follows:

(i) the complication of the more “connected” grid block system in a 2D domain;

(ii) the possibility of heterogeneity in the permeability field which leads us to the matter of how to take average properties in grid-to-grid flows between blocks of different permeability (dealt with in Chapter 4);

(iii) the issue of non-linearity for a compressible system e.g. c(P) is clearly a function of P(x,t), which is the “unknown”.

3.3.1 Discretisation of the 2D Pressure EquationWe take as the basic pressure equation for single phase slightly compressible flow, the following (see the solution for exercise 2 at the end of Chapter 5):

ck

Pt x

kPx y

Pyx

µφ ∂∂

= ∂

∂∂∂

+ ∂∂

∂∂

˜ ky

(25)

(a simplified form of equation 39, Chapter 5) where the term c

k

k

φµ

is a constant

which we will denote as β below,

ck

k

φµ

is an average permeability of the entire reservoir and ˜ ˜

˜ / ˜ / .

k and k

k = k and k

x

x x

y

y y

k

k k k=

are the local permeabilities normalised by

˜ ˜

˜ / ˜ / .

k and k

k = k and k

x

x x

y

y y

k

k k k=

; i.e.

˜ ˜

˜ / ˜ / .

k and k

k = k and k

x

x x

y

y y

k

k k k= Note that we have avoided the non-linearities for the moment (the quantities c(P), μ and ρ usually depend on pressure). However, the

Figure 8.Schematic of the implicit finite difference algorithm for solving the simple pres-sure PDE



system may be anisotropic (kx ≠ ky) and heterogeneous (the permeability may vary from grid block to grid block).

Equation 25 above can be discretised in a similar way to that applied to the 1D linear pressure PDE discussed above. However, we will need to be quite clear about our notation in 2D and, for this purpose, we refer to Figure 9. This shows the discretisation grid for the above PDE - note that this is essentially the opposite of what we did when we derived the equation in the first place! In Chapter 5, we used a control volume (or grid block) to express the mass conservation and then inserted Darcy’s law for the block to block flows; we then took limits as Δx, Δy and Δt →0. Here, we are starting with the PDE and going back to the local conservation of flows and introducing finite size Δx and Δy.

i, j

i, j - 1

i, j + 1

i - 1, j i + 1, j

∆x

∆y

(i - 1/2) (i + 1/2)

(j + 1/2)

(j - 1/2)

y (j)

x (i)

Discretising the following equation using the above notation

β ∂

∂

= ∂

∂∂∂

+ ∂∂

∂∂

Pt x

kPx y

kPyx y

˜ ˜

(26)

we obtain:

β P Pt

Px

Px

x

Py

Pyn n x

ix

i

y

j

y

j+

+ − + −−∆

≈

∂∂

− ∂∂

∆+

∂∂

− ∂

∂

11 2 1 2 1 2

k k k k˜ ˜ ˜ ˜

/ / / 11 2/

∆y

(27)

where the (i ± 1/2) and (j ± 1/2) subscripts refer to quantities at the boundaries as shown in Figure 9. We can now expand these boundary flows as follows:

Figure 9.Discretisation and notation for the 2D pressure equation.



βP P

t

P P

x

P P

xi jn

i jn

i

i jn

i jn

i

i jn

i jn

, ,

/

, ,

/

, ,˜ ˜

˜

+

+

++ +

−

+−

+−∆

= ( ) −∆

− ( ) −∆

+

1

1 2

11 1

2 1 2

11

1

2k k

x x

kk ky y( ) −∆

− ( ) −∆

+

++ +

−

+−+

j

i jn

i jn

j

i jn

i jnP P

y

P P

y1 2

11 1

2 1 2

111

2/

, ,

/

, ,˜

(28)

where the quantities ˜ , ˜ , ˜ ˜

/ / / /k k k kx x y y( ) ( ) ( ) ( )

+ − + −i i j jand

1 2 1 2 1 2 1 2 are some type of average permeabilities between the two neighbouring grid blocks - as discussed in Chapter 4. Note also we have chosen the spatial discretisation terms at the new (n+1) time level making the this an implicit finite difference scheme.

We can rearrange equation 28 above by taking all the unknown terms (at n+1) to the LHS and the known terms (at time level n) to the RHS. This gives the following:

−( )

∆

−( )

∆

+( )

∆+

( )∆

+( )

∆+

( )∆

−−

+ ++

+ − + − +˜ ˜ ˜ ˜ ˜ ˜

/,

/,

/ / / /k k k k k kx x x x y y

ii jn i

i jn i i j j

xP

xP

x x y y1 2

2 11 1 2

2 11 1 2

21 2

21 2

21 2

221

1 22 1

1 1 22 1

1

−∆

−( )

∆

−( )

∆

=∆

+

−−+ +

++

β

β

tP

yP

yP

tP

i jn

ji jn j

i jn

i jn

,

/,

/, ,

˜ ˜k kx y

(29)

Since the coefficients in equation 29 above are constants, then this defines a set of linear equations similar to those found in 1D. However, here we have up to five non-zero terms per grid block to deal with rather than the three we found for the 1D system. The matrix which arises in this 2D case is known as a pentadiagonal matrix. This set of linear equations can be written as follows:

a P a P a P a P a P bi j i jn

i j i jn

i j i jn

i j i jn

i j i jn

i j− −+

+ ++ +

− −+

+ +++ + + + =1 1

11 1

1 11 1

11 1

1, , , , , , , , , , ,. . . . .

(30)

where the constant coefficients, a a a a ai j i j i j i j i j− + − +1 1 1 1, , , , ,, , , and , are given by the coefficients in equation 29; bi, j is also a (know) constant.

3.3.2 Numbering Schemes in Solving the 2D Pressure EquationIt is quite convenient to label the pressures as Pi, j when we are working out the discretisation of the equations, but this is not helpful when we are arranging the linear equations. Here, it is useful first to consider the numbering scheme for the 2D system that allows us to dispense with the (i,j) subscripting in equations 29 or 30 above. The structure of the A - matrix in equation 30 can be made clearer by working out a specific 2D example as shown in Figure 10.



j = 5 m = 17 m = 18 m = 19 m = 20j = 4 m = 13 m = 14 m = 15 m = 16j = 3 m = 9 m = 10 m = 11 m = 12j = 2 m = 5 m = 6 m = 7 m = 8j = 1 m = 1 m = 2 m = 3 m = 4 i = 1 i = 2 i = 3 i = 4

Notation:NX = maximum number of grid blocks in x - direction, i = NX;NY = maximum number of grid blocks in y - direction, j = NYm = grid block number in the natural ordering scheme shownm = (j - 1).NX + ie.g. for i = 3, j = 4 and NX = 4, m = (4 - 1).4 + 3 = 15 (as above)

In the m-notation shown in Figure 10 (m = (j-1)NX +i), the reordered equations 30 become the following:

˜ . ˜ . ˜ . ˜ . ˜ .a P a P a P a P a P bm m m m NX m NXm

nmn

mn

m NXn

m NXn

m− + − +−+

++ +

−+

+++ + + + =

1 111

11 1 1 1

(31)

where the am are the reordered coefficients where the subscript is calculated from the m-formula in Figure 10. For example:

a a ai j j NX i j NX i NX m NX, ˜ ˜( ) ( )− → →

− − + = − + − −1 1 1 1 (32)

a a ai j j NX i j NX i NX m NX, ˜ ˜( ) ( )+ → →

+ − + = − + + +1 1 1 1 (33)

Note that, when we apply the above equation numbering scheme to the example in Figure 10 (NX = 4, NY = 5 and therefore, 1 ≤ m ≤ 20), certain “neighbours” are “missing” since a block is at the boundary (or in a corner where two neighbours are missing). For example, for block (i = 3; j = 3), that is block m = 9, the “j-1 block” is not there. Therefore, the coefficient Am-1 =0 in this case. This is best seen by writing out the structure of the 20 x 20 A-matrix by referring to Figure 10; the A-matrix structure is shown in Figure 11.

Note that the A-matrix structure in Figure 11 is sparse and has a maximum of five non-zero coefficients in a given row - it is a pentadiagonal matrix.

All implicit methods for discretising the pressure equation lead to sets of linear equations. These have the general matrix form:

A x b. = (34)

where A is a matrix like the examples shown above, x is the column vector of unknowns (like the pressures) and b is a column vector of the RHSs. This is just like equation 31 but it is in shorthand form. We will discuss methods for solving these equations later in this chapter. For the meantime, we will just assume that it can be solved. We next consider when the PDEs describing a phenomenon are non-linear PDEs.

Figure 10.Numbering system for 2D grid conversion from (i,j) → m counter.



m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

x x x

xxx x

xx x

xxx x

xx

x

x

xx

x

x

x

x

x

x

x

xx x

xx

xxx xx

x xxx

x x

x

x xx

x xxx

x x

x

x xx

x xx x

x

x

xx xx x x

x

x

xx

x

x

x

xx

x

x

x

x

m - ordering scheme used in figure 10 - shown here for reference.

j = 5 m = 17 m = 18 m = 19 m = 20j = 4 m = 13 m = 14 m = 15 m = 16j = 3 m = 9 m = 10 m = 11 m = 12j = 2 m = 5 m = 6 m = 7 m = 8j = 1 m = 1 m = 2 m = 3 m = 4 i = 1 i = 2 i = 3 i = 4

3.4 Implicit Finite Difference Approximation of Non-linear Pressure EquationsWe have noted previously that using numerical methods are usually the only way which we can solve non-linear PDEs of the type that arise in reservoir simulation. We will use the single-phase 2D equation for the flow of compressible fluids (and rocks) as the example for discussing the numerical solution of non-linear PDEs. Equation 35 below was derived in Chapter 5 and has the form:

c P

Pt x

k Px y

k Py

x y( )∂∂

= ∂

∂∂∂

+ ∂

∂∂∂

ρµ

ρµ (35)

where the non-linearities arise in this equation due to the dependence of the generalised compressibility, c(P), the density, ρ(P) and the viscosity, μ(P), on pressure, P(x,t). It is the pressure that is the main unknown in this equation and, hence, if these quantities depend on it, then they introduce difficulties into the equations. In fact, we will see shortly that the equations that arise are not linear equations - they are non-linear algebraic equations.

Figure 11.Structure of the A-matrix for the m-ordering scheme in the table shown below for reference.



Proceeding as before, we can discretise the above equation as follows:

c PP P

t

k Px

k Px

x

k Py

k Pyn n

x

i

x

i

y

j

y

( ) / / /+

+ − +−∆

=

∂∂

− ∂

∂

∆+

∂∂

− ∂

∂

11 2 1 2 1 2

ρµ

ρµ

ρµ

ρµ

∆−j

y1 2/

(36)

where we have not yet decided on the time level of the non-linearities (i.e. the time level of the pressure, Pn or Pn+1, at which we evaluate c, ρ and μ). The most difficult case would be if these were set at the “unknown” time level Pn+1 and this is what we will do as follows:

c PP P

t

k Px

k Px

x

k

nn n

x

n

i

x

n

i

yn

( ) / /++

+

+

+

−

+

−∆

=

∂∂

−

∂∂

∆

+

∂

11

1

1 2

1

1 2

1

ρµ

ρµ

ρµ

PPy

k Py

yj

yn

j∂

−

∂∂

∆+

+

−1 2

1

1 2/ /

ρ

µ

(37)

Now expanding up the ∂∂

∂∂

Px

Py

and terms gives:

c PP P

t

k P P

xk P P

n i jn

i jn

x

i

ni jn

i jn

x

i

ni jn

i jn

( ) , , /

, ,

/

, ,

++

+

++

+ +

−

+ +−

+

−∆

=

−∆

−

−∆

11

1 2

1

11 1

1 2

1 11

1ρµ

ρµ xx

x

k P P

y

k P P

yy

j

ni jn

i jn

y

j

nin

i jn

∆

+

−∆

−

−∆

+

+++ +

−

+ +−+

ρµ

ρµ

1 2

1

11 1

1 2

1

11

11

/

, ,

/

, ,

∆y (38)which can be simplified to the following:

c PP P

tk

xP P

kx

P Pn i jn

i jn

x

i

n

i jn

i jn x

i

n

i jn

i jn( )

. ., ,

/

, ,

/

, ,+

+

+

+

++ +

−

++

−+−

∆

=∆

−( ) −∆

−( )

+

11

21 2

1

11 1

21 2

1

11

1ρµ

ρµ

kk

yP P

k

y

P P

yy

j

n

i jn

i jn y

j

ni jn

i jnρ

µρ

µ. ./

, ,

/

, ,

∆

−( ) −∆

−∆

+

+

++ +

−

+ +−+

21 2

1

11 1

21 2

1 111

(39)



This can be treated as before to gather together similar unknown P-terms as follows:

−∆

−∆

+∆

+∆

+∆

−

+

−+

−

+

−+

+

+

+

kx

Pk

yP

c Pt

kx

kx

x

i

n

i jn y

j

n

i jn

nx

i

n

x

ρµ

ρµ

ρµ

ρµ

..

..

( ). .

/

,

/

,

/

21 2

1

11

21 2

1

11

1

21 2

1

2

+∆

+∆

−∆

−

−

+

+

+

−

++

+

+

++

i

ny

j

ny

j

n

i jn

x

i

n

i jn y

k

y

k

yP

kx

Pk

1 2

1

21 2

1

21 2

1

1

21 2

1

11

/ / /

,

/

,

. ..

..

ρµ

ρµ

ρµ

ρµ

..

.( ).

/

,,

∆

∆

+

+

++

+

yP

c P P

tj

n

i jn

ni jn

21 2

1

11

1

=

(40)

As before, we can write this in a compact form as follows:

α α α α α βin

j i jn

i jn

i jn

i jn

i jn

i jn

i jn

i jn

i jn

i jnP P P P P−

+−

+++

++ + +

−+

−+

++

++ ++ + + + =1

11

111

11 1 1

11

11

11

11 1

, , , , , , , , , , ,. . . . . (41)

where the elements of the A-matrix (now denoted α α α α αi jn

i jn

i jn

i jn

i jn

−+

−+ +

++

++

11

11 1

11

11

, , , , ,, , , ) and are not constants since they depend on the (unknown) value of Pn+1.

How do we go about solving the non-linear set of algebraic equations in 39 or 40 above? It turns out that we have two choices in tackling this more difficult problem as follows:

(i) We can actually use a numerical equation solver which is specifically designed to solve more difficult non-linear problems. An example of this type of approach is in using the Newton-Raphson method. We will return to this method later (once we have seen how to solve linear equations).

(i) We can choose to apply a more pragmatic algorithms such as the following:(a) Although our α-terms in equation 41 are strictly at time level (n+1), simply take them at time level n, as a first guess. So we approximate equation 41 by the following first guess:

α α α α α βi

nj i j

ni jn

i jn

i jn

i jn

i jn

i jn

i jn

i jn

i jnP P P P P− −

++ +

+ +− −

++ +

++ + + + =1 11

1 11 1

1 11

1 11

, , , , , , , , , , ,. . . . . (42)

where the quantities α α α α α βin

j i jn

i jn

i jn

i jn

i jn

i jnP

− + − +1 1 1 1, , , , , ,

,

, , , , and are evaluated at the (known) time level n - i.e. at values of pressure =

α α α α α βin

j i jn

i jn

i jn

i jn

i jn

i jnP

− + − +1 1 1 1, , , , , ,

,

, , , , and

.

(b) Solve the now linear equations 42 above to obtain a first estimate - or a first

iteration - of the

P

P

i jn

i j

i jn

,

,

,

+

+( )

1

1

αν

νat each grid point. We will use the following notation ⇒

P

P

i jn

i j

i jn

,

,

,

+

+( )

1

1

αν

ν

where ν is the iteration counter and

P

P

i jn

i j

i jn

,

,

,

+

+( )

1

1

αν

ν

is the value of the a-coefficient at the

P

P

i jn

i j

i jn

,

,

,

+

+( )

1

1

αν

ν

value after ν iterations; that is:

α αν ν

i j i j i jnP, , ,= ( )[ ]+1

(43)



(c) Use the latest iterated values of the

P

P

i jn

i j

i jn

,

,

,

+

+( )

1

1

αν

ν

to solve the (now linear) equations to go from P to Pi j

ni jn

, , + + +( ) ( )1 1 1ν ν.

(d) Keep iterating the above scheme until it converges; i.e. the difference between the sum of two successive iterated values of the pressure (Err.) is sufficiently small (< Tol, which is an acceptably small value)

Err P Pi j

ni jn

all i j

= ( ) − ( )+ + +∑ , , ,

1 1 1ν ν

(44)

Stop if Err < Tol. Otherwise continue through steps above.

The algorithm outlined in (ii) above for solving the non-linear pressure equation is represented in Figure 12.

Set the iteration counter ν = 0

α α α α α βν ν ν ν ν νi j i j i j i j i j i j− + − +1 1 1 1, , , , , ,, , , , and Calculate the at

values of pressure =

where ν = current iteration number.

Pi jn,( )ν

α αν ν

i j i j i jnP, , ,= ( )[ ]+1

α α α α α βν ν ν ν ν νi j i j

ni j i j

ni j i j

ni j i j

ni j i j

ni jP P P P P− −

++ +

+ +− −

++ +

++ + + + =1 11

1 11 1

1 11

1 11

, , , , , , , , , , ,. . . . .

Solve the set of linear equations:

to obtain

Calculate

P Pi jn

i jn

, ,+ + +

= ( )1 1 1ν

No - continue iterationsErr < Tol ?

Stop

Yes (the method has converged)

Err P Pi jn

i jn

all i j

= ( ) − ( )+ + +∑ , , ,

1 1 1ν ν

ν = ν +1

Figure 12.Algorithm for the numerical solution of the non-linear 2D pressure equations:



4 APPLICATION OF FINITE DIFFERENCES TO TWO-PHASE FLOW

4.1 Discretisation of the Two-Phase Pressure and Saturation EquationsWe now consider how finite difference methods are applied to the two-phase flow equations. We have seen how these equations were derived in Chapter 5. Recall that, in two-phase flow, we have two coupled equations to solve - a pressure equation and a saturation equation. For example, we may solve for the oil pressure, Po(x,t) and the oil saturation, So (x,t); we would then find the water saturation and water pressure by using the constraint equations, So+Sw = 1 and the capillary pressure relation, Pc(Sw) = Po - Pw, respectively.

From Chapter 5, we use the highly simplified form of the 1D pressure and saturation equations, where we choose to solve for ⇒ P(x,t) and So (x,t) as shown in Chapter 5, equations 81 and 82. Note that we have taken zero capillary pressure (therefore P = Po = Pw) and zero gravity which gives:

PRESSURE EQUATION ∂∂

( ) ∂∂

=x

SPxT oλ 0 (45)

SATURATION EQUATION φ λ∂∂

= ∂

∂( ) ∂

∂

St x

SPx

oo o (46)

(Equations 45 and 46 are the same as equations 81 and 82 from Chapter 5, respectively).

The quantity λ

λ λ λ

T o

T o o o w o

S

S S S

( )

( ) = ( ) + ( )

is the total mobility and is the sum of the oil and water mobilities;

λ

λ λ λ

T o

T o o o w o

S

S S S

( )

( ) = ( ) + ( ). The above two equations are clearly coupled together since the oil saturation appears in the non-linear coefficient of the pressure equation. Likewise, the pressure appears in the flow term on the RHS of the saturation equation.

We can now apply finite differences to each of these equations in the same way as discussed above to obtain, for the pressure equation (see notation in Figure 8):

λ λT o

iT o

i

SPx

SPx

x

( ) ∂∂

− ( ) ∂∂

∆=+ −1 2 1 2 0/ /

(47)

which can be expanded further to give:

λ λT o

in

in

iT o

in

in

i

SP P

xS

P Px

x

( ) −∆

− ( ) −

∆

∆=

++ +

+

+−

+

−

11 1

1 2

11

1

1 2 0/ /

(48)



In equation 48 above, we must now specify the time level of the non-linear mobility terms, λ λ

λ

λ

T o i T o i

on

T on

i

T on

i

S S

S

S

S

( )( ) ( )( )

( )( )

( )( )

+ −

+

+

+

+

−

1 2 1 2

1

1

1 2

1

1 2

/ /

/

/

and ; we must choose whether we take these terms at time n or at time level (n+1)? Again, we go straight to the most difficult case by defining these terms at the later time level i.e. at

λ λ

λ

λ

T o i T o i

on

T on

i

T on

i

S S

S

S

S

( )( ) ( )( )

( )( )

( )( )

+ −

+

+

+

+

−

1 2 1 2

1

1

1 2

1

1 2

/ /

/

/

and

. These terms are denoted as,

λ λ

λ

λ

T o i T o i

on

T on

i

T on

i

S S

S

S

S

( )( ) ( )( )

( )( )

( )( )

+ −

+

+

+

+

−

1 2 1 2

1

1

1 2

1

1 2

/ /

/

/

and

and

λ λ

λ

λ

T o i T o i

on

T on

i

T on

i

S S

S

S

S

( )( ) ( )( )

( )( )

( )( )

+ −

+

+

+

+

−

1 2 1 2

1

1

1 2

1

1 2

/ /

/

/

and

, in equation 48 above to obtain:

λ λT o

n

iin

in T o

n

iin

in

S

xP P

S

xP P

+

++

+ +

+

− +−

+( )( )

∆−( )

−( )( )

∆−( )

=1

1 22 1

1 1

1

1 22

11

1 0/ /

(49)

The above equation can now be arranged into the usual order as follows:

λ λ λ λT o

n

iin T o

n

i T on

iin T o

n

iin

S

xP

S

x

S

xP

S

xP

+

−−

+

+

−

+

+ +

+

++

+( )( )

∆−

( )( )∆

+( )( )

∆

+( )( )

∆=

1

1 22 1

1

1

1 22

1

1 22

1

1

1 22 1

1 0/ / / /

(50)

This is a non-linear set of algebraic equations for the unknown pressures, P P P

S

in

in

in

on

−+ +

++

+

11 1

11

1

, and , but the coefficients depend on the - also unknown - saturations,

P P P

S

in

in

in

on

−+ +

++

+

11 1

11

1

, and

. Note that equation 50 represents one of a set of equations since there is one at each grid point (and at the ends of the 1D system the values may be set by the boundary conditions).

We now consider the discretisation of the saturation equation 46. Finite differences may be applied to this equation as follows:

φ

λ λS S

t

SPx

SPx

xoin

oin o o

io o

i+

+ −−∆

=( ) ∂

∂

− ( ) ∂∂

∆

11 2 1 2/ /

(51)

Again, we can expand the derivative terms at the (i+1/2) and (i-1/2) boundaries (Figure 9) and we can take the mobility terms at the (n+1) time level to obtain:

φλ λS S

t

S

xP P

S

xP Poi

noin

o on

iin

in o o

n

iin

in

++

++

+ +

+

− +−

+−∆

=( )( )

∆−( )

−( )( )

∆−( )

11

1 22 1

1 1

1

1 22

11

1/ /

(52)

The above non-linear algebraic set of equations can be written in various ways. Two particularly useful ways to write the above equations are as follows:



Form A:

S St S

xP P

S

xP Poi

noin o o

n

iin

in o o

n

iin

in+

+

++

+ +

+

− +−

+= + ∆ ( )( )∆

−( )

−( )( )

∆−( )

1

1

1 22 1

1 1

1

1 22

11

1

φ

λ λ/ /

(53)or Form B:

S St S

xP P

S

xP Poi

noin o o

n

iin

in o o

n

iin

in+

+

++

+ +

+

− +−

+− − ∆ ( )( )∆

−( )

−( )( )

∆−( )

=1

1

1 22 1

1 1

1

1 22

11

1 0φ

λ λ/ /

(54)

The reason for writing the above two forms of the saturation equation is that each is useful, depending on how we intend to approach the solution of the coupled pressure (equation 45) and saturation (equation 46) equations.

As with the case of the solution of the non-linear single phase pressure equation for a compressible system, we have two strategies which we can use to solve the two-phase equations above, as follows:

(i) We can actually use a numerical equation solver which is specifically designed to solve more difficult non-linear problems; e.g. the Newton-Raphson method. We will return to this method later (once we have seen how to solve linear equations).

(i) We can again choose to apply a more pragmatic algorithm and this is the subject of section 4.2 below.

4.2 IMPES Strategy for Solving the Two-Phase Pressure and Saturation EquationsThe form of the discretised pressure and saturation equations which we will take are as follows (equations 50 and 53):

Pressure:

λ λ λ λT o

n

iin T o

n

i T on

iin T o

n

iin

S

xP

S

x

S

xP

S

xP

+

−−

+

+

−

+

+ +

+

++

+( )( )

∆−

( )( )∆

+( )( )

∆

+( )( )

∆=

1

1 22 1

1

1

1 22

1

1 22

1

1

1 22 1

1 0/ / / /

(50)

Saturation:

S St S

xP P

S

xP Poi

noin o o

n

iin

in o o

n

iin

in+

+

++

+ +

+

− +−

++ − ∆ ( )( )∆

−( )

−( )( )

∆−( )

1

1

1 22 1

1 1

1

1 22

11

1

φ

λ λ/ /

(53)



Pressure equation 50 would be a set of linear equations if the coefficients were known at the current time step (n), rather than being specified at the (n+1) - i.e. the unknown - time level. However, we could solve equation 50 above as if it were a linear system of equations by time-lagging the coefficients as before (see the algorithm in Figure 13). This would give us a “first guess” (or first iteration) to find the unknowns i.e. the quantities P P P

P

S

P P P

P P P

S S

in

in

in

in

on

in

in

in

in

in

in

in

in

−+ +

++

+

+

−+ +

++

−+ +

++

+++

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

11 1

11

1

1

11 1

11

11 1

11

111

,

, .

,

,

and

and

and

etc.

ν ν ν

ν ν ν

ν ν

. The same problem exists for the saturation equation 53, if we had the first guess at the

P P P

P

S

P P P

P P P

S S

in

in

in

in

on

in

in

in

in

in

in

in

in

−+ +

++

+

+

−+ +

++

−+ +

++

+++

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

11 1

11

1

1

11 1

11

11 1

11

111

,

, .

,

,

and

and

and

etc.

ν ν ν

ν ν ν

ν ν

(by the procedure just mentioned), then we could use these latest values of pressure and still time lag the coefficients (the oil mobility terms) and use the saturation equation 53 as if it were an explicit expression. This would give us an updated value of

P P P

P

S

P P P

P P P

S S

in

in

in

in

on

in

in

in

in

in

in

in

in

−+ +

++

+

+

−+ +

++

−+ +

++

+++

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

11 1

11

1

1

11 1

11

11 1

11

111

,

, .

,

,

and

and

and

etc.

ν ν ν

ν ν ν

ν ν

which could then be used back in the pressure equation and the whole process could be iterated until convergence, as discussed previously. A flow chart of this procedure has already been outlined in Chapter 5 (Figure 9) but is elaborated here in Figure 13. Note that by taking time-lagged values of the saturations, the pressure equation is linearlised and can then be solved implicitly for the pressure for that iteration, ν

⇒

P P P

P

S

P P P

P P P

S S

in

in

in

in

on

in

in

in

in

in

in

in

in

−+ +

++

+

+

−+ +

++

−+ +

++

+++

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

11 1

11

1

1

11 1

11

11 1

11

111

,

, .

,

,

and

and

and

etc.

ν ν ν

ν ν ν

ν ν

The saturations can then be obtained explicitly using the latest pressures (i.e. the

P P P

P

S

P P P

P P P

S S

in

in

in

in

on

in

in

in

in

in

in

in

in

−+ +

++

+

+

−+ +

++

−+ +

++

+++

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

11 1

11

1

1

11 1

11

11 1

11

111

,

, .

,

,

and

and

and

etc.

ν ν ν

ν ν ν

ν ν

) and the most recent iteration of the saturation (i.e. the

P P P

P

S

P P P

P P P

S S

in

in

in

in

on

in

in

in

in

in

in

in

in

−+ +

++

+

+

−+ +

++

−+ +

++

+++

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

11 1

11

1

1

11 1

11

11 1

11

111

,

, .

,

,

and

and

and

etc.

ν ν ν

ν ν ν

ν ν). Therefore, this approach is

known as the IMPES approximation, which stands for IMplicit in Pressure, Explicit in Saturation. This can be iterated until it converges although there are limitations in the size of time step, Δt, which can be taken. If Δt is too large, the IMPES method may become unstable and give unphysical results like the example in Exercise 3 above.



Set the iteration counter to zero, ν = 0

Take current values of as the ν leveliteration values: ⇒

PRESSURE EQUN. Set the total mobilities ⇒obtain the linearised pressure equation:

Solve this linear implicit pressure equation for pressure at iteration, ν, toobtain:

⇒

Calculate the error, Err., by comparing the change in pressure or saturations(or both as here) at the current (ν) and last (ν-1) iterations:

Note that the Err. term above would have some scaling factors weighing the pressure and saturation terms because of the units of pressure.

SATURATION EQUN. Now update the saturation equation as if it were explicit by taking(i) the oil mobility terms at the latest iteration, ν ⇒(ii) the latest values of the pressures just calculated implicity ⇒Update saturation explicity as follows:

to obtain latest iteration of saturations ⇒

No - continue iterationIs Err. < Tol ?

Yes

and S Pon n

Stop

and S Poν ν

λ λν νT o i T o iS S( ) ( )/ /+ −1 2 1 2 and

λ λ λ λν ν ν νT o

iin T o

i T oi

in T o

iin

S

xP

S

x

S

xP

S

xP

( )( )∆

−( )( )∆

+( )( )∆

+( )( )∆

=−−+ − + + +

++1 2

2 11 1 2

21 2

21 1 2

2 11 0/ / / /

P P Pin

in

in

−+ +

++( ) ( ) ( )1

1 111ν ν ν

, and

λ λν ν0 0 1( ) ( ) ,S So i o i etc.+

P P Pin

in

in

−+ +

++( ) ( ) ( )1

1 111ν ν ν

, and

S St S

xP P

S

xP Poi

noin o o

iin

in o o

iin

in+ +

++ + − +

−+= + ∆ ( )( )

∆ ( ) − ( )( )

−( )( )∆ ( ) − ( )( )

1 1 2

2 11 1 1 2

21

11

φ

λ λνν ν

νν ν/ /

S Sin

in+++( ) ( )111ν ν

, etc.

Err P P S Sin

i

NX

in

in

i

NX

in. = ( ) − ( ) + ( ) − ( )+

=

+ − +

=

+ −∑ ∑1

1

1 1 1

1

1 1ν ν ν ν

ν = ν +1

Figure 13.IMPES Algorithm for the numerical solution of the two-phase pressure and sat-uration equations (IMPES ⇒ IMplicit in Pressure, Explicit in Saturation)



5 THE NUMERICAL SOLUTION OF LINEAR EQUATIONS

5.1 Introduction to Linear EquationsIn all implicit finite difference methods, we end up with a set of linear equations to solve. In fact, we have shown above that the equations involved - e.g. the discretised pressure equation - may be a set of non-linear algebraic equations. However, these can usually be linearised (say, by time lagging the coefficients as discussed above) in order to obtain a set of linear equations. The solution to the original non-linear equations may then be found by repeating or iterating through the cycle of solving the linear equations. In the following section, we will address the issue of solving the non-linear equations which arise in the discretised two-phase flow equations directly.

A generalised set of linear equations may be written in full as follows:

a x a x a x a x a x a x b


a x a x a x a x a x

n n

n n

11 1 12 2 13 3 14 4 15 5 1 1

21 1 22 2 23 3 24 4 25 5 2 2

31 1 32 2 33 3 34 4 35 5

. . . . . ... .

. . . . . ... .

. . . . . ...

+ + + + + =

+ + + + + =

+ + + +

++ =

+ + + + + =

a x b


n n

n n n n n nn n n

3 3

1 1 2 2 3 3 4 4 5 5

.

.....

.....

.....

. . . . . ... .

...

...

...

(55)

where the aij and bi are known constants and the xi (i = 1, 2, ... , n) are the unknown quantities which we are trying to find. The xi in our pressure equation, for example, would be the unknown pressures at the next time step.

The set of linear equations can be written in matrix form as follows:

a a a a a a

a a a a a a

a a a a a a

a a a a a a

n

n

n

n n n n n nn

11 12 13 14 15 1

21 22 23 24 25 2

31 32 33 34 35 3

1 2 3 4 5

...

...

...

...

...

...

.....

.....

.....

...

=

x

x

x

x

b

b

b

bn n

1

2

3

1

2

3

.

.

.

.

.

.

(56)



We can write the matrix A and the column vectors x and b as follows:

A

a a a a a a

a a a a a a

a a a a a a

a a a a a a

n

n

n

n n n n n nn

=

11 12 13 14 15 1

21 22 23 24 25 2

31 32 33 34 35 3

1 2 3 4 5

...

...

...

...

...

...

.....

.....

.....

...

=

=

;

.

.

.

;

.

.

.

x

x

x

x

x

b

b

b

bn n

b

1

2

3

1

2

3 and

(57)

and the set of linear equations can be written in very compact form as follows:

A x b. = (58)

where A in an n x n square matrix and x and b are n x 1 column vectors.

A very simple example of a set of linear equations is given below:

4 2 1 13

1 3 3 14

2 1 2 11

1 2 3

1 2 3

1 2 3

. . .

. . .

. . .

x x x

x x x

x x x

+ + =

+ + =

+ + = (59)

How do we solve these? In the above case, you can do a simple trial and error solution to find that x1 = 2, x2 = 1 and x3 = 3, although this would be virtually impossible if there were hundreds of equations. Remember, there is one equation for every grid block in a linearised pressure equation in reservoir simulation (although there are lots of zeros in the A matrix). This implies that we need a clear numerical algorithm that a computer can work through and solve the linear equations. The overview of approaches to solving these equations is discussed in the next section.

5.2 General Methods for Solving Linear EquationsFirstly, let us note that there is a vast literature on solving sets of linear equations and many books on the underlying theory and on the numerical techniques have appeared. Indeed, petroleum reservoir simulation has led to the development of some of these numerical techniques. We will not cover much of this huge subject but we will cover enough that the student can appreciate - rather than understand in detail - how the linear equation are solved in a simulator.



Basically, there are two main approaches for solving sets of linear equations involving direct methods and iterative methods as follows:

(i) Direct Methods: this group of methods involves following a specific algorithm and taking a fixed number of steps to get to the answer (i.e. the numerical values of x1, x2 .. xn). Usually, this involves a set of forward elimination steps in order to get the equations into a particularly suitable form for solution (see below), followed by a back substitution set of steps which give us the answer. An example of such a direct method is Gaussian Elimination.

(ii) Iterative Methods: in this type of method, we usually start off with a first guess (or estimate) to the solution vector, say x(o) . Where we take this as iteration zero, v = 0. We then have a procedure - an algorithm - for successively improving this guess by iteration to obtain, x(1) → x(2) → x(3) .... → x(ν) etc. If it is successful, then this iterative method should converge to the correct x as ν increases. The solution should get closer and closer to the “correct” answer but, in many cases, we cannot say exactly how quickly it will get there. Therefore, iterative methods do not have a fixed number of steps in them as do direct methods. Examples of iterative methods are the Jacobi iteration, the LSOR (Line Successive Over Relaxation) method, etc.

The following two sections will discuss each of these approaches for solving sets of linear equations in turn.

5.3 Direct Methods for Solving Linear EquationsIn fact, we will not present the details of any direct method for solving linear equations. Instead, we will discuss a general outline of how these methods work. Starting with the basic linear equation in matrix form:

A x b. = (60)

where A in an n x n square matrix and x and b are n x 1 column vectors. We can write the n x (n+1) augmented matrix of A and b coefficients as follows:

a a a a a a b

a a a a a a b

a a a a a a b

a a a a a a b

n

n

n

n n n n n nn

11 12 13 14 15 1 1

21 22 23 24 25 2 2

31 32 33 34 35 3 3

1 2 3 4 5

... ...

... ...

... ...

...

...

...

.....

.....

.....

... nn

(61)

Remember that any mathematical operation we perform on one of the linear equations (e.g. multiplying through by x2) must be performed on both sides of the equation i.e.



on the aij and the bi. Therefore, suppose we can transform the above augmented matrix into the following form (we do not say how this is done, just that it can be done):

c c c c c c e

c c c c c e

c c c c e

c e

n

n

n

nn n

11 12 13 14 15 1 1

22 23 24 25 2 2

33 34 35 3 3

0

0 0

0 0

... ...

... ...

... ...

0 0 0 ...

...

...

...

.....

.....

.....

(62)

The C-matrix above is in upper triangular form i.e. only the diagonals and above are non-zero (cij ≠ 0, if j ≥ i) and all elements below the diagonal are zero (cij = 0, if i > j), as shown above. Now consider why this particular form is of interest in solving the original equations. The reason is that, if we have formed the augmented matrix correctly (i.e. doing the same operations to the A and b coefficients), then the following matrix equation is equivalent to the original matrix equation:

i.e. A x b C x e. .= = <=> (63)

Therefore,

c c c c c c

c c c c c

c c c c

c

x

x

x

n

n

n

nn

11 12 13 14 15 1

22 23 24 25 2

33 34 35 3

1

2

3

0

0 0

0 0

..

..

..

0 0 0 ...

...

...

...

.....

.....

.....

.

..

.

.

.

.

x

e

e

e

en n

=

1

2

3

(64)

This matrix equation is very easy to solve, as can be seen by writing it out in full as follows:



c x c x c x c x c x c x e

c x c x c x c x c x e

c x c x c

n n

n n

11 1 12 2 13 3 14 4 15 5 1 1

22 2 23 3 24 4 25 5 2 2

33 3 34 4 35

. . . . . .

. . . . .

. .

+ + + + + =

+ + + + =

+ +

...

...

.. .

. . .

...... ...

...... ...

. .

.

x c x e

c x c x c x e

c x c x e

c x e

n n

n n

n n n n n n n

nn n n

5 3 3

44 4 45 5 4 4

1 1 1 1 1

...

...

, ,

+ =

+ + =

+ ==

− − − − −

(65)

We can easily solve this equation by back-substitution, starting from the equation n which only has one term to find xn, this xn is then used in the (n-1) equation (which has 2 terms) to find xn-1 etc. as follows:

. now use in ..

. . .

use in ..

.

c x e x e c x

c x c x e xe c x

cx x

c x c

nn n n n n nn n

n n n n n n n nn n n n

n nn n

n n n n

= => =

+ = => =−

+

− − − − − −− −

− −−

− − −

/

,, ,,

,

,

1 1 1 1 1 11 1

1 11

2 2 2 −− − − − −

−− − − − −

− −

+ =

=> =−

2 1 1 2 2

22 2 1 1 2

2 2

, ,

, ,

,

-.

n n n n n n

nn n n n n n n

n n

x c x e

xe c x c x

c

. .

. .

etc

(66)

Hence, by working back through these equations in upper triangular form, we can calculate xn, xn-1, , xn-2.... back to x1.



EXERCISE 4.

Solve the following simple example of an upper triangular matrix:

4 2 5

2 2

3 4 1 30

2 1 3

3 15

1 2 3 4 5

2 3 4 5

3 4 5

4 5

5

1 2 3

x x x x x

x x x x

x x x

x x

x

Answer

x x x

- 1 1 1

1 2 12

= ..........; = ..........; = ..........;

− + + =− + + =

+ + =− =

=

= ..........; x x4 5 = ..........

Answer:

4 2 5

2 2

3 4 1 30

2 1 3

3 15

1 2 3 4 5

2 3 4 5

3 4 5

4 5

5

1 2 3

x x x x x

x x x x

x x x

x x

x

Answer

x x x

- 1 1 1

1 2 12

= ..........; = ..........; = ..........;

− + + =− + + =

+ + =− =

=

= ..........; x x4 5 = ..........

5.4 Iterative Methods for Solving Linear Equations As noted above, the idea in an iterative method for solving a set of linear equations is to make a first guess at the solution and then to refine it in a stepwise manner using a suitable algorithm. This procedure should gradually converge to the correct answer. We illustrate the idea of such a method using a very simple iterative scheme. Note that this simple point iterative scheme will work for some of the examples we try here but it is not one that is recommended for use in reservoir simulation. However, it adequately illustrates the main ideas which you need to know for the purposes of this part of the course.

Our simple scheme starts with the longhand version of a set of linear equations and, for our purposes, we will just take a set of four linear equations as follows:

a x a x a x a x b

a x a x a x a x b

a x a x a x a x b

a x a x a x a x b

11 1 12 2 13 3 14 4 1

21 1 22 2 23 3 24 4 2

31 1 32 2 33 3 34 4 3

41 1 42 2 43 3 44 4 4

. . . .

. . . .

. . . .

. . . .

+ + + =

+ + + =

+ + + =

+ + + = (67)

We can rearrange the above set of equations as follows:



xa

b a x a x a x

xa

b a x a x a x

xa

b a x a x a x

xa

b a x a x a x

111

1 12 2 13 3 14 4

222

2 21 1 23 3 24 4

333

3 31 1 32 2 34 4

444

4 41 1 42 2 43 3

1

1

1

1

= − + +( )[ ]

= − + +( )[ ]

= − + +( )[ ]

= − + +( )[ ] (68)

The resulting equations are precisely equivalent to the original set although, if anything, they look a bit more complicated. Why would we deliberately complicate the situation? In fact, the above reorganised equations forms the basis for an iterative scheme, which we can use to solve the equations numerically. Firstly, we need to introduce some notation as follows:

Notation: xi

(v) - the solution for xi at iteration ν; where ν is the iteration counter. xi

x x

( )

( ) ( ),

ν

ν0 - the first guess and νth iteration of the solution vector, x.

Err. - an estimate of the error in the iteration scheme from the ν to the (ν+1) iteration

Tol - some “small” quantity which determines the acceptable error in the iteration scheme; if Err. < Tol, then the scheme has converged.

Using the above notation in equations 68 above, gives the following simple iteration scheme:



xa

b a x a x a x

xa

b a x a x a x

xa

b a x a x

11

111 12 2 13 3 14 4

21

222 21 2 23 3 24 4

31

333 31 1 32 2

1

1

1

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) (

. . .

. . .

. .

ν ν ν ν

ν ν ν ν

ν ν

+

+

+

= − + +( )[ ]

= − + +( )[ ]

= − + νν ν

ν ν ν ν

) ( )

( ) ( ) ( ) ( )

.

. . .

+( )[ ]

= − + +( )[ ]+

a x

xa

b a x a x a x

34 4

41

443 41 1 42 2 43 3

1

(69)

This iteration scheme may now be applied as follows:

(i) Make an initial guess at the solution, iteration ν = 0: x x x x x

Err x xik

ik

i

( ) ( ) ( ) ( ) ( ), , ,

.

01

020

30

40

1

1

4

=

= −+

=∑

(ii) Update the solution to the next iteration ν+1 using equations 69

(iii) Estimate an Error term (Err.) by comparing the latest with the previous iteration of the unknowns, e.g.:

Err x xi

ii. = −+

=∑ ν ν1

1

4

(iv) Is Err. < Tol.

If yes - the scheme has converged; if no - go back to step (ii) and continue the iterations.

This scheme is now illustrated with a practical example.

Example: Solve the following set of equations using the iterative scheme above.

- 0.32 0.1 13.92

2.1 0.21 5.63

- 0.32 0.3 15.19

0.22 5.2 16.76

3 1 0 5

0 2 0 33

0 23 4 0

0 42 0 5

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

. .

. .

. .

. .

x x x x

x x x x

x x x x

x x x x

+ + =

+ + + =

+ + =

+ + + =



Take the first guess: = 2, = 2, = 2, = 2 x x x x10

20

30

40( ) ( ) ( ) ( )

Hint: reorganise the above equations as follows:

x x x x

x x x x

x x

11

2 3 4

21

11

3 4

31

0 5

0 20 0 33

0 23

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )

.

. .

.

ν ν ν ν

ν ν ν ν

ν

+

+ +

+

= + +( )( )

= + +( )( )

=

(1 / 3.1) * 13.92 - - 0.32 0.1

(1 / 2.1) * 5.63 - 0.21

(1 / 4.0) * 15.19 - 111

21

4

41

11

21

310 42 0 5

( ) ( ) ( )

( ) ( ) ( ) ( ). .

ν ν ν

ν ν ν ν

+ +

+ + + +

+( )( )

= + +( )( )

- 0.32 0.30

(1 / 5.2) * 16.76 - 0.22

x x

x x x x

EXERCISE 5.

Fill in the table below for 10 iterations using a calculator or a spreadsheet: POINT ITERATIVE SOLUTION OF LINEAR EQUATIONS

Iteration counter x at iter. ν ν x1 x2 x3 x4First guess = 0 2.0 2.0 2.0 2.0 1 2 3 4 5 6 7 8 9 10

Iteration counter x at iter. ν ν x1 x2 x3 x4First guess = 0 2.0 2.0 2.0 2.0 1 4.309677 1.97619 3.6925 2.784615 2 4.008926 1.411795 3.498943 2.436331 3 3.993119 1.505683 3.497206 2.503112 4 4.000937 1.500783 3.500617 2.500584 5 3.999963 1.499755 3.499965 2.499832 6 3.999986 1.500026 3.499995 2.500017 7 4.000003 1.5 3.500002 2.500001 8 4 1.499999 3.5 2.5Converged ⇒ 9 4 1.5 3.5 2.5 10 4 1.5 3.5 2.5

Note - converges fully at k = 9 iterations.

In some cases, it may be possible to develop improved iteration schemes by using the latest information that is available. For example, in the above iteration scheme, we could use the very latest value of x1

(ν+1) when we are calculating x2(ν+1) since we

already have this quantity. Likewise, we can use both x1(ν+1) and x2

(ν+1) when we calculate x3

(ν+1) etc as shown below:



x x x x

x x x x

x x

11

2 3 4

21

11

3 4

31

0 5

0 20 0 33

0 23

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )

.

. .

.

ν ν ν ν

ν ν ν ν

ν

+

+ +

+

= + +( )( )

= + +( )( )

=

(1 / 3.1) * 13.92 - - 0.32 0.1

(1 / 2.1) * 5.63 - 0.21

(1 / 4.0) * 15.19 - 111

21

4

41

11

21

310 42 0 5

( ) ( ) ( )

( ) ( ) ( ) ( ). .

ν ν ν

ν ν ν ν

+ +

+ + + +

+( )( )

= + +( )( )

- 0.32 0.30

(1 / 5.2) * 16.76 - 0.22

x x

x x x x

x x x11

21

31( ) ( ) ( ) , ν ν ν+ + + , => underlined terms, imply they are at the latest time

available).

When this is done, we find the following results:IMPROVED POINT ITERATIVE SOLUTION OF LINEAR EQUATIONS (Using latest information)

- see spreadsheet CHAP6Ex5.xls - Sheet 2

Iteration counter x at iter. ν ν x1 x2 x3 x4First guess = 0 2 2 2 2 1 4.309677 1.756221 3.540191 2.460283 2 4.021247 1.495632 3.501408 2.498333 3 3.999376 1.500005 3.500161 2.500035 4 3.999973 1.499974 3.499997 2.500004 5 3.999998 1.5 3.5 2.5Converged ⇒ 6 4 1.5 3.5 2.5 7 4 1.5 3.5 2.5 8 4 1.5 3.5 2.5 9 4 1.5 3.5 2.5 10 4 1.5 3.5 2.5

Note - converges fully at ν = 6 iterations.

Clearly, comparing this table with the previous one using the simple point iterative method, we see that this method does indeed converge quicker. Although this is just one example, it is generally true that using the latest information in an iterative scheme improves convergence.

Notes on iterative schemes: there are several point to note about iterative schemes, which we have not really demonstrated or explained here. However, you can confirm some of these points by using the supplied spreadsheets. These points are:

(i) Iterative solution schemes for linear equations are often relatively simply to apply - and to program on a computer (usually in FORTRAN). If you study the supplied spreadsheets (CHAP6Ex5.xls), you can see that they are quite simple in structure for this set of equations.



(ii) The convergence rate of an iterative scheme may depend on how good the initial guess (x(0)) is. If you want to demonstrate this for yourself, run the spreadsheet (CHAP6Ex5.xls) with a more remote initial guess. Here is the same example as that presented above with a completely absurd initial guess:

POINT ITERATIVE SOLUTION OF LINEAR EQUATIONS - Bad First Guess (you can use CHAP6Ex5.xls to confirm these results)

Iteration counter x at iter. ν ν x1 x2 x3 x4First guess = 0 900 -2000 456 23000 1 -1017.45 -2454.69 -1932.95 -28.7 2 63.79526 406.2002 -131.922 375.1144 3 55.59796 -20.1756 4.491703 -6.43019 4 1.890636 -2.67691 -0.53117 0.84584 5 4.326953 2.668945 3.538073 3.234699 6 4.090824 1.389409 3.519613 2.420476 7 3.987986 1.49622 3.491895 2.495457 8 4.001064 1.502872 3.500729 2.50191 9 4.000117 1.499593 3.500025 2.499722 10 3.999963 1.500013 3.499982 2.500005 11 4.000004 1.500006 3.500003 2.500004 12 4 1.499999 3.5 2.499999Converged ⇒ 13 4 1.5 3.5 2.5 14 4 1.5 3.5 2.5 15 4 1.5 3.5 2.5

(iii) We cannot tell in advance how many iterations may be required in order to converge a given iterative scheme. Clearly, from the results above, a good initial guess helps. In reservoir simulation, when we solve the pressure equation, a good enough guess of the new pressure is the values of the old pressures at the last time step. If nothing radical has changed in the reservoir, then this may be fine. However, if new wells have started up in the model or existing wells have changed rate very significantly, then the “pressure at last time step” guess may not be very good. However, with a very robust numerical method, convergence should still be achieved.

(iv) It helps in an iterative method to use the latest information available. This was demonstrated in the iterative schemes in spreadsheet CHAP6Ex5.xls.

5.5 A Comparison of Iterative and Direct Methods for Solving Linear EquationsIn this Chapter, we have described two general methods - direct and iterative - for solving the linear equations which arise when we discretise the flow equations of reservoir simulation. We have not indicated which of these methods is usually “best” for reservoir simulation problems. This is because, it depends on the problem. The final numerical problem which is solved in a reservoir simulator could be quite small and simple, or it could be very large and have some intrinsically difficult numerical problems within it.



In any numerical computational method, such as those for solving a set of linear equations, we need to have some way of calculating the amount of “work” involved. To some extent this depends on the computer architecture since new generation parallel processing machines are becoming available as discussed in Chapter 1. However, we will take quite a simple view based on a conventional serial processing machine and will define computational work as simply the number of multiplications and divisions (addition and subtraction is cheap!). In this way, we can define the amount of work required for any given direct and iterative scheme for solving linear equations. We present results without any proof for two such schemes called the BAND (a direct method) and LSOR (Line Successive Over-Relaxation; an iterative method) schemes. For a 2D problem with NX and NY grid blocks in the x and y directions, respectively (where we choose NY < NX)

BAND (direct), Work, WB ≈ NX.NY3 (70)

LSOR (iterative), Work, WLSOR ≈ NX.NY. Niter (71) where Niter is the number of iterations until convergence is reached. It is already quite clear that the amount of work for the direct method - which only has to be done once - is larger than that for a single iteration of the iterative method. However, we do not know in advance the number of iterations, Niter. We illustrate some typical results for these two methods with a simple exercise below.

EXERCISE 6.

Which is best method (i.e. that requiring the lowest amount of computational work), BAND or LSOR, for the following problems?

(i) NX = 5, NY = 3, Niter = 50 (a small problem)

(ii) NX = 20, NY = 5, Niter = 50

(iii) NX = 100, NY = 20, Niter = 70

(iv) NX = 400, NY = 100, Niter = 150

NX NY Niter BAND LSOR, Comment WB WLSOR5 3 5020 5 50100 20 70400 100 150



We can summarise our comparison between direct and iterative methods for solving the sets of linear equations that arise in reservoir simulation as follows:

(i) A direct method for solving a set of linear equations has an algorithm that involves a fixed number of steps for a given size of problem. Given that the equations are properly behaved (i.e. the problem has a stable solution), the direct method is guaranteed to get to the solution in this fixed number of steps. No first guess is required for a direct method.

(ii) An iterative method on the other hand, starts from a first guess at the solution (x(o)) and then applied a (usually simpler) algorithm to get better and better approximations to the true solution of the linear equations. If successful, the method will converge in a certain number of iterations, Niter, which we hope will be as small as possible. However, we cannot usually tell what this number will be in advance. Also, in some cases the iterative method may not converge for certain types of “difficult” problem. We may need to have good first guess to make our iterative method fast. Also, it often helps to use the latest computed information that is available (see example above).

(iii) Usually the amount of “work” required for a direct method is smaller for smaller problems but iterative methods usually win out for larger problems. For an iterative method, the amount of work per iteration is usually relatively small but the number of iterations (Niter) required to reach convergence may be large and is usually unknown in advance.



EXERCISE 7.

The linear equations which arise in reservoir simulation may be solved by a direct solution method or an iterative solution method. Fill in the table below:

Direct solution method Iterative solution method

Give a verybrief descriptionof each method

Main advantages 1. 1.

2. 2.

Main 1. 1.disadvantages

2. 2.



6. DIRECT SOLUTION OF THE NON-LINEAR EQUATIONS OF MULTI-PHASE FLOW

6.1 Introduction to Sets of Non-linear EquationsWe noted in Section 4 above that the equations which we obtain when we discretise the two-phase flow equations are actually non-linear in nature. However, the strategy we discussed above (the IMPES method) involved tackling the problem almost as if it were a linear set of equations since we time-lagged the coefficients to reduce the problem to a linear set of equations. Then we repeated this process - we applied repeated iterations - until it (hopefully) converged. Therefore, we took a non-linear problem and solved it as if it were a series of linear problems.

In this section, we will introduce the general idea of solving sets of non-linear equations numerically and indicate how this can be applied to the two-phase flow equations. We will do this in a simplified manner that shows the basic principles without going into too much detail. Firstly, compare the difference between solving the following two sets of two equations, a set of 2 linear equations:

2 10

3 51 2

1 2

x x

x x

+ =− = (72)

and a set of 2 non-linear equations:

x x e

x x x x

x1 2

2 12

1 2

2

2 2

5

1

+ =

( ) − ( ) = −

−.

. . (73)

It is immediately obvious that the second set of non-linear equations is more difficult. The first set of linear equations can be rearranged easily to show that x1 = 3 and x2 = 4. However, it is not as straightforward to do the same thing for the non-linear set of equations. As a first attempt to solve these, we might try to develop an iterative scheme by rearranging the equations as follows

x x e

xx x

x

x1

12

21 2 1

2

1

2 2

5

1( ) ( )

( )( ) ( )

( )

.

.

( )ν ν

νν ν

ν

ν+ −

+

= −

=( ) +

(74)

where, as before, ν denotes the iteration counter.

The actual solution in this case is, x1 =2.51068, x2 = 3.144089. Applying the above iterative scheme with a first guess, x1

(0) = x2(0) = 1.0, gives the results shown in Table

2 below (where we have removed some of the intermediate iterations).

Table 2: Non-linear scheme applied to solution of equations 73 using the point iterative scheme of equation 74 (spreadsheet, CHAP6Ex5.xls - Sheet 3)



Iteration counter x at iter. ν ν x1 x2 First guess = 0 1 1 1 2.735759 2.44949 2 2.317673 2.920421 3 2.575338 2.987627 4 2.454885 3.104133 .... .... .... 9 2.513337 3.141814 10 2.508958 3.144173 .... .... .... 25 2.510682 3.144089 26 2.510681 3.144089

You can check the results in Table 2 or experiment with other first guesses using Sheet 3 of spreadsheet CHAP6Ex5.xls. We note that the convergence rate in Table 2 is not very fast but, in this case, it does reach a solution. In general, it is usually very difficult to establish for certain whether a given scheme will converge for non-linear equations although there is a large body of mathematics associated with the solution of such systems (which is beyond the scope of this course).

6.2 Newton’s Method for Solving Sets of Non-linear EquationsWe take a step back to the solution of a single non-linear equation such as:

x e x2 0 30. .− = (75)

which has the solution x = 0.829069 (you can verify this by calculating x e x2. − and making sure it is 0.30 - to an accuracy of ~ 4.6x10-8). This equation can be represented as:

f x f x x e x( ) ( ) . .= = −−0 0 302 where (76)

and the solution we require is the value of x for which the function f(x) is zero. Expand f(x) as a Taylor series as follows:

f x f x x f x

xf xo( ) ( ) . ' ( ) . ' ' ( ) ...≈ + + +δ δ

0

2

02 (77)

Thinking in terms of an iteration scheme (x(ν) → x(ν+1)) as a basis for calculating better and better guesses of the solution of f(x) = 0, we can rewrite the Taylor series above as:

f x f x x x f x( ) ( ) ( ). ' ( )( ) ( ) ( ) ( )≈ + −+ν ν ν ν1 (78)

where we have neglected all the second order and higher terms. Since we require the solution for f(x) = 0, then we obtain:

f x x x f x( ) ( ). ' ( )( ) ( ) ( ) ( )ν ν ν ν+ − =+1 0 (79)



which can be rearranged to the following algorithm to estimate our updated guess, x(ν+1) as follows:

x x

f xf x

( ) ( )( )

' ( )

( )( )

ν νν

ν+ = −1

(80)

This equation is the basis of the Newton-Raphson algorithm for obtaining better and better estimates of the solution of the equation, f(x) = 0. Note that we need both a first guess, x(0), and also an expression for the derivative f ' (x(ν)) at iteration ν. In the simple example in equation 76, we can obtain the derivative analytically as follows:

df xdx

f x x e x ex x( ) ' ( ) . .= = −− − 2 2

(81)

Therefore the Newton-Raphson algorithm for solving equation 75 above is as follows:

x x

x e

x e x e

x

x x

( ) ( )( )

( ) ( )-

. .

. .

( )

( ) ( )

ν νν

ν ν

ν

ν ν+

−

− −=

( ) −

− ( )

1

2

2

0 30

2

(82)

The point iterative method of the previous section and the Newton-Raphson method of equation 82 have both been applied to the solution of equation 75 (see Sheet 4 of spreadsheet CHAP6Ex5.xls for details). The results are shown in Table 3 for a first guess of x(0) = 1.

Table 3: Comparison of the point iterative and Newton-Raphson methods for solving non-linear equation 75; (see spreadsheet CHAP6Ex5.xls - Sheet 4 for details)

Iteration Value of x at iteration νNumber

ν Point Newton- Iteration Raphson method method x(ν) x(ν)

0 1 11 0.903042 0.8154852 0.860307 0.8252723 0.84212 0.828064 0.834497 0.8288055 0.831322 0.8296 0.830003 0.8290517 0.829456 0.8290648 0.82923 0.8290689 0.829136 0.82906910 0.829097 0.82906911 0.82908 0.82906912 0.829074 0.82906913 0.829071 0.82906914 0.82907 0.82906915 0.829069 0.82906916 0.829069 0.829069



The correct solution to equation 75 is x = 0.829069. The results in Table 3 show that the Newton-Raphson method converges to this solution (to an accuracy better than 10-6) in 9 iterations, whereas it takes 15 iteration for the point iterative method to converge at the same level accuracy. This performance can vary quite a bit from problem to problem but the Newton-Raphson method is generally better if the derivative term, f '(x(ν)), is not too close to zero. When this derivative gets too small, it can be seen that the second term in equation 80 would start to get very large or “blow up”, as it is sometimes described. We will not go into details about the convergence properties of the Newton-Raphson but it is meant to be “quadratic” meaning that the error should decrease quite rapidly.

We now go on to see how the Newton-Raphson method can be applied to sets of non-linear equations. Returning to the set of two non-linear equations in the previous section, we note that another way of writing this set of equations in a general way is as follows:

F x x

F x x

1 1 2

2 1 2

0

0

,

,

( ) =

( ) = (83)

where, in the case of equation 73 above, these functions would be given by:

F x x x x e

F x x x x x x

x1 1 2 1 2

2 1 2 2 12

1 2

2

2 2

5

1, .

, . .

( ) = + −

( ) = ( ) − ( ) +

−

(84)

The problem is then to find the values of x1 and x2 that make F1 = F2 = 0. In this section, we present Newton’s method for the solution of sets of non-linear equations. Basically, we will simply state the Newton-Raphson algorithm without proof (although it will resemble the simple form of the Newton-Raphson method above) for sets of non-linear equations. We will then illustrate what this means for the example of the set of two non-linear equations above (equation 73).

Definitions:

As before,

x x

F x

x

F x

( ) ( )

( )

( )

ν ν

ν

ν

and

for the "true" solution of the set of non - linear equations

+

( )

( ) =

1

0

= the solution vectors at iteration levels ν and (ν+1)

New terms are:

N = the number of non-linear equations (and hence unknowns, x1, x2, .... xN)

x x

F x

x

F x

( ) ( )

( )

( )

ν ν

ν

ν

and


+

( )

( ) =

1

0

= the vector of function values, F1, F2 .. FN at x(ν)

x x

F x

x

F x

( ) ( )

( )

( )

ν ν

ν

ν

and


+

( )

( ) =

1

0



That is:

F x

F x x x x

F x x x x

F x x x x

F x x x x

N

N

N

N N

( )

, , ,.....,

, , ,.....,

, , ,.....,

.....

.....

, , ,.....,

ν

ν ν ν ν

ν ν ν ν

ν ν ν ν

ν ν ν ν

( ) =

( )( )( )

( )

1 1 2 3

2 1 2 3

3 1 2 3

1 2 3

(85)

J x ( )ν( ) = the NxN Jacobian matrix defined as follows:

J x

F x

x

F x

x

F x

x

F x

x

F x

x

F x

x

F x

x

F x

x

F x

N

N

( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

(

ν

ν ν ν ν

ν ν ν ν

( ) =

∂ ( )∂

∂ ( )∂

∂ ( )∂

∂ ( )∂

∂ ( )∂

∂ ( )∂

∂ ( )∂

∂ ( )∂

∂

1

1

1

2

1

3

1

2

1

2

2

2

3

2

3

......

......

νν ν ν ν

ν ν ν ν

) ( ) ( ) ( )

( ) ( ) ( ) ( )

.....

.....

( )∂

∂ ( )∂

∂ ( )∂

∂ ( )∂

∂ ( )∂

∂ ( )∂

∂ ( )∂

∂ ( )∂

x

F x

x

F x

x

F x

x

F x

x

F x

x

F x

x

F x

x

N

N N N N

N

1

3

2

3

3

3

1 2 3

......

......

(86)

J x ( )ν( )[ ]−1

= the inverse of the Jacobian matrix. (Recall that the inverse of a matrix A is denoted by A-1 and by definition, A-1 .A = I, where I is the identity matrix - all diagonals 1 and all other elements zero - see below). The above matrix is also an NxN matrix with the property:

J x J x I

J x J x I

( ) ( )

( ) ( )

. ,

.

ν ν

ν ν

( )[ ] ( )[ ] =

( )[ ] ( )[ ] = =

−

−

1

1

1

0

0

0

the identity matrix

0 0 0 0 ...... 0

0 1 0 0 0 ...... 0

0 0 1 0 0 ...... 0

0 0 1 0 ...... 0

0 0 0 1 ...... 0

.......

.......

0 0 0 0 ...... 1

That is →

J x J x I

J x J x I

( ) ( )

( ) ( )

. ,

.

ν ν

ν ν

( )[ ] ( )[ ] =

( )[ ] ( )[ ] = =

−

−

1

1

1

0

0

0

the identity matrix

0 0 0 0 ...... 0

0 1 0 0 0 ...... 0

0 0 1 0 0 ...... 0

0 0 1 0 ...... 0

0 0 0 1 ...... 0

.......

.......

0 0 0 0 ...... 1

(87)



Using all of the above definitions, the Newton-Raphson algorithm for a set of non-linear equations, F x( ) = 0 , is given by the following expression:

x(ν+1) = x(ν) − J x (ν)( )[ ]−1

.F x (ν)( ) (88)

We first demonstrate how this is applied to a simple example (equations 73) before going on to show how it is applied to the more complicated non-linear sets of equations which arise in the fully implicit discretisation of the reservoir simulation equations.

Example: Returning to the simple example, where N = 2 (rearranged equations 73):

F x x x x e

F x x x x x x

x1 1 2 1 2

2 1 2 2 12

1 2

2

2 2

5

1, .

, . .

( ) = + −

( ) = ( ) − ( ) +

−

(89)

In the notation developed above, these equations become:

F x

F x x

F x x

x x e

x x x x

x

( ) =( )( )

=+ −

( ) − ( ) +

−1 1 2

2 1 2

1 2

2 12

1 2

2

2 2

5

1,

,

.

. .

(90)

We note that the 2x2 Jacobian matrix is given by:

J x

F x

x

F x

x

F x

x

F x

x

( )

( ) ( )

( ) ( )

ν

ν ν

ν ν

( ) =

∂ ( )∂

∂ ( )∂

∂ ( )∂

∂ ( )∂

1

1

1

2

2

1

2

2

(91)

where each of the elements can be evaluated analytically to obtain:

J x

x e e

x x x x x x

x x

( )

( )

( ) ( ) ( ) ( ) ( ) ( )

. .

. .

( ) ( )

ν

ν

ν ν ν ν ν ν

ν ν

( ) =

−( ) ( )

− ( )( ) ( ) −( )

− −1 2 2

2

2

2 1 2

2

12

1 2

1 1

2 (92)

Hence, the Newton-Raphson method for this simple system becomes:



x x

x e e

x x x x x x

xx x

( ) ( )

( )

( ) ( ) ( ) ( ) ( ) ( )

( ). .

. .

( ) ( )

ν ν

ν

ν ν ν ν ν ν

νν ν

+

− −−

= −

−( ) ( )

− ( )( ) ( ) −( )

1

2

2 1 2

2

12

1 2

1

11 2 2

2

1 1

2

++ −

( ) − ( ) +

−

2 2

5

2

2 12

1 2

2

1x e

x x x x

x( )

( ) ( ) ( ) ( )

.

. .

( )ν

ν ν ν ν

ν

(93)

which we could solve if we knew how to invert the Jacobian matrix to find J x ( )ν( )[ ]−1

This is a detail which we don’t need to know for this course but it can be done. Indeed, for a 2x2 matrix - such as in the above case - it is quite easy to work out the inverse. The inverse of a simple 2 x 2 matrix A given by:

A = a b

c d

is well known to be

A-1 = 1( )

ad bc

d b

c a−−

−

This can easily be proven by multiplying out there matrices and showing that A-1 .A = I. The factor (ad-bc) is known as the determinant of the matrix and it must be non-zero for A-1 to exist. Rather than using equation above to write out the analytical form of the J x( )( )ν[ ] matrix we first simply numerically evaluate the matrix J at a given iteration and apply equation above no get J[ ]−1

. This is done in the spreadsheet Chap6Exxx.xls where results are shown.

J x k( )( )[ ]−1= *** (94)

which allows us to apply the Newton-Raphson method directly to our simple example. For details see Sheet 5 spreadsheet ***.xls. The results are shown in Table 6.xx. (NOTES & SPEADSHEET STILL UNDER CONSTRUCTION)



6.3 Newton’s Method Applied to the Non-linear Equations of Two-Phase FlowWe now return to the discretised equations of two phase flow which we noted at the time, formed a set of non-linear equations. The simplified form of these equations is as follows (see Section 4.1):

Pressure (equation 50):

λ λ λ λT o

n

iin T o

n

i T on

iin T o

n

iin

S

xP

S

x

S

xP

S

xP

+

−−+

+

−

+

+ +

+

+++

( )( )∆

−( )( )

∆+

( )( )∆

+( )( )

∆=

1

1 22 1

1

1

1 22

1

1 22

1

1

1 22 1

1 0/ / / /

Saturation (Form B - equation 54):

S St S

xP P

S

xP Poi

noin o o

n

iin

in o o

n

iin

in+

+

++

+ +

+

− +−

+− − ∆ ( )( )∆

−( )

−( )( )

∆−( )

=1

1

1 22 1

1 1

1

1 22

11

1 0φ

λ λ/ /

Given that the unknowns that we are trying to find are P S iin

in+ + =1 1, 1,2, NX.,

then we can write the above equations in general form as:

F S P

F S P

P in n

S in n

,

,

,

,

+ +

+ +

( ) =

( ) =

1 1

1 1

0

0 (95)

where the two non-linear equations, F S P F S P

S P

P in n

S in n

n n

, ,, ,+ + + +

+ +

( ) ( )1 1 1 1

1 1

and

and

, arise from the pressure (P) and saturation (S) equations, respectively, as given above. The vectors of unknowns,

F S P F S P

S P

P in n

S in n

n n

, ,, ,+ + + +

+ +

( ) ( )1 1 1 1

1 1

and

and , are given by :

S

S

S

S

S

S

S

S

P

P

P

P

Pn

n

n

n

in

in

in

NXn

n

n

n

n

i+

+

+

+

−+

+

++

+

+

+

+

+

−=

=1

11

21

31

11

1

11

1

1

11

21

31

...

...

...

...

...

...

...

and 111

1

11

1

n

in

in

NXn

P

P

P

+

+

++

+

...

...

...

(96)



However, the given F S P F S P

S S S P P P

P in n

S in n

in

in

in

in

in

in

, ,, ,

, , , ,

+ + + +

−+ +

++

−+ +

++

( ) ( )1 1 1 1

11 1

11

11 1

11

and

and

at grid block i only depend on the quantities

F S P F S P

S S S P P P

P in n

S in n

in

in

in

in

in

in

, ,, ,

, , , ,

+ + + +

−+ +

++

−+ +

++

( ) ( )1 1 1 1

11 1

11

11 1

11

and

and and, rather than on all of the other saturations and pressures in the system, since these are the nearest neighbours coupled together in the equations above.

We may also write one total unknows vector Xn+1 using a combination of the saturation and pressure vectors as follows:

X

S

P

S

P

S

P

S

P

n

n

n

n

n

in

in

NXn

NXn

+

+

+

+

+

+

+

+

+

=

1

11

11

21

21

1

1

1

1

and the equation to be solved is then F(Xn+1)=0

The saturations and pressures are coupled together to their nearest neighbours through the discretisation equations (50 and 54 above). The general form of the solution using the Newton iteration is then to take a starting guess (iteration, ν=0) Xn+1(0) and then apply the formulation as above to obtain:

X X J X F Xn n+ + + −= + [ ]1 1 1 1 ( ) ( ) ( ) ( )( ) . ( )ν ν ν ν

It is rather more complex to constuct the Jacobian matrix J X ( ) of derivatives but it can be done. We also need to invert this matrice to obtain J[ ]−1

in order to apply the above algorithm. However, in practice, there are various methods that try to construct the J[ ]−1

matrix more directly, often in an approximate manner. Note that the Jacobian matrix is very sparse since there are just nearest neighbour interaction (as was the case for the matrices associated with single phase flow discussed earlier in this Chapter). A consequence of this is that the inverse matrix is also quite sparse and there are many numerical techniques available to solve such problems.

In this course, we will not give any more detail on the solution of the non-linear equations which arise in reservoir simulation.



7 NUMERICAL DISPERSION - A MATHEMATICAL APPROACH

7.1 Introduction to the ProblemIn Chapter 4, we discussed the physical idea of numerical dispersion, which is also sometimes referred to as numerical diffusion. Numerical dispersion is essentially the artificial spreading or “diffusion” of a front - e.g. a waterfront in a water/oil displacement - due to the coarse grid used in the simulation. It is a numerical effect and it can be reduced by taking a finer grid. We note that in real reservoirs there are some real dispersive physical mechanisms which result in the spreading of fronts. These arise due to the effects of capillary pressure and also from the interaction of the fluid flow with small scale (cm - m) permeability heterogeneity of the reservoir rock. Indeed, there are also some quite complex interactions between the capillary forces and the small scale heterogeneity that also lead to types of physical spreading in the reservoir. In an ideal calculation, we would take a sufficiently fine grid that the level of physical (i.e. real) spreading or diffusion was correctly represented by our grid; i.e. the numerical diffusion would be less that the physical diffusion. This is almost never possible in a field scale simulation, although it can be achieved in modelling laboratory scale experiments on flows through small rock samples or bead packs. For such a fine scale simulation, the level of numerical diffusion can realistically be made much less than that from physical sources.

In this section, we return to the issue of numerical dispersion - a term which we will now use interchangeably with “numerical diffusion”. Indeed, “diffusion” is often referred to as quite a specific physical effect which is described by certain well-known equations which are, not surprisingly, known as diffusion equations. For example, the simplified pressure equation derived in Chapter 5, equation 27, is an example of a diffusion equation. This equation has the form:

∂∂

= ∂

∂

Pt

DP

xh

2

2

(97)

where Dh is the hydraulic diffusivity (Dh = k/( cf .φ.μ)) and this is a standard form of the classical diffusion equation. We now consider how the effects of a grid can lead to “diffusion-like” terms when we try to solve the flow equations numerically.

In the above equations, it is specifically the ∂∂

2

2

Px

term that is the “dispersive” or “diffusive” part.

7.2 Mathematical Derivation of Numerical DispersionIn order to show mathematically how a “diffusive” term arises when we solve certain transport equations numerically, we use a simple form of the two-phase saturation equation. In 1D, the transport equation for a simple waterfront (with Pc = 0 and zero gravity) is the well-known fractional flow equation (see Chapter 2) as follows:

φ ∂

∂

= − ∂

∂

St

fx

w wv (98)

where Sw is the water saturation and f S

f Sq

q q

f S

w w

w ww

w o

w w

( )

( ) =+

( )

is the fractional flow of water



(

f S

f Sq

q q

f S

w w

w ww

w o

w w

( )

( ) =+

( )

) which is a function only of water saturation, Sw. We can

differentiate f S

f Sq

q q

f S

w w

w ww

w o

w w

( )

( ) =+

( )

by the chain rule to obtain:

φ ∂

∂

= − ∂

∂

= − ∂

∂

∂∂

St

fx

fS

Sx

w w w

w

wv v (99)

where the term − ∂

∂

vfS

w

w is a non-linear term giving the water velocity, vw Sw( ) , which is also a function of water saturation: that is:

v vw S

fSw

w

w

( ) = − ∂∂

(100)

To simplify the problem even further for our purposes here, we take a straight line

fractional flow, f S

fS

w w

w

w

( )

∂∂

, which means that

f S

fS

w w

w

w

( )

∂∂

is a constant and, hence, so is

the water velocity, Vw. Hence, the simple 1D transport equation for a convected waterfront is:

∂∂

= − ∂

∂

St

Sx

w wvw

(101)

where, as noted above, vw is now constant. This equation describes the physical situation illustrated schematically in Figure 14.

Water Vw Oil

1

0

Sw

x1 x2

t1 t2

x

Governing equation

Vw = constant

Vw =x2 - x1

t2 - t1

∂Sw

∂t

∂Sw

∂x= - Vw

Starting from the transport equation 101 above, we can easily apply finite differences using our familiar notation (Chapter 5) to obtain:

φ S S

tS S

xError termswi

nwin

wwi wi

+−−

∆

= − −∆

+1

1v (102)

where we have used the backward difference (sometimes referred to as the upstream

difference) to discretise the spatial term, ∂∂

Sxw

. Note that we have not yet specified the time level of the spatial terms. If we take these at the n time level (known), then it

Figure 14The advance of a sharp saturation front governed by the transport equation with a constant velocity vw.



would be an explicit scheme and if we take them at the n+1 time level (unknown), it would be an implicit scheme. It does not matter for our current purposes here, since we are principally interested in the “Error terms” which are indicated in equation 102. To determine what these terms look like, we need to go back to the original finite differences based on Taylor expansion of the underlying function, Sw(x,t), in this case.

We can expand Sw (x,t) either in space (x) or in time (t) as follows:

SPACE

S x t S x xSx

x Sxw w

w w, .( ) = ( ) + ∂∂

+ ∂

∂

+0

2 2

22δ δ

higher order terms(t fixed) (103)

TIME S x t S t tSt

t Stw w

w w, .( ) = ( ) + ∂∂

+ ∂

∂

+0

2 2

22δ δ

higher order terms(x fixed) (104)

We can rearrange each of the above equations 103 and 104 to obtain the finite

difference approximations for the ∂∂

∂∂

Sx

St

and terms with the leading error terms as follows (now ignoring the higher order terms):

SPACE ∂∂

≈

( ) − ( )

− ∂∂

Sx

S x t S x

xx S

x

, 02

22δδ (105)

TIME ∂∂

≈

( ) − ( )

− ∂∂

St

S x t S t

tt S

t

, 02

22δδ

(106)

To return to our usual notation, we make the identities:

S x t S

S x S

S t S

x x

t t

in

i

in

( , )

( )

( )

⇔

⇔

⇔⇔ ∆⇔ ∆

+

−

1

0 1

0

δδ (107)

Equation 105 now becomes:

∂∂

≈ −

∆

− ∆ ∂∂

−Sx

S Sx

x Sx

i i 12

22 (108)

and equation 106 becomes:

∂∂

≈ −

∆

− ∆ ∂∂

+St

S St

t St

in

in1 2

22 (109)



We now substitute the above finite difference approximations into the governing equation 102 with their leading error terms as follows:

φ S S

tt S

tS S

xx S

xin

in

wi i

+−−

∆

− ∆ ∂∂

≈ − −∆

+ ∆ ∂∂

1 2

21

2

22 2v

vw.

(110)

Collecting the error terms together on the RHS gives:

φ S St

S Sx

x Sx

t St

in

in

wi i

+−−

∆

≈ − −∆

+ ∆ ∂∂

+ ∆ ∂∂

11

2

2

2

22 2v

v

Error term

w.

(111)

The error term in the finite difference scheme is now clear and is shown in equation 111. However, it still needs some simplifying since it is a strange mixed term with

both ∂∂

∂∂

2

2

2

2

Sx

St

and terms in it. We now want to eliminate the

∂∂

2

2

St term

and this is done by using the original governing equation.

To obtain ∂∂

2

2

St

, first note from the original equation 101 that ∂∂

= − ∂

∂

St

Sx

ww

wv

and we can then differentiate ∂∂

Stw with respect to t as follows:

∂∂

∂∂

= ∂

∂

= − ∂∂

∂∂

t

St

St t

Sx

w ww

w2

2 v (112)

Thus, we can rearrange the RHS of equation 112 as follows:

− ∂

∂∂∂

= − ∂

∂∂∂

v vw

ww

w

tSx x

St (113)

We now return again to the governing equation (equation 101) and use it for ∂∂

Stw

in equation 113 to obtain:

− ∂

∂∂∂

= − ∂

∂− ∂

∂

= ∂

∂

v v v vw2

ww

w ww w

xSt x

Sx

Sx

2

2

(114)

and hence:

∂∂

= ∂∂

2

2

2

2

St

Sx

w wvw2

(115)

We now substitute this expression into the error term in equation 111 above to obtain the following:



v v v

v v

w w

w

. .

.

∆ ∂∂

+ ∆ ∂∂

= ∆ ∂∂

+ ∆ ∂∂

= ∆ + ∆

∂∂

x Sx

t St

x Sx

t Sx

x t Sx

w

w

2 2 2 2

2 2

2

2

2

2

2

2

2 2

2

2 2

2 (116)

The finite difference equation with its error term in equation 111 now becomes:

S St

S Sx

x t Sx

in

in

wi i

+−−

∆

≈ − −∆

+ ∆ + ∆

∂∂

11

2 2

22 2v

v vw w.

(117)

In this equation, we now see that the form of the error term is exactly like a “diffusive” term i.e. it multiplies a (∂2Sw/∂x2) term. Hence we identify the level of numerical dispersion or diffusion, Dnum, arising from our simple finite difference scheme as:

D

x t x tnum = ∆ + ∆

= ∆ + ∆

v vv

vw ww

w..

2 2 2 2

2

(118)

If Dx t

t x

num = ∆ + ∆

∆ << ∆

vv

v

ww

w

.2 2

, we can take such a small time step that

Dx t

t x

num = ∆ + ∆

∆ << ∆

vv

v

ww

w

.2 2

For this case:

D

xnum ≈ ∆vw.

2 (119)

We can now solve two problems as follows

(i) Firstly, we may apply an explicit finite difference method to obtain an accurate solution of the following convection-dispersion equation.

∂∂

= − ∂

∂

+ ∂

∂

St

Sx

DSx

w w wvw .2

2

(120)

That is, where the numerical dispersion is much less than the physical dispersion due to the explicit D term. If this solution is converged (i.e. does not change on refining Δx and Δt) then we can plot this at a suitable time, t1, as shown in Figure 15.

1

0x

Sw

Governing equation:

Vw = water velocity

Sw

t

Sw

x

2Sw

x2= - Vw +Dt = t1

Figure 15.Dispersed frontal displace-ment with a known and con-verged level of dispersion.



(ii) Now take only the explicit finite difference approximation to the convection equation only, that is:

φ

φ

S St

S Sx

S St

xS S

win

win

wwin

win

win

win w

win

win

+−

+−

−∆

= − −∆

= − ∆∆

−( )

11

11

v

v .

.

which gives ⇒

φ

φ

S St

S Sx

S St

xS S

win

win

wwin

win

win

win w

win

win

+−

+−

−∆

= − −∆

= − ∆∆

−( )

11

11

v

v .

.

Solve this with a relatively coarser grid, Δx, but with a fine time step thus predicting

Dnum to be vw.∆

x2 . Choose this level of dispersion to deliberately match that in

part (i) above i.e. choose Δx such that Dnum = D. Plot out a saturation profile like that in Figure 15 at the same time t1 and compare these.

8 CLOSING REMARKS

In this Chapter, we have introduced the student to finite difference approximations of the partial differential equations (PDEs) that describe both single- and two-phase flow through porous media. These discretised equations have led to systems of either linear or non-linear equations which are then solved numerically. Methods for solving these equations have been discussed in principle and some idea has been given of how these are applied in practical reservoir simulation. At the end of the unit, some discussion was presented on grid-to-grid flows and how these inter-gridblock properties are averaged. A more mathematical of numerical dispersion was also given.



APPENDIX A: Some useful Matrix Theorems

This lists (without proof) some useful theorems from matrix algebra which underpin much of the practical application work which we have described in this section.

The central problem which we are interested in is:

A x b

A

. = where

A x b

A

. =

is an NxN square matrix

1. If A x b

A

. = actually has a solution, then

A x b

A

. =

is said to be non-singular or invertible and the following five conditions apply and are equivalent (i.e. if one of them hold, they all hold):

(i) A A A I I− − =1 1 exists where . ; is the identity matrix which has 1s on the diagonal and zeros elsewhere. I

I A A I A. .= =

is like the number “1” in that

I

I A A I A. .= = .

(ii) det A( ) ≠ 0; the determinant of the matrix is non-zero. This is a number found by “multiplying out” the matrix in a specific way.

(iii) There is no non-zero vector x such that, A x

A

. = 0 . In other words, if A x

A

. = 0, then the vector x must be zero - have 0s for all its elements.

(iv) The rows of

A x

A

. = 0

are linearly independent. And ..

(v) the columns of

A x

A

. = 0

are linearly independent. Where (iv) and (v) say that we cannot get one of the rows or columns by making some combination of the existing ones.

2. A matrix

A x

A

. = 0

which is square and symmetric ( A A where A A i e a aT T

ij ji= =, ; . . is the transpose of ) is said to be positive definite if: x A xT. . > 0

3. If the matrix

A x

A

. = 0

is positive definite then it can be decomposed in exactly one way into a product: A G G

G

T= . such that matrix

A G G

G

T= .

is lower triangular and has positive entries on the main diagonal. This is known as Cholesky decomposition.



SOLUTIONS TO EXERCISES

EXERCISE 1.

Apply finite differences to the solution of the equation:

dydt

y

= +2 42.

where, at t = 0, y(t = 0) = 1. Take time steps of Δt = 0.001 (arbitrary time units) and step the solution forward to t = 0.25. Use the notation yn+1 for the (unknown) y at n+1 time level and yn for the (known) y value at the current, n, time level.

Plot the numerically calculated y as a function of t between t = 0 and t = 0.25 and plot it against the analytical value (do the integral to find this).

Answer: is given below where the working is shown in spreadsheet CHAP6Ex1.xls. This gives the finite difference formula, a spreadsheet implementing it and the analytical solution for comparison.

SOLUTION 1.

Discretise the equation using the suggested notation as follows:

y yt

y

y y t y

duu

u

dyy

ydt t C

yt C

y t

n nn

n n n

+

+

−

−

−

−∆

= +

= + ∆ +( )

+=

+=

= = +

=

= +

( )

∫

∫ ∫

12

1 2

2 21

2 21

1

2 4

2 4

1

12 2

12 2 2

2

12 2 2

.( )

.( )

tan

( )tan

tan

α α α

α

== +( )[ ]

=

=−

2 2 2

12 2

12

0 21760491

tan

tan .

t C

C

which is rearranged to the explicit formula:

y yt

y

y y t y

duu

u

dyy

ydt t C

yt C

y t

n nn

n n n

+

+

−

−

−

−∆

= +

= + ∆ +( )

+=

+=

= = +

=

= +

( )

∫

∫ ∫

12

1 2

2 21

2 21

1

2 4

2 4

1

12 2

12 2 2

2

12 2 2

.( )

.( )

tan

( )tan

tan

α α α

α

== +( )[ ]

=

=−

2 2 2

12 2

12

0 21760491

tan

tan .

t C

C

which can be set up quite easily on a spreadsheet.

To find the analytical answer, you might need the standard form of the following integral:

y yt

y

y y t y

duu

u

dyy

ydt t C

yt C

y t

n nn

n n n

+

+

−

−

−

−∆

= +

= + ∆ +( )

+=

+=

= = +

=

= +

( )

∫

∫ ∫

12

1 2

2 21

2 21

1

2 4

2 4

1

12 2

12 2 2

2

12 2 2

.( )

.( )

tan

( )tan

tan

α α α

α

== +( )[ ]

=

=−

2 2 2

12 2

12

0 21760491

tan

tan .

t C

C

Using this standard form gives:

y yt

y

y y t y

duu

u

dyy

ydt t C

yt C

y t

n nn

n n n

+

+

−

−

−

−∆

= +

= + ∆ +( )

+=

+=

= = +

=

= +

( )

∫

∫ ∫

12

1 2

2 21

2 21

1

2 4

2 4

1

12 2

12 2 2

2

12 2 2

.( )

.( )

tan

( )tan

tan

α α α

α

== +( )[ ]

=

=−

2 2 2

12 2

12

0 21760491

tan

tan .

t C

C

where C is the constant of integration and we identify

y yt

y

y y t y

duu

u

dyy

ydt t C

yt C

y t

n nn

n n n

+

+

−

−

−

−∆

= +

= + ∆ +( )

+=

+=

= = +

=

= +

( )

∫

∫ ∫

12

1 2

2 21

2 21

1

2 4

2 4

1

12 2

12 2 2

2

12 2 2

.( )

.( )

tan

( )tan

tan

α α α

α

== +( )[ ]

=

=−

2 2 2

12 2

12

0 21760491

tan

tan .

t C

C

above. Therefore, this becomes:



y yt

y

y y t y

duu

u

dyy

ydt t C

yt C

y t

n nn

n n n

+

+

−

−

−

−∆

= +

= + ∆ +( )

+=

+=

= = +

=

= +

( )

∫

∫ ∫

12

1 2

2 21

2 21

1

2 4

2 4

1

12 2

12 2 2

2

12 2 2

.( )

.( )

tan

( )tan

tan

α α α

α

== +( )[ ]

=

=−

2 2 2

12 2

12

0 21760491

tan

tan .

t C

C

which easily rearranges to:

y yt

y

y y t y

duu

u

dyy

ydt t C

yt C

y t

n nn

n n n

+

+

−

−

−

−∆

= +

= + ∆ +( )

+=

+=

= = +

=

= +

( )

∫

∫ ∫

12

1 2

2 21

2 21

1

2 4

2 4

1

12 2

12 2 2

2

12 2 2

.( )

.( )

tan

( )tan

tan

α α α

α

== +( )[ ]

=

=−

2 2 2

12 2

12

0 21760491

tan

tan .

t C

CWe now find C from the initial conditions since y(t = 0) = 1; it is easy to see that:

y yt

y

y y t y

duu

u

dyy

ydt t C

yt C

y t

n nn

n n n

+

+

−

−

−

−∆

= +

= + ∆ +( )

+=

+=

= = +

=

= +

( )

∫

∫ ∫

12

1 2

2 21

2 21

1

2 4

2 4

1

12 2

12 2 2

2

12 2 2

.( )

.( )

tan

( )tan

tan

α α α

α

== +( )[ ]

=

=−

2 2 2

12 2

12

0 21760491

tan

tan .

t C

C

which is then used in the analytic solution for y(t) above.

This is implemented in the spreadsheet CHAP6Ex1.xls. Note that at t > 0.3, both the analytical and numerical solutions go a bit strange (very large and they start to disagree) since the tan function has a singularity y(t) → ∞.

EXERCISE 2.

Fill in the above table using the algorithm:

P P

tx

P P Pin

in

in

in

in+

+ −= + ∆∆

+ −( )12 1 1 2

Hint: make up a spread sheet as above and set the first unknown block (shown grey shaded in table above) with the above formula. Copy this and paste it into all of the cells in the entire unknown area (surrounded by red border above).

SOLUTION 2.

If you get stuck, look at spreadsheet CHAP6Ex2.xls on the disk.

EXERCISE 3.

Experiment with the spreadsheet in CHAP6Ex2.xls to examine the effects of the three quantities above - Δt, Δx (or NX) and the solution as t → ∞.



EXERCISE 4.

Solve the following simple example of an upper triangular matrix:

4 2 5

2 2

3 4 1 30

2 1 3

3 15

1 2 3 4 5

2 3 4 5

3 4 5

4 5

5

1 2 3

x x x x x

x x x x

x x x

x x

x

Answer

x x x

- 1 1 1

1 2 12

= ..........; = ..........; = ..........;

− + + =− + + =

+ + =− =

=

= ..........; x x4 5 = ..........

SOLUTION 4.

4 2 5

2 2

3 4 1 30

2 1 3

3 15

1 2 3 4 5

2 3 4 5

3 4 5

4 5

5

1 2 3

x x x x x

x x x x

x x x

x x

x

Answer

x x x

- 1 1 1

1 2 12

= ..........; = ..........; = ..........;

− + + =− + + =

+ + =− =

=

= ..........; x x4 5 = ..........

EXERCISE 5.

fill in the table below for 10 iterations using a calculator or a spreadsheet: POINT ITERATIVE SOLUTION OF LINEAR EQUATIONS




SOLUTION 5.

Answer - Exercise 5: see spreadsheet CHAP6Ex5.xls - Sheet 1

POINT ITERATIVE SOLUTION OF LINEAR EQUATIONS






EXERCISE 6.

Which is best method (i.e. that requiring the lowest amount of computational work), BAND or LSOR, for the following problems?

(i) NX = 5, NY = 3, Niter = 50 (a small problem)

(ii) NX = 20, NY = 5, Niter = 50

(iii) NX = 100, NY = 20, Niter = 70

(iv) NX = 400, NY = 100, Niter = 150

NX NY Niter BAND LSOR, Comment WB WLSOR5 3 5020 5 50100 20 70400 100 150

SOLUTION 6.

NX NY Niter BAND LSOR, Comment WB WLSOR5 3 50 135 750 For a very small problem like this, a direct method is usually better, although both would take very little time even on a PC.20 5 50 2500 5000 Again a direct method is somewhat better for a small problem.100 20 70 8x105 1.4x105 As the problem grows in size, iterative methods start to overtake direct method.400 100 150 4x108 6x106 For very large problems, iterative methods virtually always win over direct methods by more than an order of magnitude.



EXERCISE 7.

The linear equations which arise in reservoir simulation may be solved by a direct solution method or an iterative solution method. Fill in the table below:

Direct solution method Iterative solution method

Give a verybrief descriptionof each method

Main advantages 1. 1.

2. 2.

Main 1. 1.disadvantages

2. 2.





77Permeability Upscaling

CONTENTS

1 SINGLE-PHASE FLOW 1.1 INTRODUCTION 1.2 Upscaling Porosity and Water Saturation 1.3 Averaging Permeability 1.3.1 Flow Parallel to Uniform Layers 1.3.2 Flow Across Uniform Layers 1.3.3 Flow through Correlated Random Fields 1.3.4 Additional Averaging Methods 1.3.5 Summary of Permeability Averaging 1.4 Numerical Methods 1.4.1 Recap on Flow Simulation 1.4.2 Boundary Conditions 1.5 Upscaling Errors 1.5.1 Correlated Random Fields 1.5.2 Evaluating the Accuracy of Upscaling 1.5.3 Upscaling of a Sand/Shale Model 1.6 Summary of Single-Phase Upscaling

2 TWO-PHASE FLOW 2.1 Introduction 2.2 Applying Single-Phase Upscaling to a Two-Phase Problem 2.3 Improving Single-Phase Upscaling 2.3.1 Non-Uniform Upscaling 2.3.2 Well Drive Upscaling 2.4 Introduction to Two-Phase Upscaling 2.5 Steady-State Methods 2.5.1 Capillary-Equilibrium 2.6 Dynamic Methods 2.6.1 Introduction 2.6.2 The Kyte and Berry Method 2.6.3 Discussion on Numerical Dispersion 2.6.4 Disadvantages of the Kyte and Berry Method 2.6.5 Alternative Methods 2.6.6 Example of the PVW Method 2.7 Summary of Two-Phase Flow

3 ADDITIONAL TOPICS 3.1 Upscaling at Wells 3.2 Permeability Tensors 3.2.1 Flow Through Tilted Layers 3.2.2 Simulation with Full Permeability Tensors 3.3 Small-Scale Heterogeneity 3.3.1 The Geopseudo Method 3.3.2 Capillary-Dominated Flow

3.3.3 Geopseudo Example 3.3.4 When to use the Geopseudo Method 3.4 Uncertainty and Upscaling 3.5 Upscaling Summary

4 REFERENCES

2


Learning Objectives

After reading through this Chapter, the Student should be able to do the following:

• Appreciate why upscaling is necessary.

• Know how to calculate effective permeability in simple models by averaging.

• Understand how to perform numerical upscaling of single-phase fl ow.

• Be aware of the effects of heterogeneity on two-phase fl ow.

• Realise the limitations of applying single-phase upscaling to a two-phase problem.

• Know how to carry out steady-state, capillary-equilibrium upscaling for two- phase fl ow.

• Become familiar with two-phase dynamic upscaling (the Kyte and Berry Method), and understand the advantages and disadvantages of applying dynamic upscaling.

• Understand how to upscale around a well.

• Appreciate that permeability is a full tensor property.

• Know how to upscale from the core-scale to the scale of a geological model, taking account of fi ne-scale structure and capillary effects.

Institute of Petroleum Engineering, Heriot-Watt University 3


1 SINGLE-PHASE FLOW

1.1 IntroductionReservoir modelling often involves generating multi-million cell models, which are too large for carrying out fl ow simulations using conventional techniques. The number of cells must therefore be reduced by “upscaling” (Figure 1). Some quanti-ties, such as porosity and water saturation, are easy to upscale, because they may be averaged arithmetically. However, other quantities – notably permeability – are much more diffi cult to upscale.

Geological model

Full-field model

We usually refer to the upscaled permeability as the effective permeability. The effective permeability is defi ned as the permeability of a single homogeneous cell which gives rise to the same fl ow as the fi ne-scale distribution when the same pressure gradient is applied. We assume that Darcy’s Law holds at the coarse scale:

Q Ak Px

eff= − ∆∆µ (1)

where Q = total fl ow, A = area, keff

= effective permeability, eff

= effective permeability, eff

μ = viscosity, and ΔP/Δx is the pressure gradient.

True effective permeability is an intrinsic property of the model and ought to be independent of the applied boundary conditions (Section 1.4). However, in practice, the effective permeability often does depend on the boundary conditions, and on the method used for calculation. Upscaling must always be carried out with care in order to obtain “sensible” results.

In Figure 1, the geological model on the left is a fi ne-scale model with 20 million cells, and the coarse-scale model on the right consists of about 300,000 cells. Each of the coarse-scale cells contains an effective permeability. An example of fi ne-scale and coarse-scale grids is shown in the 2D model in Figure 2. An effective permeability is calculated for each coarse-scale cell, either by averaging the fi ne-grid values, or by performing a numerical simulation.

Figure 1Upscaling Example

4


fine grid coarse grid

one coarse cell

fine grid coarse grid

one coarse cell

In this section of the upscaling course, we assume that there is only one phase present – water or oil, and that we have steady-state linear fl ow. We show how simple aver-aging may sometimes be used to estimate upscaled parameters, and then move on to methods which involve numerical simulation. This is followed by a set of examples which demonstrate how errors may arise, and how to avoid them.

1.2 Upscaling Porosity and Water SaturationWe start by averaging porosity and water saturation, using a simple model (Figure 3). (Note that the water saturation is not required for a single-phase problem. However, we include it here because it is simple to upscale.) There are 10 grid blocks of size 1 m3, 4 of which have a porosity of 0.15, and 6 of which have a porosity of 0.20.

φ = 0.15 φ = 0.20

Sw = 0.40Sw = 0.50

The average porosity, φ , is given by:

φ = total pore volumetotal volume (2)

Therefore, in this case:

φ = × + × =4 0 15 6 0 2010

0 18. . . .

When averaging the water saturation, we need to take the porosity into account. In the previous example, suppose the water saturation was 0.5 in the blocks with porosity of 0.15, and 0.4 in the blocks with porosity 0.2, then the average water saturation is:

S total amount of watertotal pore volumew =

(3)

Here, the average water saturation is:

Figure 2The upscaling procedure

Figure 3Example for averaging porosity and water saturation



Sw = × × + × ×× + ×

= + = =

4 0 15 0 5 6 0 20 0 44 0 15 6 0 20

0 3 0 481 8

0 781 8

0 433

. . . .. .

. ..

..

. .

1.3 Averaging PermeabilityIn some simple models, such as parallel layers or a random distribution, the effective permeability may be calculated by averaging.

1.3.1 Flow Parallel to Uniform Layers

∆x

P1 P2

ki, tiQi

Consider a set of (infi nite) parallel layers of thickness, ti and permeability ki, where i = 1, 2, .. n (the number of layers). The effective permeability of these layers is given by the arithmetic average, ka.

k kt k

teff a

i ii

n

ii

n= = =

=

∑

∑1

1 (4) (Equation (4) may be proved by applying a fi xed pressure gradient along the layers.)

Example 1x

z

t1 = 3 mm, k1 = 10 mD

t2 = 5 mm, k 5 mm, k 5 m 2 = 100 mD

1

2

Suppose we have two layers as shown in Figure 5. The effective permeability for fl ow in the x-direction is given by Equation (4), and is:

k mk mDak mak mk m=k m× +k m× +k m×k m×k mk m= =k mk m= =k m3 1k m3 1k m× +3 1× +k m× +k m3 1k m× +k m0 5k m0 5k m× +0 5× +k m× +k m0 5k m× +k m100k m100k m3 5+3 5+

530k m530k m8

66k m66k m25k m25k m.k m.k m

Figure 4Along-layer fl ow

Figure 5A simple, two-layer model

6


1.3.2 Flow Across Uniform Layers

∆x

Q

ki, ti∆Pi

For fl ow perpendicular to the layers, the effective permeability is given by the har-monic average, kh:

k kt

tk

eff h

ii

n

i

ii

n= = =

=

∑

∑1

1

.

(5)

(Equation (5) may be proved by assuming a constant fl ow rate through each layer.)

Example 2Equation (5) may be used to calculate the effective permeability for fl ow across the two layers in the model shown in Figure 5, i.e. fl ow in the z-direction.

k mDh = ++

= =3 53 10 5 100

80 35

22 86.

.

From Examples 1 and 2, we see that the permeability is different in different direc-tions. In reservoirs with approximately horizontal layers, the arithmetic average may be used for calculating the effective permeability in the horizontal direction, and the harmonic average may be used for calculating the effective permeability in the vertical direction.

1.3.3 Flow through Correlated Random FieldsFigure 7 shows an example of a correlated random permeability distribution. Corre-lated random fi elds are described in Section 1.5.1. Basically, “correlated” means that areas of high or low permeability tend to be clustered, so that the spatial distribution is smoother than a totally random one. The “correlation length” is approximately the size of patches of high or low permeability. The longer the correlation length, the longer will be the range of the semi-variogram for the permeability distribution.

Assuming that we are averaging over many correlation lengths, permeability should be isotropic (same in the x-, y- and z-directions). The effective permeability for a random permeability distribution is proportional to the geometric average, which is given by:

Figure 6Across-layer fl ow



kk

ng

ii

n

=( )

=∑

expln

1

(6)

where i = 1, 2, .. n is the number of cells in the distribution.

Correlation Length

The results given below have been derived theoretically for log-normal distributions, with a standard deviation of σ

Y, where Y = ln(k). The results depend on the number

Y, where Y = ln(k). The results depend on the number

Y

of dimensions:

k k in D

k k in D

k k in D

eff g Y

eff g

eff g Y

= −( )=

= +( )

1 2 1

2

1 6 3

2

2

σ

σ } (7)

These formulae are approximate, and assume σY is small (< 0.5). (You are not required

Y is small (< 0.5). (You are not required

Y

to know the proof.) The 1D result is an approximation of the harmonic average. Note that the results do not depend on the correlation length of the fi eld, provided it is much smaller than the system size.

Also note that ka > k

g > k

h > k

h > k , and the effective permeability always lies between the

two extremes: ka and k

h and k

h and k .

Example 3

75 cells of 10 mD 125 cells of 100mD

Figure 7A correlated, random permeability distribution (white = high permeability, dark = low permeability)

Figure 8A random arrangement of the permeabilties in the simple example

8


Suppose that the permeability values in the simple example are jumbled up, so that there are 75 small cells of 10 mD, and 125 small cells of 100 mD. See Figure 8. The effective permeability of this model is:

k k

mD

eff g= =

=

= =

× + ×

+

10

10

10 42 17

75 10 125 100200

75 250200

1 625

log( ) log( )

. . .

1.3.4 Additional Averaging MethodsSince averaging is very quick (compared with numerical simulation), many engi-neers use this technique in more complex models. Sometimes engineers increase the accuracy by using power averaging. The power average is defi ned as:

kk

np

ii

n

=

=∑ α

α

1

1/

,

(8)

where α is the power. The value of the power depends on the type of model, and must be calibrated against numerical simulation (Section 1.4).

Also, sometimes, engineers use a combination of the arithmetic and harmonic aver-ages, e.g. they take the arithmetic average of the permeabilities in each column and then calculate the harmonic average of the columns.

1.3.5 Summary of Permeability AveragingTo summarise, there are two types of simple model in which we can calculate the effective permeability by averaging:

• Parallel layers• Correlated random fi elds

Since averaging is very quick, it is frequently used as an approximation for the effec-tive permeability in more complex models.

1.4 Numerical MethodsIn general, the permeability distribution will not be simple enough for us to be able to calculate the effective permeability analytically (i.e. by averaging), and we will have to perform a numerical simulation. We can use a fi nite difference method to calculate the pressures.

1.4.1 Recap on Flow SimulationThe continuity equation tells us that there is no net accumulation or loss of fl uid within a grid block:



q q q qxin zin xout zout+ = + . (9)

(We are assuming incompressible rock and fl uids, here.)

qxin qxout

qzin

qzout

i,j i+1,ji-1,j

i,j+1

i,j-1

x

z

Darcy’s law is used to express the fl ows in terms of the pressures and permeabilities. For example, if the grid blocks in Figure 9 are of length Δx and height Δz (and unit width in the y-direction), then:

qk z P P

xxinx i j i j i j= −

−( )− −, / , , ,1 2 1∆∆µ (10)

where kx,i-1/2,j

is the harmonic average of the permeabilities in the x-direction in blocks (i-1,j) and (i,j). (You now should know now why the harmonic average is used here.) The other fl ows are calculated in a similar manner.

It is useful to use the transmissibilities, Tx = kxΔz/Δx and T

z = kzΔx/Δz. (Assume

the width, Δy = 1.) We can therefore derive the pressure equation:

T T T T P

T P T P

T P T P

x i j x i j z i j z i j i j

x i j i j x i j i j

z i j i j z i j i j

, / , , / , , , / , , / ,

, / , , , / , ,

, , / , , , / ,

− + − +

− − + +

− − + +

+ + +( )− −

− −

1 2 1 2 1 2 1 2

1 2 1 1 2 1

1 2 1 1 2 1

== 0 (11) An equation is set up for each P

ij, i = 1, 2, .. nx and j = 1, 2, .. nz. The transmissibilities

are known, and using the appropriate boundary conditions, we can solve this set of linear equations to obtain the pressure in each grid block. The effective permeability is than calculated from the total fl ow and the total pressure drop, as described below.

Note that the boundary conditions are applied to each coarse grid cell in turn, and they may not be a good approximation to the pressures which would arise in a fi ne-grid simluation. This leads to errors in the results. Upscaling errors are discussed in Section 1.5.

1.4.2 Boundary ConditionsBoundary conditions are required to specify what happens at the edges of the model.

Figure 9Recap on numerical fl ow simulation

10


a) No-Flow Boundaries

no flow through the sides

no flow through the sides

P1 P2

The pressure is fi xed on two sides of the model, and no fl ow is allowed through the others sides of the model. This type of boundary condition is suitable for models where there is little cross-fl ow: for example, models with approximately horizontal layers, or a random distribution. These are the most commonly applied boundary conditions. Figure 11 illustrates how an effective permeability may be calculated in the x-direction.

on left facePressure= P2Pressure= P1on right face

L

Area, AFlow Rate, Q

x

y

z

1. Solve the steady-state equation to give the pressures, Pij, in each grid block.2. Calculate the inter-block fl ows in the x-direction using Darcy’s Law. (See

Equation 10.)3. Calculate the total fl ow, Q, by adding the individual fl ows between two y-z

planes. (Any two planes will do, because the total fl ow is constant.)4. Calculate the effective permeability for fl ow in the x-direction, using the

equation:

Qk A

Leffk Aeffk Axk Axk A

= , ( )P P( )P P( )1 2( )P P( )P P1 2P P( )P PP P−P P( )P P−P P1 2P P−P P( )P P−P Pµ (12)

Repeat the calculation for fl ow in the y- and z-directions, to obtain keff,y

and keff,z

.

(b) Periodic Boundary ConditionsPeriodic boundary conditions are useful for calculating the effective permeabilities in models where there are infi nitely repeated geological structures in each direction. (See Section 3.2.) The use of periodic boundary conditions ensures that we also

Figure 10Fixed pressure, or no-fl ow, boundary conditions

Figure 11The calculation of effective permeability using no-fl ow boundary conditions



have an infi nitely repeated pattern of fl ows and pressure gradients. In the example shown in Figure 12, there is a net pressure gradient in the x-direction. The blocks are numbered i = 1, 2, ..nx in the x-direction, and j = 1, 2, .. nz in the z-direction.

x

z

P(i,0) = P(i,nz)

P(i,nz+1) = P(i,1)

P(0,j) = P(nx,j)+∆P P(nx+1,j) = P(1,j)-∆P

One advantage of using periodic boundary conditions, is that fl uid can fl ow through the sides of the model. This method can be used to calculate a full tensor effective permeability (Section 3.2).

(c) Linear Pressure Boundary ConditionsIn linear pressure boundary conditions (Figure 13), the pressure is fi xed at each end, as in the fi xed-pressure boundary conditions. Then, the pressure at the edges of the model is interpolated linearly from one side to the other. Like the periodic boundary conditions, the linear pressure boundary conditions allow fl ow through the edges.

P1 P2

P1 P2

P1 P2

(d) Flow Jacket, or SkinTo reduce the effect of boundary conditions when calculating the effective permeability, some engineers perform the simulations on a larger grid than necessary. The extra grid blocks round the edges are referred to as a “jacket” or “skin”. See Figure 14.

Figure 12Periodic boundary conditions

Figure 13Linear pressure boundary conditions

12


keff calculated forthis block

boundary conditions appliedto outer edges of model

1.5 Upscaling ErrorsIn the process of upscaling, information about the fi ne-scale structure is lost, and upscaling usually gives rise to errors. However, in some cases, the errors will be larger than in others. In this section, we examine a series of models which show examples of where upscaling is successful and where it is not.

1.5.1 Correlated Random FieldsWe introduce the concept of a correlated random fi eld. Although the permeability distribution in real rocks may not follow this type of model, it is a useful way to parameterise heterogeneity. We assume that the probability density function (pdf) of the model is normal or log normal, as shown in Figure 15. In an isotropic model (i.e. same in all directions), the fi eld is then characterised by three parameters: the mean, μ, the standard deviation, σ, and correlation length, λ. The standard deviation determines the width of the pdf (i.e. the permeability contrast), and the correlation length determines approximately the distance over which the permeability values are similar. Figure 16 shows examples of the 4 models with varying σ and σ and σ λ.

0.00

0.01

0.02

0.06

0.5 1.0 1.5 2.0 2.5 3.0 3.5log(permeabil ity)

b )

0.00

0.01

0.02

0.06

40 60 80 100 120 140 160Permeabil ity (mD)

a )

0.01

0.02

0.06

0 500 1000 1500 2000Permeability (mD)

0.00

c )

0.00

0.01

0.02

0.06


0.00

0.01

0.02

0.06


b )

0.00

0.01

0.02

0.06

40 60 80 100 120 140 160Permeabil ity (mD)

0.00

0.01

0.02

0.06

40 60 80 100 120 140 160Permeabil ity (mD)

a )

0.01

0.02

0.06

0 500 1000 1500 2000Permeability (mD)

0.000.01

0.02

0.06

0 500 1000 1500 2000Permeability (mD)

0.00

c )

0.05

0.04

0.03

Freq

uenc

y

0.05

0.04

0.03

Freq

uenc

y

0.05

0.04

0.03

Freq

uenc

y

Figure 14Example of a fl ow jacket round a model. In this case the jacket is 4 cells thick

Figure 15Normal and log-normal permeability distributions.a) Normal distribution with mean = 100 mD, and standard deviation = 20 mD.b) Log-normal distribution with mean = 2.0, and standard deviation = 0.5.c) Log-normal distribution as above, but with permeability plotted on the x-axis, rather than log(permeability)



Figure 16Models with different standard deviations and correlations lengths

a) small σ, small λ b) large σ, small λ

a) small σ, large λ b) large σ, large λ

Permeability (mD)

100500 150 200

1.5.2 Evaluating the Accuracy of UpscalingOne way to evaluate the accuracy of upscaling is to compare upscaling in two stages,

keff2

, with upscaling in a single stage, keff1

, as shown in Figure 17. If upscaling is

accurate, then k keff eff2 1= . This is the case for upscaling along and across parallel

layers in the idealised models of Sections 1.3.1 and 1.3.2.

fine-scale model

single cellk1

eff

k2eff

The accuracy of scale-up is affected by the correlation length and standard deviation of the distribution, and we use the method shown in Figures 10 and 11 to demonstrate this effect. Instead of generating many fi ne-scale models with different correlation lengths, we create 1 fi ne-scale model, but upscale by different factors – so that the

Figure 17Comparison of one-stage and two-stage upscaling

14


coarse block size varies relative to the correlation length. Figure 18a shows a fi ne-scale model with a correlated random permeability distribution. The model has 400 x 400 grid cells, each of size 1 m3. The permeability distribution is ln-normal, with a mean of 4.6 (corresponding to 100 mD), and a correlation length of 10 m. Three different versions of the model were created with different standard deviations: 0.5, 0.75 and 1.0. The following scale-up factors were tested:

4 4 5 5 8 8 10 10 16 16 20 20 40 40 80 80× × × × × × × ×, , , , , , ,In terms of the correlation length, this gives coarse-scale cells of size:

0.4, 0.5, 0.8, 1.0, 1.6, 2.0, 4.0, 8.0.

Figure 18b and c show examples of coarse-scale models with scale-up factors of 5 and 50. In each case the ratio of two-stage upscaling to single-stage upscaling was calculated. The results are plotted in Figure 19. The results are least accurate when the scale-up factor is 10 – 50, i.e. when the coarse block size is 1 – 5 times the cor-relation length. Also, the error increases with the standard deviation of the model, as one might expect.

a) b) c)

Permeability (mD)

1010.1 100 1000 10000

1.00

1.01

02

1.03

0 20 40 60 80Scale-up Factor

sigma = 0.75sigma = 1.00

sigma = 0.50

1.00

1.

1.1.

01

02

1.03

0 20 40 60 80Scale-up Factor

sigma = 0.75sigma = 1.00

sigma = 0.50sigma = 0.75sigma = 1.00

sigma = 0.50keff2

/kef

f1

The conclusions from these examples are:

• Upscaling will be least accurate when the coarse cell size is comparable to, or slightly larger than the correlation length.

• Upscaling errors increase as the standard deviation of the model increases.

Figure 18a) Fine-scale model with 400 x 400 cells; b) coarse-scale model with 80 x 80 cells; c) coarse-scale model with 8 x 8 cells

Figure 19The ratio of k keff eff

2 1 for

different scale-up factors



1.5.3 Upscaling of a Sand/Shale ModelUpscaling errors are largest in models where there are high permeability contrasts. Unfortunately, high permeability contrasts frequently occur in reservoir rocks. For example, we often require to model the following:

• Low permeability shales in a high permeability sandstone• Low permeability faults in a high permeability sandstone• High permeability channels in a low net/gross reservoir• High permeability fractures in a low permeability reservoir

All these cases are diffi cult to model. As an example, we consider a sand/shale model, where the shale has zero permeability. Figure 20 shows the fi ne-scale model, which has to be upscaled to 3 coarse blocks, as shown. Since there is a shale lying across each coarse block, each coarse block will have zero permeability in the z-direction (vertical). However, fl uid can fl ow through the model vertically, as shown. This error arises because the coarse block size is similar to the characteristic length of the shales. Upscaling would be more accurate, if the coarse block size was much larger, or much smaller than the shales. Alternatively, using a “skin” or “fl ow jacket” will increase the accuracy of upscaling (Section 1.4.2, Figure 14).

1.6 Summary of Single-Phase UpscalingThe main points to remember from this section are:

• Fine-scale geological models usually require upscaling for full-field simulation.

• Upscaled permeability is generally referred to as effective permeability.• Some quantities, such are porosity and water saturation are easy to upscale,

because they may be averaged arithmetically.• In some simple models, permeability may also be upscaled using averaging, as

follows: − the arithmetic average for along-layer fl ow; − the harmonic average for across-layer fl ow; − the geometric average for a random distribution.• In more complex models, the effective permeability is calculated using a

numerical simulation.• Different boundary conditions may be used when calculating the effective

permeability numerically: constant pressure (or no-fl ow), periodic and linear.• Upscaling is least accurate when the coarse cell size is comparable to, or slightly

larger that the correlation of the permeability distribution.• Upscaling errors increase as the standard deviation of the model increases.

Figure 20Sand/shale model

16


2 TWO-PHASE FLOW

2.1 IntroductionOften we need to simulate two-phase systems, e.g. a water fl ood or a gas fl ood of an oil reservoir, or an oil reservoir with a gas cap or an aquifer. The aim of upscaling in this case is to calculate a coarse-scale model which can reproduce the fl ow rates of the different fl uids. The coarse model should also provide a good approximation to the saturation distribution in the reservoir with time.

The paths which the injected fl uid takes through the reservoir depends on the forces present:

• Viscous – due to injection of a fl uid• Capillary• Gravity

Therefore, the balance of forces should be taken into account during upscaling. Before learning how to upscale two-phase fl ow, we show the effects which geologi-cal heterogeneity may have on hydrocarbon recovery.

Consider the following simple model (Figure 21), with alternating horizontal layers of 100 mD and 10 mD (referred to as facies 1 and facies 2). We assume that the model is fi lled with oil and connate water initially, and simulate a water fl ood, by injecting at uniform rate at the left side, and producing from the right side (at constant bottom-hole pressure). The density of the two fl uids is the same for this example, so that there are no gravity effects. Figure 22 shows the relative permeabilities and capillary pressures, and Table 1 lists the properties of the 3 cases simulated with this model.

100 mD10 mD100 mD10 mD

0

2

4

6

8

10

12

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Water Saturation

Cap

Pre

ssu

re

Rel

Per

m

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Water Saturation

krw 1kro 1krw 2kro 2

Pc 1Pc 2

Case Porosity Rel Perm/Pc Curve No. Flow Regime facies 1 facies 2 facies 1 facies 21 0.2 0.2 1 1 viscous2 0.2 0.05 1 1 viscous3 0.2 0.1 1 2 visc+capillary

Figure 21Simple layered model for demonstrating viscous and capillary effects

Figure 22The relative permeability and capillary pressure curves

Table 1Properties of the fi rst set of examples



Figure 23 shows the distribution of oil saturation after the injection of 0.2 PV, for Cases 1 and 2. Both cases use only a single relative permeability curve and the fl ow regime is viscous-dominated. Water fl ows faster along the high permeability layers, as one would expect. However, notice that this effect is reduced when the porosity of facies 2 is reduced.

Oil Saturation

0.50.40.3 0.6 0.7

In Case 3, both relative permeability and capillary pressure tables are used. These curves are typical of a water-wet rock: the capillary pressure is much higher in the low permeability facies, and the connate water saturation is higher. In this case, water is imbibed along the low permeability layers, and also there is cross-fl ow from the high permeability layers to the low permeability ones (Figure 24). Due to the effects of capillary pressure, the front is nearly level in the two facies.

Oil Saturation

0.50.40.3 0.6 0.7

Figure 25 shows the recovery as a function of pore-volumes injected, and demon-strates that models may have the same effective absolute permeability, but different recoveries.

Figure 23The oil saturation for Cases 1 and 2, after the injection of 0.2 PV

Figure 24The oil saturation for Case 3, after the injection of 0.2 PV

18


0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6

Pore Volumes Injected

Case 1Case 2Case 3

In the next example, graded models are used, as shown in Figure 26. There are two versions: Case 4 with permeability increasing upwards (referred to as coarsening-up) and Case 5 with permeability decreasing upwards (referred to as fi ning-up). The permeabilities range from 200 mD to 1000 mD, the porosity was kept constant at 0.2, and the fi rst relative permeability curve was used. In this model, however, the densities of the fl uids were different: the density of water was set to 1000 kg/m3 and that of oil was set to 200 kg/m3.

200 mD

1000 mD

1000 mD

200 mD

a) Coarsening-up b) Fining-up

200 mD

1000 mD

1000 mD

200 mD

200 mD

1000 mD

200 mD

1000 mD

1000 mD

200 mD

1000 mD

200 mD

a) Coarsening-up b) Fining-up

Again a waterfl ood was performed, and the results are shown in Figure 27. Since water is more dense than oil, water has a tendency to slump down. In Case 4 (coars-ening-up), this tendency is reduced by the fact that the viscous forces tend to move the fl uid faster in the upper layers. However, in Case 5 (fi ning-up), the slumping effect is reinforced by the viscous force moving fl uid faster along the lower layers. This means that the breakthrough time is earlier in Case 5 than in Case 4, as shown in Figure 27. The effective absolute permeability in these two models is the same, but the two-phase fl ow effects are different, due to the effect of gravity. (If the den-sity of water equalled the density of oil, the recovery and watercut curves would be identical.)

Oil Saturation

0.50.40.3 0.6 0.7

a) Coarsening-up a) Fining-up

Figure 25Cumulative recovery and watercut for Cases 1 - 3

Figure 26The graded layer models for Cases 4 and 5

Figure 27The oil saturation after the injection of 0.2 PV in the graded layer models



Figure 28Cumulative recovery and watercut for the graded layer models

0.00.0 0.1 0.2 0.3 0.4 0.5 0.6


0.00.0 0.1 0.2 0.3 0.4 0.5 0.6


Case 4Case 5

0.00.0 0.1 0.2 0.3 0.4 0.5 0.6


0.00.0 0.1 0.2 0.3 0.4 0.5 0.6


0.00.0 0.1 0.2 0.3 0.4 0.5 0.6


0.00.0 0.1 0.2 0.3 0.4 0.5 0.6


Case 4Case 5Case 4Case 5

Frac

tiona

l Rec

over

y

Wat

er C

ut

0.1

0.2

0.3

0.4

0.5

0.2

0.4

0.6

0.8

1.0

In summary, in a viscous-dominated fl ood, permeability heterogeneity disperses the fl ood front (Cases 1 and 2), so that breakthrough occurs earlier, and the water cut curve rises less steeply. However, the effect of heterogeneity also depends on the balance of fl uid forces. In a water-wet system, capillary pressure can help the front to advance more evenly along the layers (Case 3). (More information on capillary effects is given in Section 3.3.) Gravity effects may increase or reduce the viscous effects, depending on permeabilities in the model (Cases 4 and 5).

2.2 Applying Single-Phase Upscaling to a Two-Phase ProblemMost engineers only perform single-phase upscaling although, as shown above, het-erogeneities give rise to a variety of effects in two-phase fl ow. The reason for this is that two-phase upscaling is time-consuming and the results are not always robust (i.e. they may contain large errors). We deal with two-phase upscaling in Sections 2.4 – 2.6.

Figure 29 (left side) shows a very heterogeneous 2D model, with correlated random permeabilities. The permeability distribution was ln-normal (i.e. natural logs), with a standard deviation of 2.0 The model is assumed to be in the horizontal plain. The details of the model are given in Table 2. The model was upscaled using the pres-sure solution method with no-fl ow boundaries. Three different scale-up factors were used, and the coarse-scale models 1 and 3 are also shown in Figure 29 (middle and right side).

log10(k)

-2 -1 0 1 2 3 4 5 6

Model Number of Cells Cell dimension (m) Scale-up FactorFine 105 x 105 5 -Coarse 1 21 x 21 25 5 x 5Coarse 2 15 x 15 35 7 x 7Coarse 3 7 x 7 75 15 x 15

Figure 29Fine- and coarse-scale models used for demonstrating the effects of applying single-phase upscaling to a two-phase problem

Table 2

20


A waterfl ood with a quarter 5-spot well pattern was performed (i.e. 2 wells in diagonally opposite corners). The same relative permeability curve was used for the whole model (Figure 30), and the capillary pressure was set to zero. The fl ood was therefore viscous-dominated. The viscosity of water was 0.3 and that of oil was 3.0. The resulting recovery curves are shown in Figure 31. As one might expect, the error increases with the scale-up factor.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1Water Saturation

Rel

ativ

e Pe

rmea

bilit

y

krwkro

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1


Frac

tiona

l Rec

over

y

fineups 5x5ups 7x7ups 15x15

The model was modifi ed by reducing the standard deviation to 0.2 (lowering the permeability contrast), and the simulations were repeated with the low heterogeneity model. The recovery is shown in Figure 32 for the scale-up factor of 15 x 15. As expected, the errors are smaller for the low heterogeneity model.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1Pore Volumes Injected

Frac

tiona

l Rec

over

y

lo het finelo het coarsehi het finehi het coarse

In addition, the errors caused by using only single-phase upscaling, are larger when the coarse block size similar to the correlation length. Also, the upscaling error tends

Figure 30The relative permeability curve used for the random model. (Capillary pressure was set to zero)

Figure 31Comparison of recovery for different scale-up factors

Figure 32Comparison of recovery for models with different levels of heterogeneity



to be larger in unstable fl oods (injected fl uid is of lower viscosity than the in situ fl uid) than in stable fl oods.

In summary, single-phase upscaling may be adequate for upscaling two-phase sys-tems, provided that:

• The scale-up factor is small• The permeability contrasts are small• The correlation length is very large, or very small compared to the coarse cell

size• The fl ood is stable (favourable mobility ratio)• The fl ood is not capillary-dominated (See Section 3.3)• The fl ood is not gravity-dominated

2.3 Improving Single-Phase UpscalingThere are two approaches which may make single-phase more accurate when applying it to two-phase problems. The fi rst is to use non-uniform upscaling, and the second is to perform a global single-phase simulation (i.e. over the whole fi ne-scale model) using the correct boundary conditions, including wells. We refer to this second method as Well Drive Upscaling (WDU).

2.3.1 Non-Uniform UpscalingConsider a model with horizontal layers, as shown in Figure 33. There is a high permeability streak running across the model. The model details are given in Table 3. In both coarse-scale models there are 3 coarse cells in the vertical direction. In model Coarse 1, the cells are each 5 m thick. However, in model Coarse 2, the thicknesses are: 7 m, 1 m, 7 m, so that the high permeability streak is still resolved.

b)b)b) C C Coaoaoarsrsrse e e 1 c1 c1 c) C) C) Coaroaroarsesese 2 2 2

Model No. of cells Cell size (m)Fine 100 x 15 1 x 1Coarse 1 20 x 3 5 x 5Coarse 2 20 x 3 variable

Figure 34 shows the recovery for these models. It can be seen that the model with uniform coarse cells (Coarse 1) gives very inaccurate results, and the model which maintains the high permeability streak (Coarse 2) is much more accurate.

Figure 33Model with a high permeability streak

Table 3

22


0.00

0.05

0.10

0.15

0.20

0.25

0.0 0.1 0.2 0.3 0.4


Frac

tiona

l Rec

over

y

finecoarse 1coarse 2

In a practical upscaling application, much attention is paid to upgridding the model – i.e. deciding how to amalgamate the layers, so that upscaling is as accurate as pos-sible. The coarse grid may also be non-uniform in the x- and y-directions.

Several methods for performing non-uniform upscaling have been developed. For example, Durlofsky et al. (1996, 1997) fi rst carry out a single-phase simulation. Then they use the inter-block fl ows to determine the coarse block boundaries. Smaller coarse blocks are assigned to regions where there are high fl ow rates.

2.3.2 Well Drive UpscalingWhen upscaling using numerical simulation, boundary conditions are applied to each coarse-scale cell in turn (Figure 2.) These are called local boundary condi-tions. However, the boundary conditions described in Section 1.4.2 may be quite different from the pressures which actually occur in a fi ne-scale simulation, lead-ing to inaccuracies in the upscaled model. To overcome this problem, we may use global boundary conditions. Figure 35 demonstrates the difference between local and global boundary conditions.

Boundary conditionsapplied to coarse cell

Producer

Injector

P1 P2

Local Global

In the Well-Drive Upscaling method (WDU), a single-phase simulation is performed on the whole fi ne grid (Zhang et al, 2005). (It is feasible to perform a single pressure solve on grids with several million grid cells.) Then the effective transmissibility between coarse-scale cells is calculated, rather than the effective permeability. The upscaled transmissibility is give by:

Figure 34Recovery and watercut for fi ne- and coarse-scale models

Figure 35Local and global boundary conditions



Tq

P P=

−∑Ι ΙΙ (13)

Where q denotes the fi ne-scale fl ows (Figure 36) and PI and PII are the (pore-volume weighted) average pressures in coarse cells I and II.

Ι ΙΙΙ ΙΙ

Upscaling is also performed at the wells, using the method described in Section 3.1. This method produces very accurate single-phase upscaling, which leads to an increase in the accuracy of two-phase fl ow at the coarse scale.

Many tests on upscaling methods have been carried out using the model generated for the 10th SPE comparative solution project, which was on upscaling (Christie and Blunt, 2001). This model is referred to as the SPE 10 model (Figure 37a). We use layer 59 (Figure 37b) as an example of well drive upscaling, because this layer is particularly heterogeneous. There is an injection well at the centre and production wells in each corner.

a) b) P1 P2

P4 P3

log10(k)

-2-3 -1 0 1 2 3 4 5

Layer 59 was upscaled using the WDU method and also the conventional method with local boundary conditions. Figure 38 shows the oil saturation distribution. It can clearly be seen that the WDU method reproduces the results of fi ne-scale model much better than the conventional approach. This is because appropriate boundary conditions have been applied to the model.

Figure 36Upscaling transmissibility

Figure 37The SPE 10 model, and layer 59

24


Soil

0.500.350.20 0.65 0.80

fine local WDU

2.4 Introduction to Two-Phase UpscalingSo far, when performing upscaling, we have assumed that there is only one phase present, and that the fl ow is in a steady state. We only need to upscale the absolute permeability. However, when there are two phases fl owing, such as water displac-ing oil, the system is not, in general, in a steady state. We need to simulate fi ne-scale fl oods in order to upscale relative permeability and capillary pressure. This is referred to as dynamic upscaling, and the upscaled relative permeabilities are known as pseudo relative permeabilities, or pseudos. Pseudos can be calculated to take account of physical dispersion, and also to compensate for numerical dispersion (Section 2.6.3).

When upscaling, we should use the phase permeabilities:

k k kf abs rf= (14)

Where “f” stands for fl uid – oil, gas or water. Generally, we assume that both the absolute and the relative permeabilities are homogeneous and isotropic at the smallest

scale ( k kx z= ). As we upscale, the absolute and relative permeabilities may become

anisotropic ( k krx rz≠ ). To obtain effective (or pseudo) relative permeabilities, the absolute permeability must be scaled-up separately. Then the pseudo relative perme-ability is calculated as follows:

k k krf x f x abs x, , ,= (15)

Similar equations are used for fl ow in the y- and z- directions.

2.5 Steady-State MethodsIf fl uids are in a steady state, the saturation does not change with time and the fractional fl ow (fl ow of water/total fl ow) is constant. Although fl oods are dynamic processes, sometimes a fl ood may approach a steady state. For example, over small scales (20 cm, or less), oil and water may come into capillary equilibrium.

Figure 38Comparison of oil saturation distribution in fi ne- and coarse-scale simulations of Layer 59 of the SPE 10 model



In a steady-state upscaling method, we assume that within a short interval of time the zone of interest is in a steady-state, but we allow the fl uid saturation to change gradu-ally, so that a full range of saturation is obtained. At steady-state, the water saturation does not change with time, i.e. ∂Sw/∂t = 0, so the continuity equation becomes:

∇ ⋅ =uf 0, (16)

where u is Darcy velocity, and f is fl uid. From Darcy’s law:

∇ ⋅( )⋅ ∇( )⋅ ∇ =( )k P( )⋅ ∇( )⋅ ∇k P⋅ ∇( )⋅ ∇( )f f( )⋅ ∇( )⋅ ∇f f⋅ ∇( )⋅ ∇( )k P( )f f( )k P( )⋅ ∇( )⋅ ∇k P⋅ ∇( )⋅ ∇f f⋅ ∇( )⋅ ∇k P⋅ ∇( )⋅ ∇ 0. (17)

There are several steady-state methods, depending on the balance of forces:

• Capillary equilibrium,• Vertical equilibrium (gravity-dominated fl ood)• Viscous-dominated steady-state

We concentrate here on the capillary equilibrium method.

The advantage of steady-state methods is that they turn two-phase upscaling into a series of single-phase upscaling calculations. This means that steady-state methods are feasible for models with large numbers of grid cells. (See, for example, Pickup and Stephen, 2000; and Pickup et al, 2000.)

2.5.1 Capillary-EquilibriumAssume that the injection rate is very low, gravity forces are negligible, and that the fl uids have come into capillary equilibrium with a coarse-scale cell. This means that the saturation distribution is determined by the capillary pressure curves.

The method is as follows:

1. Choose a Pc level.

2. Determine the water saturations, and then the relative permeabilities.

3. Calculate the pore volume-weighted average water saturation.

4. Calculate the phase permeabilities: ko = k

absk

ro, k

w, k

w, k = k

absk

rw.

5. Calculate the effective water phase permeability, kw

6. Calculate the effective oil phase permeability, ko

7. Calculate the relative permeabilities, krw

= kw/k

abs, etc.

8. Repeat the process with another value of Pc.

Steps 5 and 6 may be carried out analytically or numerically, depending on the distribution.

26


Example 4Consider a model with two layers of equal thickness, as shown in Figure 39. The absolute permeabilities are 100 mD and 20 mD. Assume that the porosity in each layer is equal to 0.2. The relative permeability and P

c curves for each layer are

shown in Figure 40.

100

20

kabs (mD)

100

20

kabs (mD)

0

0.2

0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Water Saturation

hi

hi

lo

lo0

2

4

6

8

0

12

0.2 0.3 0.4 0.5 0.6 0.7 0.8Water Saturation

lo

hi

0

0.2

0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Water Saturation

hi

hi

lo

lo0

0.2

0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Water Saturation

hi

hi

lo

lo0

2

4

6

8

0

12


lo

hi

0

2

4

6

8

0

12


lo

hi

Cap

Pre

ssur

e

Rel

Per

m

0.4

0.6

Using the arithmetic and harmonic averages (Section 1.3), the effective permeability is:

k kx z= =60 00 33 33. .

Suppose we choose a capillary pressure of Pc = 0.45.

In the high perm layer: Sw = 0.34,

krw = 0.0013, kw = 0.13, kro = 0.5, ko = 50.

In the low perm layer: Sw = 0.44,

krw = 0.0016, kw = 0.032, kro = 0.48, ko = 9.6.

Figure 41 shows the phase permeabilities.

0.13

0.032

kw (mD)

50.0

9.6

ko (mD)

0.13

0.032

kw (mD)

0.13

0.032

kw (mD)

50.0

9.6

ko (mD)

50.0

9.6

ko (mD)

Since the layers are of equal width, the average saturation is Sw = 0.39. The effective phase permeabilities are then calculated using the arithmetic and harmonic averages. Then the relative permeabilities are calculated using Equation 15.

Figure 39Model with horizontal layers

Figure 40Relative permeability and capillary pressure curves

Figure 41Phase permeabilities



k kk kk kk k

wx rwx

wz rwz

ox rox

oz roz

= = == = == = == = =

0 081 0 081 60 00 0 001350 051 0 051 33 33 0 0015329 8 29 8 60 00 0 5016 1 16 1 33 33 0 48

. . / . .

. . / . .. . / . .. . / . .

Note that the kv/kh ratio ( = k kz x ) is different for oil and water:

k kw z w x, , .= 0 63 k ko z o x, , .= 0 54

Effective relative permeability curves may be derived by repeating this calculation for a range of capillary pressure values (Figure 42). The capillary-equilibrium method is useful as a quick method for upscaling small-scale models (Section 3.3). However, it is only valid in cases where the fl ow rate is very low.

00.10.20.3

0.70.80.9


krox

kroz krwzkrwx

00.10.20.3

0.70.80.9


0.10.20.3

0.70.80.9


krox

kroz krwzkrwx

0.4

0.5

0.6

Rel

Per

m

2.6 DYNAMIC METHODS

2.6.1 IntroductionFor dynamic (or non steady-state methods), we need to perform a two-phase fl ow simulation on a fi ne grid. There are basically two types of dynamic method:

a) Weighted Pressure MethodsAs in single-phase numerical upscaling, a common approach in two-phase upscaling is to sum the fl ow, average the pressure gradient and use Darcyʼs Law to obtain the pseudo phase permeability. However the pressure may be averaged in different ways. Here, we shall concentrate on the Kyte and Berry (1975) method.

b) Total Mobility MethodsIn total mobility methods, we avoid averaging the pressure, and scale-up the total mobility. Then the average fractional fl ow is used to calculate the pseudo relative permeabilities.

The total mobility is:

λ λ λµ µt o w

ro

o

rw

w

k k= + = + . (18)

Figure 42Effective relative permeability curves

28


The fractional fl ow is the fl ow of water divided by the total fl ow:

f qq q

qqw

w

o w

w

t

=+

= (19)

Again there are a number of variations of this method, the most commonly used being that of Stone (1991).

2.6.2 The Kyte and Berry MethodA simple version of the Kyte and Berry (1975) method is presented here, using the grid shown in Figure 43.

i=1 2 3 4 5 6 7 8 9 10j=1

2345 z

x

DX DZ

i=1 2 3 4 5 6 7 8 9 10j=1

2345 z

x

∆Z∆X

∆

∆

Pseudo calculatedfor this coarse block

The diagram shows two coarse grid blocks, each of which is made up of 5 x 5 fi ne blocks. The equations below show how to calculate the pseudo relative permeabilities and capillary pressure for the left coarse block.

The fi rst step is to perform a fi ne-scale, two-phase simulation (e.g. in ECLIPSE), saving the pressures and inter-block fl ows at specifi ed intervals of time (in the re-start fi les). The method proceeds as follows:

1. Calculate the effective absolute permeability in the area shown in Figure 44, i.e. half way between the two coarse blocks.

i i i = = = 3 i3 i3 i = = = 7 7 7

j j j = = = 111

j j j = = = 555

Kyte and Berry approximate the effective permeability using the arithmetic average in each column, and then taking the harmonic average of the columns. The area between the two coarse blocks is used, for reasons explained below.

Figure 43Model used for describing the Kyte and Berry Method. The thickness of the model is ΔyΔyΔ

Figure 44The area used for calculating the effective absolute permeability



kz k

Zi

j ijj=

∆

∆=∑

1

5

(20)

where Δzj

zj

z and ΔZ are the thicknesses of the fi ne and coarse blocks, respectively. (In this case, all the blocks are of equal size.)

k X

x kI

i ii

=∆

=∑

∆

3

7

(21)

where Δxi and ΔX are the lengths of the fi ne and coarse blocks, and kI is the required

effective absolute permeability.

The pseudos are then calculated, at certain times during the simulation. (These are the times at which the restart fi les are written in the Eclipse simulation.)

2. Calculate the average water saturation:

SS x z

x zw

w ij ij i jij

ij i jij

= ==

==

∑∑

∑∑

, φ

φ

∆ ∆

∆ ∆

1

5

1

5

1

5

1

5

(22)

where φij is the porosity.

3. Calculate the total fl ow of oil and water out of the left coarse block (Figure 45).

q qf f jj

==∑ 5

1

5

, , (23)

where qf5,j

is the fl ow of fl uid “f” from fi ne block number (5,j).

i = 5

j = 1

j = 5

i = 5

j = 1

j = 5

4. Calculate the average phase pressures in the central column of each coarse block. In this example, we use the fine blocks in columns 3 and 8, the shaded areas in Figure 46.

Figure 45Calculation of the total fl ow

30


i = 3 i = 7

j = 1

j = 5I II

i = 3 i = 7

j = 1

j = 5I II

In the Kyte and Berry method, the pressures are weighted by the phase permeabilities times the height of the cells (which in this case are all the same size). This is so that more weight is given to regions where there is greater fl ow. However, there is no scientifi c justifi cation for using this weighting. In the fi rst coarse block (numbered, I), the average pressure is:

Pk k z P g D D

k k zfI

j rf j f j f jj

j rf jj

=− −( )

=

=

∑

∑

3 3 3 3 31

5

3 3 31

5

∆

∆

ρ ( )

(24)

where D3j is the depth of cell (3,j) and D is the average depth of coarse cell I. The

term gρf(D

f(D

f 3j - D ) is to normalise the pressure to the grid block centre. The average

pressure for coarse block II is calculated in the same manner, but using column 8 instead of column 3. The pressure difference is then calculated as:

∆P P Pf fI fII= − . (25)

5. The pseudo rel perms are then calculated using Darcy’s law. Firstly, calculate the pseudo potential difference. (Potential is defi ned as Φ = P-ρgz, so that the fl ow rate is proportional to ∇Φ.)

∆Φ ∆ ∆f f∆ ∆f f∆ ∆f∆ ∆f∆ ∆P g∆ ∆P g∆ ∆f fP gf f∆ ∆f f∆ ∆P g∆ ∆f f∆ ∆D= −∆ ∆= −∆ ∆f f= −f f∆ ∆f f∆ ∆= −∆ ∆f f∆ ∆∆ ∆P g∆ ∆= −∆ ∆P g∆ ∆∆ ∆f f∆ ∆P g∆ ∆f f∆ ∆= −∆ ∆f f∆ ∆P g∆ ∆f f∆ ∆ρ∆ ∆ρ∆ ∆ , (26)

where ΔD is the depth difference between the two coarse grid centres. Then:

k q Xrf

f fq Xfq XI I

= −µ ∆q X∆q X∆ ∆Zk∆ ∆ZkI I∆ ∆I IΦI IΦI I (27)

6. Calculate the pseudo capillary pressure using:

P P Pc oP Pc oP P I wPI wP I= −P P= −P Pc o= −c oP Pc oP P= −P Pc oP P I w= −I w (28)

The Eclipse PSEUDO package can be used for calculating Kyte and Berry pseudos.

Figure 46The cells used for averaging the phase pressures



2.6.3 Discussion on Numerical DispersionOne advantage of pseudo-isation methods, such as that of Kyte and Berry is that they can take account of numerical dispersion. When a simulation is carried out using a larger grid, the front between the oil and water becomes more spread out. However, the Kyte and Berry method counteracts this effect by calculating the fl ows on the down-stream side of the coarse block, instead of the middle. This is illustrated by a simple example. Figure 47 shows an example of input relative permeability curves (“rock” curves).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Water Saturation

Rel

Per

m

If the water saturation is Sw = 0.5, the rock curves show that there is a small amount of oil and water fl owing. However, when the average saturation, Sw, is 0.5 in the coarse block, the distribution could be as shown in Figure 48.

coarseblockwater

oil

Since the water has reached only half way across the coarse block, there should be no water fl owing out of the right side. The Kyte and Berry method calculates the pseudo relative permeabilities using the fl ow on the downstream side of the coarse block, to prevent water breaking through too soon. The pseudo water relative permeability curve is moved to the right, relative to the rock curves, as shown in Figure 49.

Figure 47Example of “rock” curves

Figure 48Example of the water saturation in a coarse block

32


0

0.1

0.2

0.3

0.4

0.5

0

0

.6

.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8

Water Saturation

Rel

ativ

e Pe

rmea

bilit

y

pseudos - solid linesave. rock curves - dashed lines

2.6.4 Disadvantages of the Kyte and Berry MethodThere are certain problems with the Kyte and Berry method.

• Negative rel perms are produced, if ∆Φf has the same sign as qf .

• Infi nite rel perms occur if ∆Φf is zero.• The method of averaging the pressures, using relative permeability as a weighting

function, may cause errors when the fl uids are separated due to gravity. For fl oods which are gravity-dominated, the TW method works better (Section 2.6.5).

• Non-zero pseudo capillary pressure may be produced, even if there is no capillary pressure in the fi ne-scale simulation. This is because a different weighting is used for calculating the average pressure in each phase.

• The capillary pressure may be different in different directions, because only the central column is used for averaging the pressures.

Because of the fi rst two disadvantages, i.e. negative, or infi nite rel perms, pseudos obtained from packages like the PSEUDO must be vetted before using at the coarse scale. Often “odd” values of relative permeability are set to zero.

Good reviews of various methods for calculating pseudos are presented in Barker and Dupouy (1999) and Barker and Thibeau (1997).

Note that dynamic upscaling methods, such as that of Kyte and Berry are diffi cult to apply in practice. Ideally, a fi ne-scale two-phase fl ow simulation is required for each coarse-scale cell (plus a “fl ow jacket”), and this is time consuming. Also, it is diffi cult to determine the correct boundary conditions to use, so the results may not be accurate.

If a pseudo is calculated for each coarse cell, in each direction, there may be 10,000s of pseudos in the coarse-scale model. The number of pseudos must be reduced, by grouping similar pseudos together.

Figure 49Example of pseudo relative permeability curves



Pseudo relative permeability curves depend on a number of factors, including:

(a) The balance of forcesThe shape and end points of a pseudo depend on the ratio of viscous/capillary and viscous/gravity forces. These ratios may be different in different parts of the res-ervoir.

(b) The well locationsThe wells determine the fl ow rate and direction. If a new well is drilled, the pseudos ought to be re-calculated.

Because of these problems, two-phase upscaling is rarely used for upscaling from a geological model to a full-fi eld simulation.

2.6.5 Alternative MethodsThere are a number of similar methods to the Kyte and Berry Method.

(a) The Pore-Volume Weighted MethodThe problems of non-zero capillary pressure and directional capillary pressure, mentioned in Section 2.6.4, may be overcome by using a pore volume weighted average of the pressures over the entire coarse block. ECLIPSE uses this method for calculating the average capillary pressure in the Kyte and Berry method. Also, pore volume weighting may be used for averaging the pressures when calculating the pseudo relative permeabilities. In this case, the method is called the Pore Volume Weighted Method. It is available in the Eclipse PSEUDO package.

(b) The TW MethodThis method was developed by Nasir Darman at Heriot-Watt University (Darman et al, 1999). It is similar to the Kyte and Berry method, except transmissibility weighting is used when calculating the average pressure. The method works better than the Kyte and Berry method in cases where gravity effects are signifi cant (e.g. a gas fl ood).

Both these methods share the same problems discussed in Section 2.6.4, namely, they are diffi cult to apply in practice.

2.6.6 Example of the PVW MethodIn two-phase dynamic upscaling methods, pseudo relative permeabilities are cal-culated, so that (hopefully) the results of a coarse-scale simulation provide a good approximation to the fi ne-scale results. Layer 59 of the SPE 10 upscaling study (Figure 37b) is used here as an example. A global simulation was performed on the fi ne grid, i.e. the whole of the fi ne grid was included in the fi ne-scale fl ow simulation. (Note that this would not be done, in practice, because there is no point in upscaling, if you can simulation the whole fi ne grid.) The fi ne-scale model had 60 x 220 cells and the coarse-scale model had 10 x 22, which corresponded to a scale-up factor of 6 x 10. Pseudo relative permeabilities were calculated for each coarse cell, in each direction.

Figure 50 shows the oil saturation for the fi ne-scale simulation, a coarse-scale simulation using the WDU method (Section 2.3.2), and a coarse-scale simulation using pseudos

34


from the PVW method. Both the WDU and the PWV methods give reasonable oil saturation distributions. The oil recovery rate, for well P4, for these models is shown in Figure 51, along with the oil rate for a coarse-scale simulation using single-phase upscaling with local boundary conditions (Sections 2.2. and 2.3).

Soil

0.500.350.20 0.65 0.80

fine WDU PVW

0

10

20

30

40

50

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8


FineLocalWDUPVW

Wel

l Rat

e (m

3 /day

)

It can be seen in Figure 51 that both the WDU and the PVW methods agree well with the fi ne-scale simulation. However, the results from the single-phase upscaling with local boundary conditions are poor. This example shows that two-phase upscaling is not necessarily always more accurate than single-phase upscaling.

2.7 Summary of Two-Phase FlowThe main points on two-phase fl ow are:

• Two-phase upscaling is time consuming and not always robust, so is rarely used by engineers.

• Usually, only single-phase upscaling is performed.• But, heterogeneity interacts with two-phase fl ow, and tends to produce dispersion of

the fl ood front, which is not taken into account using single-phase upscaling.• So, single-phase upscaling may give rise to errors, especially when there is

a large scale-up factor, and the reservoir model is very heterogeneous (large standard deviation).

Figure 50The oil saturation distribution for the fi ne-scale model of layer 59 and the coarse-scale models obtained using the WDU and the PVW methods

Figure 51The oil recovery rate for well P4 for the fi ne-scale model and coarse-scale models obtained using the WDU, PVW and local upscaling methods



• Errors in single-phase upscaling may be reduced by using non-uniform upscaling, or well-drive upscaling.

• Ideally, two-phase upscaling should be performed to take account of two-phase fl ow.

• Steady-state upscaling is relatively quick to apply, and is feasible for large models. However, it is only valid in limited cases, e.g. when the fl uids are approximately in capillary equilibrium.

• Dynamic methods are potentially more accurate.• The Kyte and Berry (1975) Method was described as an example.• Dynamic methods can compensate for the effects of numerical dispersion.• Dynamic methods are diffi cult to apply in practice.

3 ADDITIONAL TOPICS

This course has, so far, focussed mainly on common methods for upscaling a geo-logical model for full-fi eld simulation. Most of the single-phase upscaling methods presented may be found in geological packages, such as IRAP/RMS and Petrel. (The WDU and TW methods which were developed at Heriot-Watt are not available in commercial packages.) However, there are a number of other important issues which should be taken into account when upscaling. In this section, we cover these issues in a variety of additional topics:

• Upscaling as Wells• Permeability Tensors• The Geopseudo Method• Uncertainty and Upscaling

3.1 Upscaling at WellsIn the single-phase upscaling methods described in Chapter 1, we assumed that the fl ow was linear. This means that the upscaling methods were not appropriate for regions containing wells, where there is radial fl ow. We start with a brief overview of simulation in blocks containing a well.

A grid block in the simulator is much larger than the diameter of a well, and the pressure calculated for a block containing a well is different from the actual bottom hole pressure. These are related by:

q Iw= −= −w= −w ( )P P( )P Pw b( )w bP Pw bP P( )P Pw bP P= −( )= −P P= −P P( )P P= −P PP Pw bP P= −P Pw bP P( )P Pw bP P= −P Pw bP Pµ (29)

where Pw is the well-bore pressure and Pb is the pressure of the block. Iw is the well index, given by:

I k zw = ∆k z∆k z

( )( )r r( )r rr r( )r ro w( )o wo w( )o wr ro wr r( )r ro wr rr ro wr r( )r ro wr r2π

ln (30)

36


where rw is the well-bore radius and ro is the equivalent radius, given by Peacemanʼs equation (Peaceman, 1978; Peaceman, 1983). Iw is also referred to as the well con-nection factor, or the connection transmissibility factor.

Durlofsky et al. (2000) put forward a method for upscaling in the near-well region. Others have put forward similar methods. The method is only approximate, but improves the accuracy of coarse-scale simulations. The fi rst step is to calculate effective single-phase permeabilities, using one of the conventional methods (e.g. periodic boundary condition applied to each block in turn). Then, a fi ne-scale single-phase simulation of the well block and surrounding blocks is carried out (Figure 52). From the results, the total fl ows out of the coarse-scale well block, and the average pressures in the coarse blocks are calculated. These are used to calculate upscaled transmissibilities between the coarse-scale well block and the surrounding blocks, and a coarse-scale well index.

q

T1

T2

T3

T4

This method improves the accuracy of upscaling at well, and it is also incorporated into the well drive upscaling method (WDU), described in Section 2.3.2.

3.2 Permeability TensorsSuppose that we have layers which are tilted at an angle to the horizontal, as in Figure 53.

∆P

net flow in x-dir

net flow in z-dir

x

z

A pressure gradient has been applied in the x-direction. This will obviously give rise to a fl ow in the x-direction. The fl uid takes a path through the medium, so that it expends a minimum amount of energy. There will be a component of fl ow up the high permeability, and only a small amount of fl ow across the low permeable layer, as shown. This gives rise to a net fl ow in the z-direction, or cross-fl ow. Here, the term cross-fl ow is used to describe fl ow perpendicular to the applied pressure gradi-ent. When calculating the effective permeability of this model, we need to take this cross-fl ow into account. This may be done using a tensor effective permeability, k,where:

Figure 52Near-well upscaling (after Durlofsky et al., 2000)

Figure 53Cross-fl ow due to tilted layers. Light-coloured layers represent high permeability and dark-coloured layers represent low permeability



kk k kk k kk k k

xx xy xz

yx yy yz

zx zy zz

=

(31)

The fi rst index applies to the fl ow direction, and the second to the direction of the pressure gradient. For example kxy is the fl ow in the x-direction caused by a pres-sure gradient in the y-direction. The terms kxx, kyy, kzz are known as the diagonal terms. These are the terms which are usually considered – the horizontal and vertical permeabilities, kh and kv, respectively. The other terms, which describe the cross-fl ow, are the off-diagonal terms.

With tensor permeabilities, Darcyʼs Law becomes:

u k P= − ⋅ ∇µ

, (32)

where u is the Darcy velocity (vector) and P is pressure (scalar).

u ku k Px

k Py

k Pz

u ku k Px

k Py

k Pz

u ku k Px

k Py

k Pz

x xu kx xu k x xkx xk y xyy xyky xk z

y yu ky yu k x yxx yxkx yk y yyy yy

ky yk z

z zu kz zu k x zkx zk y zyy zyky zk z

u k= −u k ∂∂x x∂x x+x x+x x

∂∂y x∂y x+y x+y x

∂∂

u ku k

u k

u kx xx xu kx xu k

u kx xu ku ku ku ku ku ku k

u k

u ku k

u kx xx xx xx xu kx xu k

u kx xu ku kx xu k

u kx xu k

u k= −u k ∂∂x y∂x y+x y+x y

∂∂y y∂y y+y y+y y

∂∂

u ku k

u k

u ky yy yu ky yu k

u ky yu ku ku ku ku ku ku ku k

u ku k

u ky yy yy yy yu ky yu k

u ky yu ku ky yu k

u ky yu k

u k= −u k ∂∂x z∂x z+x z+x z

∂∂y z∂y z+y z+y z

∂∂

u ku k

u k

u kz zz zu kz zu k

u kz zu ku ku ku ku ku ku k

u k

u ku k

u kz zz zz zz zu kz zu k

u kz zu ku kz zu k

u kz zu k

1u k1u k

1u k1u k

1u k1u k

µ

µy yµy y

µ (33)

In Sections 1.3.1 and 1.3.2, we studied fl ow along and across horizontal layers. The model in Figure 5 is repeated here (Figure 54), showing the arithmetic average for the effective permeability for along-layer fl ow and the harmonic average for across layer fl ow.

ttttttttttttttt111111111111111 = = = = = = = = = = = = = = = 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 mm,mm,mm,mm,mm,mm,mm,mm,mm,mm,mm,mm,mm,mm,mm, k k k k k k k k k k k k k k k111111111111111 = = = = = = = = = = = = = = = 101010101010101010101010101010 m m m m m m m m m m m m m m mDDDDDDDDDDDDDDD

mDmDmDttttttttttttttt222222222222222 = = = = = = = = = = = = = = = 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 mmmmmmmmmmmmmmmmmmmmmmmmmmmmmm, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k222222222222222 = = = = = = = = = = = = = = = 11111111111111100 00 00 00 00 00 00 00 00 00 00 00 00 00 00 mmDmmDmmDmmmmDmmDmmDmDmDDmDDmDDDDmDDmDDmDD

xxxxxxx

zzzzzzz

66666666666666.2.2.2.2.2.2.25 5 5 5 5 5 5 mDmDmDmDmDmDmD

22222222222222.......86 86 86 86 86 86 86 mDmDmDmDmmDmDmDmmDmDDmDmDmDDmD

fififine-ne-ne-scascascalelele cococoarararsesese---scascascalelele

In this model, there is no cross-fl ow, so we may write the effective permeability in tensor form with zero off-diagonal terms, as follows:

k =

66 25 00 22 86.

.2 8.2 8

or in 3D:

Figure 54The simple two-layer model

38


k =

66 25 0 00 66 25 00 0 22 86

..

.

3.2.1 Flow Through Tilted Layers

θ

z

x

z'x'

This model is essentially the same as the one in Figures 4 and 6, although the layers have are repeated, and they have been tilted. In the frame of reference defi ned by the

′x and ′z axes, the effective permeability may be calculated using the arithmetic and harmonic averages as before. However, in the x-z co-ordinate system, the effective permeability should be represented by a full tensor. The terms of the tensor may be calculated from the arithmetic and harmonic averages, as follows:

kk k k kk k k ka h a h

a h a h

=+ −( )

−( ) +

cos sin sin cossin cos sin cos

2 2

2 2

θ θ θ θθ θ θ θ

(34)

This formula is obtained by rotating the co-ordinate axes through an angle θ. (You are not required to know the proof.) This example is in 2D, so only the k

xx, k

xz, k

zx

and kzz

are shown. Further rotations may be carried out around the ′x or ′z axes to obtain a full 3D tensor.

Note that:

• The tensor is symmetric (kxz

= kzx

).• Depending on the sign of θ, the off-diagonal terms may be positive or negative.

Example 5Suppose the example in Figure 54 is rotated by 30º (Figure 56), and calculate the effective permeability tensor.

Figure 55Layers tilted at an angle of θ to the horizontal



Figure 56Layered model tilted by 30º

x

z 30o

x

z

x

z 30o

'z

cos230 = 0.75, sin230 = 0.25, sin30.cos30 = 0.433.

From before, ka = 66.25 mD, and k

h = 66.25 mD, and k

h = 66.25 mD, and k = 22.86 mD.

k mD

k mD

k mD

k

xx

xz

zz

= × + × =

= −( ) × =

= × + × =

=

66 25 0 75 22 86 0 25 55 40

66 25 22 86 0 433 18 79

66 25 0 25 22 86 0 75 33 71

55 40 18 7918 79 33 71

. . . . . ,

. . . . ,

. . . . . .

. .

. ..

Full tensor permeabilities may also be calculated from numerical simulations. It is useful to use periodic boundaries, as described in Section 1.4.2. When a pressure gradient is applied in the x-direction, there will be fl ow in the x-direction, and also fl ow in the z-direction due to internal heterogeneity. These fl ows can be used to calculate the k

xx and k

zx tensor terms. Then a pressure gradient is applied in the z-

direction to obtain the kzz

and kxz

terms. (In 3D, a pressure gradient should also be applied in the y-direction.)

3.2.2 Simulation with Full Permeability TensorsHaving calculated full effective permeability tensors, we need special software to handle them at the larger scale. Conventional fi nite difference simulators use a 5-point scheme in 2D and a 7-point scheme in 3D, and only take diagonal tensors – e.g. when running ECLIPSE, you usually specify PERMX, PERMY and PERMZ. Simulation with full tensors is more complicated and more time-consuming, but some packages allow the user to input full tensors. In Eclipse, there is a full tensor option which allows you to specify terms such as PERMXY.

In 2D, a 9-point scheme is required to take account of cross-fl ow. This means that there are 9 terms in each of the pressure equations, as illustrated in Equation (35).

a P a P a P a P a P

a P a P a P a Pi j i j i j i j i j

i j i j i j i j

1 2 1 3 1 4 1 5 1

6 1 1 7 1 1 8 1 1 9 1 1 0, , , , ,

, , , , .

− − − −

− − − − =− + − +

− − − + + − + + (35)

The coeffi cients, ai, in Equation (35) depend on the transmissibilities between the blocks. There are several different methods of discretisation which give slightly dif-

40


ferent results. To extend this to 3D, we need either a 19-point scheme or a 27-point scheme. See Figure 57. (The 19-point scheme leaves out the 8 corners of the cube.) Obviously, it takes longer to solve equations with a larger number of terms.

i-1,j-1 i,j-1 i+1,j-1

i-1,j i,j i+1,j

i-1,j+1 i,j+1 i+1,j+1

x

z

a) b)

Often the off-diagonal elements of the permeability tensor (kxy, etc) are negligible, so the limitations of using a 5-point (2D) or a 7-point (3D) scheme are not serious. In layered systems, the size of the off-diagonal term may be gauged from Equation (34) in Section 3.2.1:

k k kxz a h= −( )sin cos .θ θ (36)

This is a maximum for θ = 45º, and increases as (ka – k

h – k

h – k ) increases. Therefore, full

permeability tensors become more important as the angle of the lamination or bed-ding increases, and as the permeability contrast increases.

3.3 Small-Scale HeterogeneityMost reservoirs are modelled using, what is commonly termed a “fi ne-scale geological model”. This is a stochastic model with grid cells of size approximately 50 m in the horizontal directions, and about 0.5 m in the vertical. There are typically about 107

such cells in a full fi eld model. These cells must be reduced in number to about 104 for full-fi eld simulation. However, each of the grid cells in the geological model is likely to be heterogeneous, containing, for example, sedimentary structures. Petrophysical data (permeabilities, relative permeabilities, and capillary pressures) are acquired from core plugs, which are only a few cm long. When small-scale structure is present, petrophysical data should be upscaled before being applied to the grid blocks of the geological model. Figure 58 shows the ranges of scales of sedimentary structures, along with the scales of measurements and typical sizes of models.

0.001 0.01 0.1 1.0 1 10 100 1000 10000

Horizontal length (m)

0.001

0.01

0.1

1

10

100

Laminae

Beds

Para-sequences

Geological model

Flow model

Core

Probe

Log

Seismic data

0.001 0.01 0.1 1.0 1 10 100 1000 10000

Horizontal length (m)

0.001

0.01

0.1

1

0

100

Laminae

Beds

Para-sequences

Geological model

Flow model

Geological model

Flow model

Core

Probe

Log

Seismic data

Core

Probe

Log

Seismic data

Verti

cal t

hick

ness

(m)

Figure 57a) 9-point scheme for 2D. b) 27-point scheme for 3D

Figure 58Length scales



For convenience, we consider upscaling as two separate stages (Figure 59). Stage 1 is upscaling from the smallest scale at which we may treat the rock as a porous medium (rather than a network of pores), up to the scale of the stochastic geologi-cal model, i.e. from the mm – cm scale to the m – Dm scale. Stage 2 is upscaling from the stochastic geological model to the full-fi eld simulation model, which has already been described.

Core plug

Stage 1Upscaling

Stage 2Upscaling

SedimentaryStructure

Geological Model~ 107 blocks

Simulation Model~ 104 blocks

3.3.1 The Geopseudo MethodUpscaling from the core-scale to the scale of the geological model (Stage 1 in Figure 59) is frequently ignored by engineers, who apply core plug permeabilities and “rock” relative permeability curves directly to the geological model. However, work carried out at Heriot-Watt University has demonstrated that small-scale structures, such as sedimentary lamination may have a signifi cant effect on oil recovery (Corbett et al., 1992; Ringrose et al., 1993; Huang et al., 1995). For example, in a waterfl ood of a water-wet rock, water is imbibed into the low permeability laminae, and oil may become trapped in the high permeability laminae.

The Geopseudo Method is an approach, where upscaling is carried out in stages, using geologically signifi cant length-scales (Figure 60). Models of typical sedi-mentary structures are created and permeability values are assigned to the laminae (from probe permeameter measurements, or by analysing core plug data). Relative permeabilities and capillary pressure curves are also assigned to each lamina-type (by history matching SCAL experiments on core plugs). Flow simulations are carried out to calculate the effective single-phase permeability and the two-phase pseudo parameters. Additional stages of modelling and upscaling may be required – e.g. upscaling from beds to bed-sets.

In the fi nest-scale model, the grid cells may be a mm cube, or less. If we upscale to blocks of 50 m x 50 m x 0.5 m, we are upscaling by a factor of at least 5 x 104 in the horizontal directions and 500 in the vertical.

Figure 59Two separate stages of upscaling. (Geological model taken from “Tenth SPE Comparative Solution Project: A Comparison of Techniques”, by Christie and Blunt, 2001.)

42


Low Perm High Perm

Individual Rel. Perm Curves

Pseudo Rel. Perm CurvesEffective Perm

3.3.2 Capillary-Dominated FlowAt small scales the fl ow is often capillary-dominated. Figure 24 showed a moderate capillary effect: the imbibition of water along the low permeability layer made the fl ood front approximately level in the two layers (instead being ahead in the high permeability layer, in the case of a viscous-dominated fl ood). In that model (Figure 21), the layers were 1 m thick, and the grid cells were 10 cm square. If the size of the model is reduced by a factor 100, so that the layers are 1 cm thick, and represent sedimentary laminae, the effects of capillary pressure are much stronger, as shown in Figure 61. In this case, strong capillary imbibition draws water into the low perme-ability layers (black) so that the front advances faster in this layer. Notice that, in this fi gure, there is little lateral variation in the shading, showing that the water saturation is almost constant in each layer. This is because the front has been spread out by the effects of capillary pressure, and the model is almost in capillary equilibrium.

Oil Saturation

0.50.40.3 0.6 0.7

When the fl ow is across the layers, as in Figure 62, the effects of capillary pressure are even more striking. This fi gure shows the same small-scale model of sedimen-tary lamination. When the injection rate is low (average frontal advance rate of 0.3 m/day), the fl ood is capillary-dominated (Figure 62a), water (black) has been imbibed into the low permeability layers leaving oil trapped in the high permeability laminae (grey). As the injection rate is increased, the oil has more viscous force and can overcome the capillary forces leading to less trapping of oil (Figures 61b). In the case of Figure 62c, the fl ood is viscous dominated and all the movable oil has been displaced by water.

Figure 60Illustration of the Geopseudo Method

Figure 61Example of capillary-dominated fl ood in a layered model



Oil Saturation

0.50.40.3 0.6 0.7

a) b) c)a) b) c)a) b) c)

The examples shown in Figures 61 and 62 demonstrate the signifi cance of capillary effects at the small-scale. When upscaling from the lamina-scale, these effects should not be ignored, and two-phase upscaling should be performed.

3.3.3 Geopseudo ExampleAll the upscaling methods described in the previous sections may be used in the Geopseudo approach, depending on the type of heterogeneities and the fl uids fl owing – averaging, single-phase numerical methods, two-phase steady-state methods, or two-phase dynamic methods. Since a fl ood is often capillary-dominated, as shown above, steady-state upscaling using the capillary equilibrium method is often appro-priate. Two-phase dynamic upscaling may also be used, and we show an example of the Kyte and Berry method below.

Figure 63 shows a model of sedimentary ripples. Kyte and Berry pseudos were cal-culated for the model using four different fl ow rates. There is a factor of 10 between each fl ow rate, with rate 1 being the fastest. Figure 64 shows the resulting pseudos (from Pickup and Stephen, 2000).

200 mD

10 mD

1 cm,18 cells

3 cm, 54 cells

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Water Saturation

rate 1rate 2rate 3rate 4

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Water Saturation

rate 1rate 2rate 3rate 4

Rel

ativ

e P

erm

eab

ility

Rel

ativ

e P

erm

eab

ility

Figure 62Examples of across-layer fl ow. a) capillary dominated, b) intermediate, c) viscous-dominated

Figure 63A model of ripples (based on the Ardross Outcrop, near St. Monance in Fife, Scotland)

Figure 64Pseudo relative permeabilities for different fl ow rates, for oil (left) and water (right)

44


Note the following:

1. At high rates, the pseudos are shifted to the right. This is to compensate for numerical dispersion.

2. At very low fl ow rates (rate 4), the fl ood is capillary-dominated, and the oil is trapped. The pseudo oil relative permeability goes to zero around S

w = 0.46.

3.3.4 When to use the Geopseudo MethodGeopseudo upscaling may be time-consuming, and there is no point in upscaling from the smallest scales, unless cores are available for the fi eld. Cores must be studied to identify the sedimentary structures present, and probe permeability measurements should be taken to populate the small-scale models. Additionally, reliable SCAL data is also required.

Ringrose et al. (1999) give a list of guidelines for when Geopseudo upscaling may be necessary:

1) Are immiscible fl uids fl owing?2) Are signifi cant small-scale heterogeneities present? Specifi cally: • Is the permeability contrast greater than 5:1? • Is the layer thickness less than 20 cm? • Is the mean permeability less than 500 mD?3) What is the large-scale structure of the reservoir? In many cases, large-scale

connectivity may be the dominant issue, in which case, small-scale structure may have to be ignored. The Weber and van Geuns (1990) classifi cation may be used to describe the large-scale structures:

• Layer cake reservoirs – small-scale structure will usually have primary importance.

• Jigsaw puzzle reservoirs – small-scale structure may be important.• Labyrinth reservoirs – small-scale structure will usually be of secondary

importance.

3.4 Uncertainty and UpscalingDuring the 1990s, reservoir modelling developed (along with computing power) so that geologists could create models containing millions of grid cells. Such models are often time-consuming to generate, and only a few are created for each reservoir. These detailed models are too large for full-fi eld fl ow simulation, and must be upscaled to reduce the number of cells to, about 104 or 105. Research into upscaling has focussed on trying to develop methods to accurately upscale these types of models.

However, it is now recognised that there are many uncertainties in the reservoir modelling, and instead of concentrating on a few detailed models, geologists and engineers are starting to generate thousands of models in order to characterise the effects of uncertainty. These models must be coarse so that the simulations can run very quickly.

These changes mean that, in future, people are less likely to follow the “traditional” upscaling approach. However, if the effects of fi ne-scale structure are ignored, this



will lead to errors in the predicted recovery. It is therefore very important to understand the effects of possible sub-grid heterogeneity on absolute and relative permeability, and to include these effects, when necessary. This is an area of active research at Heriot-Watt University.

3.5 Upscaling SummarySeveral reviews have been published on upscaling. These give an overview of some of the methods described in this chapter: e.g. Christie (1996), Renard and Marsily (1997) and Christie (2001).

Here is a summary of the main points:

• The effective permeability of simple permeability models (layered or random) may be calculated using averaging.

• In general, effective permeability should be calculated using a numerical simulation, along with suitable boundary conditions.

• Permeability upscaling is often inaccurate, particularly when the coarse cell size is comparable, or slightly larger than the correlation length of the permeability distribution, and when the standard deviation is large.

• Usually only single-phase upscaling is used in two-phase systems. However, this can give rise to errors, especially when the scale-up factor is large and when the standard deviation of the permeability distribution is large.

• Single-phase upscaling for two-phase systems may be made more accurate by using a non-uniform coarse grid, or by using the Well Drive Upscaling method, which increases the accuracy of single-phase upscaling by using the “correct” boundary conditions.

• The capillary equilibrium method is useful, particularly for small-scale models. It is feasible, even for models with a relatively large number of grid cells.

• Two-phase dynamic upscaling methods should be able to reproduce two-phase fl ow on a coarse scale. The Kyte and Berry (1975) method is an example of a pressure averaging approach.

• In general, two-phase upscaling is diffi cult to apply. It is more time consuming than single-phase upscaling, and the results are not robust (negative or infi nite values may be obtained).

• The regions around wells should be treated as a special case, because the fl ow is radial. The well index and the transmissibilities around the well block should eb upscaled.

• Permeability is actually a tensor quantity (4 terms in 2D, 9 terms in 3D). Full tensors may be used to take account of cross-fl ow within a grid cell. However, in general, only the diagonal terms are used (k

xx, k

yy, k

yy, k and k

zz, often referred to

as kx, k

y, k

y, k and k

z).

46


• It is important to take account of small-scale (mm – m) heterogeneity in some reservoirs. This may be done using the Geopseudo Method, in which models of sedimentary structures are generated and upscaled.

• Capillary effects are often signifi cant at small-scales, and it is important to take these into account using two-phase upscaling (steady-state or dynamic).



4 REFERENCES

Barker, J. W. and Thibeau, S., 1997. “A Critical Review of the Use of Pseudo Relative Permeabilities for Upscaling”, SPE Reservoir Engineering, May, 1997, 138-143.

Barker, J. W. and Dupouy, P., 1999. “An Analysis of Dynamic Pseudo-Relative Per-meability Methods for Oil-Water Flows”, Petroleum Geoscience, 5 (4), 385 - 394.

Christie, M. A., 1996. “Upscaling for Reservoir Simulation”, J. Pet. Tech., November 1996, 48, 1004-1008.

Christie, M. A., 2001. “Flow in Porous Media – Scale Up of Multiphase Flow”, Current Opinion in Colloid and Interface Science”, 6, 23 – 241.

Christie, M. A. and Blunt, M. J., 2001. “Tenth SPE Comparative Solution Project: A Comparison of Upscaling Techniques”, presented at the SPE Reservoir Simulation Symposium, Houston Texas, 11 – 14 February, 2001.

Corbett, P. W. M., Ringrose, P. S., Jensen, J. L. and Sorbie, K. S., 1992. “Laminated Clastic Reservoirs - The Interplay of Capillary Pressure and Sedimentary Architec-ture”, SPE 24699, presented at the 67th Annual Technical Conference of the SPE, Washington, DC, 4 - 7 October, 1992.

Darman, N. H., 2000. “Upscaling of Two-Phase Flow in Oil-Gas Systems”, Ph.D. Thesis, Heriot-Watt University.

Darman, N. H., Pickup, G. E. and Sorbie, K. S., 2002. “A Comparison of Two-Phase Dynamic Upscaling Methods Based on Fluid Potentials”, Computational Geosciences, 6, 5 – 27.

Durlofsky, L. J., Behrens, R. A., Jones, R. C. and Bernath, A., 1996. “Scale Up of Heterogeneous Three Dimensional Reservoir Descriptions”, SPEJ, 1, 313-326.

Durlofsky, L. J., Jones, R. C. and Milliken, W. J., 1997. “A Nonuniform Coarsening Approach for the Scale Up of Displacement Processes in Heterogeneous Porous Media”, Advances in Water Resources, 20, 335 – 347.

Durlofsky, L. J., Milliken, W. J. and Bernath, A., 2000. “Scaleup in the Near-Well Region”, SPEJ, 5 (1), 110 – 117.

Huang, Y., Ringrose, P. R. and Sorbie, K. S., 1995. “Capillary Trapping Mechanisms in Water-Wet Laminated Rocks”, SPE RE, 10 (4), 287 – 292.

Kyte, J. R. and Berry, D. W., 1975. “New Pseudo Functions to Control Numerical Dispersion”, SPEJ, August 1975, 269-276.

Peaceman, D. W., 1978. “Interpretation of Well-Block Pressures in Numerical Res-ervoir Simulation”, SPEJ, June 1978, 183-194.

48

Peaceman, D. W., 1983. “Interpretation of Well-Block Pressures in Numerical Reservoir Simulation with Nonsquare Grid Blocks and Anisotropic Permeability”, SPEJ, June 1983, 531-543.

Pickup, G. E. and Stephen, K. D., 2000. “An Assessment of Steady-State Scale-Up for Small-Scale Geological Models”, Petroleum Geoscience, 6 (3), 203 – 210.

Pickup, G. E., Ringrose, P. S. and Sharif, A., 2000.”Steady-State Upscaling: From Lamina-Scale to Full-Field Model”, SPEJ, 5 (2), 208 – 217.

Renard, P. and de Marsily, G., “Calculating Equivalent Permeability: A Review”, Advances in Water Resources, 20 (5/6), 253 – 278.

Ringrose, P. S., Sorbie, K. S., Corbett, P. W. M. and Jensen, J. L., 1993. “Immiscible Flow Behaviour in Laminated and Cross-bedded Sandstones”, J. Petroleum Science and Engineering, 9(2), 103-124.

Ringrose, P. S., Pickup, G. E., Jensen, J. L. and Forrester, M. M., 1999. “The Ardross Reservoir Gridblock Analog: Sedimentology, Statistical Representivity, and Flow Upscaling”, in Reservoir Characterization – Recent Advances, eds R. Schatzinger and J. Jordan, AAPG Memoir 71, p 256 – 276.

Stone, H. L. 1991. “Rigorous Black Oil Pseudo Functions”, SPE 21207, presented at the 11th SPE Symposium on Reservoir Simulation, Anaheim, CA, February, 17-20, 1991.

Weber, K. J. and van Geuns, L. C., 1990. “Framework for Constructing Clastic Res-ervoir Simulation models”. JPT, October 1990, p 1248 – 1297.

Zhang, P., Pickup, G. E. and Christie, M. A., 2005. “A New Upscaling Approach for Highly Heterogeneous Reservoirs”, presented at the SPE Reservoir Simulation Symposium , February 2005.

1 INTRODUCTION

2 MODELLING SINGLE-PHASE FLOW AT THEPORE-SCALE - A BRIEF OVERVIEW

2.1 Deviations from Darcy’s Law2.2 Empirical Models2.3 Probabilistic Models2.4 Capillary Bundle Models2.5 First Principles Derivation of Carmen-

Kozeny Model2.6 Network Modelling Techniques

3 MODELLING MULTIPHASE FLOWAT THE PORE-SCALE3.1 Capillary Pressure — What Does it Mean

and When is it Important?3.2 Steady-and Unsteady-State Flow3.3 Drainage at the Pore-Scale3.4 Imbibition at the Pore-Scale3.5 The Pore Doublet Model3.6 Introduction to Percolation Theory3.7 Network Modelling of Multiphase Flow

4 EXPERIMENTAL DETERMINATION OFPETROPHYSICAL DATA

4.1 Laboratory Measurement of CapillaryPressure

4.2 Laboratory Measurement of RelativePermeability

5 EMPIRICAL AND THEORETICAL APPROACHESTO GENERATING PETROPHYSICALPROPERTIES FOR RESERVOIR SIMULATION5.1 Methods for Generating Capillary PressureCurves and Pore Size Distributions5.2 Methods for Generating Relative

Permeabilities5.3 Hysteresis Phenomena

6 WETTABILITY - CONCEPTS ANDAPPLICATIONS6.1 Introductory Concepts6.2 Wettability Measurement and Classification6.3 The Impact of Wettability on PetrophysicalProperties6.4 Network Modelling of Wettability Effects

7 CONCLUDING REMARKS

8 APPENDIX A: Some Useful Definitionsand Concepts

9 APPENDIX B: Unsteady-State RelativePermeability Calculations

10APPENDIX C: Details of the Heriot-Watt MixWetSimulator

88Petrophysical Input

2

LEARNING OBJECTIVES

Having worked through this chapter the students should be able to...

Modelling Single Phase Flow• Appreciate the different types of model used to predict single phase permeability• Use a Carman-Kozeny equation to calculate absolute permeability given values

for the remaining variables

Modelling Multiphase Flow• Explain the meaning of capillary pressure and use Laplace’s equation to relate

capillary pressure to pore entry radius, contact angle and interfacial tension• Identify the difference between steady- and unsteady-state flow• Describe the pore-scale physics characterising drainage processes in porous

media• Describe the pore-scale physics characterising imbibition processes in porous

media• Describe a network model and explain how it can be used to investigate

multiphase flow in porous media

Experimental Determination of Petrophysical Data• Identify several different methods for measuring capillary pressure and relative

permeability

Generating Petrophysical Properties for Reservoir Simulation• Calculate oil-water capillary pressure curves from mercury injection data• Derive a pore size distribution from mercury injection data• Generate several capillary pressure curves from a single curve using a Leverett-

J function• Determine the Brooks and Corey _-parameter from capillary pressure data and

use this to predict relative permeabilities given the relevant equations• Identify three causes of capillary pressure hysteresis

Wettability - Concepts and Applications• Explain how wettability variations affect waterflooding at the pore-scale• Identify two wettability measures used routinely in industry• List “Craig’s Rules of Thumb” in the context of water-wet and oil-wet relative

permeability curves

3Institute of Petroleum Engineering, Heriot-Watt University


1 INTRODUCTION

Outline of the purpose of this chapter:

• To inform the student of the types of petrophysical data that are used in reservoirsimulation (k, φ, k

ro, k

rw, P

c). k and φ are generally measured or determined by

correlation or model, so, the central focus here will be on multi-phase properties (kro,krw, Pc);

• To briefly review relative permeability and capillary pressure are measuredexperimentally.

• To explain the underlying pore-scale physics of two-phase flow and show how thisbehaviour leads to the results we see at the macroscopic scale.

• To review empirical and theoretical models used to generate relative permeabilitiesand their application in Reservoir Simulation;

• To review wettability measures (such as USBM and Amott tests) and to explainhow wettability modifies relative permeability and capillary pressure.

The notion of a "porous medium" immediately conjures up an intuitive picture: putin its crudest terms, a porous medium may be thought of as a solid with holes in it.Unfortunately, such a superficial definition is of little use when trying to describe suchmaterials objectively, and a more precise formulation must be attempted. A cylindricalpipe, for example, would not generally be considered a porous medium, nor would asolid containing isolated holes. There is a tacit understanding that "real" porous mediashould be capable of sustaining fluid transport, implying a certain degree ofinterconnectedness within the underlying pore structure. In short, a truly porousmaterial should have a specific permeability associated with it.

There are countless examples of porous materials in everyday life, each with its ownparticular pore structure and transport potential. These range from leather, wood,paper and textiles, to bricks, concrete and sand; even animal tissue and bones containintricate pore networks. The need to understand such a vast array of permeablematerials has consequently fostered a great deal of scientific interest from manydiverse fields: soil mechanics, groundwater hydrology, industrial filtration, andpetroleum engineering, to name but a few. Although the theme of fluid flow throughporous media is a common feature of all of the disciplines listed above, each has itsown technical terminology associated with the subject. For example, "dewetting","desaturation" and "drainage" are all terms synonymous with the displacement of awetting phase from the interstices of a porous material by a nonwetting phase.Throughout this section, however, the terminology used will be that generallyencountered in the petroleum industry. A variety of fundamental concepts relating toboth pore structure and solid/fluid interactions can be found in Appendix A.

4

2 MODELLING SINGLE-PHASE FLOW AT THE PORE-SCALE- A BRIEF OVERVIEW

Examination of any photomicrograph immediately demonstrates why the modellingof fluid flow through a porous medium is such a formidable task: the underlying porestructure is extremely complex, with tortuous channels embedded throughout thesolid matrix. Nevertheless, over the years there have been many attempts toencapsulate this structure into a simple, idealised analogue. Such models cangenerally be divided into four broad categories; (i) those which attempt to reduce theporous medium to a single representative conduit, (ii) probabilistic models, (iii)empirical correlations, and (iv) network models, where the medium is approximatedby a lattice of connected conduits with distributed radii. A brief discussion of suchanalogues will be presented below; more detailed descriptions can be found in themonographs of Scheidegger (1963) and Dullien (1979).

2.1 Deviations from Darcy’s LawAlthough Darcy’s Law has been validated countless times experimentally, we shouldnevertheless be aware of some possible difficulties. For example, liquid permeabilitiescan be greatly affected by clay distribution, brine composition, and brine pH. This isevident in Table 1, which shows variations in absolute permeability with increasedbrine salinity — in some cases, decreased salinity leads to a decreased permeabilityestimate, whilst other samples exhibit the reverse behaviour. There is also experimentalevidence for so-called lubrication effects, where oil permeability measured at Swiactually exceeds k

abs (this is rather surprising when one considers the fact that isolated

water islands within a sample should actually form effective baffles to flow).

Field Zone Ka K1000 K500 K300 K200 K100 KW

S 34 4080 1445 1380 1290 1190 885 17.2 S 34 24800 11800 10600 10000 9000 7400 147 S 34 40100 23000 18600 15300 13800 8200 270 S 34 39700 20400 17600 17300 17100 14300 1680 S 34 12000 5450 4550 4600 4510 3280 167

S 34 4850 1910 1430 925 736 326 5.0 S 34 22800 13600 6150 4010 3490 1970 19.5 S 34 34800 23600 7800 5460 5220 3860 9.9 S 34 27000 21000 15400 13100 12900 10900 1030 S 34 12500 4750 2800 1680 973 157 2.4

S 34 13600 5160 4640 4200 4150 2790 197 S 34 7640 1788 1840 2010 2540 2020 119 S 34 111000 4250 2520 1500 866 180 6.2 S 34 6500 2380 2080 1585 1230 794 4.1 T 36 2630 2180 2140 2080 2150 2010 1960

T 36 3340 2820 2730 2700 2690 2490 2460 T 36 2640 2040 1920 1860 1860 1860 1550 T 36 3360 2500 2400 2340 2340 2280 2060 T 36 4020 3180 2900 2860 2820 2650 2460 T 36 3090 2080 1900 1750 1630 1490 1040

Table 1Effect of water salinity onpermeability of naturalcores (grains per gallon ofchloride ion as shown). K ameans permeability to air;K500 means permeabilityto 500 grains per galchloride solution; K wmeans permeability tofresh water



There are two other well-known limitations of Darcy’s Law:

(i) At high injection rates, inertial effects can become important and Darcy’s Lawshould be replaced with the Forcheimer Equation:

dpdx

a q b q= +µ ρ 2

where η is the fluid viscosity and the second term on the right-hand side correspondsto inertial effects.

(ii) Low pressure gas measurements must be corrected by using the KlinkenbergEquation:

k kpapp = +

1α

where kapp

is the apparent (measured) permeability, k the actual permeability, p the gaspressure, and α a parameter that depends upon the properties of the gas being used.The correction is needed because the mean free path of a gas molecule at low pressureis of the order of a pore radius and the continuum concept begins to break down.

2.2 Empirical ModelsThere have been numerous attempts to derive correlations between permeability andother sample properties, such as porosity, capillary pressure, grain size distribution,and electrical properties. The porosity/permeability relationship has perhaps been themost widely studied (see, for example, Mavis and Wilsey, 1936; Büche, 1937; Rose,1945; Habesch, 1990), but correlations vary so much that no universally acceptedformula can be adopted successfully. Consequently, most empirical correlationscontain "geometrical factors" which serve to fit experimental measurements withoutany real consideration of the underlying pore structure.

2.3 Probabilistic ModelsAs their name suggests, probabilistic models involve the use of some kind ofprobability law. One of the most popular of these is the "cut-and-random-rejoin"model of Childs and Collis-George (1950), which has been further extended byMarshall (1958) and Millington and Quirk (1961). The underlying theory involves thesectioning of the porous sample into two parts perpendicular to the direction of flow.These are then joined together again in a random fashion, and statistical analysis isused to approximate the permeability of the subsequent hybrid.

2.4 Capillary Bundle ModelsCapillary bundle models characterise the porous medium using systems of capillarytubes with well-defined properties (Figure 1). Each different model contains tubeswith different characteristics: they may be uniform and identical, for example, oruniform but with distributed radii, or periodically constricted and identical, etc.

6

k = φ=

D2

f(D) dD / D2

f(D) dD / D6

2D2

D 2

s2

+ σ 2

f(s)ds

Dmax

Dmin

D

32

k = φD2

96

k =96

φ∫ ∫∫

< >[ ]

a b c

If all tubes are identical (diameter=D) and lie parallel to the flow direction, then acombination of Poiseuille's law and Darcy's law gives:

kD= φ

2

32(1)

as a permeability predictor, where φ is the porosity (see Appendix B for mathematicaldetails).

An interesting aspect of this simple result is that the quantity (k/φ)1/2 can be thoughtof as a sort of average pore diameter.

2.5 First Principles Derivation of Carmen-Kozeny ModelThis popular approach, based upon hydraulic radius theory, relies upon two mainassumptions: (i) that a porous medium can be adequately characterised by a singletortuous channel having a characteristic radius, usually called the hydraulic radius,and (ii) that the effect of the interconnected pore structure can be contained within anempirical constant known as the tortuosity factor. The mathematical details of thisapproach are given in Appendix B.

Carmen-Kozeny Model - One of the more notable correlations based upon conduitflow is that developed by Carmen and Kozeny (Carman, 1937, 1938, 1956; Kozeny,1927). The basic premise of this modelling approach is that particle transit times in theactual porous medium and the equivalent tortuous rough conduit must be the same.After some analysis (see Appendix B), we arrive at the relationship:

kT Ss

=−φ

φ

3

2 2 22 1( ) (2)

See Appendix A for definitions

The permeability can be written in terms of an average grain diameter (Dp) by noting

that, for spherical particles, Ss=6/D

p. Hence,



kD

Tp=−

φφ

3 2

2 272 1( )(3)

Although a whole family of similar models exist, they differ only in the method ofcalculating an hydraulic radius R

H and shape factors.

2.6 Network Modelling TechniquesModels that account for the interconnected nature of porous media constitute a groupof analogues which can truly be referred to as network models. Here, the medium ismodelled using a system of interconnected capillary elements, which generallyconfigure to some known lattice topology. A variety of lattices are shown in Figure2. Although these network structures are somewhat idealised, the capillary radii areassigned randomly from a realistic pore size distribution in an effort to partiallyreconstruct the actual porous medium under investigation.

Hexagonal Square Kagome

Triagonal Cubic Crossed square

A fully interconnected network was first used by Fatt (1956) for primary drainagestudies. Although the two-dimensional lattice used was extremely small (200-400tubes), the novelty of the approach encouraged great interest in the subject, andimprovements on Fatt's primitive model were soon forthcoming (Rose ,1957; Dodd& Kiel ,1959). However, simulations using small lattices could never hope to capturethe full behaviour of microscopic flow processes. With the advent of high speedcomputers, however, much larger 3D systems can now be constructed, with the resultthat microscopic flow behaviour can be more accurately simulated. Figure 3 showsa capillary dominated drainage process using an 80 x 80 square lattice, the details ofwhich will be presented later: the resulting picture is somewhat more reassuring. Thefractal capillary fingering is in excellent agreement with experimental observation(see Lenormand et al, 1988).

Figure 2

8

Only recently has it been computationally feasible to carry out studies using largethree-dimensional networks (Figure 4).

The Basic ModelMany network models attempt to distinguish between "pores" and "throats", bybuilding networks consisting of hollow spheres connected by thin capillary tubes. Insuch models, all of the liquid volume is assumed to be contained in the spherical poreswith pressure differences being maintained by the throats (Lenormand, 1986). Theapproach taken here is somewhat more straightforward. The porous medium isinitially modelled using a three-dimensional cubic network of what will be referredto as pore elements; unlike many previous studies, no distinction is made here betweenpores and throats. The model consists of a three dimensional cubic network of

Figure 3

Figure 4



capillary elements. This simple lattice has dimensions Nx x N

y x N

z where N

x, N

y, N

z

are the number of nodes in the x, y and z directions respectively. Periodic boundaryconditions are assumed in the y and z directions in order to simulate larger systems andeliminate surface effects. The pressure gradient is taken to be in the x direction.

Now, for a single element of radius r and length L, the flow Q is given by Poiseuille’slaw:

Qr P

L= π 4

8µ∆

(4)

where µ is the viscosity and ∆P the pressure difference acting across the capillary. Ateach node (i, j, k), the sum of the flows Q

i must add up to zero (conservation of mass),

and so:

Qii=∑ =

1

6

0

Consideration of the whole network leads to a set of Nx.N

y.N

z linear pressure

equations, the solutions to which can then used to calculate the elemental flows.Summing the outlet flows yields the total network flow, which can then be substitutedinto Darcy's equation to give a value for the total network permeability. If the systemcontains more than one fluid, then the process is carried out for each fluid in turn; theresulting phase conductivities now being referred to as effective permeabilities. Themodelling of multiphase flow is discussed more fully in later sections.

3 MODELLING MULTIPHASE FLOW AT THE PORE-SCALE

3.1 Capillary Pressure - What Does it Mean and When is it Important?For many, the term "capillary pressure" is a rather difficult concept to grasp, especiallyin the context of a flowing hydrocarbon reservoir. For example, we could be shownthe schematic in Figure 5(a) and wonder why no oil flow occurs even though P

oil>P

water.

Athough capillary pressure may seem problematic, we shall soon see that it is actuallya very straightforward measure to interpret.

10

Poil>Pwater - why no flow?

P0

rPi

Oil Water

Projectedarea= πr2

(b)

(a)

(c)

(Pi - P0)πR2 = 2πRΣ

=> (Pi - P0) =2ΣR

Let us first consider a rubber balloon that has been inflated to a certain pressure (Pi)

and then tied (we will assume that this "experiment" is taking place in an atmosphereat pressure P

o, usually atmospheric pressure). If a force balance is considered for one

half of the balloon (Figure 5(c)), then we can show that, at equilibrium, the elasticforce acting around the circular perimeter of the balloon must counterbalance thedifference in pressure projected onto the shaded cross section.

This leads to a relationship between the pressure difference across the balloon surfaceand the radius of the balloon:

( )P PRi o− = 2Σ

(5)

where Σ is the elastic tension characterising the balloon wall (dimensions of Force/Length). Here is an example where a pressure difference exists between two regionsof fluid but no flow occurs — the elastic membrane of the balloon counteracts this.Now, instead of thinking of a rubber balloon (where Σ is actually a function of R itself),we can carry out a similar analysis for a gas bubble at pressure P

gas floating in oil at

pressure Poil

. We can now write immediately:

( )P PRgas oil

go− =2σ

(6)

where σgo

represents the interfacial tension between gas and oil. This relationship tellsus that the pressure difference across a small spherical bubble is larger than that acrossa large spherical bubble.

This type of analysis can be applied to the more general case of an interfacecharacterised by two different principal radii of curvature (eg a sausage-shapedballoon Figure B2). The details are slightly more involved (see Dullien (1979), butthe final result is simply:

Figure 5



( ) ( )P PR Rgas oil go− = +σ 1 1

1 2(7)

where R1 and R

2 are the principal radii of curvature characterising the interface.

Return now to the idealized (and some what unrealistic) situation where we have twofluids at equilibrium in a capillary tube separated by a curved interface (this curvedinterface appears at the microscopic scale when one of the fluids preferentially wetsa solid surface, Figure 6). The pressure difference across the fluid-fluid interface isknown as the capillary pressure. Note the difference between tube radius (r

A) and the

interface radius (RA) when the contact angle is non-zero.

RA rA

θ

A little bit of trigonometry can be used to derive an equation for two fluids atequilibrium in a circular cylinder, known as the Young-Laplace equation. This relatesthe pressure difference across a curved interface (i.e. capillary pressure P

c) in terms

of the associated contact angle, interfacial tension and pore (tube) radius:

P P PR R Rc gas oil go

go= − = + =( ) cos ( )cos

σ θσ θ1 1 2

(8)

Although we have used gas displacing oil in our example, results for any nonwettingfluid displacing a wetting fluid can be inferred immediately.

Another consequence of the equation is that larger pressure differences are needed fora nonwetting phase to displace a wetting phase from smaller tubes ((P

gas-P

oil)∝1/R).

We can therefore go on to examine the very simple drainage process shown in Figure7, where oil (dark) displaces water (light) from a set of parallel tubes (R

1>R

2>R

3) as

oil pressure is gradually increased. Once the oil pressure exceeds the water pressureby an amount 2σ

owcosθ/R

1, then the largest tube fills and the system settles down to

a new equilibrium configuration. Subsequent displacements occur at (Poil

-P

wat)>2σ

owcosθ/R

2, and (P

oil-P

wat)>2σ

owcosθ/R

3. The corresponding plot of water

saturation vs capillary pressure is shown in Figure 8. This could be thought of as a verybasic capillary pressure curve and we will return to this concept later.

Figure 6

12

P1 P2

P3 P4

Sw

Pc

2σ cos θR3

2σ cos θR2

2σ cos θR1

Simulators use capillary pressure curves to relate oil and water pressures at a givenwater saturation (see earlier chapters regarding this). Generally, capillary pressure ismostly important at the small scale (~cm), although it should also be included inreservoir-scale models involving transition zones.

3.2 Steady-and Unsteady-State FlowThe aim of this section is to broaden the understanding of two-phase flow in porousmedia; more specifically, to the simultaneous flow of oil and water through reservoirrock. To this end, concepts from percolation theory will be introduced towards the endof this section, where the physical porous medium will be approximated using aninterconnected capillary network and each element may contain a different fluid:either water or oil. Before dealing directly with percolation issues, however, it will bebeneficial to first discuss some of the more general aspects of two-phase flow, suchas phase distributions and relative permeabilities.

When dealing with any process involving multiphase flow, it is important to distinguishbetween steady-state and unsteady-state regimes (Figure 9). With regard to flow inporous media, the former is characterised by phase saturations that are invariant withtime; that is, the volume flux of each fluid entering the system is the same as thatleaving. Experimentally, this would be achieved by fixing the inlet flows of oil andwater at a certain ratio and leaving the system to equilibrate (such that the individualfluid fluxes exiting the sample are the same as those entering). There are consequentlyno pore-scale displacements of any kind once steady-state has been reached - each

Figure 7

Figure 8



phase tends to flow within its own tortuous network of pores. Once measurementshave been completed at this ratio of fluxes, a new ratio could be set and the processrepeated.

Core-Scale

Steady-state Unsteady-state

Steady-state Unsteady-state

Pore-Scale

In contrast to this, unsteady-state flow is accompanied by almost continuous saturationchanges, implying continuous displacement of one fluid by another. This would occurif one fluid was injected into a sample containing a second "resident" phase. Note thatthe two flow regimes are seldom independent, however, as steady-state conditions areonly achieved once some degree of transient unsteady-state displacement has takenplace to redistribute the phases.

Although the "channel flow concept" of steady-state flow outlined above appears tobe generally valid (Craig, 1971), there are certain conditions under which thisassumption may be questioned. If there is no strong wetting preference for the matrix,for example, or if the interfacial tension between the two fluids is very small, then aslug-like flow regime may develop. Alternatively, if the flow channels are rough andhave an irregular cross-section (which is almost always the case in natural rockmaterial), then the wetting phase will tend to line the channel walls and the nonwettingphase will usually reside in the centre of pores. Such phenomena can play a vital rolein determining phase distributions during a variety of displacements, and their effectscannot always be overlooked. We shall return to these Issues later.

3.3 Drainage at the Pore-ScaleIn order to better understand the following discussion on two phase displacements, webegin with the idea of a pore size distribution, PSD — a schematic representation ofthe range and frequency of pore radii characterising a given sample (Cf probabilitydistributions from statistics). A schematic PSD and 3D network are shown in Figure10, where the network of "pores" are simply capillary tubes of different radius (butequal length in this case). The radius r of each pore is the capillary entry radius (i.e.the radius characterising the pressure difference required for a nonwetting fluid toinvade a tube containing a wetting fluid, see Section 3.1).

We will now discuss drainage capillary pressure and relative permeability in twodifferent arrays of capillary tubes. First, we will return to a capillary bundle model andthen we will go on to examine drainage in an interconnected network.

Figure 9

14

Single Pore

PSD(r)

r

(a) (b)

Drainage in a Capillary BundleSuppose we now start with a water wet (θ = 0) porous medium - and return to the caseof an ideal parallel bundle of tubes - filled with 100% water (Sw = 1) and then considerthe physics of oil (the non-wetting fluid) displacing this water. For simplicity, we willtake a fully connected capillary bundle of tubes with a uniform PSD and a minimumradius, Rmin, and a maximum pore size, Rmax (Figure 11). Oil cannot spontaneouslyinvade water-wet pores and requires an increase in pressure for a displacement tooccur (see earlier discussion).

PSD

Rmin Rmax

f(r)

1

Rmax - Rmin

radius, r ->

The steps in a Primary Drainage process - and the corresponding drainage capillarypressures - would be as follows and these are illustrated schematically in Figures12(c). From the pore occupancies, we calculate the water saturation Sw by summingthe volume of the water-filled pores, divided by the volume of all pores. Similarly, wemay calculate the relative permeabilities (described in a later section).

Step 1: To enter the water filled porous network the oil pressure must be such that Po1

- Pw > Pc,1 = 2σ/σ/σ/σ/σ/Rmax - this is the minimum entry pressure before the oil can displacethe water from the largest pore where r = Rmax. Figure 12(a). Because Pc,1 is thesmallest of all entry pressures, oil enters the biggest pore first.

Figure 10

Figure 11



Step 2: The oil pressure increases such that, Pc1

< Pc2

= 2σσσσσ/r2 where r

2 < R

max. At this

higher capillary pressure, the oil displaces water from all the pores that have Rmax

>r > r

2. This leads to a finite oil saturation in the (fully accessible) capillary bundle.

Figure 12(b).

Step 3: The oil pressure increases again such that, Pc1 < P

c2 < P

c3 = 2σσσσσ/r

3 where r

3 <

r2 < R

max. At this even higher capillary pressure, the oil displaces more water from all

the pores that have Rmax

> r > r3. This leads to a increased oil saturation in the (fully

accessible) capillary bundle. Figure 12(c).

Step 4 (not shown): In principle, for a fully accessible system, we can increase thecapillary pressure to P

cmax = 2σσσσσ/ R

min and this would displace all the water with 100%

oil (So = 1).

For this Primary Drainage process, the corresponding drainage relative permeabilities(rel perms) are shown in Figures 13(a) – 13(c).

Step 1: At this point, there is only water flow (Sw = 1) and therefore krw = 1 and kro= 0. Figure 13(a).

Step 2: Now, the water saturation is Sw = S

w2 and k

rw falls quite rapidly since the water

is now flowing in the smaller pores. Correspondingly, kro rises more rapidly since theoil is flowing in the larger pores. Figure 13(b).

Step 3: The water saturation is now Sw = S

w3 and k

rw is relatively low since the water

is now flowing in the smallest pores. Again, kro rises very rapidly since the oil is

flowing in the larger pores as shown in Figure 13(c).

In fact, there are analytical expressions for the relative permeabilities of a capillarybundle model and these will be introduced later in the course. Also, for a capillarybundle, the sum of the relative permeabilities turns out to be unity, but this is a resultspecific to simple fully accessible models and is not of great importance for realporous media.

16

(a) Step 1: primary drainage, Pc1 = 2σ/Rmax

(b) Step 2: primary drainage, Pc1 < Pc2 = 2σ/r2 (r2 < Rmax)

(c) Step 3: primary drainage, Pc1 < Pc2 < Pc3 = 2σ/r3 (r3 < r2 < Rmax)

PSD

Rmin Rmax

f(r)

radius, r ->

water oil

r2

PSD

Rmin Rmax

f(r)

radius, r ->

water oil

Pc

Pc2 = 2σ

Sw -->

r2

PSD

Rmin Rmax

f(r)

radius, r ->

water oil

r3

Pc

Pc3 = 2σ

Sw -->

r3

Pc

Pc1 = 2σ

Sw -->

Rmax

Figure 12



(a) Step 1: corresponding drainage relative permeabilities

(b) Step 2: drainage relative permeabilities at Sw = Sw2

(c) Step 3: drainage relative permeabilities at Sw = Sw3

PSD

Rmin Rmax

f(r)

radius, r ->

water oil

r2

PSD

Rmin Rmax

f(r)

radius, r ->

water

Sw2

PSD

Rmin Rmax

f(r)

radius, r ->

water oil

r3

kr

krw

kro

Sw -->

kr

krw

kro

Sw -->

Sw3

kr

krw

kro

Sw -->

Drainage in connected networksDrainage physics: We now illustrate what happens when drainage occurs in aconnected network of pores such as that shown in Figure 10(a). Without loss ofgenerality we can again assume a uniform pore size distribution. The Young-Laplaceequation still applies and the steps shown in Figures 12 and 13 still broadly occur butthere are some important differences from the (fully connected) capillary bundlemodel as follows. Although a pore could be occupied by the oil (non-wetting phase)at a given capillary pressure, Pc1 say where Pc1 =2σσσσσ/r 1, there are two reasons why theoil may be prevented from invading this “target” pore:

Figure 13

18

(i) The particular pore may be inaccessible to the invading oil, i.e. the invading oilphase may not be able to "see" the pore of radius r1, (this would be the case if theinvading oil cluster had not yet reached the target pore). This is the issue ofaccessibility and a water filled pore may not be occupied by oil unless the Pc isabove the entry pressure and the oil can access the pore in question;

(ii) The water residing in the target pore could be trapped. Water can be trapped ina pore when two conditions are satisfied; (a) when. there is no chain of water-filledpores from the target pore to the outlet of the network (the target pore is thenhydraulically disconnected) and (b) when there are no wetting films coating thepore surfaces that could allow water to "leak away" from the pore. These films fillthe "corners" of pores as shown for the example of a simple triangular pore inFigure 14.

oil

water

The drainage process in an interconnected network as described above is illustratedin Figure 15 and it is known as invasion percolation (with or without trapping). In thisfigure, the oil (yellow) is displacing the water (blue) from the left. As noted above,this is governed by the Young-Laplace equation, but for the oil to displace the waterfrom a given pore, this pore must also be accessible. There are probably pores in thelarge areas of trapped water which are large enough to be occupied by oil but they areinaccessible.

A sequence of invasion percolation (drainage) calculations in a 2-D network showingthe fluid distributions are shown in Figure16 for co-ordination number, z = 2.667 andin Figure17 for z = 4 (here, red is oil, blue is water and spaces denote missing poresi.e. lower co-ordination number). The characteristics of the overall displacementpattern in these two cases are quite similar but in the lower z case, the initial percolatingcapillary finger of non-wetting phase is somewhat more "spindly" or "ramified"because of lower accessibility. Just at breakthrough, a spanning cluster of non-wettingphase is formed, i.e. a continuous cluster that goes from the inlet to the outlet. Thisspanning cluster has a flowing "backbone" but it also has some dead end branches.After, breakthrough, the main cluster of non-wetting phase continues to grow andwater is displaced. In the primary drainage simulations shown, water is first displaceddirectly - the water is displaced by oil and escapes through a direct route of bulk-filledpores to the outlet of the network. Later in the flood, at higher oil saturations, the oil"surrounds" some water filled pores, such that the water appears trapped (see forexample Figures 16 (d) and 17(d)). However, it is evident from the later figures(Figures 16 (e) and 17(e)) that some of this hydraulically disconnected water hasescaped. This water has escaped through the water films in the corners of the oil-filledpores (e.g. see the triangular pore in figure above).

Figure 14Invasion percolation whereoil (yellow) displaces thewater



(a) drainage; z = 2.667, So =0.1; (b) drainage; z = 2.667, So =0.3;

(c) drainage; z = 2.667, So =0.5; (d) drainage; z = 2.667, So =0.7;

(e) drainage; z = 2.667, So =0.9.

Figure 15Invasion percolation whereoil (yellow) displaces thewater

Figure 16Oil/water drainage floodat various stages in a 2-D(20 x 20) network for awater wet system withcoordination number, z =2.667. This is essentiallyinvasion percolation withperiodic boundaryconditions (i.e. if theinvading oil leaves the topof the network, it re-enters at the bottom).Observe that water filmsin oil-filled pores are notvisualised

20

(a) drainage; z = 4, So =0.1; (b) drainage; z = 4, So =0.3;

(c) drainage; z = 4, So =0.5;

(e) drainage; z = 4, So =0.9.

(d) drainage; z = 4, So =0.7;

Accessibility and the accessibility function A(r): The meaning of accessibility and thedefinition of the accessibility function, A(r), are shown in Figure 18, together with thepore filling sequence for a connected network. Compare this with the drainage processin a fully accessible (capillary bundle) model.

Figure 17Oil/water drainage floodat various stages in a 2-D(20 x 20) network for awater wet system withcoordination number, z =4. This is essentiallyinvasion percolation



p(r)

(a) (b)

(c) (d)

Accessibility A(r)=A2/(A1+A2)

Rmin

A1

A2

Rmaxr

Pores filled with Hg

Mercury

Air

10

13

4 1

8

3

6

5 2

11

12

7

9

Mercury

Air

10

13

4 1

8

3

6

5 2

11

12

7

9

Mercury

Air

10

13

4 1

8

3

6

5 2

11

12

7

9

Mercury

Air

10

13

4 1

8

3

6

5 2

11

12

7

9

Shielded

Shielded Shielded

The accessibility function is defined as A(r) = A2/(A1+A2) and we can explain thisconcept physically as follows..Consider a mercury (Hg)→ air invasion percolationprocess (Figure 18). For the mercury to displace the air from a given pore, this poremust be accessible. In the sequence of Hg-pore filling in (a) to (d), at certain stages(e.g. (c)) some pores are “shielded”, i.e. they are large enough to be invaded but cannot (yet) be seen by the invading mercury. How accessibility is related to theaccessibility function for a drainage displacement is shown in Figure 19(a) – 19(c).where the “low occupancy” value of the accessibility is seen to be 0 and the highoccupancy is 1 i.e. the phase is at a sufficiently high saturation that it can effectively“see” all pores that can be entered at a given (high) capillary pressure.

The underlying theory of the drainage process in an interconnected network is a topicknown as Percolation Theory (See Stauffer and Aharony, Introduction to PercolationTheory, 2nd Edition, Taylor and Francis, 1992) and we will give a fuller expositionof this topic later in this chapter.

Figure 18The idea of accessibilityand the accessibilityfunction for mercury(black) invasion into air

22

(a) Accessibility in a connected network - above percolation radius, rp

(b) Accessibility in a connected network - just below percolation radius, r < rp

(c) Accessibility in a connected network - well below percolation radius, r << rp

◆PSD

Rmin Rmax

f(r)

radius, r ->

water

rp

◆rp

Rmax > r > rp

A(r) A(r) = 0 for

r -->0

1

Accessibility function

Rmax

◆

rp

r < rp

A(r) A(r) > 0 for

r -->0

1


Rmax

◆PSD

Rmin Rmax

f(r)

radius, r ->

water

r < rp

Notoccupied= A1

Oil occupied= A2

◆

rp

r << rp

A(r) A(r) = 1 for

r -->0

1


Rmax

PSD

Rmin Rmax

f(r)

radius, r ->

water oil

r << rp

◆

Oil occupied= A2

(A1 = 0)=> A(r) = 1

a)

b)

Figure 19The meaning ofaccessibility and theaccessibility function,A(r), in a primary drainageprocess

Figure 20Comparison of a drainageflood (oil displacing water)in (a) a 2-D water wetglass micromodel whereoil is the light fluid and theetched “geometrical” porepattern can be seen; and(b) the corresponding 2-Dtheoretical network modelcalculation of the sameflood where oil is in lighterblue and the red pores arewater filled (McDougalland Sorbie, Heriot-WattU., unpublished)



A comparison of an experimental drainage flood (oil displacing water) in a glassmicromodel and the corresponding network model calculation is shown in Figure 20.Figure 20(a) shows the 2D water wet glass micromodel where oil is the light fluid andthe etched “geometrical” pore pattern can be seen. The corresponding 2D theoreticalnetwork model calculation of the same flood is shown in Figure 20(b) where oil is inred and the lighter blue pores are water filled. The agreement between these twofigures is sufficiently good to be confident that we have captured the main pore scalephysics of the drainage process in our network model.

3.4 Imbibition at the Pore-ScaleWe now consider waterflooding of the same strongly water-wet 2D network modelsused in the primary drainage processes described above (i.e. cosθθθθθ = 1 in all pores). Aprocess where the wetting phase increases such as water injection in this system isknown as imbibition . The pore scale physics of imbibition is not simply the reverseof the drainage process although there are some features that are common to each. Wenoted that in drainage (oil → water; or o→w), the oil displaced the water by a piston-like displacement mechanism governed by the Young-Laplace equation. At the porelevel, imbibition - or water displacing oil (w→o) in this case - can actually take placeby two distinct mechanisms viz. by piston-like displacement and by snap-off.

Piston-like displacement of oil by water is the reverse of drainage except that it occurswhen one of the phase pressures change such that Po - Pw < Pc = 2σσσσσ/r. The secondmechanism, snap-off, is associated with the flow of wetting phase (water) throughfilms, which swell around the oil in a pore to form a “collar” which eventually - at anappropriate capillary pressure - causes the oil to snap off thus occupying the space withwater. This snap-off process is shown schematically in Figure 21 for a single pore andin Figure 22 for a 2D interconnected network. A given waterflood will generallyconsist of a mixture of the two displacement mechanisms outlined above.

Non-wetting fluid

Wetting fluid

Σ

Swelling of wetting phaseto form "collar"

Figure 21

24

(a) (b)

(c) (d)

Water

Oil

10

13

4 1

8

3

6

5 2

11

12

7

9

Water

Oil

10

13

4 1

8

3

6

5 2

11

12

7

9

Water

Oil

10

13

4 1

8

3

6

5 2

11

12

7

9

Water

Oil

10

13

4 1

8

3

6

5 2

11

12

7

9

Trapped

How the snap-off process relates to oil recovery is shown schematically in Figure 23,which shows a “ganglion” of oil being snapped-off by water in a water-wet rock. Theoil could only escape through the connected cluster of oil filled pores (since there areno oil films in a water -wet porous medium). We also note that the isolated blob ofoil left behind in this process in Figure 23 is “residual oil” since it is “trapped” andcannot now move (unless viscous rather than capillary forces are invoked).

Without proof, we note here that the capillary pressure for snap-off is lower than thatfor piston-like displacement. Indeed, in a strongly water-wet circular capillary (cosθθθθθ=1), the snap-off capillary pressure is approximately, P

c = σσσσσ/r i.e. half the value for

piston like displacement. Hence, if a capillary is occupied by oil and the oil pressureis lowered, the capillary entry pressure for piston-like displacement will be reachedfirst and - if water is freely available, in adjacent pores for example - then piston-likedisplacement will occur first. On the other hand, if the capillary entry pressure dropsbelow the piston-like entry pressure but no water front is available, then it will not fillwith water. However, if the oil phase pressure, P

o, drops sufficiently (or P

w increases

sufficiently) that the snap-off capillary pressure is reached and there are water filmsto carry water to that pore, then snap-off will occur. Hence, imbibition is a morecomplex process than drainage and the balance between piston-like and snap-offevents that occurs depends on a range of factors such as the range of pore sizes, pore-geometry (aspect ratios), the connectivity of the network (z) and the presence/absenceof wetting films (wettability). Clearly, if we have a wide range of pore sizes, then aswe drop the oil phase pressure, P

o, the pore that fills next with water is that for which

Po - P

w < P

c; i.e. where the P

c refers to either P

c piston-like in an accessible pore or

snap-off in a smaller but isolated pore which can be supplied wetting phase throughfilms.

Figure 22Schematic of the snap-offmechanism in imbibition.A 3-D view of a pore withwetting films in thecorners is given in Figure14



Oil Trapping by Filament Snap Off

Flow

TrappedOil

this oil filament isunstable and "snaps"

oil escapesthroughcontinousoil phase

continous oil

SandGrains

Oil Trapping by Filament Snap Off

Flow

TrappedOil

"snap-off"

oil escapesthroughcontinousoil phase

continous oil

SandGrains

Such a model of imbibition has been implemented in the 2D network of water wetpores discussed above where we take:

=> Pc for piston-like displacement = 2σσσσσ/r

=> Pc for snap-off events = σσσσσ/r

The phase distribution patterns for imbibition in this network is shown for z = 2.667in Figures 24(a) - (d) and for z = 4 in Figures 25(a) - (d). Observe that in this case nodirection of flow can be observed, as pores fill simultaneously throughout thenetwork. In the two cases above, we can determine that the percentages of snap-offand piston like events in each flood are:

♦ z = 2.667 (Figure 24)Piston-like = 79 % Snap-off = 21 %

♦ z = 4 (Figure 25)Piston-like = 92.5% Snap-off = 7.5%

Figure 23Schematic of snap-off ofan oil ganglion within aporous medium

26

These results are as we expect since, if we considered a fully accessible capillarybundle model (as discussed earlier in this section), then only piston like displacementswould occur (for the reasons discussed above). For such a case, the imbibitioncapillary pressure for a parallel bundle of tubes would be identical to the drainagecurve i.e. no hysteresis should be observed. For z = 4, the system is moderatelyaccessible and hence there is only a small fraction of snap-off events (7.5%). For z =2.667, we find that the fraction of snap-off events grows to 21%. The corollary of thisis that the more snap-off events we observe in the imbibition process, the greatershould be the hysteresis between the primary drainage and the imbibition Pc curvesi.e. the lower the imbibition Pc curve should drop below the primary drainage curve.This is precisely what is seen in the simulations (Figure 26). Notice that there are moresources for hysteresis than just snap-off — the lower coordination number also leadsto more residual oil trapping and so magnifies the hysteresis effect.

Obviously, we can also derive the relative permeabilities for the drainage andimbibition processes in the connected networks (see later), which show hysteresiseffects similar to those for the Pc

curves of Figure 26.

(a) imbibition z = 2.667, Sw = 0.05; (b) imbibition, z = 2.667, Sw =0.15

(c) imbibition z = 2.667, Sw = 0.25; (d) imbibition, z = 2.667, Sw =0.281

Figure 24Oil/water imbuition floodat various stages in a 2-D(20 x 20) network for awater wet system withcoordination number, z =2.667



(a) imbibition z = 4, Sw = 0.05; (b) imbibition, z = 4, Sw =0.15

(c) imbibition z = 4, Sw = 0.25; (d) imbibition, z = 4, Sw =0.366

15000

10000

0000

00 10.25 0.750.5

Sw Sw

Pc(

Pa)

3

2

1

15000

10000

0000

00 10.25 0.750.5

Pc(

Pa) drainage

imbibition

In reality, of course, pore geometries in rock samples are far more complicated thanthose characterising capillary networks and imbibition is somewhat more complex(drainage is still relatively straightforward). Micromodel studies by Lenormand andco-workers (Lenormand and Zarcone, 1984) has uncovered other possible mechanisms,as illustrated in Figure 27. Their different mechanisms have been termed I1, I2, I3, etc,where the integer corresponds to the number of pores surrounding a junction that arefilled with nonwetting fluid. We can determine the stability of each meniscus in Figure27 as the capillary pressure is lowered — remember, that the imbibition process ischaracterized by increasing meniscus radii as the flood proceeds. Hence, as long asthe next stage of the waterflood results in an increase in meniscus radius, thedisplacement remains stable. So, in Figure 27(a) (an I1 mechanism), the displacement

Figure 25Oil/water imbuition floodat various stages in a 2-D(20 x 20) network for awater wet system withcoordination number, z = 4

Figure 26

28

begins steadily as the capillary pressure is decreased, with the meniscus movinggradually from position 1 to position 2 to position 3. However, the next step involvesthe meniscus leaving three—“grain” corners and the radius of curvature”decreases”—this is unstable and the water immediately flows to position 4 and continues to displaceoil from the corresponding pore. Similar reasoning holds for the case shown in—Figure 27(b) (an I2 mechanism) — here, however, the meniscus remains stable up toposition 4 and only becomes unstable after grain edge A has been reached and themeniscus separates. The relative importance of each mechanism during an imbibitiondisplacement once again depends upon pore geometry, pore size distribution, flowrateand supply.

x

1

1

1

15

2

3

4

5

x

1234

01 02

p0nw

p0w

(a)

(b) (d)

(c)

x' = √2x

x'

R

A

3.5 The Pore Doublet ModelThe simplest connected pore system is known as the pore doublet (see Appendix B,Figure B3). Here, the displacing fluid is introduced to the inlet of the doublet at acharacteristic flowrate q. When the displacing fluid is nonwetting, the widest branchof the doublet fills first as expected (lowest capillary entry pressure) and wetting phaseis trapped in the thin branch. The situation is far more interesting however when weconsider imbibition in the doublet. A little analysis shows that the frontal velocity ineach branch depends upon the ratio of branch radii (β), the supply rate (q), and adimensionless number known as the capillary number (N

vc=µLq/πR

13σcosθ), which

is a ratio of viscous to capillary forces. At low flow rates (small q and small Nvc), the

velocity ratio v2/v

1 shows that the thinnest tube fills first and the meniscus in the wider

tube actually retracts. At high rates (large Nvc), we see that v

2/v

1=R

22/R

12 and the wider

tube fills first. Fuller treatments of the problem can be found in Moore and Slobod(1956), Chatzis and Dullien (1983), Laidlaw and Wardlaw (1983), and Sorbie, Wu andMcDougall (1995).

Figure 27



3.6 Introduction to Percolation TheoryIn the previous two sections, the process of fluid flow in a porous medium has beenconsidered from the point of view of the fluid. However, it is also possible to considerthe process as being determined by the geometry of the porous medium itself. Oneapproach, which has since become known as percolation theory, was first used byBroadbent and Hammersley (1957) to investigate the flow of gas through carbongranules (for the design of gas masks to be used in coal mines). As well as describingthe flow of fluids and gases through porous media, their theory has subsequently beenused to describe many other diverse processes; such as, the electrical properties ofamorphous semiconductors, the behaviour of crystalline semiconductors containingimpurities, and magnetic phase transitions. Phenomena that are best described bypercolation theory are critical phenomena: characterised by a critical point at whichsome property of the system changes abruptly.

In the present context of fluid flow, percolation theory emphasises the topologicalaspects of problems, dealing with the connectivity of a very large number of elementalpores and describing the size and behaviour of connected phase clusters in a well-defined manner.

The primary focus of this section is to discuss how ideas from percolation theory canbe applied more specifically to flow in a porous medium. As a simple, instructiveanalogue, consider first a two-dimensional square lattice of capillary elements asshown in Figure 28(a). The most pertinent concepts from percolation theory will nowbe discussed using this geometry. Consider the critical behaviour of the network.Assume that, initially, all of the tubes are blocked and that they are then opened atrandom. For any given geometry there is a unique fraction of tubes that must be openbefore flow across the network can commence; this critical fraction is called thepercolation threshold (P

th) and for a simple square lattice has the value P

c=0.5 exactly

(Figure 28(b) shows the distribution of closed and open tubes and Figure28(c) showsthe flowing, spanning cluster of open tubes). One of the most incredible aspects of thisresult is that it is independent of the radius distribution; it only depends upon thetopological structure of the network (actually, the co-ordination number (z) and theEuclidean dimension (d)). In fact, the percolation threshold and system topology arelinked by the equation:

z Pd

dc.( )

=− 1

(9)

(see Stauffer and Aharony, 1992). Table B1 in Appendix B shows percolationthresholds for a variety of two- and three-dimensional geometries.

30

(a)

(b) (c)

Now, if instead of random opening, the pores are opened systematically beginningwith those of largest radius (top-down filling), it is clear that flow will commence oncea cluster of large open pores spans the system. The radius at which this occurs is knownas the percolation radius, R

p, and is defined implicitly by the equation:

P f r drth R

R

P= ∫ ( )max (10)

where f(r) is the normalised tube radius distribution function and Pc the percolation

threshold. However, the simulation of low-rate drainage processes is carried out usinga top-down invasion percolation model with hydraulic trapping of the wetting phase.In this case, the injected nonwetting phase first fills the largest pores connected to theinlet face of the network, and then proceeds along progressively narrower pathways,sequentially occupying smaller and smaller pores. Although this process appears tobe very different from the pure top-down pore filling, the resulting flowing clustersare, in fact, identical. Hence, the invasion percolation spanning cluster also appearsat R=R

p.

How does this apply to flow in porous media? Well, many imbibition and drainageprocesses exhibit critical behaviour: porous media contain clusters of oil-filled,water-filled and gas-filled pores and we would only see flow of a particular phasewhen the corresponding phase clusters span the system (Figure 29).

Figure 28



8 Isolated clusters3 Isolated clusters +1 spanning cluster

Moreover, we see no flow below certain critical saturations (Sor, Swi) and thesecritical saturations will largely be determined by the connectedness of the porousmedium under investigation (see equation 10 above, where P

th is a function of z).

3.7 A Brief Introduction to Network Modelling of Multiphase FlowThe last section described the concept of percolation theory. The aim of this sectionis to broaden the understanding of two-phase flow in porous media; more specifically,to the simultaneous flow of oil and water through reservoir rock. To this end, conceptsfrom percolation theory are utilised more fully. The physical porous medium is onceagain simulated using a capillary network, but now each element may contain adifferent fluid: either water or oil, according to the pore-scale physics of drainage andimbibition discussed earlier. In the parlance of percolation theory, this type ofanalogue is commonly known as a bond percolation model.

The porous medium is initially modelled using a three-dimensional cubic network ofwhat will be referred to as pore elements; unlike many previous studies, no distinctionis made here between pores and throats. We consider it debatable as to whatconstitutes a pore and a throat in a real porous medium, and propose a more abstractapproach based upon effective flow cylinders (the 3R’s approach; McDougall, 1994).In this formulation, which closely follows the analysis of Heibaet al (1982), the“radius”derived from the pore-size distribution is the radius governing the capillary entrypressure of the pore element and is related to the capillary pressure by:

p rrc ( )α 1

(11)

On the basis of geometrical analysis, Heiba et al postulated an approach for theestimation of volume and conductance of each pore element in the network. Theyreported that the volume, v(r), and conductance, g(r), of each pore element can berelated to the radius governing the pore capillary entry pressure by the followingrelationships.

v(r) r 0 3α νν ≤ ≤ (12)

Figure 29

32

g(r) r 1 4α λλ ≤ ≤ (13)

Different combinations of these exponents correspond to different “facies”; e.g.ν=3andλ=1 would be most appropriate for unconsolidated media, whilst ν=1 and λ=4would apply primarily to consolidated samples. These ideas are central to the 3R’sapproach. One can further dispense with pore/throat arguments by incorporatingeffective contact angles into the model formulation (see Dixit et al, 1997). Differentcontact angle ranges can be used to mimic the effects of different pore/throat aspectratios and the competition between snap-off and pistonlike displacement duringimbibition. In addition, the rich variety of hysteresis phenomena observed duringmultiphase flow in consolidated and unconsolidated porous media can be reproducedand interpreted.

When more than one fluid is flowing through a network, phase occupancy during agiven process (e.g. primary drainage, secondary imbibition etc.) may be characterisedby a set of rules (based upon the physics discussed earlier) which, when combined withtopological considerations (accessibility), give realistic saturation distributionsthroughout a displacement. This approach is particularly well-suited to the modellingof capillary-dominated flow. However, since any increase or decrease in thesaturation of a particular phase depends upon the spatial distribution of that phase, thecomputational effort saved in dispensing with unsteady state calculations is more thanoffset by the implementation of a clustering algorithm (after Hoshen and Kopelman,1976). This algorithm is essentially a “book keeping” exercise which locates andlabels the phase clusters which are distributed throughout the network. These clustersare continually changing their structure during a displacement, and so the efficacy ofthe simulation as a whole is intrinsically linked to the efficacy of the clusteringalgorithm itself. A great deal of time and effort has been invested in achieving theoptimum performance of this element of the network simulator. The precisecomputational details are dealt with more fully in McDougall (1994).

Although we have rules that tell us how a given displacement proceeds as a functionof capillary pressure, we still need a method of calculating the flow of each phase atany given stage. In fact, this turns out to be fairly straightforward: we can set a certaincapillary pressure in the model, determine the phase occupancies, effectively “freeze”each phase, and use the numerical approach from Section 2.6 to calculate the fractionalflows . This facilitates the calculation of effective permeabilities for the two intertwinednetworks (one for each phase) over a range of saturation values. Inherent in this is theassumption that the flow is stationary, thus steady-state relative permeabilities areconsidered here; for work on dynamic relative permeabilities see Blunt and King,1990. Measured relative permeabilities depend upon the saturation histories, saturationof the fluids, pore space morphology, wetting characteristics of the fluids, the ratio ofthe fluid viscosities and the capillary number. Many of these factors are alreadyincluded in the model and will be described later in Section 5.2 when we discuss theuse of network models in calculating capillary pressure curves and relativepermeabilities.

To summarise, network modelling is a powerful tool for increasing our understandingof multiphase flow in porous media ——the key element of such an approach lies inthe nterconnected nature of the underlying model. It is particularly well-suited to



qualitative sensitivity studies of petrophysical parameters (examining the effects ofconnectivity, pore size distribution, dead-end pore space, co-operative filling events,&c. upon P

c and K

rel), although more quantitative studies can also be considered.

z

y x

Imbibition

f(R)

R

0.02

0.015

0.005

0.01

20 40 50 80 100

Pores filledwith wetting

phase

Pores filledwith nonwetting

phase

Drainage

f(R)

R

0.02

0.015

0.005

0.01

20 40 50 80 100

Pores filledwith wetting

phase

Nonwettingphase

4 EXPERIMENTAL DETERMINATION OF PETROPHYSICAL DATA

4.1 Laboratory Measurement of Capillary PressureThere are a number of ways in which capillary pressure can be measured experimentallyand a few will be briefly discussed here — almost all rely upon highly specialisedequipment of varying degrees of sophistication.

Mercury Injection — Perhaps the most straightforward approach to capillary pressuremeasurement utilises a mercury porosimeter (although manual pumps are alsoavailable, Figure 31(a)(iii). Typically, a 1” diameter cylindrical plug of the poroussample is inserted into a glass penetrometer (essentially a close-fitting glasscontainer), which is then put into the machine, evacuated and contacted with mercuryat zero pressure. A table of pressure values ranging from 0-60,000psia is then inputinto the porosimeter and the pressure of the mercury in contact with the porous plugis raised sequentially in a number of discrete steps. The volume of mercury penetratingthe sample is measured at each pressure step (once a given equilibration interval haselapsed) and the resulting mercury-vacuum capillary pressure curve is output to a file.Mercury extrusion curves can be similarly measured, although the sub-atmosphericpart of the curve is difficult to obtain without experimental artefact. Note, that themercury-vacuum curve cannot be used directly in a reservoir simulator — it must bere-scaled appropriately (the procedure for doing this is described in section 5.1).

Porous Diaphragm — The porous diaphragm method is shown in Figure 31(a)(ii). Awater-filled porous plug is placed in an oil-filled (or gas-filled) container, with itslower face in hydraulic contact with a water-wet glass frit (a glass disc containing

Figure 30

34

micron-scale holes). The pressure in the surrounding nonwetting phase (oil or gas) isthen increased incrementally, and the displaced water leaves the container via the frit.Saturations are measured at each pressure (once the system has equilibrated) and theoil-water or gas-water P

c curve can subsequently be determined. Note that the

topology of the invading nonwetting clusters could be different from those obtainedduring mercury intrusion, as nonwetting fluid is unable to enter the lower face of thesample. Note also that a residual wetting phase saturation is also measured using thismethod, whereas mercury injection ends with the sample completely filled withmercury (there is no wetting phase in this case, as the sample is initially evacuated).

Unfortunately, the porous diaphragm method is rather slow (and therefore expensive),as equilibration times for an oil-water system are generally high. This problem can beameliorated somewhat by using water/air systems and rescaling the resulting data.One final point of note relates to the operating range of the experimental equipment.The maximum possible driving pressure is determined by the displacement pressureof the glass frit (diaphragm) upon which the sample is placed. Once the displacementpressure of the diaphragm has been exceeded, the wetting phase is no longer able to leavethe sample and the experiment ends (Figure 31(b)).

Centrifuge Method — The centrifuge method is quick but relies on highly specialistequipment and analytical treatment of the data. The sample is put into a centrifugesurrounded by a displacing phase (Figure 31(a)(ii)). The centrifuge is spun at differentspeeds and the volume of displaced fluid measured on each occasion. Whilst anaverage capillary pressure can be inferred corresponding to each rotational speed, thelocal capillary pressure and saturation values vary with distance from the centre ofrotation. Hence, some analysis is required to back-out an appropriate average Pccurve.

Dynamic Method — This method is used in conjunction with steady-state relativepermeability measurements (see below). Here, semi-permeable membranes can beused to measure wetting-phase and non-wetting phase pressures independently at anumber of different fixed fractional flows. Hence, a direct measure of P

c can be

achieved over a range of saturation values.



0-200 psi Pressure Guage

0-2,000 psi Pressure Guage

Regulating Valve

CylinderLucite Window

Lucite Window

U-TubeManometer

ToAtmosphere

Mercury Pump

Centrifuge

I

H

P

MN L

E

F

C

D

GJ K

Nitrogen Pressure

Neoprene Stopper

Saran Tube

Nickel-PlatedSpringCore

Seal of Red Oil

Scale of Squared Paper

Kleenex Paper

Ultra-FineFritted GlassDisk

Crude Oil

Brine

Porous Diaphragm

(i) (ii)

(iii)

Figure 31a

36

Pc

Pc diaphragm

Pc core

Operatingrange

0100%σW

4.2 Laboratory Measurement of Relative PermeabilityRelative permeabilities can be measured using either steady-state or unsteady-statemethods: unsteady-state measurements can usually be completed in a much shortertime, and have consequently become the oil industry standard. The question as towhether such rapid measurements are representative of conditions in the reservoir,however, is a matter of some debate.

Steady-State MethodsSteady-state methods are characterised by simultaneous injection of the two phases ata fixed ratio and known flowrates. Steady-state conditions are assumed to have beenreached once the inlet and outlet fluxes of each phase have equilibrated and/or aconstant pressure drop is seen to exist across the sample. This may take many hoursor even days, depending upon the type of material under investigation and themeasurement technique being used. Once equilibration has been reached, Darcy’slaw can be used for each phase in turn, resulting in a pair of relative permeability valuesvalid at that particular saturation. The fluid flux ratio is then changed (whilst keepingthe total flowrate constant), yielding a second set of data once a new steady-state hasbeen achieved. Similar measurements for a number of different flux ratios can thenbe used to give a set of relative permeability curves which span the entire saturationrange. Although the time involved in extracting such data is clearly an importantconcern, steady-state measurements performed at low rates should be considered mostindicative of reservoir behaviour. Typical laboratory relative permeability curves areshown in Figures32 and 33. Some of the more popular experimental techniques willnow be discussed.

Figure 31b



100

80

60

40

20

00 20 40 60 80 100

Oil saturation, %

Rel

ativ

e pe

rmea

bilit

y, %

Core No. 0-2-ABerea outcropK = 120 mdL = 2.30 cm

★

★

★ ★

★ ★

★

★

★

★

★

★

★★

kg , k0 , Penn State

kg , k0 , single-core dynamic

kg , k0 , dispersed feed

kg , k0 , Hafford technique

kg , k0 , gas-drive techniquekg , k0 , Hassler method

100

80

60

40

20

00 20 40 60 80 100

Oil saturation, %

Rel

ativ

e pe

rmea

bilit

y, %

Core No. 0-2Berea outcropK = 118 mdL = 7.23 cm

★

★

★

★

★

★

★

★

★

★

★

★

★

★★

kg , k0 , Penn State

kg , k0 , single-core dynamic

kg , k0 , dispersed feed

kg , k0 , Hafford technique

kg , k0 , gas-drive techniquekg , k0 , Hassler method

Figure 32

Figure 33

38

The Penn-State Method — Originally designed by Morse et al (1947), this techniquehas recently become one of the most popular. A typical apparatus is shown in Figure85a and essentially consists of three similar core plugs mounted in a core holder witha pair of pressure tappings. Only the central plug is considered for measurementpurposes; one of the two outer units acting as a mixer at the inlet, whilst the other servesto alleviate capillary end effects at the outlet. The three-plug assembly also facilitatesdismantling of the test section for saturation measurements to be carried out. Themethod can be used for both liquid-liquid and liquid-gas measurements and has beenused over a wide range of wettability conditions.

The Hassler Method — The laboratory apparatus used for this type of steady-state testis shown in Figure 35d. Semi-permeable membranes are positioned at both the inletand outlet ends of the core sample, which serve to separate the test fluids outwith thesample whilst permitting two-phase flow to take place within it. The importance ofthis is that allows the two pressure drops to be regulated independently, enablingequilibration of the capillary pressure at both ends of the core. This procedure isdesigned to provide a uniform saturation distribution throughout the system, and thuseliminate any capillary end effects.

The Dispersed-Feed Method— This technique was devised by Richardson et al (1952) in an attempt to introducethe fluids to the test sample in a more appropriate manner. It utilises an upstreamdispersing medium in order to spread the wetting phase uniformly across the face ofthe test sample before entering it. The nonwetting phase is injected into radial groovesthat are machined into the downstream end of the dispersing section, at its boundarywith the core plug. A similar concept lies behind the Hafford apparatus, which injectsnonwetting fluid directly into the sample, whilst the wetting phase must first passthrough a central semi-permeable membrane. Again, the idea is to obtain morerealistic injection behaviour.

Richardson et al (1952) have compared a variety of steady-state methods, andconclude that all four of the techniques outlined above should give very similardrainage relative permeability-saturation curves (Figures 32 and 33).

Unsteady-State MethodsAlthough unsteady-state relative permeability measurements can be made much morerapidly than those requiring steady-state equilibration, analysis of the resulting datais more difficult and open to a certain degree of ambiguity. In unsteady-state tests, onephase is displaced directly from the core sample by the injection of another, at a ratehigh enough such that capillary pressure remains negligible. An analysis of theresulting production data, based upon the Buckley-Leverett frontal advance theory(Buckley and Leverett, 1942), then permits the determination of a relative permeabilityratio (k

rw/k

ro). The relevant theory was given by Welge (1952), who demonstrated that:

f kk

oo rw

w ro

=+

1

1µµ

(14)



where fo is the fraction of oil in the outlet stream, and the mi are viscosities. This was

later extended by Johnson et al (1959), who developed a technique for calculating theindividual relative permeabilities from the permeability ratio. Several other methodsalso exist (Saraf and McCaffery, 1982; Jones and Roszelle, 1978; inter alia). A samplecalculation is shown in Figure 34 —note that we only get relative permeability afterbreakthrough of the displacing phase.

kri

1.0

0.8

0.6

0.4

0.2

0.00.0 0.2 0.4 0.6 0.8 1.0

Sw

krwkro

In the past, much work has been carried out to compare these unsteady-state resultswith their steady-state equivalents. Unfortunately, although the outcomes appear tobe broadly in agreement, there have been a number of exceptions (e.g. Schneider andOwens, 1970; Owens et al, 1965; Loomis and Crowell, 1962; Archer and Wong,1971). The limitations of the unsteady-state approach for determining water-oilrelative permeabilities have been discussed more fully by Craig (1971): the mainconcerns relate to the high pressure differentials involved (in excess of 50psi), and thefact that large viscosity ratios are often used in an attempt to extend the range of two-phase flow. The general applicability of the resulting data must consequently becalled into question.

Centrifuge MethodsCentrifuge techniques can provide useful results extremely rapidly, with the addedbonus that they are thought to be free of the viscous fingering problems thataccompany many other unsteady-state methods. They are, however, subject toproblems associated with capillary end effect and cannot be used to quantify therelative permeability of the displacing phase. Such methods involve monitoring theproduction from samples that are initially saturated with one or two phases, withanalytical techniques being used to back-out relative permeability values (see VanSpronsen, 1982, for a fuller account).

Figure 34

40

Thermometer

Electrodes

Copperorificeplate

Packingnut

Inlet

Gas

Gaspressureguage

Gas meter

Oilpressure

Oil

Oil pressure pad

Oil burette

Oil burette

Gas

Outlet Intlet

Oil

Endsection

Testsection

Mixingsection

Bronzescreen

Differentialpressure taps

Highly permeable disk

Penn-State Hafford

Dispersed Feed Hassler

Porous end plate

Gas meter

Dispersingsection face

Gas-pressureguage

Lucite

Lucite-mountedcore

Dispersingsection

Flowmeter

Core

Corematerial

vacuum

(a) (b)

(c) (d)

The Effects of Flowrate, Viscosity Ratio, and Interfacial Tension.At sufficiently low flowrates, microscopic flow behaviour should always be capillary-dominated if the system has a strong wetting preference. At such rates, relativepermeabilities in the medium saturation range should consequently be independent ofthe viscosity ratio, and this conclusion appears to have been borne out by manysubsequent investigations (Leverett et al, 1939, 1941; Wyckoff and Botset, 1936;Saraf and Fatt, 1967; Levine, 1954). Near the endpoints, however, and in cases wherethe wetting phase is flowing only through thick films, there sometimes exists a stronghydraulic coupling between the two phases. As a result, the nonwetting phase mayexperience hydraulic slip and any analysis using Darcy’s Law becomes invalid. IfDarcy’s law is applied regardless, however, nonwetting phase relative permeabilitiesgreater than unity can often be the result: the experimental results of Odeh (1959)clearly demonstrate this effect (Figure 36a). If high rates are used in relativepermeability measurements, the subsequent flow behaviour will no longer remaincapillary-dominated, and viscous forces will tend to take over. Moreover, if the

Figure 35



associated viscosity ratio is high, then the less viscous invading fluid will begin tofinger through the sample and a condition of uniform saturation will be impossible toachieve. With this in mind, it is clear that such considerations should not beoverlooked when attempting to interpret a wide variety of unsteady-state coreflood data.

The effect of interfacial tension upon relative permeabilities can also be significant.In cases where the experimental flowrate is high and the interfacial tension is small,the capillary forces become less significant and slugs of both fluids may begin to flowthrough the same network of pores. In fact, as the interfacial tension approaches zero,the relative permeability curves actually become straight diagonal lines: i.e. the totaleffective mobility of the system remains constant over the entire saturation range.Experimental results showing this effect are reproduced in Figure 36b.

240

200

100

0 00

.25

.25

.75

.75

1.0

1.0

.5

.50 50 100

Sw Sq

???? m=82.2

???? m=74.5

???? m=5.7

???? m=5.2

???? m=0.9

???? m=0.6

???? m=42.0 ????

????

????

????

(a) (b)

QUESTIONS WE SHOULD NOW BE ABLE TO ANSWER

• What is capillary pressure?• At what scale is it most important?• What things affect meniscus curvature?• How do drainage and imbibition processes differ at the pore scale?• Why can the roughness of pore walls be important?• Is flowrate important in laboratory tests and, if so, why?• What are the advantages of network models over other pore-scale models?

Now, how do we use the knowledge we’ve obtained so far ?

Figure 36

42

5 EMPIRICAL AND THEORETICAL APPROACHES TO GENERATINGPETROPHYSICAL PROPERTIES FOR RESERVOIR SIMULATION

5.1 Methods for Generating Capillary Pressure Curves and Pore Size Distributions

BackgroundIn this section, we will describe a number of approaches to generate capillary pressurecurves for use in reservoir simulation. We often have to undertake a simulation studywithout possessing all of the data we would ideally require — perhaps we have onlyone experimental capillary pressure curve at our disposal and perhaps it comes froma part of the reservoir not directly related to our current study. We need some way ofinferring reasonable data from those available to us at the time and, although we mayhave to make some gross assumptions, we should be able to invoke our knowledge offlow at the pore-scale to help us.

Let us begin by reminding ourselves of the drainage case (drainage curves arefrequently used in simulators — often erroneously). Remember that, in drainage, weare increasing the pressure difference between the nonwetting and wetting phases. AsPc increases, the radius of interface curvature decreases and, using a capillary tubeanalogue, the Young-Laplace equation tells us that the nonwetting phase begins toinvade the porous medium when Pc>2σcosθ/R

max — i.e. at the so-called displacement

pressure of the sample. As Pc continues to increase, the nonwetting phase invadesprogressively smaller pores. Hence, the nonwetting phase saturation graduallyincreases with Pc. The resulting plot of Pc vs Sw is the drainage capillary pressurecurve (Figure 37).

Pc

Pc

D.P.

Displacementpressure

%1000

0 σ0100% σw

Pendular

Funicular Insular

Funicular

Water wetsand

Figure 37



We can now use this discussion to infer the type of capillary pressure curve that mayresult from different samples. For example, how would the pore size distributionaffect the curve? We already know that the displacement pressure is affected by R

max

— the largest pore in the sample — but it should also be clear that the range of poresizes in a sample can greatly affect the shape of the corresponding Pc curve (Figure88). Flat plateaux indicate samples that are fairly homogeneous at the pore scale,whilst steeper curves indicate a large variance in the distribution. Moreover, finetextured rocks with small cemented grains can be expected to exhibit higher capillarypressures at a given saturation that coarse-textured media — also higher displacementpressures. So, from our basic understanding of capillary pressure at the microscopicscale, we have been able to infer a great deal about how we would expect capillarypressure curves to vary at the continuum (macroscopic) scale.

Pc

D.P.

100%0 100%

0σ0

σw

Water wetsand

1

2

3

Irredu-cible watersaturation

All same radii

Rescaling Mercury Injection DataOne of the most common ways to derive oil-water or gas-oil capillary pressure curvesis to rescale mercury injection data (which is routinely measured and relatively cheapto obtain). Consider mercury injection into a porous medium containing a largeexterior pore of radius R

extmax. The capillary pressure required for this pore to be

invaded by mercury is given by the Young-Laplace equation:

PRce mercury air

mercury air

ext

( ) =/

/max

( cos )2 σ θ(15)

Figure 38

44

If we now consider the same medium, filled with water, undergoing oil injection, thenthe requisite oil-water capillary pressure is now:

PRce oil water

oil water

ext

( ) =/

/max

( cos )2 σ θ(16)

Now, eliminating Rext

max from equations 15 and 16we arrive at the scaling relationship:

P Pce oil water ce mercury airoil water

mercury air

( ) = ( )/ //

/

( cos )

( cos )

σ θσ θ

(17)

This clearly holds for any Pc-value, and so a complete oil-water capillary pressure

curve can be obtained from the mercury injection data by applying equation 17at eachsaturation.

Experience shows that the ratio of interfacial tensions and cosines on the right handside of 17(often difficult to measure accurately) should have a value of approximately6. However, different ratios have sometimes been needed to reconcile experimentalmercury-air and oil-water data from different rock-types (Figure 39 — the ratio isreferred to as—“Factor” in these plots). Such differences may be due to interactionsbetween contact angle and pore geometry.

60

50

40

30

20

10

0

348

290

232

174

116

58

00

0

20

20

40

40

60

60

80

80

100

100

Restored state

Mercury injection

Limestone core

Porosity - 23.0%Permeability - 3.36 mdFactor - 5.8

Wat

er/n

itrog

en c

apili

ary

pres

sure

, psi

H20

Hg

Liquid saturation, %

30

25

20

15

10

5

0

225.0

187.5

150.0

112.5

75.0

37.5

00

0

20

20

40

40

60

60

80

80

100

100

Sandstone core

Porosity - 28.1%Permeability - 1.43 darcysFactor - 7.5

Wat

er/n

itrog

en c

apili

ary

pres

sure

, psi

Mer

cury

cap

iliar

y pr

essu

re, p

si

Mer

cury

cap

iliar

y pr

essu

re, p

si

H20

Hg

Liquid saturation, %

Restored state

Mercury injection

Figure 39



Derivation of Pore Size DistributionsIn addition to providing raw data for the derivation of capillary pressure curves,mercury injection data can also be used to derive so-called pore-size distributions(PSDs). The theory for deriving PSDs from intrusion data comes from a paper byRitter and Drake (1945), which makes a number of simplifying assumptions: (i) thepores are circular cylinders in shape, and (ii) all pores of a given radius are accessibleto the mercury (i.e. no accessibility issues). By using the Young-Laplace equation toconvert Pc to pore radius, the PSD (D(r)) can be derived from the capillary pressuredata via the relationship:

D rPr

dS

dPc mercury

c

( ) =

(18)

In fact, some algebra shows that this can be rewritten as:

D rdSdr

air( ) =

(19)

The procedure can be tested using the data shown in Table 2 and you are encouragedto follow the example.

Unfortunately, the distributions that are produced by this method (and therefore bycommercial porosimeters) are not pore size distributions but pore volume distributions.The spike (very typical) is caused by accessibility effects prevalent during drainageprocesses and the volume associated with some large “shielded” pores can beincorrectly assigned to small pores (see Figure 40). To get a better idea of the true PSD(that is the frequency distribution of pore radii), the Ritter and Drake distribution D(r)should be divided by r2 for each r-value — this generally shows that there are far moresmaller pores than D(r)would suggest. That said, the volume-weighted distributionsprovided by commercial porosimeters are still useful lithological fingerprints.

(Pc) mercury/air sair 0 0 5 0.97 6 0.92 8 0.8 9 0.7 11 0.6 13 0.5 16 0.4 18 0.3 25 0.2 50 0.15

1. Plot the mercury capilliary pressure curve

2. Derive the oil-water curve (σcosθ=40mN/m for oil-water and 375mN/m for mercury-air)

3. Use Y-L equation to derive Sw vs r plot (this is the cumulative intrusion curve)

4. Plot PSD (D(r)) using Ritter and Drake formula (Hint: use Y-L equation to get a formula for dSw/dr)

Table 2

46

D

(a) (b)

De De

The Leverett J-FunctionThe foregoing discussion has shown that Pc curves are clearly affected by the samplepore-size distribution, interfacial tension, contact angle, and pore structure. Leverett(1941) developed a dimensionless group — called a J-function—— to account forthese effects. The J-function has become an important tool for capillary pressureinterpolation and extrapolation and increases the utility of a single capillary pressurecurve. It allows us to adapt a single data set to other areas of a reservoir where data maybe unavailable.

We saw in section 3.4 (Capillary Bundle Models) that a “representative pore-radius”can be inferred from sample permeability and porosity via the relation:

˜ ~Rkφ (20)

Hence, a dimensionless group can be formed by defining:

JP kc=

σ θ φcos

1

2(21)

This can be applied at a number of different saturation values — at each saturation,P

c(S

w) is determined and (21) used to find J(S

w). Often, the contact angle is difficult

to ascertain with any accuracy, and the cosθ term is often set to unity.

Having found J(Sw), capillary pressure curves corresponding to a range of different

permeability and porosity values (and different fluid combinations via the σcosθterm) can be determined by simply inverting the J-function, viz:

P SJ S

kc w

w( )( ). cos=

σ θ

φ

1

2

(22)

Figure 40



Example J-functions are shown in Figure41 together with a derived set of Pc-curves.The procedure can also be used to examine the relationship between permeability(and/or porosity and/or fluid combinations) and water saturation at different capillarypressure cut-off values ——in general, at a given Pc, we find that lower k => higherS

w (Figure 42).

1.5

1.4

1.3

1.2

1.1

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 10 20 30 40 50 60 70 80 90 100

Cap

illia

ry p

ress

ure

func

tion,

√ (

Sw

) =

Pc σ()k φ

1 2

Water saturation, Sw

Reservoir Formation

Hawkins Woodbine 0.347Rangely Weber 0.151El Robie Moreno 0.18Kinsella Viking 0.315Katie Deese 0.116Leduc Devonian 0.114

Alundum (consolidated) 0.371Leverett (unconsolidated) 0.419

Lim√(Sw)Sw

-1

Hawkins

Kinsella

Leverett

AlundumTheoretical limitingvalue for regularpacked spheres

0.447

Leduc

Katie

Kinsella Shale

Morano Rangely

√ (S

w)

=P

c

σ co

s θ

()k φ

1 2

4

3

2

1

00 20 40 60 80 100

Liquid saturation, %(b)

√ (S

w)

=P

c

σ co

s θ

()k φ

1 2

4

3

2

1

00 20 40 60 80 100

Liquid saturation, %(d)

√ (S

w)

=P

c

σ co

s θ

()k φ

1 2

4

3

2

1

00 20 40 60 80 100

Liquid saturation, %(c)

After Amyx Bass and Whiting 1960

0 10 20 30 40 50 60 70 80 90 100

Water saturation, %

Oil

wat

er c

apill

iary

pre

ssur

e, p

si(r

eser

voir

cond

ition

s)

Hei

ght a

bove

zer

o ca

pilli

ary

pres

sure

, ft

30

27

24

21

18

15

12

9

5

3

0

90

81

72

63

54

45

36

27

18

9

0

200

180

160

140

120

100

80

60

40

20

0

900

md

500

md

200

md

100

md

50 m

d

10 m

d

Figure 41

48

Capillary pressure = 5 psi

Brine saturation SPorosity φ20 25 30

log

k

log

perm

eabi

lity,

md

Transition ZonesWe conclude this section on capillary pressure curves with a brief discussion oftransition zones. Transition zones are usually described using the water-wet verticalbead pack idealization shown in Figure 43. A water-saturated bead pack is allowedto drain under gravity until capillary equilibrium is reached. At the top of the column,the water is in the form of discrete rings connected via thin wetting films over grains— called the pendular regime. Moving down the column, the rings become larger andultimately coalesce — the funicular regime. Near the bottom of the pack, water iscontinuous.

Figure 42



(b)

yrr

(i)

θθ

σ

R

(a)r

d

(ii)

ry

θθ

σ

R

(i)

θc

b

a

δ

(ii)

c

b

a

δθ

(iii)

c

b

a

δ

(iv)

c

b

a

δ

θ

Whilst this idealization gives some limited insight into transition zone behaviour, amore enlightening picture emerges if we use our knowledge of drainage capillarypressure at the pore scale — this can give us a mental picture of fluid distributions aswe move through a transition zone. Referring to Figure 44a, we see that thehydrostatic oil and water gradients meet at the OWC (P

o=P

w). Above this contact,

Po>P

w and capillary pressure increases as we go further up the reservoir. Figure 44b

shows how the phases would be distributed in the pore network, with oil present insmaller and smaller pores as we move up through the transition zone.

Figure 43

50

ExplorationWell

Gas

Oil

Water

OWC 5500'

GOC 5200'

Test Resultsat 5250 ftpo = 2402 psiadpo

dD= 0.35 psi/ft

Pressure (psia)

2250

2265 23695000

5500

5250

2375 2500

Depth(feet)

GOC: po = pg = 2385

OWC: po = pw = 2490

Pcow

Pcgo

Water Oil

CappilaryPressureIncreasing

(a) (b)

Capillary pressure data can also be used to infer fluid saturations as a function of depthwithin a transition zone. Assuming that oil and water pressures remain continuousthroughout the entire height of the zone (doubtful under some circumstances), we canwrite equations for each hydraulic gradient (z measures height above OWC andincreases vertically upwards):

dPdz

gww= − ρ (23)

dPdz

goo= − ρ (24)

Subtracting (23) from (24) leads to:

dPdz

dPdz

dPdz

go w cw o− = = −( )ρ ρ (25)

which can be integrated to give:

Figure 44



P g zc w o= −( )ρ ρ (27)

Hence, any capillary pressure curve can be re-plotted to give saturation with depthinformation. An example is shown in Figure 45.

830

840

850

860

870

880

890

900

910

920

930

940

950

960

970

980

990

1000

1010

10200 10 20 30 40 50 60 70 80 90 100

Dep

th, M

bel

ow s

ea le

vel

Minimum of 22% connate water

Approximate gas-oil contact

Water saturation scale %

Data derrived fromcapillary pressureData obtained fromseismic logs

5.2 Methods for Generating Relative PermeabilitiesMultiphase flow experiments are costly, difficult and time-consuming to carry out andit is often infeasible to perform a large number of laboratory sensitivity studies. Itwould therefore be highly desirable if a cheap, predictive model for relative permeabilitycould be developed that only required cheap, easily-accessible data as inputs. In fact,a number of models are already currently available (although some still lack a degreeof refinement) and we will discuss three such approaches below.

Purcell ModelThe Purcell model seeks to utilise cheap capillary pressure data in order to predictmore expensive relative permeabilities. The model revisits the capillary bundlemodel and initially uses a little algebra to develop a predictor for absolute permeability.

Figure 45

52

The development is shown in Figure B4 in Appendix B.

Extending the approach to deal with relative permeabilities is relatively straightforward— we simply need to replace the summation in (equation B6 with integrals over watersaturation. The wetting phase relative permeability integral runs from 0 to S

wt (the

wetting phase saturation of interest), whilst the nonwetting integral runs from Swt

to1. The equations to be applied at successive values of water saturation are given inFigure B5.

A graphical “recipe” for relative permeability prediction via capillary pressure data isessentially the following:

(i) Obtain capillary pressure data Pc(S

w)

(ii) Plot the curve 1/Pc2 vs Sw (see Figure 46)

(iii)Find the area under the entire curve (=k)

(iv)For any chosen value of water saturation (Swt

, say), calculate the area under the1/Pc2 curve from 0 to S

wt. This gives k

wt

(v) For the same value of water saturation, calculate the area under the 1/Pc2 curvefrom S

wt to 1. This gives k

nwt

(vi)Use the values obtained from (iii) – (v) to determine relative permeabilities

14

12

10

8

6

4

2

0

0.56

0.48

0.40

0.32

0.24

0.16

0.08

0100 80 60 40 20 0

Percent of total pore space occupied by mercury

Cap

illar

y pr

essu

re, P

c, a

tm

Pc

(Pc)21

1/(c

apill

ary

pres

sure

)2 , 1/P

c2 , atm

-2

There is one clear drawback using this approach, however: the relative permeabilitiesadd up to unity over the full saturation range and are therefore completely symmetrical.This limitation led Burdine to add the following improvement.

Figure 46



Burdine ModelThe Burdine model simply takes the Purcell model and re-scales the endpointsthrough the prefactors:

λ λw ww wi

wi ornw w

w or

wi or

SS S

S SS

S SS S

( ) ( )= −− −

= − −− −1

11

(27)

The curves are therefore defined over the wetting-phase saturation range (1-Sor) to (1-

Swi

). The full equations are given in Appendix B (Figure B6). Although a number ofgross assumptions have been made along the way, the Burdine method offers a cheapalternative to laboratory measurement of relative permeabilities. It is easily codedinto a spreadsheet and is often carried out by practicing reservoir engineers —experience appears to show that the wetting phase prediction is often fairly good,whilst the nonwetting curve is less well reproduced.

Brooks and Corey ModelAlthough the models described above yield relative permeability curves that arequalitatively reasonable, reproduction of experimental data is only achieved via theintroduction of some form of empiricism. Many studies have subsequently reliedentirely upon empirical curve fitting techniques. One of the most popular empiricalcorrelations is that due to Corey (1954), who proposed the following:

k Srw eff∝ 4 (28)

k S Srnw eff eff∝ − −( ) ( )1 12 2 (29)

However, it was soon discovered (not surprisingly) that the exponents in Corey’soriginal equations would have to be varied in order to fit different materials. Theywere consequently generalised by Brooks and Corey (1964) to:

k Srw eff= +( )2 3λ λ (30)

k S Srnw eff eff= − − +( ) ( )( )1 12 2 λ λ (31)

S P P P Peff cb c c cb= ( ) ≥λ( ) (32)

with Pcb

representing the breakthrough capillary pressure and λ the “pore sizedistribution index”. Both of these parameters have to be determined experimentally:S

eff vs P

c data is plotted on a log-log scale and a straight line fitted, the slope gives λ

and the intercept with Seff

=1 is assumed to give Pcb

.

Network ModelsIt is quite apparent from this discussion that no satisfactory predictive model currentlyexists which can adequately account for the complex jumble of channels that pervadea porous medium. One of the most promising developments of recent years, however,has been the possibility of applying interconnected network models to the study of

54

microscopic flow. A detailed discussion now follows to determine the extent to whichthe capillary network model is an analogue for two-phase relative permeabilityexperiments.

McDougall and Sorbie have developed an approach for “predicting” relativepermeability based on “anchoring” to capillary pressure data. They have recentlydeveloped a PC-based software package known as MixWet, which allows a widerange of multiphase simulations to be undertaken at the pore-scale under a variety ofdifferent wettability conditions — the interface is shown in Figure 47. Certain porescale parameters required as input to MixWet are estimated using the inverse mercurycapillary pressure curve - which we call the R-plot since 1/P

c ~ R, where R is the pore

radius. The full methodology is explained by McDougall et al (SCA, Edinburgh,2001).

The model calculates the total flow through two intertwined networks (one for eachphase) over a range of saturation values. Inherent in this is the assumption that the flowis stationary, thus steady-state relative permeabilities are considered here. In all thatfollows, the wetting phase is assumed to be water and the non-wetting phase oil. Eachpore in the network is assigned a capillary entry radius from a distribution. At eachstage of the displacement, the total flow is calculated separately for each of the

Figure 47



intertwined networks. The results are then converted into normalised permeabilitiesas functions of saturation. Additional details can be found in the earlier discussionpresented in section 2.6.

Primary Displacements — Consider first the case of a strongly water-wet rock whichis initially 100% saturated with oil. Several possible displacement mechanisms havebeen identified in two dimensions (Lenormand and Zarcone, 1984), one of which isknown as “snap-off”. When brought into contact with the water phase, a stronglywater-wet rock will spontaneously imbibe the wetting fluid via film flow alongirregularities on the pore surfaces. In effect, the water slowly wets all internal grainsurfaces and is essentially present everywhere in the matrix. As this imbibitionprocess continues, the thin films begin to swell. Eventually, the thinnest pores willbecome completely filled with water and the original oil will be displaced if an escaperoute exists (see earlier discussion in section 3.4). This continues with the gradualfilling of progressively wider pores until no further displacement is possible since theoil phase becomes disconnected. The snap-off mechanism is thought to be the mostprevalent imbibition mechanism governing capillary-dominated waterfloods ofconsolidated media (low aspect ratio pores). More pistonlike displacements mayoccur in media containing high aspect ratio pores (e.g. unconsolidated beadpacks) butsuch systems are not considered here (the competition between snap-off and pistonlikedisplacements is discussed more fully in Dixit et al, 1997). A typical set of imbibitionrelative permeability curves obtained from 3D network modelling are shown in Figure48 and are seen to correlate well with experimental observations

Kri

1.0

0.8

0.6

0.4

0.2

0.00.0 0.2 0.4 0.6 0.8 1.0

Sw

The simulation of low-rate drainage processes is carried out using an invasionpercolation model with hydraulic trapping of the wetting phase. In this case, theinjected non-wetting phase first fills the largest pores connected to the inlet face of thenetwork, and then proceeds along progressively narrower pathways, occupyingsuccessively smaller pores (see section3.3). The drainage displacement is terminatedat a pre-determined limiting capillary pressure — if an unrealistically high capillarypressure were applied to the network, no irreducible water saturation would remainat the end of the flood. This drainage model, first proposed by Chandler et al (1982)and Wilkinson and Willemson (1983), is basically the displacement mechanismgoverning mercury porosimetry experiments. A set of simulated drainage relative

Figure 48Imbibition relativepermeability curves frompore-scale simulation

56

permeability curves are shown in Figure 49 and again correlate well with experimentalobservation.

Kri

1.0

0.8

0.6

0.4

0.2

0.00.0 0.2 0.4 0.6 0.8 1.0

Sw

Secondary Displacement Processes ——Secondary displacement processes aredisplacements carried out on a porous medium which is already partially saturatedwith the displacing phase. For example, if a primary drainage experiment is followedby a waterflood, the process is called secondary imbibition. This cycle exhibits anhysteresis effect between the primary drainage and secondary imbibition non-wettingrelative permeability curves, but very little variation is evident in the wetting phasecurves and the hysteresis effect is usually considered to be negligible. Various theoriesrelating to pore size distribution and matrix cementation have been put forward in anattempt to explain this phenomenon, but network modelling of the various secondaryprocesses provides important clues as to the real causes of hysteresis in porous media.

Both secondary drainage and secondary imbibition have been studied, but for brevityonly the primary drainage-secondary imbibition scenario will be discussed here. Thesimulation is shown in Figure 50. The similarity to experimental curves fromconsolidated media is striking and the hysteresis effect is clearly duplicated in the caseof the non-wetting phase. The wetting phase curve shows little sign of deviation whichis also in general agreement with experimental findings. In order to explain thisphenomenon, a step by step analysis of the distribution of invaded pores must beconsidered. It is found that the hysteresis effect is due to the different physicsgoverning imbibition and drainage processes; in particular, the associated issue ofaccessibility during drainage (McDougall and Sorbie, 1992). Different hysteresispatterns are exhibited by unconsolidated media and possible causes of this have beendiscussed by Jerauld and Salter (1990). More recently, the full range of experimentally-observed hysteresis phenomena has been modelled and interpreted by Dixit et al(1997). Both studies show that pore aspect ratio and the competition between snap-off and pistonlike displacement affect the hysteresis trend, although differentmethodologies were used to capture the effects.

Figure 49Drainage relativepermeability curves frompore-scale simulation



Kri

1.0

0.8

0.6

0.4

0.2

0.00.0 0.2 0.4 0.6 0.8 1.0

Sw

krwint

krwsec

Primary Drainage

Secondary Imbibition

Comparison with Experiment — Although we have presented a number of idealisedrelative permeability simulations, the acid test of such a model is to compare relativepermeability prediction with experiment. This is an on-going exercise but somepreliminary examples are shown in Figure 51 together with the matching inversecapillary pressure data (R-Plots)..

Figure 50Non-wetting phasehysteresis after a primarydrainage -> secondaryimbibition cycle

58

Comparing Sample F with z3.3nu0.9n0lmb4

Sg compensated

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.10 0.2 0.4 0.6 0.8

kr

Kro sim

Krg sim

Kro expt

Krg expt

Sample F with best-fit

SHg

7

6

5

4

3

2

1

00 0.2 0.4 0.6 0.8 1

R, m

icro

ns

Experimental Analytic



Sample F with best-fit

SHg

35

40

30

25

20

25

10

5

00 0.2 0.4 0.6 0.8 1

R, m

icro

ns

Sample H with best-fit

SHg

70

8090

100

60

50

40

30

20

10

00 0.2 0.4 0.6 0.8 1

R, m

icro

ns

Comparing Sample G with z3.3nu1.2n0lmb3

Sg compensated

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.10 0.2 0.4 0.6 0.8

kr

Kro sim

Krg sim

Kro expt

Krg expt

Comparing Sample H with z4nu0.6n0lmb3

Sg compensated

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.10 0.2 0.4 0.6 0.8

kr

Kro sim

Krg sim

Kro expt

Krg expt

5.3 Hysteresis PhenomenaWe have already seen that both capillary pressure and relative permeability curvesexhibit hysteresis — that is, they depend upon saturation history (Figure 52 and 53).There are a number of possible causes for hysteresis but the three main effects are thefollowing:

(i) Contact angle hysteresis — advancing and receding contact angles differ(reflected in Pc through the Young-Laplace equation)

Figure 51Gas-oil relativepermeabilities (predictionvs experimental) andinverse capillary pressuredata for reservoir samples



(ii) Pore structure hysteresis — sloping pore walls mean that pores fill and empty atdifferent capillary pressures

(iii)Topological hysteresis — imbibition and drainage processes are differenttopologically (film-flow versus fingered invasion)

There is often such a difference between imbibition and drainage curves (bothcapillary pressure and relative permeability) that we must make sure that we are usingthe correct set of curves in our reservoir simulation studies (for instance, we shouldnot be using drainage curves as simulation input when modelling a waterflood in awater-wet reservoir). Built-in numerical models are available in Eclipse and othercommercial reservoir simulators to account for flow reversals at intermediatesaturations.

Pc

Pc 1Swi

Sw

b d c

a

1.00.0

1.0

0.0Swi

Sor

Sw

kr

Sw

Imbibition

Drainage

1.00.0

1.0

0.0

kro

Intermediatepath

Secondarydrainage

Figure 52

60

Water Wet

Venango Core VL-2k = 28.2 md

48

40

32

24

16

8

00 20 40 60 80 100

Cap

illar

y P

ress

ure

- C

m o

f Hg

Water Saturation - percent

2 1

6 WETTABILITY — CONCEPTS AND APPLICATIONS

6.1 Introductory ConceptsWhat is Wettability?The term wettability refers to the wetting preference of a solid substrate in the presenceof different fluid combinations (liquids and/or gases). We have already seen in Section3,1 and Appendix A6 that the wetting preference of a solid can be characterized by acontact angle, as shown in Figure 54. The wetting phase is defined as the fluid thatcontacts the solid surface at an angle less that 90o.

θ = 158º

θ = 30º

θ = 30º θ = 48º

θ = 54º θ = 106º

θ = 30º

θ = 35º

Water

Water

Silica Surface

Isooctane Isoquinoline NaphthenicAcid

Isooctane+

5.7% Isoquinoline

Calcite Surface

WaterWater

Figure 53

Figure 54



This figure clearly demonstrates an important issue, namely, that the wetting preferenceof a rock depends not only upon the fluids involved but also upon the mineralogy ofthe rock surface. For instance, we see that, in the presence of isoquinoline, water isnonwetting on a silica substrate but wetting on a calcite substrate. In cases when a solidhas no wetting preference, (i.e. when the angle separating the two fluid interfaces isclose to 90o) the system is said to be of “neutral” wettability.

The importance of rock wettability cannot be over-emphasised, as it affects almost alltypes of core analysis: capillary pressure, relative permeability, waterflood behaviour,and electrical properties (see Anderson, 1987, for an excellent series of reviewarticles). The simple reason for this is that wettability affects the location, flow, anddistribution of fluids in a porous medium — hence, most measured petrophysicalproperties must be affected. Moreover, we shall see later how some core handlingprocedures can drastically alter the wettability state of a core: this would invariablylead to the measurement of SCAL data inappropriate to the reservoir under investigation.

In most of what has been described in this chapter, the assumption has been made thatthe system under consideration was water-wet. Indeed, historically, all reservoirswere believed to be strongly water-wet and almost all clean sedimentary rocks are ina water-wet condition. An additional argument for the validity of the water-wetassumption was the following: the majority of reservoirs were deposited in an aqueousenvironment, with oil only migrating at a later time. The rock surfaces wereconsequently in constant contact with water and no wettability alterations werepossible as connate water would prevent oil contacting the rock surfaces. However,Nutting (1934) realised that some producing reservoirs were, in fact, oil-wet (the rocksurface was preferentially wetted by oil in the presence of water) and it is nowgenerally accepted that water-wet reservoirs are the exception rather than the rule(Table 3 and 4).

Contact Angle Percent of (degrees) ReservoirsWater-wet 0 to 80 8Intermediate wet 80 to 100 12Oil-wet 100 to 160 65Strongly oil-wet 160 to 180 15

Contact Angle Silicate Carbonate Total (degrees) Reservoirs Reservoirs ReservoirsWater-wet 0 to 75 13 2 15Intermediate wet 75 to 105 2 1 3Oil-wet 105 to 180 15 22 37 Total 30 25 55

We now know that wettability is a rather complicated issue (still actively beingresearched) and that there are a number of different factors affecting reservoirwettability, including:

Table 3

Table 4

62

(i) Surface-active compounds in the crude oil. These are generally believed to bepolar compounds (polar head and hydrocarbon tail), mostly prevalent in theheavier crude fractions — resins and asphaltenes — that form an organic filmor adsorb onto pore walls

(ii) Brine chemistry

(iii) Brine salinity

(iv) Brine pH

(v) The presence of multivalent metal cations (Ca2+, Mg2+, Cu2+, Ni2+, Fe3+)

(vi) Pressure and temperature

(vii) Mineralogy (including clays)

In short, surface chemistry determines the wettability state of a reservoir (or, morecorrectly, any given region of a reservoir; as wettability can be expected to varyspatially — and possibly temporally——throughout a reservoir).

Having discovered that most reservoirs are not water-wet, we now have to modifyeverything that we have learned so far — pore-scale physics, capillary pressuremodels, relative permeability models, and network models. However, this is not asdifficult as it may first appear; we already understand drainage and imbibitionprocesses at the pore scale, we know how to define capillary pressure, and weappreciate the dynamics underlying relative permeability measurements. The followingdiscussion should help you apply your previous water-wet knowledge to systems thatare not water-wet.

Pore-Scale EffectsThe effect of wettability at the pore-scale is shown in Figure 55. If a rock is water-wet,we have already seen that there is a tendency for water to reside in the tighter poresand to form a film over the grain surfaces. Oil (the nonwetting phase) resides in thelarger pores. In this case, the term “imbibition””— a process whereby a wetting phasedisplaces a nonwetting phase — would refer to the displacement of oil by water. Theterm “drainage” would apply to oil displacing water . In an oil-wet system, however,the situation is reversed— oil now forms a thin film over the grain surfaces and waterfills the larger pores. Consequently, in an oil-wet medium,“imbibition” refers to thedisplacement of water by oil, whilst “drainage” refers to the displacement of oil bywater. These differences clearly have major implications for waterflooding reservoirs(if a reservoir is oil-wet, a waterflood is a drainage displacement, oil is flushed fromsmall pores, and oil production via film-flow also becomes important).



Water Oil Rock Grains

(a)

Water Oil Rock Grains

(b)

Oil

Water

Oil

Water

Oil

Water

Oil

Water

Oil

Water

Oil

Water

The foregoing discussion has clarified some of the pore-scale physics affecting water-wet and oil-wet media. However, given the complex nature of wettability alterationsin reality, a more realistic representation of wettability at the pore scale may be thatshown in Figure 56. Here, a combination of water-wet and oil-wet pathways exists thatallows film-flow access to the displacing phase and film-flow escape for the displacedphase. Hence, a waterflood would now consist of a combination of imbibition anddrainage events: water initially imbibing along water-wet pathways (displacing oilfrom small pores) and then requiring an overpressure to drain oil from the larger oil-wet pores. It is clear that the underlying mineralogy and surface chemistry of thesystem would determine the connectivity and topology of the “wettability pathways”but how can we determine these pathways”— in short, how can we quantify thewettability of a porous medium?

Figure 55

64

Water-Wet Oil-Wet

6.2 Wettability Classification and MeasurementWhen we come to define the type of wettability associated with a particular poroussample, we come up against a minefield of inconsistent terminology. It will bebeneficial, therefore, to give some definitions that will appear periodically throughoutthis section.

Wettability can be classed as being uniform or non-uniform as follows:

Uniform — the wettability of the entire porespace is the same (100% water-wet, 100%oil-wet, or 100% “intermediate-wet”) and the contact angle is essentially the same inevery pore;

Non-Uniform — this is more characteristic of hydrocarbon reservoirs. The porespaceexhibits “heterogeneous” wettability, with variations in wetting from pore to pore(and possibly within a pore) — say 70% water-wet pores and 30% oil-wet pores. Wecan introduce 2 subdivisions (Figure 57):

Mixed-wet — a certain fraction of thelargest pores are oil-wet (there are valid depositional arguments for how this maycome about).Fractionally-wet — no size preference for oil-wetness (there are valid mineralogicalarguments for this).

Given our knowledge of pore-scale displacements, it is clear that each type ofwettability distribution will yield different capillary pressures and relative permeabilities(and recoveries), and we should be aware of this when we come to interpret SCAL andwettability test data.

Figure 56



F(R)

R

0.02

0.015

0.005

0.01

20 40 60 80 100

Mixed-Wet

Oil-Wet

Water-Wet

α

F(R)

R

0.02

0.015

0.005

0.01

20 40 60 80 100

Fractionally-Wet

Oil-Wet

Water-Wetα

Wettability MeasurementCore wettability can be determined in a number of ways, although three mainquantitative methods are used most frequently:

(i) Contact angle measurement — this measures the wettability of a mineral surfaceand is mainly used by specialist researchers

(ii) Amott method — this measurement gives valuable information regarding theextent and connectivity of wettability pathways of a core

(iii)U S Bureau of Mines (USBM) method — yields an average wettabilitymeasurement, is less informative, but much faster (and therefore cheaper) thanthe Amott method

Table 5 gives some expected values for water-wet, neutral-wet, and oil-wet mediafrom each test (we will explain each of these shortly).

Figure 57

66

Water-Wet Neutrally Wet Oil-WetContact angle Minimum 0º 60 to 75º 105 to 120º Maximum 60 to 75º 105 to 120º 180ºUSBM wettability index W near 1 W near 0 W near -1Amott wettability index Displacement-by-water ratio Positive Zero Zero Displacement-by-oil ratio Zero Zero PositiveAmott-Harvey wettability index 0.3 ≤ / ≤ 1.0 -0.3 < / < 0.3 -1.0 ≤ / ≤ -0.3

Other qualitative methods exist, such as rate-of-imbibition tests, NMR methods anddye adsorption methods, but these are not routinely undertaken. Let us now examinethe three main methodologies in more detail.

Contact angle methodThis is shown in Figure 58 and is mainly applicable to pure fluids and isolated mineralsurfaces. A sessile drop (single surface) or modified sessile drop (two surfaces) canbe used. A drop of fluid (oil, say) is generally placed between the mineral surfaces ina bath of a second fluid (water). The surfaces are then moved in opposite directionsparallel to one another and advancing and receding contact angles can be determined.These angles usually differ from one another (hysteresis effect) and this is onecomponent of the hysteresis observed in capillary pressure and relative permeabilitycurves. Notice also that the early-time behaviour of the measurement is oftenmisleading — you have to wait for the system to equilibrate. Whilst this technique isoften used in wettability research laboratories, it is not carried out routinely in theindustry.

Crystal

Crystal

Oil OilWater

WaterAdvancingContactAngle

180

150

120

90

60

30

00 200 400 600 800 1000 1200 1400 1600 1800

Age of the oil-mineral interface (hours)

Con

tact

Ang

le (

degr

ees)

Curve "E" Kareem

Curve "D" San Andres

Curve "B" Deosol

Curve "C" Tertiary Kenai

Curve "A" Pure Grade C10

Table 5

Figure 58



Amott TestThe Amott test is slow but informative.It combines both imbibition and forced displacement and is usually carried out via thefollowing “recipe”:

(1) Centrifuge core down to Sor

(2) Immerse in oil and measure volume of oil imbibed after 20 hours (Voi)

(3) Centrifuge the core down to Swi and measure the total oil volume (Vot)

(4) Immerse in water and measure volume of water imbibed after 20 hours (Vwi)

(5) Centrifuge the core down to Sor and measure the total water volume (Vwt

)

(6) Calculate the displacement-by-oil ratio (Io) and the displacement-by-water ratio(I

w), i.e.

IVV

IVVo

oi

otw

wi

wt

= =

If Iw ->1 and Io->0, the core is water-wet

If Io ->1 and Iw->0, the core is oil-wet

If Iw=0 and Io=0, the core is said to be neutrally-wet (neither phase has imbibed)

If Iw>0 and Io>0, the core is said to be heterogeneously-wet (mixed- or fractionally-wet)

Often, the indices are combined to give a single index; the Amott-Harvey Index =Iw-Io, yielding a number between -1 (oil-wet) and 1 (water-wet). If possible, however,it is better to have access to both I

o and I

w, as these individual indices tell far more than

the single Amott-Harvey index. Unfortunately, there are two main drawbacks to theAmott test: firstly, it takes a long time for fluids to equilibrate, making the test time-consuming and costly; and secondly, the ratios used in the calculations are verysensitive to errors in saturation measurement. A far quicker, but less informative,measure of wettability is given by the USBM test.

USBM TestThe USBM test uses thermodynamic arguments to ascertain the approximate workdone on the system durig centrifuge displacements. This essentially boils down tocalculating areas under Pc curves (= work done during a displacement) as follows(Figure 59).

68

*10

0

0 100-10

Average Water Saturation, Percent

a

Cap

illar

y P

ress

ure,

PS

I

Water Wet Log A1/A2 = 0.79

II

IA2

A1

*10

0

0 100-10

Average Water Saturation, Percent

b

Cap

illar

y P

ress

ure,

PS

I

Oil Wet Log A1/A2 = -0.51

II

I

A2

A1

(1) Core driven down to Swi

by centrifuge

(2) Core is centrifuged under brine at incremental speeds until Pc=-10psi (NB. waterpressure greater than oil pressure, so Pc=Po-Pw is negative). Measure expelledoil volume

(3) Core is centrifuged under oil at incremental speeds until Pc=+10psi. Measureexpelled water volume

(4) Plot the two capillary pressure curves

(5) Calculate the areas under the oil-drive (A1) and water-drive(A2) curves

(6) USBM Index (W) = log(A1/A2)

Figure 59



Although W theoratically goes from -infinity to +infinity, -1<W<1 is usually reported(W>0 indicates some degree of water-wetness, W<0 some degree of oil-wetness).Note, that other methods exist that combine elements of the Amott and USBM tests(see the literature for details, an example is shown in Figure 60).

6.3 The Impact of Wettability on Petrophysical Properties

Core HandlingThe previous discussion has highlighted the importance of wettability at the pore-scale and its implications for oil recovery (as demonstrated by the Amott and USBMtests). We should therefore endeavour to preserve the natural state of a core as muchas possible if we wish to derive SCAL data that has relevance to the reservoir underconsideration. There are a number of issues of which we should be aware: (i)wettability alterations can occur during drilling due to complex mud chemistry, (ii)pressure and temperature change as a core is brought to surface, (iii) asphaltenes maysubsequently precipitate and light ends may be lost. Pressure coring can help alleviatesome of these problems but this is not always available.

Uniform wetting-contact angle(Morrow and McCafferty, 1976)

oil wet intermediate water wet

180º 133º 62º 0º

Amott-IFP Index(Cuiec, 1991)

oil wet

slightlyoil wet

neutral slightlywater wet

intermediate water wet

-1 -0.3 -0.1 +0.3+0.1 +1

Three types of core are generally used in core analysis:

Native-state core— ideal if available as it minimises losses, wrap cores in polyethylene film and foil,then seal in liquid paraffin

Cleaned core — should only be used if reservoir is strongly water-wet (rare).Cleaning procedure should depend upon the crude oil/brine/rock system underinvestigation

Restored-state core — Only course of action if wettability has been altered. Clean thecore, flow reservoir fluids in correct order, age the core (1000 hours/40 days) and hopethat reservoir conditions have been re-established. Often, of course, this is not the caseand recoveries and relative permeabilities are badly affected (Figure 61).

Figure 60

70

NATIVE CORECRUDE OILCLEANED CORECRUDE OILCLEANED COREREFINED OIL

Water Injected, Pore Volumes

Oil

Rec

over

y, P

erce

nt P

V

0.01

60

50

40

30

20

10

00.1 1.0 10 100

Contaminated 452, mO 27.6% 25.8% 780, mO

254, mO 31.8% 24.8% 338, mO

202, mO 29.2% 24.8% 338, mO

k0 kaSwi φ

Cleaned

Restored-state

1

1

.8

.8

.6

.6

.4

.4

.2

.20

0

Rel

ativ

e P

erm

eabi

lity

Water Saturation, PV

The following discussion, regarding the sensitivity of a number of petrophysicalproperties to wettability, will highlight the importance of proper core handling andcleaning procedures still further.

The Effect of Wettability Upon Electrical PropertiesWe begin our discussion of petrophysical parameters with a brief look at Archie’sequation (Figure 62). This is routinely used by petrophysicists to determine watersaturation in a formation from wireline log data and the exponent (n) has beendetermined experimentally from plugs and has a value of about 2 for water-wet/cleaned cores. Under non-water-wet conditions, however, the fluids are distributeddifferently in the pore-space (fewer water films are also present) and we should notbe surprised to learn that Archie’s equation breaks down (although it is often usedregardless). Consequently, if n=2 is used regardless of the wettability conditionpertaining in the reservoir, then saturation predictions will be wrong (see Pirson andFraser, 1960, for an example of how expensive this assumption has been in the past).In order to demonstrate how inappropriate it would be to assume n=2 for non-water-wet material, we could take reservoir core and actually measure saturations directly.When taken together with direct resistivity measurements, we could then use Archie’sequation to infer what the appropriate exponent should be over a range of saturationvalues. The table in Figure 62 shows just such a case: we see that the exponent is noteven constant, implying the breakdown of the assumptions underpinning the modelitself - so use the Archie equation with care

Figure 61



Archie Saturation Exponents as a function ofSaturation for a Conducting Nonwetting Phase

Air/NaCi Solution

BrineSaturation

(% PV)66.2 1.9765.1 1.9863.2 1.9259.3 2.0151.4 1.9343.6 1.9939.5 2.1133.9 4.0630.1 7.5028.4 8.90

Oil/NaCi Solution

BrineSaturation

(% PV)n n64.163.160.255.350.744.240.536.834.333.931.0

2.352.312.462.372.512.462.612.814.007.159

S_w

n =R

RR

t

o

I≡

Sw, and

Sw

Ro

R t

where:

=

=

= brine saturation in the porous medium

resistivity of the porous medium atsaturation resistivity of the 100%brine-saturated formation

The Effect of Wettability Upon Capillary PressureWe have already seen an oil/water interface will become curved in order to balancethe pressure jump across it opposing interfacial tension forces. The pressure jump atwhich this balance is attained is given by Laplace’s equation:

p p pr rc o w= − = +

σ 1 1

1 2

(33)

where s is the interfacial tension between the two fluids, and r1 and r

2 are the two

principal radii of curvature (see section 3.1). The pressure difference Pc is thecapillary pressure and, in oil-water systems, is conventionally taken to be the pressurein the oil phase minus the pressure in the water phase. For drainage of a water-wetcircular capillary (radius R) and zero contact angle, this relationship becomes:

p p pRc o w= − = 2σ

(34)

Notice, however, that for water to invade an oil-wet capillary, the capillary pressuremust become negative (i.e. p

w must become greater than p

o) The combination of the

words “negative” and “pressure” may at first seem confusing, but the term is merelyan artefact of conventional terminology. The concept of negative capillary pressureis central to displacements in heterogeneously-wet media.

Figure 62

72

When discussing displacements in porous media of heterogeneous wettability, theterms imbibition and drainage become somewhat confused. For example, waterfloodsin heterogeneously-wet media may be a combination of conventional imbibition anddrainage processes. When the pore network contains a mixture of water-wet and oil-wet pores, the waterflood initially proceeds as an imbibition, with water spontaneouslyimbibing along water-wet pathways displacing oil from the smallest pores. Eventually,however, the water has to be forced into oil-wet pores and the displacement becomesone of drainage.

Pc(+ve)

Swi(Initial Water)

Primary Drainage

Age + Change Wettability

0 100Sw%

Pc(+ve)

Pc(-ve)

Swi(Initial Water)

0 100Sw%

Primary Drainage

Forced Oil Drive

Water Imbibition(Spontaneous)

(Secondary Drainage)

Forced Water Drive

OilImbibition(Spont-aneous)

32

24

16

8

0

-8

-16

-240 20 40 60 80 100

2

1

3

A

B

C

CURVE

1.DRAINAGE2.SPONTANEOUS IMBIBITION3.FORCED IMBIBTION

BEREA COREk = 184.3 md

POINTA. IRREDUCIBLE WETTING SATURATIONB. ZERO - CAPILLARY-PRESSURE NONWETTING SATURATIONC. IRREDUCIBLE NONWETTING SATURATION

WATER SATURATION, PERCENT P.V.

WATER WET

WATER SATURATION - PERCENT

CA

PIL

LAR

Y P

RE

SS

UR

E -

Cm

of H

g

48

40

32

24

16

8

00 20 40 60 80 100

VENANGO CORE VL - 2k = 28.2 md

Capillary pressure characteristics, strongly water-wet rock.Curve 1 - Drainage. Curve 2 - Imbition.

2 1

OIL WET

-48

-40

-32

-24

-16

-8

00 20 40 60 80 100

OIL SATURATION, PERCENT

CA

PIL

LAR

Y P

RE

SS

UR

E, C

M H

G

Oil-water capillary pressure characterisitics. Ten-sleepsandstone. oil-wet rock (after Ref.29). Curve 1 - drainage.Curve 2 - Imbition.

2

1

20

16

12

8

4

00 20 40 60 80 100

CA

PIL

LAR

Y P

RE

SS

UR

E, C

M.H

G

WATER SATURATION, PERCENT

Drainage capillary pressure characteristics (after Ref.30)

CA

PIL

LAR

Y P

RE

SS

UR

E -

Cm

of H

g

32

24

16

8

0

-8

-16

-240 20 40 50 60 100

WATER SATURATION - PERCENT

Oil-water capillary pressure characteristics,intermediate wettability. Curve 1 - drainage. Curve 2- spontaneous imbition. Curve 3 - forced imbition.

INTERMEDIATEWET

BEREA CORE 2-MO16-1k - 184.3 md

1

2

3

Figure 63Experimental capillarypressures in cores forvarious processes(drainage, imbibition) andwettability conditions(water-wet, oil-wet,intermediate-wet )



So, in general, a negative leg in the capillary pressure curve indicates some fractionof oil-wet porespace. The concept is shown both schematically and for experimentaldata in Figure 63.

The Effect of Wettability Upon Relative PermeabilityAs wettability controls the distribution of phases within the porespace, it is hardlysurprising that wettability has a majorimpact upon relative permeability curves andsubsequent reservoir performance. We can use our pore-scale modeling knowledgeto help explain the differences seen between the slopes of water-wet and oil- wetcurves (Figure 64). In the water-wet case, water imbibes via the smallest pores in thesystem, leaving oil resident in large, fast-flowing pores. Consequently, we wouldexpect the oil relative permeability curve to decrease slowly with increasing watersaturation and the water relative permeability endpoint to remain low — this is exactlywhat is observed. Conversely, waterflooding an oil-wet medium should lead to a rapiddecrease in oil relative permeability, together with a high water endpoint—— onceagain, this is what is observed.

RE

LAT

IVE

PE

RM

EA

BIL

ITY,

PE

RC

EN

T

WATER SATURATION, PERCENT P.V.

100

80

60

40

20

00 20 40 60 80 100

OIL WETWATER WET

OIL

OIL

WATER

WATER

CRAIG'S RULES OF THUMBS FOR CETERMINING WETTABILITY

Interstitial water saturation

Saturation at which oil and water relativepermeabilities are equal.

Relative permeability to water at themaximum water saturation (i.e.,floodout): based on the effective oilpermeability at reservoir interstitial watersaturation.

Water-Wet

Usually greater than20 to 25% PV.

Greater than 50%water saturation.

Generally less than30%

Oil-Wet

Generally less than15% PV.Frequently lessthan 100%.

Less than 50% watersaturation.

Greater than 50%

and approaching100%.

100

60

20

00 40 80 100

100

60

20

00 40 80 100

Water saturation % Water saturation %

Oil

Water-wetreservoir

WaterA

S wi

Rel

ativ

e pe

rmea

bilit

y, %

of a

ir pe

rmea

bilit

y

In water-wet system:S mostly > 20%At point A: k = k ;S > 50%k at S / k at W < 30%

w

ro rw w

rw or ro wi

A

Oil

WaterS wi

Oil-wetreservoir

In oil-wet system:S < 15%At point A: k = k ;S < 50%k at S / k at S > 50%

w

ro rw w

rw or ro wi

Influence of wettability on relative permeability; after Fertl, OGJ, 22 May 1978

Figure 64

74

In fact, the key features of such curves were presented by Craig (1971) who indicatedthe differences between the two in the form of several rules of thumb (see Table inFigure 64). Many subsequent experimental studies have agreed with these ideas(Donaldson and Thomas, 1971; Shankar and Dullien, 1981; inter alia) (Figure 65 ).However, as always, there are a few exceptions that prove the rule(s) — pore geometryand connectivity can change things quite a lot. Nevertheless, the trends are often useful

[Influence of wettability on relative permeability; after Fertl, OGJ, 22 May 1978]

.9

10

.8

.7

.6

.5

.4

.3

.2

.1

010 20 30 40 50 60 70 80

0

1

2

3

4

6

7

5

8

9

10

WATER SATURATION, PERCENT

RE

LAT

IVE

PE

RM

EA

BIL

ITY

TO

OIL

RE

LAT

IVE

PE

RM

EA

BIL

ITY

TO

WAT

ER

CORENO.

PERCENTSILANE

USBNWETTABILITY

1

2

3

4

5

0

0.02

.2

2.0

10.0

0.649

0.176

-0.222

-1.250

-1.333

5

5

4

3

2

1

54

3

2

1

RE

LAT

IVE

PE

RM

EA

BIL

ITY,

FR

AC

TIO

N %

1.0

0.5

0.10

0.05

0.01

0 0.2 0.4 0.6 0.8 1.0DISPLACED PHASE SATURATION, FRACTION P.V.

DISPLACING PHASE

DISPLACED PHASE

Nitrogen

Nitrogen

Dioclyl Ether

Heptone, Dodecone

Heptone, DodeconeDioclyl Ether

Nitrogen

Nitrogen

Water

up to 49

108

131

138 and greater

o

o

o

q

DISPLACEDPHASES

108

131

UP TO 49

138AND GREATER

DISPLACINGPHASES

o

o

o

o

Figure 65



CONTACT ANGLE04790138180

0 WOR= 25

WATER-WET

OIL-WET

80

70

60

50

40

30

20

10

00 0.2 0.4 0.6 0.8 10

o

o

oo

o

WATER INJECTED, PORE VOLUMES

OIL

RE

CO

VE

RY,

PE

RC

EN

T O

IL-I

N P

LAC

E

OIL

SAT

UR

ATIO

N. %

P.V

.

1007050

3020

10

6

3

12 5 10 20 50 100 200 500 1000 2000 5000

PORE VOLUMES OF FLOOD WATER

The Effect of Wettability Upon Waterflood PerformanceClearly, relative permeability affects waterflooding performance through the fractionalflow equation (as does the viscosity ratio) but what type of wettability distributionleads to the optimum recovery? In the relatively sparse literature on non-uniformsystems there is much disagreement regarding this question (Figures 66 and 67).

Figure 66

76

70

60

50

70

60

50

1.0 0.5 0.0 0.5 1.0

WATER-WET OIL-WET

DISPLACEMENT BY WATER RATIO DISPLACEMENT BY OIL RATIO

AMOTT WETTABILITY INDEX

OHIO SANDSTONE

0 20 40 60 80 100

RE

SID

UA

L W

ATE

R S

ATU

RAT

ION

, PE

RC

EN

T P

.V.

WEIGHT - PERCENT OIL - WET SAND

OIL

RE

CO

VE

RY

AT

24

P.V.

TH

RO

UG

HP

UT,

% O

RIG

INA

L

1.0

0.8

0.6

0.4

0.2

0.0

Kri

0.0 0.2 0.4 0.6 0.8 1.0

SW

100% WW

75% WW

50% WW

25% WW

0% WW

krw1

kro2/25

krw 1/0

krw1/25

krw2

kro2

Figure 67

Figure 68Relative permeabilitycurves from mixed-wetsystems: a is the oil-wetpore fraction



Donaldson et al (1969) and Emery et al (1970) performed waterflood experimentsusing core plugs which had been aged in crude for varying periods of time. Bothstudies showed that the more water-wet the rock, the more efficient the displacement.Conversely, Kennedy et al (1955), Amott (1959) and Salathiel (1973) have all shownthat the most efficient recovery takes place at close to neutral conditions. The precisewettability details are unclear from study to study however and so some sort ofmodelling approach is appropriate to understand the issue more fully. In the nextsection, we therefore go on to examine how network modelling techniques may beapplied in this context.

6.4 Network Modelling of Wettability Effects

IntroductionThe wettability characteristics of a porous medium play a major role in a diverse rangeof measurements including: capillary pressure data, relative permeability curves,electrical conductivity, waterflood recovery efficiency and residual oil saturation.This section describes the development and implementation of a pore-scale simulatorcapable of modelling multiphase flow in porous media of nonuniform wettability.This has been achieved by explicitly incorporating pore wettability effects into thesteady-state models described earlier.

Results are presented which show how ααααα (the fraction of pores which are assigned oil-wet characteristics) affects resulting relative permeability curves. These have beenused to calculate waterflood displacement efficiencies for a range of wettabilityconditions, and recovery is shown to be maximum at close to neutral conditions.Moreover, simulated capillary pressure data have demonstrated that standard wettabilitytests (such as Amott-Harvey and free imbibition) may give spurious results when thesample is fractionally-wet in nature (McDougall and Sorbie, 1993a).

Here, attention is restricted to mixed-wet systems. The term “mixed” wettability wasfirst introduced by Salathiel (1973) to describe systems where the oil-wet porescorrespond to the largest in the sample, the small pores remaining water-wet. Suchsituations may arise when oil migrates to water-wet reservoirs and preferentially fillsthe larger interstices. The wettability characteristics of these pores may then be alteredby the adsorption of polar compounds and/or the deposition of organic matter from theoriginal crude, thereby rendering them oil-wet. Fractional wettability, however, isgenerally related to the rock matrix itself and is due to the differences in surfacechemistry of the constituent minerals. Because of these variations, crude oilcomponents may adsorb onto some pore walls whilst ignoring others. This, in effect,means that fractionally-wet rock contains oil-wet pores of all sizes. Attention here,however, will be restricted to mixed-wet systems.

Waterflood Simulation DetailsOne of the advantages of using microscopic network simulators is that physicalproperties can easily be ascribed to each pore individually. Here, the wettability ofeach pore is controlled so that some are preferentially wetted by water and others byoil; the fraction of pores wetted by oil is denoted by α. The wetting phase contact angleis taken to be zero in both oil-wet and water-wet pores, i.e. pore walls are very stronglywetted by the corresponding wetting phase. In all that follows, the term “cluster”refers to any group of connected pores containing the same phase, whilst a “spanning

78

cluster” is a cluster which spans the network — connecting the inlet face of thenetwork to the outlet face.

Each simulation begins with the network 100% saturated with oil, and water is thenintroduced at the inlet face. Each waterflood consists of the following two stages:

(1) Water is first allowed to spontaneously imbibe via film flow but only alongcontinuous water-wet pathways which have access to the inlet. Accessiblewater-wet pores are then assumed to become filled via a “snap-off” mechanism,whereby the smallest pores are filled first followed by the next smallest and soon. The defending oil phase can escape from a pore by draining along a pathwayof oil-filled pores which connect it to the outlet.

(2) Once spontaneous imbibition has ceased, the invading water is over-pressured(i.e. a negative capillary pressure is applied) and now acts as a nonwetting fluid.The displacement is modelled using an invasion percolation process, and thewater next fills the largest oil-wet pores connected to either the inlet face of thenetwork or the invading water cluster. If, at any time during the forcedimbibition, water-wet pores are contacted by the invading cluster, then they arefilled spontaneously if the defending oil can escape. Throughout forcedimbibition, oil may escape in two different ways: either

(a) by draining along a pathway of oil-filled pores which connect it to the outlet,or

(b) by draining via film flow along a pathway of oil-wet pores to the outlet.

Clustering algorithms (following Hoshen and Kopelman, 1976) have been developedwhich permit the labelling of both oil clusters and water clusters as well as clusters ofoil-wet and water-wet pores. Note that the fluid clusters are a dynamic phenomenon,whilst the “wettability clusters” remain static during a given process.

The relative permeability curves from mixed-wet networks, computed for a variety ofα values, are shown in Figure 68. It is apparent that the oil curve loses curvature andthe water curve gains curvature as the oil-wet pore fraction increases. Furthermore, thecrossover point does not steadily move towards lower water saturations (as is oftensupposed). For a between 0 and 0.5, it actually shifts to higher saturations; only whenα> 0.5 does it begin to moves back towards lower values.

The precise structure of relative permeability curves plays a vital role in determiningreservoir performance and efficiency. The results described above show thatexperiments performed on unrepresentative core samples may yield inaccurate curvesand subsequently lead to incorrect field predictions. The precise effect of reservoirwettability on waterflood performance is now examined in more detail.

Modelling Waterflood PerformanceThe relative permeability curves described above can now be used in the conventionalfractional flow equations enabling the construction of a family of fractional flowcurves. Buckley-Leverett analyses can then be carried out to uncover how themicroscopic displacement efficiency is expected to be influenced by the wettabilityof the system. The results from the pore-scale simulations are shown in Figure 69 and



support the conclusion that optimum recovery occurs at some intermediate wettabilitystate. Indeed, comparisons with a laboratory study on wettability effects (Figure 70)are extremely encouraging: the simple rule-based simulator implemented herereproduces the experimental observations very satisfactorily.

0.8

0.7

0.6

0.5-20 0 20 40 60 80 100

Rec

over

y E

ffici

ency

20PV

3PV

BT

% Water-Wet Pores

0.8

0.7

0.6

0.5

0.4

0.3-1.0 -0.6 -0.2 0.2 0.6 1.0

Rec

over

y E

ffici

ency

BT

3PV

20PV

Amott-Harvey Wettability Index

Current ResearchRecent development work in the Institute of Petroleum Engineering at Heriot-WattUniversity has focussed upon on a new PC-based Visual C++ mixed-wet simulator.The advantages of the Visual C++ approach are twofold: (i) PC-based material is farmore accessible to the general user than raw Unix-based research code, and (ii) C++facilitates the creation of dynamic arrays that can be created “on the heap” and deletedat the end of function calls — this utilises memory far more efficiently, leading to thepossibility of producing far larger networks than before.The latest version of the mixed-wet simulator (MixWet) is now fully-functional(Appendix C) and a large number of new features are available. The full cycle ofprimary drainage — aging — water imbibition — water drainage — oil imbibition— oil drainage are all included. Aging after primary drainage is controlled via the

Figure 69Recovery efficiency vs %water-wet pores using amixed-wet simulator

Figure 70Experimental observationsfrom Jadhunandan andMorrow (1991)

80

“Wettability Parameters” group, which can be used to vary the percentage and sizedistribution of oil wet pores. Additional checkboxes are available to turn film flow offand on, calculate relative permeabilities, change boundary conditions from uni-directional flooding to intrusion from all sides, and calculate NMR T

2 signals

automatically. Moreover, the random number seed can be set explicitly by the user inorder to examine different realisations of statistically similar networks. A step-by-stepdescription of the new interface is given in Appendix C.

7 CONCLUDING REMARKS

We conclude with some final thoughts as to the importance of understandingmultiphase flow at the microscopic scale:

• The continuum approach fails to explain a great many observations

• A lot of the confusion surrounding petrophysics and petrophysical simulatorinput can be cleared up by considering the associated small-scale physics

• Remember — all displacements ultimately occur pore-by-pore

• So, by understanding the controlling physics at the pore-scale, we can look atways of improving recovery in the future (IFT reduction, depressurisation,gravity drainage, something more novel?)

• The underlying physics can be complicated and a number of controllingphenomena are intrinsically coupled (wettability, pore structure, capillarity, etc)

• BUT, if we don’t attempt to understand petrophysics at a fundamental level, thenwe run the risk of ever increasing uncertainty in our reservoir predictions



REFERENCES

Baker, L. E., 1988, “Three-Phase Relative Permeability Correlations”, SPE17369,Proceedings of the Sixth SPE/DOE Symposium on Enhanced Oil Recovery, Tulsa,OK, April 1988.

Bartell, F.E. and Osterhof, H.J., 1927,”“Determination of the Wettability of a Solidby a Liquid”, Ind. Eng. Chem., 19 (11), 1277-1280.

Blunt, M.J., 1999,”“An Empirical Model for Three-phase Relative Permeability”,SPE56474, Proceedings of the SPE Annual Technical Conference and Exhibition,Houston, TX, October 1999.

Cuiec, L., 1991, “Evaluation of Reservoir Wettability and Its Effects on Oil Recovery”,in Interfacial Phenomena in Oil Recovery, N.R. Morrow, ed., Marcel Dekker, Inc.,New York City, 319.

Dixit, A.B., McDougall, S.R., Sorbie, K.S. and Buckley, J.S., 1999, “Pore-ScaleModelling of Wettability Effects and their Influence on Oil Recovery,“SPE Res. Eval.and Eng., 2(1), 1-12.

Hui, M.-H. and Blunt, M.J., 2000, “Pore-Scale Modeling of Three-Phase Flow and theEffects of Wettability”, SPE59309, Proceedings of the SPE/DOE Improved OilRecovery Symposium, Tulsa, OK, April 2000.

Johnson, R.E. and Dettre, R.H., 1993, “Wetting of Low Energy Surfaces”,in”Wettability, Surfactant Science Series, Volume 49, J. C. Berg (editor), MarcelDekker, New |York.

Kalaydjian, F. J.-M., 1992, “Performance and Analysis of Three-Phase CapillaryPressure Curves for Drainage and Imbibition in Porous Media”, SPE24878, Proceedingsof the 67th Annual Technical Conference and Exhibition of the SPE, Washington,DC, October 1992.

Landau, L.D. and Lifschitz, E.M., 1958, Statistical Physics, Pergamon, London, 471-473.

Li, K. and Firoozabadi, A., 2000, “Experimental Study of Wettability Alteration toPreferentially Gas-Wetting in Porous Media and Its Effects”, SPE Res. Eval. andEng., 3 (2), 139-149.

Morrow, N.R., 1990, “Wettability and Its Effects on Oil Recovery”, J. Pet. Tech., 42,1476-1484.

Morrow, N.R. and McCaffery, F., 1978, “Displacement Studies in Uniformly WettedPorous Media ”, in”Wetting, Spreading and Adhesion, G.F. Padday (ed.), AcademicPress, New York City, 289-319.

82

flren, P.E and Pinczewski, W.V., 1995, “Fluid Distributions and Pore-ScaleDisplacement Mechanisms in Drainage Dominated Three-Phase Flow”, Transport inPorous Media, 20, 105-133.

Rowlinson, J.S. and Widom, B., 1989, Molecular Theory of Capillarity, ClarendonPress, Oxford.

Stone, H.L., 1970, “Probability Model for Estimating Three-Phase RelativePermeability”, J. Pet. Tech., 20, 214-218.

Stone, H.L., 1973, “Estimation of Three-Phase Relative Permeability and ResidualData”, J. Can. Pet. Tech., 12,53-61.

van Dijke, M.I.J., Sorbie K.S. and McDougall, S.R., 2000, “A Process-BasedApproach for Capillary Pressure and Relative Permeability Relationships in Mixed-Wet and Fractionally-Wet Systems”, SPE59310, Proceedings of the SPE/DOEImproved Oil Recovery Symposium, Tulsa, OK, April 2000

van Dijke, M.I.J. and Sorbie K.S., 2000, “A Probabilistic Model for Three-PhaseRelative Permeabilities in Simple Pore Systems of Heterogeneous Wettability”,Proceedings of ECMOR 7, Baveno, Italy, September 2000.

van Dijke, M.I.J., Sorbie K.S. and McDougall, S.R., 2001a, “Saturation-Dependenciesof Three-Phase Relative Permeabilities in Mixed-Wet and Fractionally-Wet Systems”,Adv. Water Resour., 24, 365-384.

Dixit, A. B., McDougall, S. R., Sorbie, K.S. and Buckley, J.S.: “Pore-Scale Modelingof Wettability Effects and their Influence on Oil Recovery”, SPE Reservoir Eval. andEng., 2 (1), pp. 25-36, February 1999.

Fenwick, D.H. and Blunt, M.J.: 1998, “Three-Dimensional Modeling of Three PhaseImbibition and Drainage, Advances in Water Resources, 21, 121-143.

Hui, M.-H. and Blunt, M.J.: 2000, “Pore-Scale Modeling of Three-Phase Flow and theEffects of Wettability”, SPE59309, Proceedings of the SPE/DOE Improved OilRecovery Symposium, Tulsa, OK, April 2000

Lerdahl, T.R., FLren, P.E. and Bakke, S.: “A Predictive Network Model for Three-Phase Flow in Porous Media”, SPE59311, Proceedings of the SPE/DOE Conferenceon Improved Oil Recovery, Tulsa, OK, April 2000.

Mani, V. and Mohanty, K.K.: 1997, “Effect of Spreading Coefficient on Three-PhaseFlow in Porous Media”, J. Colloid and Interface Science, 187, 45.

Mani, V. and Mohanty, K.K.: 1998, “Pore-Level Network Modeling of Three-PhaseCapillary Pressure and Relative Permeability Curves”, SPE Journal, 3, 238-248.

Moulu, J.-C., Vizika, O., Egermann, P. and Kalaydjian, F.: 1999 “A New Three-PhaseRelative Permeability Model for Various Wettability Conditions”, SPE56477,



Proceedings of the SPE Annual Technical Conference and Exhibition, Houston, TX,October 1999.

McDougall, S.R., Dixit, A.B. and Sorbie, K.S.: 1996,”“The Use of CapillaritySurfaces to Predict Phase Distributions in Mixed-Wet Porous Media”, Proceedings ofECMOR V conference, Loeben, Austria, September 1996.

Flren, P.E. and Pinczewski, W.V.: 1995, “Fluid Distributions and Pore-ScaleDisplacement Mechanisms in Drainage Dominated Three-Phase Flow”, Transport inPorous Media, 20, 105-133.

Pereira, G.G., Pinczewski, W.V., Chan, D.Y.C., Paterson, L. and FLren, P.E.: 1996,“Pore-Scale Network Model for Drainage-Dominated Three-Phase Flow in PorousMedia”, Transport in Porous Media, 24, 167-201.

Pereira, G.G.: 2000, “Numerical Pore-Scale Modelling of Three-Phase Fluid Flow:Comparsion between Simulation and Experiment”, Phys. Rev. E., 59, 4229-4242.

WAG Report 6: “Water Alternating Gas (WAG) Injection Studies Progress ReportNo. 6, Heriot-Watt University, Edinburgh, December 2000.

WAGrep5:““Water Alternating Gas (WAG) Injection Studies Progress Report No. 5,Heriot-Watt University, Edinburgh, June 2000.

WAGrep6:““Water Alternating Gas (WAG) Injection Studies Progress Report No. 6,Heriot-Watt University, Edinburgh, December 2000.

Sohrabi, M., Henderson, G.D., Tehrani, D.H. and Danesh, A.: “Visualisation of OilRecovery by Water Alternating Gas (WAG) Injection using High Pressure Micromodels- Water-Wet System”, SPE63000, Proceedings of the 2000 SPE Annual TechnicalConference, Dallas TX, October 2000.

From AWR paperAleman, M.A. and Slattery, J.C.: 1988, “Estimation of three-phase relativepermeabilities”, Transport in Porous Media, 3, 111-131.

Aziz, K. and Settari, T.: 1979, Petroleum Reservoir Simulation, Applied SciencePublishers, London.

Baker, L. E.: “Three-Phase Relative Permeability Correlations”, SPE17369,Proceedings of the 1988 Sixth SPE/DOE Symposium on Enhanced Oil Recovery,Tulsa, OK, April 1988.

Baker, L. E.: “Three-Phase Relative Permeability of Water-Wet, Intermediate-Wetand Oil-Wet Sandstone,” Proceedings of the 7th European IOR -Symposium, Moscow,October 1993.

Bear, J.: 1972, Dynamics of fluids in porous media, Elsevier, New York, 1972.

84

Bradford, S.A., Abriola, L.M. and Leij, F.J.: 1997 “Wettability Effects on Two- andThree-Fluid Relative Permeabilities”, J. Contaminant Hydrology, 28, 171-191.

Burdine, N.T.: 1953 “Relative Permeability Calculations from Pore-Size DistributionData”, Trans AIME, 198, 71-77.

Blunt, M.J.: “An Empirical Model for Three-phase Relative Permeability”, SPE56474,Proceedings of the SPE Annual Technical Conference and Exhibition, Houston, TX,October 1999.

Corey, A.T., Rathjens, C.H., Henderson, J.H. and Wyllie, M.R.J.: 1956 ”Three-PhaseRelative Permeability”, J. Pet. Tech., 8, 3-5; Trans. AIME, 207, 349-351.

Cuiec, L.: 1991, “Evaluation of Reservoir Wettability and Its Effects on Oil Recovery”,in Interfacial Phenomena in Oil Recovery, N.R. Morrow, ed., Marcel Dekker, Inc.,New York City, 319.

Delshad, M. and Pope, G.A.: 1989, “Comparison of Three-Phase Oil RelativePermeability Models”, Transport in Porous Media, 4, 59-83.

DiCarlo, D.A., Sahni, A., and Blunt M.J.: “Effect of Wettability on Three-PhaseRelative Permeability”, SPE40567, Proceedings of the SPE Annual TechnicalConference and Exhibition, New Orleans, September 1998.

Dixit, A.B., Buckley, J.S., McDougall, S.R. and Sorbie, K.S.: 2000,”“EmpiricalMeasures of Wettability in Porous Media and the Relationship Between Them”,Transport in Porous Media, in press.

Fayers, F.J. and Mathews, J.D.: 1984, ”Evaluation of Normalised Stone’s Methods forEstimating Three-Phase Relative Permeabilities”, SPE Journal, 20, 224-232.

Fayers, F.J.: “Extension of Stone’s Method 1 and Conditions for Real Characteristicsin Three-Phase Flow”, SPE Reservoir Engineering, 4, 437-445, November 1989.

Heiba, A.A., Davis, H.T and Scriven, L.E.: “Effect of Wettability on Two-PhaseRelative Permeabilities and Capillary Pressures”, SPE12172, Proceedings of the SPEAnnual Technical Conference and Exhibition, San Francisco, October 1983.

Heiba, A.A., Davis, H.T and Scriven, L.E.:”“Statistical Network Theory of Three-Phase Relative Permeabilities”, SPE12690, Proceedings of the 4th DOE/SPESymposium on Enhanced Oil Recovery, Tulsa, OK, April 1984.

Hustad, O.S. and Hansen, A.-G.: 1996 “A Consistent Formulation for Three-PhaseRelative Permeabilities and Phase Pressures Based on Three Sets of Two-PhaseData”, in RUTH: A Norwegian Research Program on Improved Oil Recovery -Program Summary, S.M. Skjaeveland, A. Skauge and L. Hinderacker (Eds.), NorwegianPetroleum Directorate, Stavanger.

Jerauld, G.R. and Rathmell, J.J: ”Wettability and Relative Permeability of Prudhoe



Bay: A Case Study in Mixed-Wet Reservoirs”, SPE Reservoir Engineering, 12, 58,February 1997.

Jerauld, G.R.: “General Three-Phase Relative Permeability Model for Prudhoe Bay“,SPE Reservoir Engineering, 12, 255-263, November 1997.

Kalaydjian, F. J.-M.: “Performance and Analysis of Three-Phase Capillary PressureCurves for Drainage and Imbibition in Porous Media”, SPE24878, Proceedings of the67th Annual Technical Conference and Exhibition of the SPE, Washington, DC,October 1992.

Kalaydjian, F. J.-M., Moulu, J.-C., Vizika, O., and Munkerud, P.K.: “Three-phaseFlow in Water-Wet Porous Media: Determination of Gas/Oil Relative PermeabilitiesUnder Various Spreading Conditions”, SPE26671, Proceedings of the 68th AnnualTechnical Conference and Exhibition of the SPE, Houston, TX, October 1993.

Killough, J.E.:”“Reservoir Simulation with History-Dependent Saturation Function”,SPE Journal, February 1976; Trans. AIME, 261, 37-48.

Land, C.S.: 1968, “Calculation of Imbibition Relative Permeability in Two- andThree-Phase Flow”, SPE Journal, June 1968; Trans. AIME, 243, 149-156.

Larsen, J.A. and Skauge, A.: “Methodology for Numerical Simulation with Cycle-Dependent Relative Permeabilities”, SPE Journal, 3, 163-173, June 1998.

Leverett, M.S. and Lewis, W.B.: 1941, ”Steady Flow of Gas-Oil-Water MixturesThrough Unconsolidated Sands”, Trans. AIME, 142, 107.

Mani, V. and Mohanty, K.K.: 1997, “Effect of Spreading Coefficient on Three-PhaseFlow in Porous Media”, J. Colloid and Interface Science, 187, 45.

Morrow, N.R.: 1990, “Wettability and Its Effects on Oil Recovery”, J. Pet. Tech., 42,1476-1484.

Moulu, J.-C., Vizika, O., Kalaydjian, F. and Duquerroix, J.-P.: “A New Model forThree-Phase Relative Permeabilities Based on a Fractal Representation of the PorousMedium”, SPE38891, , Proceedings of the SPE Annual Technical Conference andExhibition, San Antonio, TX, October 1997.

Moulu, J.-C., Vizika, O., Egermann, P. and Kalaydjian, F.: “A New Three-PhaseRelative Permeability Model for Various Wettability Conditions”, SPE56477,Proceeding of the SPE Annual Technical Conference and Exhibition, Houston, TX,October 1999.

Naar, J. and Wygal, R.J.: 1961, “Three-Phase Imbibition Relative Permeability”, SPEJournal, 1, 254-258, 1961; Trans. AIME, 222, 254-258.

Oak, M.J.: “Three-Phase Relative Permeability of Intermediate-Wet Berea Sandstone”,SPE22599, Proceedings of the SPE Annual Technical Meeting and Exhibition,Dallas, TX, October 1991.

86

Flren, P.E. and Pinczewski, W.V.: 1995, “Fluid Distributions and Pore-ScaleDisplacement Mechanisms in Drainage Dominated Three-Phase Flow”, Transport inPorous Media, 20, 105-133.

Parker, J.C., Lenhard, R.J. and Kuppusamy, T.: 1987, “A Parametric Model forConstitutive Properties Governing Multiphase Flow in Porous Media”, Water ResourcesResearch, 23, 618-624.

Soll, W.E. and Celia, M.A.: 1993 “A modified percolation approach to simulatingthree-fluid capillary pressure-saturation relationships”, Advances in Water Resources,16, 107-126.

Stone, H.L.: 1970, “Probability Model for Estimating Three-Phase RelativePermeability”, J. Pet. Tech., 20, 214-218.

Stone, H.L.: 1973, “Estimation of Three-Phase Relative Permeability and ResidualData”, J. Can. Pet. Tech., 12, 53-61.

Temeng, K.O.: “Three-Phase Relative Permeability Model for Arbitrary WettabilitySystems”, Proceedings of the 6th European IOR-Symposium, Stavanger, May 1991.

Van Dijke, M.I.J., McDougall, S.R. and Sorbie K.S.: 2000a, “Three-Phase CapillaryPressure and Relative Permeability Relationships in Mixed-wet Systems”. Transportin Porous Media, in press.

Van Dijke, M.I.J., Sorbie K.S. and McDougall, S.R.: 2000b, “A Process-BasedApproach for Three-Phase Capillary Pressure and Relative Permeability Relationshipsin Mixed-Wet Systems”, SPE59310, Proceedings of the SPE/DOE Symposium onImproved Oil Recovery, Tulsa, OK, April 2000.

Vizika, O. and Lombard, J.-M.: 1996, “Wettability and Spreading: Two Key Parametersin Oil Recovery with Three-Phase Gravity Drainage”, SPE Reservoir Engineering,11, 54-60.

Zhou, D. and Blunt, M.: 1997, “Effect of spreading coefficient on the distribution oflight non-aqueous phase liquid in the subsurface”, J. Contam. Hydrol. 25,1-1



8 APPENDIX A: SOME USEFUL DEFINITIONS AND CONCEPTS

A1Sphere PackingsSphere packings are very useful for qualitative understanding of pore shape and canalso give some insight into porosity measurements. Examples are shown in FigureA.1 — the cubic packing gives the largest porosity (47.6%) whilst the rhombohedralpacking gives the smallest (26%). Let us analyse a simple packing in more detail.

Case 1 Case 2 Case 3

Case 4 Case 5 Case 6

A2Specific SurfaceAn important measure characterising the grain surface of a porous medium is knownas the specific surface. This plays an important role in the adsorption capacity of asample and affects a number of petrophysical measures, including electrical resistivity,initial water saturation, and absolute permeability. There are two definitions ofspecific surface;

(i) The surface area of the pores per unit volume of solids (Ss), and;

(ii) The surface area of the pores per unit bulk volume (Sv). Mathematical relationships

for (i) and (ii) can be derived by noting that the surface area of a sphere is 4πR2.For this idealized system, we find that S

s=3/R and S

v=π/2R — hence, the specific

surface of a porous medium is inversely proportional to the (mean) grain size.

A3The Pore Size DistributionPhotomicrographs show that pore structure is extremely complex and we may askourselves whether there is any point in trying to impose conformity by attemptinganalysis using idealised pore geometries (eg cylinders, etc.). Although it is possibleto partially reconstruct real pore geometries using microtomographical techniques,these methods are extremely computer-intensive and we really have no option but touse idealized geometries if we wish to model petrophysical parameters in a cost-effective way.

Figure A1

88

In fact, mercury porosimetry analysis still relies upon such models to define so-calledpore-size distributions — probability distribution functions defined for the radiusrange (R

min, R

max) that act as fingerprints for different rock samples. However, we need

to be careful about what we mean by “pore-size”. Mercury porosimeters assume poresto be circular cylinders and experimental pore size distribution functions are derivedunder these assumptions (in fact, these distributions are actually volume-weightedthroat-size distributions — see notes).—

A4 The Coordination NumberOne measure of the interconnectedness of pore structure is the so-called co-ordinationnumber (z): it is defined as the average number of branches meeting at a point (FigureA2). The co-ordination number plays an important role in determining such things asbreakthrough and residual saturations during multiphase displacements, and isprobably one of the most important parameters governing flow processes at the pore-scale. Co-ordination numbers can vary greatly from system to system: from z=6 fora simple cubic sphere packing to z ≈ 2.8 for Berea sandstone (Doyen, 1988; Dullien,1979). This wide variation should clearly be taken into account when attempting tointerpret flow behaviour.

x

z

y

Z = 6 Z = 4 Z = 2

A5 Surface and Interfacial TensionIt is well-documented that the molecules of a liquid are closely bound together byforces of molecular attraction, which serve to keep it as one cohesive assemblage ofparticles. Although these forces of cohesion act to cancel one another in the interiorof the liquid, the situation is somewhat different at the surface: at an air/liquidinterface the cohesive forces of the underlying liquid far exceed those of thecompeting air molecules, resulting in a net inward pull. The system then behaves asif the liquid and air were separated by a uniformly stretched membrane, characterisedby a surface tension (σ). If, instead of a liquid/air system, a liquid/liquid system isconsidered, the tensile force is referred to as interfacial tension and is one of the mostimportant parameters governing multiphase flow in porous media.

A6 Wettability, Contact Angle and Spreading PhenomenaIf a solid surface is contacted by a pair of fluids, one of them will tend to have a greateraffinity for that surface than the other. This phase is identified as the wetting phase,whilst the other is known as the nonwetting phase. The wetting preference of a flatsolid surface can be quantified by inspecting the contact angle and the associated forcebalance (Figure A3). This is due to the fact that the magnitude of the contact angle atequilibrium is intrinsically linked to the free surface energies of the system viaYoung’s equation:

Figure A2



σ σ σ θS S1 2 12− = cos

where σS1

represents the solid/fluid 1 surface free energy, σS2

the solid/fluid 2 surfacefree energy, and σ

12 the fluid-fluid interfacial tension. The value of the contact angle

may lie anywhere from 0o to 180o, and is strongly dependent upon the fluid pair andsurface material involved (Figure A4). If θ=90o, then σ

s1=σ

s2 and neither fluid is

wetting; the system is then described as neutral.

σ - σ = s1 s2 σ12 θcos

σ s1

σ12

θ σs2

θ = 158º

θ = 30º

θ = 30º θ = 48º

θ = 54º θ = 106º

θ = 30º

θ = 35º

Water

Water

Silica Surface

Isooctane Isoquinoline NaphthenicAcid

Isooctane+

5.7% Isoquinoline

Calcite Surface

WaterWater

A7Spreading PhenomenaConsideration of the trigonometric term in Young’s equation shows that mechanicalequilibrium between two fluids and a solid surface is only possible if:

σ σ σS S1 2 12− <

i.e. cosθmust always be <1. If this condition is violated, however, the system is unstable andthe wetting phase will spontaneously spread on the solid. Although quantification ofthis “spreadability” in fluid/solid systems is not possible at present (there is no currenttechnique available for measuring either σ

S1 or σ

S2), this is not a problem if the solid

is replaced by a third fluid or a gas.

Figure A3

Figure A4

90

The spreadability of an oil can be quantified with recourse to a spreading co-efficient,defined by:

So wa ow oa= − +σ σ σ( )

and so initial spreading occurs if this coefficient is either zero or positive. It is evidentfrom the above discussion that the spreading characteristics of oil/water/gas systemsmust have serious implications for a wide range of hydrocarbon recovery processes.The effect of the spreading coefficient upon three-phase displacements is dealt withmore fully in the notes.

A8CapillarityConsider the situation where a liquid is in contact with a glass capillary tube. If theadhesive forces of the liquid to the glass are greater than the cohesive forces in theliquid, then the interface will curve upwards towards the tube, forming a meniscuswhich intersects the tube wall at an angle θ (Figure A5). The fact that the meniscusis curved, means that there is now a non-zero vertical component of surface tension,which acts to pull liquid up the capillary. This continues, until the vertical componentof surface tension is exactly balanced by the weight of fluid below, i.e. when:

π ρ π σ θR h g R2 2= cos

Total Upward Force = 2πRσ WA cosθ

σ WA

2R

θ

where h is the height of the liquid column, R the capillary radius, and r the density ofthe fluid. Although the application of this simplistic example to flow in porous mediamay not be immediately obvious, it should serve to demonstrate how surface andinterfacial tension forces can play a crucial role in determining fluid distributions atthe pore scale. The full implications of capillary phenomena are apparent in the notes.

Figure A5



9 APPENDIX B: MATHEMATICAL BACKGROUND ANDDERIVATIONS

B1Capillary Bundle PermeabilityThis can be derived as follows:

Consider the system shown in Figure 1a, which consists of n capillary tubes per unitcross-sectional area. The length of the system is taken to be L. The flow through asingle cylindrical capillary of radius R is given by Poiseuille’s law:

qR P

L= π

µ

4

8∆

where µ is the fluid viscosity and ∆P the pressure drop across the tubes. Hence, thetotal flow (Q) through the porous medium is:

Q nqn R P

L= = π

µ

4

8∆

Now, the porosity (φ) of the medium is nπR2L/(AL)= nπR2 (as cross-sectional areaA=1). Hence,

QR P

L= φ

µ

2

8∆

Setting this equal to Q given by Darcy’s Law finally leads to:

kR D= =φ φ

2 2

8 32

An interesting aspect of this simple result is that the quantity (k/φ)1/2 can be thoughtof as a sort of average pore diameter.

B2 Carman-Kozeny EquationThe basic premise of this modelling approach is that particle transit times in the actualporous medium and the equivalent tortuous rough conduit must be the same. Particlevelocity in a rough conduit (v

t) is given by the equation:

vR P

Ltt

=2

2∆

µ

where Lt is the conduit length and particle velocity through the porous medium (v

r)

is given by:

92

vk P

Lrt

= ∆φµ

For travel times to be the same in both systems, we require Lt/v

t=L

r/v

r, and this leads

to the relationship:

kRT

H= φ2

22

where T is called the tortuosity of the sample (Lt/Lr). We now need to determine a

hydraulic radius. The theory utilises established hydraulic practice, in that the“equivalent” conduit is assumed to have a radius, RH which takes the form:

Rof pores SH

s

= =−

Volume of poresSurface area

φφ( )1

and hence we can write:

kT Ss

=−φ

φ

3

2 2 22 1( )

The permeability can be written In terms of an average grain diameter (Dp) by noting

that, for spherical particles, Ss=6/D

p. Hence,

Lr

Lt

Lt

Lr

T =

v = t

v = r

u = rR ∆P2

2µL t

u = v φ r r

µLr

µL φrk∆P

k∆P

For travel times to be equal

L t

v t

Lr

v r= => k =

R φ2

2T2

What do we take for R ?H

R =H

Volume of poresSurface area of pores

= φ(1 - φ)Ss

Ss=specific surface / solid volume

=> k =φ3

2(1- φ) SsT2 2 2

Figure B1Carman-Kozeny details



R1R2

N

φ

B

D

A

P

C

ρρ

σδl

δl

a b

q2r2

r1q1

qw

LL

r2

r1

Currently accepted values of the percolation thresholds of some two-dimensional networks

HoneycombSquareKagoméTriangular

3446

1-2 sin(π/18) ~ 0.6527*1/2*0.5222 sin(π/18) ~ 0.3473*

1.9622.0882.84

0.69620.59270.6521/2*

Currently accepted values of the percolation thresholds of some three-dimensional networks

Network Z pcb Bc = Zpcb pcs

Network Z pcb Bc = Zpcb pcs

DiamondSimple cubicBCCFCC

46812

0.38860.24880.17950.198

1.551.491.441.43

0.42990.31160.24640.119

*Exact result

Figure B2Generalised Derivation ofCapillary Pressure Acrossa Curved Interface

Figure B3Pore Doublet Details

Table B1Percolation Thresholds Fora Variety of NetworkGeometries

94

kwt

kkrwt

dS/ (P )

dS/ (P )

knwt

kkrnwt

dS/ (P )

dS/ (P )

c2

s 0

S S

c2

s 0

S

c2

s

S

c2

s 0

S

wt

= =

= =

=

=

=

=

=

=

=

=

∫∫

∫∫

1

1

1swt

Figure B4Purcell Method forAbsolute PermeabilityPrediction

Figure B5Purcell Extension toRelative PermeabilityPrediction



k S S

dSP

dSP

k S S

dSP

dSP

SS S

S SS

S S

rw w w wcS

S Sw

cS

S rnw w nw wcS Sw

S

cS

S

w ww wi

wi ornw w

w or

( ) ( ( )) ( ) ( ( ))

( ) ( )

= =

= −− −

= − −−

=

=

=

==

=

=

=

∫

∫

∫

∫λ λ

λ λ

22

0

20

12

2

1

20

1

111

where the prefactorsaredefined by:

SS Swi or−

Figure B6Burdine Extension toPurcell Model of RelativePermeabilities

96

10 APPENDIX C: DETAILS OF THE HERIOT-WATT MIXWETSIMULATOR

Description of Interface Controls

(1)-(3) The number of nodes (junctions) in the X- Y- and Z-direction

(4) Coordination Number (Z): the average number of pore elements meetingat a node

(5) Distortion Factor used to distort the network, leading to a distribution ofpore lengths (under development). 0.0 gives a regular cubic (3D) or square(2D) network. 0.5 is the maximum allowed value

(6) Fraction of oil-wet pores: this refers to the fraction of the total number ofpores in the network that become oil-wet after primary drainage

(6a) Theta button —used to read in water-wet and oil-wet contact angle ranges

(7)-(8) The mixed-wet and fractionally-wet radio buttons are mutually exclusive.Mixed-wet assigns oil-wet characteristics to a fraction of the largest porescontaining oil after primary drainage. Fractionally-wet assigns oil-wetcharacteristics to a size-independent fraction of the pores containing oilafter primary drainage

(9) Primary drainage (PD) checkbox: oil displaces water from a 100% water-wet network. This can also be used to examine other generic 2-phaseincompressible drainage displacements (e.g. gas-oil drainage). Successivelyhigher (positive) capillary pressures are applied to the system and thisdrives the displacement

(10) Water imbibition (WI) checkbox: water imbibes along water-wet pathwaysin the system and snaps-off in the smallest oil-filled pores. Aging willalready have taken place before this part of the cycle. The displacement iscontrolled by reducing the pressure in the oil phase (i.e. successively lower(positive) capillary pressures are applied to the system

(11) Water drainage (WD) checkbox: successively higher (negative) capillarypressures are applied to the system and water is forced into successivelynarrower oil-wet pores

(12) Oil imbibition (OI) checkbox: oil imbibes along oil-wet pathways, snapping-off the smallest water-filled pores. This is driven by a gradual reduction inwater pressure and this drives the oil imbibition

(13) Oil drive (OD) checkbox: oil pressure is increased once again and oil isforced into water-wet pores

(14) Random number seed — changing this value produces a new networkrealisation with the same average properties as others that use the same



parameter set. In order to get statistically meaningful results, several runsshould be performed using different random number seeds and the resultsaveraged

(15) PSD refers to the pore-size distribution exponent (n), where f(r)~rn. n=0refers to a Uniform distribution, n=3 gives a Cubic distribution, etc. For aLog-Uniform distribution, this parameter should be set to n=–0.999 (notn=–1, as this leads to a singularity). N=10 gives a truncated normaldistribution

(16) Volume exponent (n) — the volume of a pore element is taken to beV(r)~r n. n =2 gives cylinders

(17) Conductivity exponent (λ) — the conductance of a pore element is takento be G(r)~r l . l =4 gives cylinders

(18) Rmin

is the minimum capillary entry radius of the sample

(19) Rmax

is the maximum capillary entry radius of the sample

(20) Graphics Options: these radio buttons allow the user to view differentaspects of the simulation

(21) Calculate Kri — 2-phase relative permeabilities are calculated when thischeckbox is checked. An SOR algorithm is used to solve the pressure fieldand elemental flows can be subsequently calculated

(22) Mercury? — this checkbox is used to allow invasion of mercury from allsides of the network instead of unidirectional flooding

(23) Water Films? — if this is checked, then water will leave the system throughthin films. This, in effect, leads to no trapping of the water phase during oilinvasion in a 100% water-wet network. This option is checked whensimulating mercury injection, as the intrusion in this case is essentiallymercury-vacuum

(24) Oil Films? — if this is checked, then oil will leave the system through thinfilms. This, in effect, leads to no trapping of the oil phase during waterimbibition in a 100% oil-wet network

(25) “NMR Calculations” GroupBox — partial T2 signals are calculated if this

is checked. This information is then used to calculate the AccessibilityFunction (A(R)). Values for “Rho” and”“Mag. Density” should be leftunchanged at present as the NMR section is still an area of active research

(26) RUN — this button is pressed to set the simulation running

98

(27) Exit—— used to leave the package. At present, an error box appears upontermination: this should simply be cancelled and does not affectperformance

The main output file contains information regarding nonwetting phase saturation(Snw) (oil or gas), wetting phase relative permeability (krw) (water or oil), nonwettingphase relative permeability (krnw) (oil or gas), current capillary entry radius (R),number fraction of pores filled with nonwetting phase (p).

Ultimately, it should be possible to simply import an experimental capillary pressuredata set, interpret this automatically and “instantiate” a network that would serve asa pre-anchored numerical representation of an experimental sample. This could thenbe used for a wide variety of sensitivity studies.

1

2

3

4

5

6

78

6a

910111213

14 19

18

22

21

24

23

17

16

1520

26

27

25

Figure 78Annotated interface (seetext for details offunctions)

CONTENTS

1 GLOSSARY OF COMMON TERMS AND CONCEPTS IN RESERVOIR SIMULATION AND FLOW THROUGH POROUS MEDIA 1.1 Some General Defi nitions 1.2 Reservoir Fluid Properties 1.3 Single Phase Rock Properties 1.4 Multi-Phase Rock/Fluid Properties 1.5 Wettability and Fluid Displacement Processes 1.6 Oil Recovery Methods, Waterfl ood Patterns and Sweep Effi ciency 1.7 Terms Used in Numerical Reservoir Simulation 1.8 Numerical Solution of the Flow Equations in Reservoir Simulation 1.9 Pseudo-Isation and Upscaling 1.10 Numerical Simulation of Flow in Fractured Systems 1.11 Miscellaneous - Vertical Equilibrium, Miscible Displacement and Dispersion

Glossary of Terms

2

Glossary of Terms


Glossary of Terms

1 GLOSSARY OF COMMON TERMS AND CONCEPTS IN RESERVOIR SIMULATION AND FLOW THROUGH POROUS MEDIA

This glossary is intended for use by the reader as a quick reference to terms used commonly in reservoir engineering in general and in reservoir simulation in particular. The student is not expected to work through this from begining to end in a systematic manner. However, the students should make sure that he or she is quite familiar with all the technical terms that appear in the main text of this unit. It is hoped that this is of particular use for distance learning students who may have studied the reservoir engineering distance leasrning unit some time ago but hopefully it will also be of use to our residential students.

1.1 Some General Defi nitionsOilfi eld Units volumes in oilfi eld units are barrels (bbl or B); 1 bbl = 5.615 ft3 or 0.159 m3. A stock tank barrel (STB) is the same volume defi ned at some surface standard conditions (in the stock tank) which are usually 60oF and 14.7 psi. A reservoir barrel (RB) is the same volume defi ned at reservoir conditions which can range from ~ 90oF and 1500 psi for shallow reservoirs to > 350oF and 15,000 psi for very deep (high temperature - high pressure, HTHP) reservoirs. Note that when 1RB of oil is produced it gives a volume generally less than 1B at the surface since it loses its gas. (See formation volume factor.)

Oil Types: Dry gas; Wet gas; Gas Condensate; Volatile oil; “Black” oil; Heavy (viscous) oil; Tar - see Tables 1 and 2 below.

DDew PPooiint

BubblePoiinntt

LLiqquidVooVoV

llume

Paat

hof

Pro

dduct

ion

Bubble PointBubble Pointor

Dissolved GasDissolved GasReservoirsReservoirs

0 50 100 150 200

Reservoir Temperature, Temperature, T ºF

Res

ervo

ir P

ress

ure,

PS

IA

250 300 350

Dew PointDew Pointor

RetrogradeRetrogradeGas-CondensateGas-Condensate

ReservoirsReservoirs

B1

C

C1

B

B2

D

B3A2

AA

AA1

Single PhaseSingle PhaseGas ReservoirsGas Reservoirs

4000

3500

3000

2500

2000

1500

1000

500

Pat

h of

Res

ervo

ir F

lui

Pat

h of

Res

ervo

ir F

luid

Cric

onde

nthe

rm =

250

C

ricon

dent

herm

= 2

50 º

F

TTc =

127

ºFF

CriticalCriticalPointPoint

80%

40%

20%

10%

5%

0%Figure 1Pressure Temperature Phase Diagram of a Reservoir Fluid

4

Glossary of Terms

Reservoir Surface GOR API Typical Composition, Mole % Fluid Appearance Range Gravity

C1 C2 C3 C4 C5 C6+

Dry gas Colourless gas Almost no liquids - 100

Wet gas Colourless gas - >100 Mscf/bbl 60o -70o 96 2.7 0.3 0.5 0.1 0.4 some clear or straw-coloured liquid

Condensate Colourless gas - 3-100 50o-70o 87 4.4 2.3 1.7 0.8 3.8 significant amounts Mscf/bbl of light coloured (900-18000 m3/m3) liquid

“Volatile” or Brown liquid - "3000 40o-50o 64 7.5 4.7 4.1 3.0 16.7high shrinkage various yellow, red, scf/bbl oil or green hues (500 m3/m3)

“Black” or Dark brown 100- 2500 30o-40o 49 2.8 1.9 1.6 1.2 43.5 low shrinkage to black viscous scf/bbl oil liquid (20 - 450 m3/m3)

Heavy oil Black viscous liquid Almost no gas 10o-25o 20 3.0 2.0 2.0 12.0 71 in solution

Tar Black substance No gas < 10o - - - - - 90+ viscosity > 10,000cp

Component Black Oil Volatile Oil Gas-Condensate Dry Gas Gas

C1 48.83 64.36 87.07 95.85 86.67 C2 2.75 7.52 4.39 2.67 7.77 C3 1.93 4.74 2.29 0.34 2.95 C4 1.60 4.12 1.74 0.52 1.73 C5 1.15 2.97 0.83 0.08 0.88 C6 1.59 1.38 0.60 0.12 .... C7

+ 42.15 14.91 3.80 0.42 ....

Mol. Wt. C7+ 100.00 100.00 100.00 100.00 100.00 GOR, SCF/bbl 225 181.00 112 157 .... Tank gravity, 625 2000 18,200 105,000 Inf. 0API Liquid 34.3 50.1 60.8 54.7 .... color Greenish Medium Light Water Black Orange Straw White

1.2 Reservoir Fluid PropertiesPhase: A chemically homogeneous region of fl uid which is separated from another phase by an interface e.g. oleic (oil) phase, aqueous phase (mainly water), gas phase, solid phase (rock). There is no particular symbol but frequently subscripted o, w, g; phases are immiscible.

Table 1Describing various oil types from dry gas to tar

Table 2Mole Composition and Other Properties of Typical Single-Phase Reservoir Fluids


Glossary of Terms

Inter Facial Tension (IFT): The IFT between two phases is a measure of energy required to create a certain area of the interface. Indeed, the IFT is given in dimensions which are energy per unit area. The symbol for IFT is σ and units are dyne/cm in σ and units are dyne/cm in σc.g.s. units and N/m (newtons per m) in S.I. units. For example, if both gas and oil are present in a reservoir then the gas/oil IFT may be in the range, σgo ~ 0.1-10 mN/m; likewise. The oil/water value may be in the range, σ0w ~ 15 - 40 mN/m. Note that numerically 1mN/m = 1dyne/cm.

Component: A single chemical species that may be present in a phase; e.g. in the aqueous phase there are many components - water (H2O), sodium chloride (NaCl), dissolved oxygen (O2) etc.; in the oil phase there can be hundreds or even thousands of components - hydrocarbons based on C1, C2, C3, etc. Some of these oil components are shown in Table 2.

Viscosity: The viscosity of a fl uid is a measure of the (frictional) energy dissipated when it is in motion resisting an applied shearing force; dimensions [force/area.time] and units are Pa.s (SI) or poise (metric). The most common unit in oilfi eld applications is centiPoise (cP or cp). Typical example are:- water viscosity at standard conditions, μw ~ 1 cP; typical light North Sea oils have μo ~ 0.3 - 0.6 cP at reservoir conditions (T ~ 200oF ; P ~ 4000 - 6000 psi); at reservoir conditions, medium viscosity oils have μo ~ 1 - 6 cP; moderately viscous oils have μo ~ 6 - 50 cP; very viscous oils may have μo ~ 50 - 1000s cP and tars may have μo ~ up to 10000 cP.

Formation Volume Factor: The factor describing the ratio of volume of a phase (e.g. oil, water) in the “formation” (i.e. reservoir at high temperature and pressure) to that at the surface; symbols Bw, Bo etc. For oil, a typical range for Bo is ~1.1 - 1.3 since, at reservoir conditions, it often contains large amounts of dissolved gas which is released at surface as the pressure drops and the oil shrinks; oilfi eld units [reservoir barrels/stock tank barrel (RB/STB)].

API Gravity (°API): Defi nition =

Gas Solubility Factors (or Solution Gas/Oil Ratios): These factors describe the volume of gas (usually in standard cubic feet, SCF) per volume of oil (usually stock tank barrel, STB); symbol, Rso and Rsw; units SCF/STB.

Compressibility: The compressibility (c) of a fl uid (oil, gas, water) or rock formation can be defi ned in terms of the volume (V) change or density (ρ) change with pressure as follows:

c

VVP P

= − ∂∂

P PP PP PP P

P PP P = ∂

∂P P∂P PP PP PP PP P

P PP P

1 1V1 1V∂1 1∂1 1

1 1 1 11 1

ρP PρP Pρ

Note that this quantity is normally expressed in units of psi-1.

Typical ranges of compressibilities are presented below (from Craft & Hawkins (Terry revision), 1991):

6

Glossary of Terms

Compressibilities (units of 10-6 psi-1)

Formation rock 3 - 10 Water 2 - 4 Undersaturated Oil 5 - 100 Gas at 1000psi 900 - 1300 Gas at 5000psi 50 - 200

Compressibilities are used in reservoir engineering for Material Balance Calculations.

Material Balance Equations: Material Balance applied to a reservoir is simply a volumetric balance. It may be expressed as an equation which relates: • The quantities of oil, gas and water produced.

• The reservoir (average) pressure.

• The quantity of water infl ux (e.g. from the aquifer).

• The initial oil and gas content of the reservoir.

Essentially the material balance equations described how the energy of expansion and infl ux “drive” production in the reservoir. If there is a suffi ciently low (or zero) fl uid infl ux, the reservoir pressure will decline. One form of the Material Balance Equation is given below where each term on the left-hand side described a mechanism of fl uid production (from Craft & Hawkins (Terry revision), 1991):

N B BN m B

BB B m N B

c S c

Sp Wp W

N B B B W

t tN Bt tN B itiBtiB

giBgiB g gB Bg gB B i tm Ni tm N Bi tBiw wc Sw wc S i f

wiSwiS ep Wep W

p tN Bp tN B g wB Bg wB B pWpW

.(N B.(N B ). .N m. .N m

.( ) (i t) (i t).m N).m Ni t).i tm Ni tm N).m Ni tm N. .i t. .i tBi tB. .Bi tBi. .ic S.c Sc Sw wc S.c Sw wc S

.

. (N B. (N Bp t. (p tN Bp tN B. (N Bp tN B .

− +B− +Bt t− +t tBt tB− +Bt tBi− +i )− +) − +B B− +B Bg g− +g gB Bg gB B− +B Bg gB B i t− +i t) (− +) (i t) (i t− +i t) (i t+i t+i t

+−

p W+p W

= +N B= +N BN Bp tN B= +N Bp tN BN B. (N B= +N B. (N BN Bp tN B. (N Bp tN B= +N Bp tN B. (N Bp tN B[ ]R R[ ]R R B B[ ]B Bp t[ ]p t p[ ]pR RpR R[ ]R RpR Rsoi[ ]soiR RsoiR R[ ]R RsoiR R g w[ ]g wB Bg wB B[ ]B Bg wB B. ([ ]. (N B. (N B[ ]N B. (N Bp t. (p t[ ]p t. (p tN Bp tN B. (N Bp tN B[ ]N Bp tN B. (N Bp tN B ).[ ]).. (= +. ([ ]. (= +. (N B. (N B= +N B. (N B[ ]N B. (N B= +N B. (N Bp t. (p t= +p t. (p t[ ]p t. (p t= +p t. (p tN Bp tN B. (N Bp tN B= +N Bp tN B. (N Bp tN B[ ]N Bp tN B. (N Bp tN B= +N Bp tN B. (N Bp tN B R R−R R[ ]R R−R R B B+B BB Bg wB B+B Bg wB B

1i t1i t 1∆p W∆p W

Where the terms have the following meaning: N = initial reservoir oil, STB; Np = cumulative produced oil, STB Boi = initial oil formation volume factor, bbl/STB Bo = oil formation volume factor, bbl/STB Bgi = initial gas formation volume factor, bbl/STB Bg = gas formation volume factor, bbl/STB Bw = water formation volume factor, bbl/STB Rsoi = initial solution gas-oil ratio, SCF/STB Rp = cumulative produced gas-oil ratio, SCF/STB Rso = solution gas-oil ratio, SCF/STB We = water infl ux into the reservoir, bbl Wp = cumulative produced water, bbl cw = water isothermal compressibility, psi-1 cf = formation isothermal compressibility, psi-

∆ p = change in average reservoir pressure, psi


Glossary of Terms

Swi = initial water saturation m = (Initial hydrocarbon vol. of gas cap)/(Initial hydrocarbon vol. of oil)

In practice the material balance equation is often applied in the “linear form” of Havlena and Odeh (J. Pet. Tech., pp896-900, Aug. 1963; ibid, pp815-822, July 1964); see discussion in Craft & Hawkins (Terry revision, 1991).

In the above formulation of the Material Balance Equation, the various terms have the following interpretation.

Left-Hand Side of the Material Balance Equation

• The following terms account for the expansion of any oil and/or gas zones that may be present in the reservoir:

N B BN m B

BB Bt ti

ti

gig gB Bg gB B i.(N B.(N B )

. .N m. .N m.( )− +B− +Bt t− +t tBt tB− +Bt tB i− +i )− +) B B−B B

• The following term accounts for the change in void space volume which is the expansion of the formation and the connate water:

( ). . . .( )1( )

1( )+( )

+−

m N( )m N( ). .m N. . Bc S.c S. c

SptiBtiB w wc Sw wc S.c S.w w.c S. i f

wiSwiS∆p∆p

• The next term is the amount of water influx that has occurred into the reservoir:

We

Right-Hand Side of the Material Balance Equation

• The fi rst term of the RHS represents the production of oil and gas:

= +[ ]N B= +N B= +[ ]R R[ ]−[ ]−R R−[ ]−[ ]B[ ]p t[ ]p t[ ]N Bp tN B= +N B= +p t= +N B= +[ ]p[ ][ ]R R[ ]p[ ]R R[ ][ ]soi[ ][ ]R R[ ]soi[ ]R R[ ][ ]g[ ][ ]B[ ]g[ ]B[ ]. ([ ]. ([ ]= +[ ]= +. (= +[ ]= +N B. (N B= +N B= +. (= +N B= +[ ]N B[ ]. ([ ]N B[ ]= +[ ]= +N B= +[ ]= +. (= +[ ]= +N B= +[ ]= +p t. (p t[ ]p t[ ]. ([ ]p t[ ]= +[ ]= +p t= +[ ]= +. (= +[ ]= +p t= +[ ]= +N Bp tN B. (N Bp tN B= +N B= +p t= +N B= +. (= +N B= +p t= +N B= +[ ]N B[ ]p t[ ]N B[ ]. ([ ]N B[ ]p t[ ]N B[ ]= +[ ]= +N B= +[ ]= +p t= +[ ]= +N B= +[ ]= +. (= +[ ]= +N B= +[ ]= +p t= +[ ]= +N B= +[ ]= +[ ]).[ ]

• The second term of the RHS represents the production of water:

Bw.Wp

1.3 Single Phase Rock PropertiesPorosity: the fraction of a rock that is pore space; common symbol, φ Porosity varies from φ ≈ 0.25 for a fairly permeable rock down to φ ≈ 0.1 for a very low permeability rock; there may be an approximate correlation between k and φ.

Pores & pore throats: The tiny connected passages that exist in permeable rocks; typically of size 1μm to 200 μm; they are easily visible in s.e.m. (scanning electron microscopy). Pores may be lined by diagenetic minerals e.g. clays. The narrower constrictions between pore bodies are referred to as Pore Throats. See Figure 2:

8

Glossary of Terms

~1mm

quartz

illite

10mm

illite

quartz

Permeability: The fl uid (or gas) conducting capacity of a rock is known as the permeability; symbol k ; units Darcy (D) or milliDarcy (mD); dimensions -> [L]2. Permeability is found experimentally using Darcy's Law (see below). Permeability can be anisotropic and show tensor properties (see below) - denoted tensor properties (see below) - denoted tensor k . Probably the most important quantity from the point of view of the reservoir engineer since its distribution dictates connectivity and fl uid fl ow in a reservoir. Timmerman (p. 83, Vol. 1, Practical Reservoir Engineering, 1982) presents the rule:

Classifi cation Permeability Range (mD) poor to fair 1 - 15 moderate 15 - 50 good 50 - 250 very good 250 - 1000 excellent > 1000

k/φ Correlations: It has been found in many systems that there is a relationship between permeability, k, and porosity, φ. This is not always the case and much scatter can be seen in a k/φ crossplot. Broadly, higher permeability rocks have a higher porosity and some of the relationships reported in the literature are shown below. Some examples of k/φ correlations which have appeared in the literature are shown in Figure 3:

Figure 2


Glossary of Terms

100.0

10,000

1,000

100

10

1

50.0

10.0

5.0

1.0

0.5

0.1

0.05

0.01

6

0 10 20 30

8 10 12 14 16 18 20 22Core Porosity (%)

Core Porosity (%)

Cor

e P

erm

eabi

lty (

md)

Cor

e P

erm

eabi

lty (

md)

100.0

10,000

1,000

100

10

1

50.0

10.0

5.0

1.0

0.5

0.1

0.05

0.01

6

0 10 20 30

8 10 12 14 16 18 20 22Core Porosity (%)

Core Porosity (%)

Cor

e P

erm

eabi

lty (

md)

Cor

e P

erm

eabi

lty (

md)

Darcy’s Law: Originally a law for single phase fl ow that relates the total volumetric fl ow rate (Q) of a fl uid through a porous medium to the pressure gradient (∂P/∂x) and the properties of the fl uid (μ = viscosity) and the porous medium (k = permeability; A = cross-sectional area): Note that Darcy's law can be used to defi ne permeability using the quantities defi ned in Figure 5 as follows:

Figure 3Permeability/Porosity Correlation for Cores from the Bradford Sandstone

Figure 4Permeability/Porosity Correlation for Cores from the Brent Field

10

Glossary of Terms

Q

k A Px

= −

∂∂

.k A.k A

µ

Note that the β in the equation in Figure 5 is a factor for units conversion (see Chapter 2).

Darcy Velocity: This is the velocity, u, calculated as, u = Q/A; this may be expressed as,

uQA

k Px

= == = −

k Pk P

k Pk P

k Pk P

∂k P∂k P∂

k Pk P

k Pk P

k Pk P

µ

Pore Velocity: This is the fl uid velocity, v, given by,

v = == =Q

Au

.φ φ

∆P

L

QQ

Q = β.k.A

µ.

∆P∆P∆L

Permeability Anisotropy: Since permeability can be directional, then it is possible for kx ≠ ky ≠ kz in a given system. This is often seen in practice when comparing the horizontal permeability, (kh), with the vertical permeability, (kv) - usually as the ratio, (kv/kh). It is often found (kv/kh) < 1, i.e. there is more resistance to vertical fl ow than horizontal fl ow. The value of (kv/kh) must be evaluated with respect to the scale (i.e. the size) of the sample or system. The value of this quantity will be different in a core plug or in a large simulator grid block in which the core plug was a small part. The origin of the anisotropy may be quite different in each case.

At the small (core plug) scale, anisotropy may come from fabric anisotropy at the grain level or from lamination at the slightly larger scale (laminaset scale). At the larger scale (grid block), the anisotropy may arise from larger scale heterogeneity, even though locally the component rock facies are completely isotropic. This is illustrated in Figure 6.

Figure 5Schematic of the Single Phase Darcy Law


Glossary of Terms

Fabric Anistropy

Heterogeneity Anistropy

Lamination

Small Scale

Grid Block Scale (100s m)

Core Plug

Rock GrainsHi k lamina

Lo k lamina

For Whole System

Low Perm Lensesor Shales

kh

kv

kv

(kv/kh) = 1

Low Perm Sand(kv/kh) = 1

kh

1.4 Multi-Phase Rock/Fluid PropertiesSaturation: The saturation of a phase (oil, water, gas) is the fraction of the pore space that it occupies (not of the total rock + pore space volume); symbols Sw, Soand Sg ; saturation is a fraction, where Sw + So + Sg = 1. Multi-phase fl ow functions such as relative permeability and capillary pressure (see below) depend strongly on the local fl uid saturations.

Residual Saturation: The residual saturation of a phase is the amount of that phase (fraction pore space) that is trapped or is trapped or is trapped irreducible; e.g. after many pore volumes of water displace oil from a rock, we reach residual oil saturation, Sor; the corresponding connate (irreducible) water level is Swc (or Swi); the related trapped gas saturation is Srg; at the residual or trapped phase saturation the corresponding relative permeability(see below) of that phase is zero. Strictly, we should refer to the phases in terms of wetting and non-wetting phases - the residual phase of non-wetting phase is trapped in the pores by capillary forces. Typically, in a moderately water wet sandstone, Sor~ 0.2 - 0.35. The amount of trapped or residual phase depends on the permeability and wettability of the rock and a large amount of industry data is available on this quantity: For example, the relationship for k vs. Swc (or Swi) is shown for a range of reservoir formations, Figure 7.

Figure 6Permeability anistropy at different scales

12

Glossary of Terms

1

3 2

12

1310

117

9

8

645

11A

10,000

5,000

1,000

500

100

50

10

5

1.0

1 = Hawkins2 = Magnolia3 = Washington4 = Elk Basin5 = Tangely6 = Creole7 = Synthetic Alundum8 = Lake St John

0 10 20 30 40 50 60 70 80 90 100

% Connate water

Air

Per

mea

bilit

y (m

D)

9 = Louisiana Gulf Coast Miocene Age-Well A10 = Ditto-Wells Band C11 = North Belridge California11A = North Belridge California Core Analysis Data12 = Dominguez Second Zone13 = Ohio Sandstone

Relative Permeability: A quantity (fraction) that describes the amount of impairment to fl ow of one phase on another. It is defi ned in the two phase Darcy law (see notes); depends on the Saturation of the phase; e.g. in two phase fl ow -> krw and kro depend on Sw (since Sw + So = 1).

A schematic of the Two Phase Darcy Law showing the defi nition of Relative Permeability is presented in Figure 8.

At steady-state fl ow conditions, the oil and water fl ow rates in and out, Qo and Qw, are the same:

Figure 7Correlation between (air) permeability and the connate water (Swc) for a wc) for a wc

range of reservoir rocks


Glossary of Terms

Qw

Qo

Qw

∆Po

∆Pw

LQo

At steady-state flow conditions, the oil and water flow rates in and out, Qo and Qw are the same

NB the Units for the two-phase Darcy Law are exactly the same as those in Figure 2

The two-phase Darcy Law is as follows:

kro

krw

Sw

1

00 1

Scematic of relative permeability,krw and kro

Rel.Perm.

Qw and Qo = volumetric flow rates of water and oilA = cross-sectional areaL = system lengthµw and µo = system lengthk = absolute permeabilities∆Pw and ∆Po = the pressure drops across the water and oil phases at steady-state flow conditionskrw and kro = the water and oil relative permeabilities

Where:

Qk k A P

L

Qk k A P

L

wrw

w

w

oro

o

o

=

=

. . .

. . .

µ

µ

∆

∆

The two-phase Darcy Law is as follows:

Where: Qw and Qo = Volumetric fl ow rates of water and oil; A = Cross-sectional area; L = System length; μw and μo = Water and oil viscosities; k = Absolute permeabilities; ΔPw and ΔPo = The pressure drops across the water and oil phases at steady-state fl ow conditions krw and kro = The water and oil relative permeabilities

Several further examples of relative permeabilities and capillary pressure are given later in this glossary. Note that the Units for the two-phase Darcy Law are given in Figure 2, Chapter 2.

Capillary Pressure: The difference in pressure between two (immiscible) phases; defi ned as the non wetting phase pressure minus the wetting phase pressure; depends on the saturation - for two phases capillary pressure, Pc(Sw) = Po- Pw (for a water wet porous medium). The following fi gures show schematic fi gures for Capillary Pressure (Pc(Sw)) and Relative Permeability (krw(Sw) and kro(Sw)) for a water wet system:

Figure 8The two-phase Darcy Law and relative permeability

14

Glossary of Terms

Swc

Pc

Sw

krel

Sor

Capillary Pressure

Swc

kro

krw

Sw

Sor

Relative Permeability

Mobility and Mobility Ratio: the mobility of a phase (e.g. λw or λo) is defi ned as the effective permeability of that phase (e.g. kw = k . krw ; ko = k . kro) divided by the viscosity of that phase;

λ

µλ

µwrw

wo

ro

o

k k k k=

=

.k k.k k;

.k k.k k

Mobility ratio, M, is given by:

M

kk

o

w

ro w

rw o

= == =o= =o

λλ

µµ

..

Fractional Flow: The Fractional Flow of a phase is the volumetric fl ow rate of the phase under a given pressure gradient, in the presence of another phase. The symbols for water and oil fractional fl ow are fwsymbols for water and oil fractional fl ow are fwsymbols for water and oil fractional fl ow are f and fo and they depend on the phase saturation, Sw:

f

QQ

fQQ

Q Q Qwfwfw

To

o

TT oQ QT oQ Q w= == =w= =w = +Q Q= +Q QT o= +T oQ QT oQ Q= +Q QT oQ Q; ;; ;f; ;f

Q; ;

QQ

; ;Qo; ;ofof; ;fof

o; ;o= =; ;= =f= =f; ;f= =f; ; ; ;where

The fractional fl ows play a central part in Buckley-Leverett (B-L) theory of linear displacement which starts from the conservation equation:

∂∂

= − ∂

∂

∂

∂

= − ∂

∂

S

tfx

St

fx

w ww ww w = −w w= − ∂w w∂w ww wfw wf o o∂o o∂o oo oo oo o = −o o= −o o

fo of;

Buckley-Leverett Theory: This mathematical theory of viscous dominated water → oil displacement is based on the fact that the velocity, vSw, of a fi xed saturation value Sw is given by:

v vSwv vSwv v w

w

fwfw

Sv v=v v

∂∂

.

Figure 9Schematics of capillary pressure and relative permeability for a water-wet system


Glossary of Terms

where v is the fl uid velocity, v = Q/(Aφ) and (dfw) and (dfw) and (df /dSw) is the slope of the fractional fl ow curve. The relationship between the fractional fl ow and Buckley Leverett theory is illustrated in Figure 10.

Sor

Swc Swf

Sw Swf

Sor

Swc

FractionalFlow

of Water,

Water Saturation, Sw

Flood FrontHeight

Length, (x/L)

Fractional Flow Welge Tangent

fw

"Buckley-Leverett" Saturation Profile

1.5 Wettability and Fluid Displacement ProcessesWettability: This is a measure of the preference of the rock surface to be wetted by a particular phase - aqueous or oleic - or some mixed or intermediate combination. The Wettability of a porous medium determines the form of the relative permeabilities and capillary pressure curves; a very complex subject which is still the subject of much research. We refer to: Water wet, Oil wet and Intermediate wet systems in the following defi nitions.

Water-Wet: Water-wet formation: Where water is the preferential wetting phase. Water occupies the smaller pores and forms a fi lm over all of the rock surface - even in the pores containing oil. A Waterfl ood in such a system would be an imbibitionprocess (see below). Water would spontaneously imbibe (see below) into a water-wet core containing mobile oil at Sor, hence displacing the oil.

Oil-Wet: Oil-wet formation: Where oil is the preferential wetting phase. Oil occupies the smaller pores and forms a fi lm over all of the rock surface - even in the pores containing water. A Waterfl ood in such a system would be a drainage process (see below). Oil would spontaneously imbibe into an oil-wet core containing mobile water at Swr, displacing the water.

Intermediate-Wet: An Intermediate wet formation is where some degree of water wetness and oil wetness is shown by the same rock. Some different types of intermediately wet system have been identifi ed known as Mixed wet and Mixed wet and Mixed wet Fractionally wet. Both water and oil may spontaneously imbibe (see below) into such a system to some degree and indeed this forms the basis for certain Wettability Tests known as the Amott Test and the Amott Test and the Amott USBM Test (USBM => United States Bureau of Mines). USBM Test (USBM => United States Bureau of Mines). USBM Test

Drainage: A Drainage displacement process is when the non-wetting phase is increasing. For example, in a water wet porous medium, drainage would be oil displacing water. The drainage and imbibition capillary pressure curves and relative permeabilities are different since these petrophysical functions depend on the saturation history. A simple schematic of a drainage process is shown in Figure 11.

Figure 10Relationship between the fractional fl ow function and the Buckley-Leverett front height

16

Glossary of Terms

Qo

Qw

Oil Injection Water Wet Coreat 100% Water

Water Injection

Water imbibes intocore displacingoil-water wet orintermediate wetsystem

Water Wet Coreat sor

Core at swc

Oil

Water

Imbibition: An Imbibition displacement process is when the wetting phase is increasing. For example, in a water wet porous medium, drainage would be water displacing oil as shown in Figure 12. The drainage and imbibition capillary pressure curves and relative permeabilities are different since these petrophysical functions depend on the saturation history.

Qo

Qw


Water Injection



Core at swc

Oil

Water

Spontaneous Imbibition: This process occurs when a wetting phase invades a porous medium in the absence of any external driving force. The wetting fl uid is “sucked in” under the infl uence of the surface forces. For example, if we have a water wet core at irreducible water saturation, Swr, then water may spontaneously imbibe and displace the oil as shown in Figure 13.

The observed behaviour in a system of Intermediate Wettability is shown in Figure 14 where we see that both phases can spontaneously imbibe under certain conditions.

Qo

Qw


Water Injection



Core at swc

Oil

Water

Core at swc Core at sorOil

Water

Pc

Swc

kro

krw

Sor

d

di

i

Sw

Drainage

Drainage and ImbibitionCapillary Pressure

Drainage (d) and Imbibition (i)Relative Permeabilities

Imbibition

Sw

krel

Primary and Secondary Recovery Processes: Primary and Secondary processes refer to the stage in the fl uid displacement when one phase displaces another. For example, in a water wet porous medium (--> means displaces) :

Figure 14Intermediate wettability.Both water and oil may spontaneously imbibe into the core displacing the other phase. Shows both water wet and oil wet character

Figure 11Drainage

Figure 12Imbibtion

Figure 13Spontaneous Imbibition


Glossary of Terms

Primary drainage: is oil --> water from a core at 100% water saturation to Swr. Secondary imbibition: is water --> oil from a core at Swr and mobile oil to Sor.

Examples: Figures 15 and 16 show schematics of typical Drainage and Imbibition capillary pressure (Pc) and relative permeability (krw and kro) curves for a water wet system. Primary Drainage (oil --> water from core at 100% water) and Secondary Imbibition (water --> oil from core at Swr) processes are illustrated:

Pc

Swc

kroro

krw

Sor

d

di

i

Sw

Drainage

Drainage and ImbibitionCapillary Pressure

Drainage (d) and Imbibition (i)Relative Permeabilities

Imbibition

Sw

krel

DrainageDrainage

ImbibitionImbibition

100

100

80

80

60

60

40

40

20

200

0Wetting Phase Saturation, %PV

Rel

ativ

e P

erm

eabi

lity,

%

Examples: Further examples of experimental capillary pressures and relative permeabilities in cores are shown for various processes (Drainage and Imbibition) and wettability conditions (Water wet and Intermediate wet) in fi gure 17,18,19 and 20 on the following pages.

Figure 15Drainage and imbibition capillary pressures

Figure 16Drainage and imbibition relative permeability char-acteristics

18

Glossary of Terms

liO

retaW

100

100

80

80

60

60

40

40

20

200

0Water Saturation, %PV

Water Saturation, %PV

Rel

ativ

e P

erm

eabi

lity,

Fra

ctio

nR

elat

ive

Per

mea

bilit

y, F

ract

ion

liO

retaW

1.0

0.1

0.01

0.001

0.00010 20 40 60 80 100

liO

retaW

100

100

80

80

60

60

40

40

20

200



Rel

ativ

e P

erm

eabi

lity,

Fra

ctio

nR

elat

ive

Per

mea

bilit

y, F

ract

ion

liO

retaW

1.0

0.1

0.01

0.001

0.00010 20 40 60 80 100

100

100

80

80

60

60

40

40

20

200



Rel

ativ

e P

erm

eabi

lity,

Fra

ctio

nR

elat

ive

Per

mea

bilit

y, F

ract

ion

1.0

0.1

0.01

0.001

0.00010 20 40 60 80 100

retaW

liO

retaW

li

O

Figure 17Typical water-oil relative permeability characteristics, strongly water-wet rock

Figure 18Typical water-oil relative permeability characteristics, strongly water-wet rock

Figure 19Typical water-oil relative permeability characteristics, strongly oil-wet rock


Glossary of Terms


Rel

ativ

e P

erm

eabi

lity,

Fra

ctio

n

1.0

0.1

0.01

0.001

0.00010 20 40 60 80 100

rretaW

li

O

Examples: Experimental Capillary Pressures in cores for various processes (Drainage and Imbibition) and wettability conditions (Water wet, Oil Wet and Intermediate Wet). Figure 21.

Figure 20Typical water-oil relative permeability characteristics, strongly oil-wet rock

20

Glossary of Terms

Water Wet48

40

32

24

16

8

0

20

16

12

8

4

0

0 20 40 60 80 100

12

Water Saturation - %

0 20 40 60 80 100Water Saturation - %

0 20 40 60 80 100Water Saturation - %

Cap

illar

y P

ress

ure

- C

m o

f Hg

Cap

illar

y P

ress

ure

- C

m o

f Hg

Oil Wet

Intermediate Wet

48

40

32

24

16

8

00 20 40 60 80 100

Oil Saturation - %

Cap

illar

y P

ress

ure

- C

m o

f Hg

32

24

-24

16

-16

8

-8

0

Cap

illar

y P

ress

ure

- C

m o

f Hg

Venango core VL-2k = 28.2 md

1

2

1

2

3

Capillary Pressure Characteristics,Strongly Water-Wet Rock.

Curve 1 - DrainageCurve 2 - Imbibition

Capillary Pressure Characteristics,(After Ref. 30)

Oil-Water Capillary Pressure Characteristics,Ten-Sleep Sandstone, Oil-Wet Rock

(After Ref. 29).Curve 1 - DrainageCurve 2 - Imbibition

Oil-Water Capillary Pressure Characteristics,Intermediate Wettability.

Curve 1 - DrainageCurve 2 - Spontaneous Imbibition

Curve 3 Forced Imbibition

Figure 21


Glossary of Terms

Examples: Relative PermeabilititesExamples of experimental relative permeabilities in cores for Water Wet and Water Wet and Water Wet Oil Wet systems. Figure 22.Wet systems. Figure 22.Wet

100

100

80

80

60

60

40

40

20

200



Rel

ativ

e P

erm

eabi

lity,

Fra

ctio

nR

elat

ive

Per

mea

bilit

y, F

ract

ion

1.0

0.1

0.01

0.001

0.00010 20 40 60 80 100

retaW

liO

retaW

li

O

liO

retaW

100

100

80

80

60

60

40

40

20

200



Rel

ativ

e P

erm

eabi

lity,

Fra

ctio

nR

elat

ive

Per

mea

bilit

y, F

ract

ion

liO

retaW

1.0

0.1

0.01

0.001

0.00010 20 40 60 80 100

Typical Water-Oil Relative PermeabilityCharacteristics,

Strongly Water-Wet Rock


Strongly Oil-Wet Rock


Strongly Water-Wet Rock


Strongly Oil-Wet RockFigure 22

22

Glossary of Terms

EXAMPLES: Relative PermeabilititesA simple table summarising the typical characteristics of water-wet and oil-wet relative permeabilities is given below.

WATER WET OIL WET

Swc mostly > 20% < 15% (Often < 10%)

Sw wherekrw = kro Sw > 50% Sw < 50%(Points A on Figure 23)

krw at Sro krw < 0.3 krw > 0.5 (approaching 1)(Points B on Figure 23)

This is shown schematically in Figure 23:

retaW

liO

retaW

liO

100

100

60

60

40

40

80

80

20

200

0

100

100

60

60

40

40

80

80

20

200

0

A

A B

B

SwiSwi

Water-WetReservoir

Oil-WetReservoir

Water Saturation, %

Rel

ativ

e P

erm

eabi

lity,

% o

f Air

Per

mea

bilit

y

Water Saturation, %

In Water-Wet System

Sw mostly > 20%

At Point A: kro = krw ; Sw > 50%

krw at Sor / kro at Swi < 30%

In Oil-Wet System

Sw < 15%

At Point A: kro = krw ; Sw < 50%

krw at Sor / kro at Swi > 50%

1.6 Oil Recovery Methods, Waterfl ood Patterns and Sweep Effi ciencyHere we refer to the method used to develop the reservoir as follows:

• Primary Depletion - allowing the reservoir to produce under the original reservoir energy i.e. by natural expansion. If the pressure falls below the bubble point (Pb), then gas appears and the primary depletion process is known as solution gas drive:

• Secondary Recovery - where reservoir pressure is supported by injection, usually of water in waterfl ooding but early gas injection may be considered also as waterfl ooding but early gas injection may be considered also as waterfl oodingsecondary recovery In addition to supporting the pressure (maintaining reservoir energy), water or gas injection also displaces oil directly:

Figure 23Infl uence of wettability on relative permeability (after Fertl, OGJ, 22 May 1978)


Glossary of Terms

• Tertiary Recovery or Enhanced Oil Recovery (EOR) or Improved Oil Recovery (IOR) - this refers to a range of methods which are designed to recover additional oil that would not be recovered by either Primary or Secondary Recovery.

Such methods include:

Thermal Methods - steam, in-situ combustion,.. Gas Injection - N2, CO2, hydrocarbon gas injection (usually after a waterfl ood) Chemical Methods - surfactant, polymer, alkali,.. Microbial Methods - using bugs to recover oil!

Waterfl ood Pattern: On-land Waterfl ooding is often carried out with the producers and injectors in a particular pattern. This is known as pattern fl ooding and examples of such patterns are: Five Spot, Nine Spot, Line Drive etc. as shown schematically in Figure 24.

Injection WellProduction WellPattern Boundary

Regular Four-Spot Skewed Four-Spot Normal Nine-Spot

Direct Line Drive Staggered Line Drive

Inverted Nine-Spot

Inverted Seven-SpotSeven-SpotFive-Spot

Areal Sweep Effi ciency: The Areal Sweep Effi ciency refers to the fraction of areal reservoir that is swept at a given pore volume throughput of displacing fl uid as shown schematically in Figure 25. For example, the Areal Sweep Effi ciency at Breakthrough for various processes (Waterfl ooding, Gas Displacement and Miscible fl ooding) is shown as a function of mobility ratio in Figure 26:

Figure 24Examples of areal patterns of injectors and producers (pattern fl ooding)

24

Glossary of Terms

High Areal Sweep Poor Areal Sweep

100

90

80

70

60

500.1 1.0 10.0

Mobility Ratio

Bre

akth

roug

h A

real

Sw

eep

Effi

cien

cy, %

Water→OilGas→OilMiscible

High Areal Sweep Poor Areal Sweep

100

90

80

70

60

500.1 1.0 10.0

Mobility Ratio

Bre

akth

roug

h A

real

Sw

eep

Effi

cien

cy, %

Water→OilGas→OilMiscible

Vertical Sweep Effi ciency: The Vertical Sweep Effi ciency refers to the fraction of vertical section (or cross-section) of reservoir that is swept at a given pore volume throughput of displacing fl uid. This is function of the heterogeneity of the system (e.g. stratifi cation), the fl uid displacement process (e.g. waterfl ooding, gas injection) and the balance of forces (e.g. importance of gravity). It is shown schematically in Figure 27.

Figure 26Areal sweep effi ciency at breakthrough in a fi ve spot pattern for various displacement processes

Figure 25 Schematic of areal sweep effi ciency


Glossary of Terms

Good Vertical Sweep

Poor Vertical Sweep(By gravity over-ride or the presenceof a high-k streak in this case)

1.7 Terms Used in Numerical Reservoir SimulationMass Conservation: This is a general principle which is used in many areas of computational fl uid dynamics. It says that:

(mass fl ow rate into a block) - (mass fl ow rate out) = (the rate of mass accumulation in that block)A reservoir simulation model (for 1, 2 or 3 phases) is basically: A mass conservation equation combined with Darcy’s law.

Black Oil Model: Different types of formulation of the transport equations for multiphase/multicomponent fl ow are used to simulate the various recovery processes; by far the most common is the “Black Oil Model” which can simulate primary “Black Oil Model” which can simulate primary “Black Oil Model”depletion and most secondary recovery processes. A black oil simulation model is one of the most common approaches to modelling immiscible two and three phase (o, w, g) fl ow processes in porous media; it treats the phases rather like components; it does not model full compositional effects; instead, it allows the gas to dissolve in the other two phases (described by Rso and Rsw); however, no “oil” is allowed to enter the gas phase.

Grid Structure: This refers to the geometry of the grid being used in the numerical simulation of the system. This grid may be Cartesian, radial or distorted and may be 1D, 2D or 3D (see notes).

Spatial Discretisation: This is the process of dividing the grid in space into divisions of Δx, Δy and Δz. In reservoir simulation, we always “chop up” the reservoir into blocks as shown in the gridded examples below and then we model the block →block fl ows.

Figure 27 Schematic showing vertical sweep effi ciency

26

Glossary of Terms

Temporal Discretisation: This is the process of dividing up the time steps into divisions of Δt.

2D Areal Grid: This is a 2D grid structure as shown in Figure 28 which is imposed looking down onto the reservoir. For a Cartesian system, it would divide up the x and y directions in the reservoir into increments of Δx and Δy.

W1

W2

W3

y

x

∆x

∆z

∆y

Just 1 x z-block in 2D Areal Grid

2D Cross-Sectional Model: This is a 2D grid structure which is imposed on a vertical slice down through the reservoir. For a Cartesian system, it would divide up the x and z directions in the reservoir into increments of Δx and Δz. Cross-sectional calculations are carried out to asses the effects of vertical stratifi cation in the system and to generate pseudo-function for upscaling. (Figure 29).

3D Cartesian GridThe 3D Cartesian Grid is the most commonly used grid when constructing a relatively simple model of a reservoir or a setion of a reservoir. This is shown in Figure 30.

Figure 29

Figure 28Perspective view of a 2D areal (x/y) reservoir simulation grid: W = well


Glossary of Terms

Water Injector

Producer

∆x

∆y

∆z

(Variable)

Transmissibility: The transmissibility between two grid blocks is a measure of how easily fl uids fl ow between them. The mathematical expression for two phase fl ow between grid blocks i and (i+1) is (for water):

Block i Block i+1

Sw i Qw Sw i+1

krw i (kA)i krw i+1 (kA) i+1

(i+1/2) Boundary

Q k

kB xwQ kwQ k irw

w wBw wBi

Q k=Q k( )Q k( )Q kA( )A

( )P P( )P Pi i( )i iP Pi iP P( )P Pi iP PP P−P P( )P P−P P

∆++

i i( )i i+i i( )i iP Pi iP P( )P Pi iP P+P Pi iP P( )P Pi iP P1 2

1 2

i i( )i i1i i( )i iP Pi iP P( )P Pi iP P1P Pi iP P( )P Pi iP P/1 2/1 2

/1 2/1 2

.µ

where the inter-grid block quantities are averages at the interfaces (where i+1/2 denotes this block to block interface. The single phase Transmissibility, Ti+1/2, Ti+1/2, T , is given by:

T

xiTiT i+

+=( )kA( )kA

∆1 21 2

/1 2/1 2/1 2/1 2

and the full Water Transmissibility, Tw,i+1/2, Tw,i+1/2, T , between the two grid blocks is given by:

T

xk

BT

kBw iTw iT i rw

w w i

iTiT rw

w wBw wBi

, /w i, /w i/

/

/

/

., /+, /+

++

+

=( )kA( )kA

∆

=

1 2, /1 2, /

1 2/1 2/

1 2/1 2/

1 2/1 2/

1 2/1 2/µ µBµ µBw wµ µw wBw wBµ µBw wB

iµ µ

i /µ µ

/

/µ µ/

+µ µ

+µ µµ µ µ µµ µ1 2

µ µ1 2/1 2/

µ µ/1 2/

Figure 31

Figure 30A 3D Cartesian grid for reservoir simulation

28

Glossary of Terms

Therefore, we may write:

Q Tw wQ Tw wQ T i i i= −Q T= −Q T + +i i+ +i i, /i i, /i i= −, /= −i i+ +i i, /i i+ +i i( )P P( )P Pi i( )i iP Pi iP P( )P Pi iP Pi( )iP PiP P( )P PiP P= −( )= −P P= −P P( )P P= −P P+ +( )+ +P P+ +P P( )P P+ +P Pi i+ +i i( )i i+ +i iP Pi iP P+ +P Pi iP P( )P Pi iP P+ +P Pi iP P1 2i i1 2i i= −1 2= −i i+ +i i1 2i i+ +i i, /1 2, /i i, /i i1 2i i, /i i= −, /= −1 2= −, /= −i i+ +i i, /i i+ +i i1 2i i+ +i i, /i i+ +i i 1( )1( )P P( )P P1P P( )P PP P= −P P( )P P= −P P1P P= −P P( )P P= −P P

The water transmissibility is clearly made up of two parts each of which is an average between the blocks. The single phase part is (k.A)av and the two phase part is [krw/(μw.Bw)]av

(k.A)av - a Harmonic Average between blocks is taken for the single phase part of the transmissibility (see Chapter 4; Section 3.2).

[krw/(μw.Bw)]av - this term is more complicated. For the average relative permeability term, [krw]av an Upstream Weighting is used; For [(Upstream Weighting is used; For [(Upstream Weighting μw.Bw)]av the Arithmetic Average between blocks is taken. (See Chapter 4; Section 3.3).

Numerical Dispersion: The spreading of a fl ood front in a displacement process such as waterfl ooding, which is due to numerical effects, is known as Numerical Dispersion. It is due to both the spatial (Δx) and time (Δt) discretisation or truncation error that arises from the gridding. This spreading of fl ood fronts tends to lead to early breakthrough and other errors in recovery. How bad the error is depends on several factors including the actual fl uid recovery process being simulated e.g. waterfl ooding, water-alternating-gas (WAG) etc. (See Chapter 4; Section 2.2).

Grid Orientation: The Grid Orientation problem arises when we have fl uid fl ow both oriented with the principal grid direction and diagonally across this grid as shown schematically in Figure 32. Numerical results are different for each of the fl uid “paths” through the grid structure. This problem arises mainly due to the use of 5-point difference schemes (in 2D) in the Spatial Discretisation. It may be alleviated by using more sophisticated numerical schemes such as 9-point schemes (in 2D).

II

I = InjectorP = Producer

PP

PP

Local Grid Refi nement: Local Grid Refi nement is when the simulation grid is Local Grid Refi nement is when the simulation grid is Local Grid Refi nementmade fi ne in a region of the reservoir where (LGR) quantities (such as pressure or saturations) are changing rapidly. The idea is to increase the accuracy of the simulation in the region where it matters, rather than everywhere in the reservoir. E.g. LGR ==> close to wells or in the fl ood pilot area but coarser grid in the aquifer.

Figure 32Flow arrows show the fl uid paths in oriented grid and diagonal fl ow leading to grid orientation errors


Glossary of Terms

Hybrid Grid LGR: Hybrid Grids are mixed geometry combinations of grids which are used to improve the modelling of fl ows in different regions. The most common use of hybrid grids are Cartesian/Radial combinations where the radial grid is used near a well. Hybrid Grid LGR can be used in a similar way to other LGR scheme.

Examples: A simple example of LGR and Hybrid Grid structure is shown in fi gure Hybrid Grid structure is shown in fi gure Hybrid Grid33.

Coarse Gridin Aquifer

Hybrid Grid

Producer

Injector

Distorted Grids: A Distorted Grid is a grid structure that is “bent” to more closely Distorted Grid is a grid structure that is “bent” to more closely Distorted Gridfollow the fl ow lines or the system geometry in a particular case.

Corner Point Geometry: In some simulators (e.g. Eclipse), the option exists to enter the geometry of the vertices of the grid blocks. This allows the user to defi ne complex geometries which better match the system shape. This option is known as Corner Point Geometry and it requires that the block → block transmissibilities are modifi ed accordingly.

The idea of Corner Point Geometry is illustrated schematically in fi gure 34:

Figure 33Schematic of local grid refi nement (LGR)

30

Glossary of Terms

Coordinates ofVertices ( ) Specified

Block ( ) centres

Block <-> BlockTransmissibiltyHighly Distorted Grid Blocks

Fault

Distorted Grid

Corner Point Geometry

L1

L1

L2

L2

L3

L3L4

L4

Figure 34Grid structures for Faults and Distorted Grids


Glossary of Terms

1 Extented Refinement

2 Imbedded Refinement (Rectangular)

3 Variable Refinement (Radial)

4 Hybrid (Radial in Rectangular) Local Grid Refinement Around Vertical Wells

5 Hybrid Local Grid Refinement (Horizontal Wells)

Figure 35Types of local Grids

32

Glossary of Terms

History Matching: History Matching in numerical simulation is the process of adjusting the simulator input in such a way as to achieve a better fi t to the actual reservoir performance. Ideally, the changes in the simulation model should most closely refl ect change in the knowledge of the fi eld geology e.g. the permeability of a high perm streak, the presence of sealing faults etc. The observables which are commonly matched are the fi eld and individual well cumulative productions, watercuts and pressures.

Examples: Examples of history matching of pressure and water-oil-ratio (WOR) in two reservoirs are Figure 36. Note that in the WOR match the “fi rst pass” was very inaccurate but that eventually a suitable match was found. A vitally important point is that a good history match must be obtained for the right reason. It may be possible to get a satisfactory match for the wrong reason i.e. by adjusting a variable that is not the primary cause of the mis-match (indeed, this is very often the case). However, such a model will eventually have very poor predictive properties.

Final Match

CalculatedField Data

First Trial

Observed Data

Time, Days x 10

Wat

ercu

t

1.0

0.8

0.6

0.4

0.2

0.00 1 3 5 7 9 11

Year

Pre

ssur

e (p

si)

3600

3400

3200

3000

2800

2600

2400

2200

20000 1 2 3 4 5 6 7 8 9 10 11 12

(a)


Year

Pre

ssur

e (p

si)

3600

3400

3200

3000

2800

2600

2400

2200

20000 1 2 3 4 5 6 7 8 9 10 11 12

(b)


Year

Pre

ssur

e (p

si)

3600

3400

3200

3000

2800

2600

2400

2200

20000 1 2 3 4 5 6 7 8 9 10 11 12

(c)

Final Pressure Matches of Typical Khursaniyah Field Wells:(a) Reservoir AB(b) Reservoir C(c) Reservoir D (MP)

Initial and Final Matches of WOR of a Well in aHighly Stratified Reservoir

1.8 Numerical Solution of the Flow Equations in Reservoir SimulationFinite Differences: When the derivative in a differential equation is approximated as a difference equation as follows:

Figure 36A fi eld of a history match of watercut and well pressures; redraw from Mattax and Dalton (1990)


Glossary of Terms

∂∂

≈ ( )

∆( )−( )S

x( )S S( )−( )−S S−( )−

xi

( )i i( )( )S S( )i i( )S S( )( )1( )

then this is referred to as a Finite Difference approximation. In this example, which is illustrated below, (∂S/∂x)i is the derivative of Saturation (S) with respect to x at grid point i ; Si and Si-1 are the discrete values of S at grid points i and i-1, respectively; Δx is the size of the spatial grid.

Slope = Slope = ∂s∂∂x i

SSi-1

SSi

SSi+1

∆x ∆x ∆x ....x ....

i-1 i+1ix

S(x

,t)

Linear Equations: When fi nite difference methods are applied to the differential equations of reservoir simulation, a set of linear equations results. These have the form:

A. x bx b=x b

where A is a matrix of coeffi cients, x is the vector of unknowns and b is the (known) “right hand side”. Expanded up, this set of linear equations has the form:

a11 x1 + a12 x2 + a13 x3 .... + a1n xn = b1a21 x1 + a22 x2 + a23 x3 .... + a2n xn = b2a31 x1 + a32 x2 + a33 x3 .... + a3n xn = b3a41 x1 + a42 x2 + a43 x3 .... + a4n xn = b4

............an1 x1 + an2 x2 + an3 x3 .... + ann xn = bn

Direct Solution Of Linear Equations: A Direct Solution method is when the linear equations are solved by an algorithm which has a fi xed number of operations (given fi xed number of operations (given fi xedN, the number of linear equations [unknowns]). If the equations have a solution, then, in principle, a direct method will give the exact answer, x(true), to the machine accuracy. E.g. Gaussian Elimination

Iterative Solution Of Linear Equations: An Iterative Solution method is when the linear equations are solved by an algorithm which has a variable number of operations. A fi rst estimate of the solution vector x(0) is made and this is successively refi ned to converge to the true solution. In a convergent iterative method, then x(v) → x(true) as v → ∞. It is because of this iterative process that a variable number of steps may

Figure 37

34

Glossary of Terms

be required depending on how accurate the answer must be. Normally, the iterative method would be carried out until:

| x(v) - x(true)| < Tol.

where Tol. is some small pre-specifi ed tolerance. E.g. Line Successive Over-relaxation (LSOR)

Grid Ordering Schemes: When the simulation grid blocks are ordered in various ways, the structure of the non-zeros in the sparse matrix, A, is different. Advantage can be taken of the precise structure when solving these equations. E.g. Schemes known as D2 and D4 ordering.

1.9 Pseudo-Isation and UpscalingUpscaling: The process of reproducing the results of a calculation which is carried out on a “fi ne grid” on a coarser grid is known as Upscaling. The basic idea of upscaling is shown schematically in Figure 38. The input properties at the coarser scale must take into account the fl ow effects of the smaller scale structure. These coarser scale properties then become pseudo-properties.


Glossary of Terms

High Sw

Low Sw

Oil

Fine Grid Layered Model

Upscaling or Pseudo-Isation

Coarse Grid Layered Model

Fine GridCoarse Grid

Time

Oil Recovery

% O

OIP

Pseudo-Property: This refers to the value of a property or function (e.g. permeability, relative permeability..) which is an average or effective value at a certain scale - usually the grid block scale. For example, we might put the value kx = 150 mD in a simulator grid block which is 200 ft x 200 ft x 30 ft. Clearly, this incorporates a large amount of geological substructure and permeability may vary very signifi cantly in different parts of this block.

Pseudo-Relative Permeability: This is probably the most important pseudo-property that is used in reservoir simulation. It refers to the effective relative permeability in the simulation model at the grid block scale and is shown schematically in Figures 39 and 40. It must incorporate the effects of all the smaller scale geological heterogeneity, the balance of forces (viscous/capillary/gravity) and certain numerical effects (numerical dispersion). Methods for calculating the Pseudo-rel perm include: Jacks et al, Kyte and Berry, Stone etc. Newer methods are based on tensor pseudo-relative permeabilities (Pickup and Sorbie, 1994).

Figure 38Basic idea of upscaling

36

Glossary of Terms

Geopseudos: When the fl uid fl ow upscaling is performed in the correct context of the sedimentary structure up from the lamina, laminaset, bedform.. scales, then the approach is known as the Geopseudo Methodology. This has been developed in work at Heriot-Watt which has extended more conventional approaches by “putting in the geology”.

Figure 40 shows a simple example of a pseudo relative permeability showing “holdup of fl uid”.

Fine Grid Layered Model

Upscaling or Pseudo-Isation

Coarse Grid Layered Model

Fine GridCoarse Grid

Time

Oil Recovery

% O

OIP

krel

Sw

krel

Sw

"Rock"Relative Permeabilities

Pseudo-RelativePermeabilities

High Sw

Low Sw

Oil

Figure 39Basic idea of upscaling or pseudo-isation


Glossary of Terms

Sor

Sw

Sw

Sor

Swc

Swc

Water Flow

No Water Flow

krokrw

Sw

Rel Perms.

Rel Perms. ?

1.10 Numerical Simulation of Flow in Fractured SystemsFractured System: In this context, we imply systems (such as in many carbonate reservoirs) where small scale conductive fractures occur but most of the oil is in the rock matrix. In certain non-porous fractured rock reservoirs (e.g. fractured volcanics), it is possible to have all the oil in the fractures but these are much less common.

Typically, in porous fractured systems: fracture porosity, φf = 0.1 - 1% of bulk volume f = 0.1 - 1% of bulk volume f(i.e. as a fraction φf = 0.001 - 0.01).f = 0.001 - 0.01).f

Features of Fracture Geometry: The main geometric features of fractures which are thought to affect fl uid fl ow are the: fracture orientation, width, conductivity, connectivity and spacing (or fracture density). The “interface” between the fracture and the matrix will also play a very important role in multiphase fl ow and fl uid displacement processes.

Stylolites: Stylolites are frequently found in limestones. They are laterally extensive features formed by grain-to-grain sutured contact caused by the phenomenon of pressure solution. These features may signifi cantly reduce vertical permeability thus causing systems containing them to have very low (kv/kh) ratios (at certain scales).

Vugs: Vugs are dissolution “holes” in a carbonate rock caused by diagenetic reactions.

Dual Porosity Models: These are the most widely used simulation models for modelling fl ow in fractured systems. They have separate conservation/fl ow equations for the matrix and the fractures and matrix → fracture fl ow is represented by Transfer Functions. They are most frequently used to model multiphase fl ow in fractured carbonates.

Variants of this model allow for; (i) fl ow only in fractures and (ii) fl ow in both fractures and matrix.

Discrete Models of Fracture Systems: A more recent approach to fl ow in fractured systems tries to represent the fractures explicitly as oriented planes with various shapes in 3D. Single phase (tracer) fl ow models of this type are used to model radio-

Figure 40A simple example of a pseudo relative permeability

38

Glossary of Terms

nuclide transport in fractured media. However, multi-phase models of this type are not commercially available at present.

Transfer Function: The function which describes the oil fl ow rate between the matrix and the fractures is known as the Transfer Function. Approximate analytical equations for this function have been suggested by Birk, Boxerman and Ahronovsky.

Sudation: When oil is recovered from the matrix blocks in a fracture by a combination of gravity and capillary forces, the recovery mechanism is sometimes referred to as Sudation.

1.11 Miscellaneous-Vertical Equilibrium, Miscible Displacement and DispersionVertical Equilibrium: The concept of Vertical Equilibrium (VE) is quite widely used in reservoir engineering. It takes several forms, two of which are listed below (and illustrated schematically in Figure 41:

A: the pressure gradients in a particular direction (x, say), ∂P/∂x, are all equal locally in a long system. See Figure 41a.

B: there is virtually instant crossfl ow vertically - nearly infi nite - compared with the horizontal fl ow. See Figure 41b.

The VE assumption is often made in order to simplify the mathematical analysis of certain fl uid fl ow problems in reservoir engineering.

Vertical Equilibrium (VE) is known to apply in “long thin” systems (where Δx >> Δz, in Figure 41b). More accurately, the VE limit is approached as the scaling group, RL → ∞ ; where:

R

xz

kkL

z

x

= ∆∆

./1 2/1 2/

and kz and kx are the vertical and horizontal permeabilities, respectively. In practice, if RL is > 10, then VE is a very good assumption.L is > 10, then VE is a very good assumption.L


Glossary of Terms

∆z

∆z

∆x

∆x

A: Equality of Layer Pressure Gradients (Where ∆x >> ∆z)

B: Instantaneous/Infinite Vertical Crossflow

Layer 1

Layer 2

Layer 3

∂P∂x 1

∂P∂x 2

∂P∂x 3

Miscible Displacement: Whereas oil and water are immiscible fl uids (i.e. they do not mix and are separated by an interface), some fl uids are fully Miscible (i.e. they mix freely in all proportions). When a gas (or other fl uid) is injected into an oil reservoir and the fl uids are miscible, this is referred to as a Miscible Displacement. When two fl uids (e.g. gas and oil) are fully miscible (σgo = 0), the local pressure and the pressure gradients are the same (there is no capillary pressure since there is no interface).

The mixing between the solvent and the oil can occur locally by Dispersion and by Fingering (see Fingering (see Fingering Viscous Fingering below). The displacement is described by a Viscous Fingering below). The displacement is described by a Viscous Fingeringgeneralised Convection-Dispersion Equation where the mixing viscosity, μ(c) is a function of the concentration of the solvent, c (or oil). Often, the solvent viscosity is below that of the oil (i.e. μs < μo) which tend to cause an instability to develop in the displacement known as Viscous Fingering.

The Miscible Flow Equations: These comprise of a Pressure Equation and a Transport Equation. The pressure equation is derived by inserting Darcy’s Law (with a viscosity dependent on solvent concentration) into the continuity equation. The transport equation is a generalised convection-dispersion (or convection-diffusion) equation.

Continuity equation:

∇ =∇ =.u∇ = 0 (assuming incompressible flow)

Darcy Equation:

uk

P= −( )c( )c

∇P∇Pµ

. P. P∇. ∇P∇P. P∇P

where u is the Darcy velocity, c is the miscible solvent concentration and k is the

Figure 41Schematic views of vertical equilibrium

40

Glossary of Terms

permeability tensor.

The pressure equation is then given by:

∇ = −∇

( )∇

∇( )

∇

. .∇ =. .∇ = −∇. .−∇ . ˜

. .( )

. .( ). . ˜

∇ =u∇ =∇ =. .∇ =u∇ =. .∇ =k( )c( )

P qP q∇P q∇

P q

P q

P q

P q

P q

=P q= ˜P q

k( )c( )

P q∇P q∇

P q

P q

P q

P q

P q

= −P q= − ˜P q

µ( )µ( )

µ( )µ( )

or

where q represent any source/sink terms

The transport equation is the generalised convection-dispersion (diffusion) equation:

∂∂

= ∇ ( )( )∇( )∇ − ∇c

t( )D c( )∇( )∇D c∇( )∇ c= ∇ = ∇ − ∇ − ∇v − ∇v − ∇cv c. .( ). .( )( )D c( ). .( )D c( ) − ∇.− ∇− ∇v − ∇.− ∇v − ∇

where D is the dispersion tensor and v is the pore velocity ( v v v =v uφ

).

Dispersion and Dispersivity: Hydrodynamic dispersion in a porous medium at the small (core) scale is a frontal spreading or mixing which is due to various fl ow paths which the fl uid can fl ow along at the pore scale. This mixing is a “diffusive” process since the growth of the mixing zone, Lf , tends to grow in proportion to t .In a tracer core fl ood experiment, the Dispersion Coeffi cient, D, may be measured by fi tting the effl uent profi le to an analytical solution of the Convection-Dispersion Equation (see fi gure 42). Units of D (cm2/s - at lab scale).

Dispersive mixing behaviour can also be seen from the “mixing” effect of heterogeneities at larger scales in a porous medium. D, has been found to depend linearly on velocity through the relationship: D = α . v; where α is the Dispersivity and has dimensions of length. Dispersion is actually a tensor in rather the same way that permeability ( k ) in its general form is a tensor.


Glossary of Terms

Figure 43Experimental demonstration of viscous fi ngering

Figure 42Schematic illustration of dispersion and the convection-dispersion equation for simple tracer fl ow

Lf

DispersiveMixing Zone Effluent

ConcentrationCe(1,t)C(x,t1)

time (pv)

Ce 1

C

0 1x1

1

In situ concentration profile at time, t1; C(x,t1)

Described by Convection Dispersion Equation:

C = Dimensionless Concentration (C/Co)D = Dispersion Coefficientv = Pore Velocity (Q/Aφ)

∂C∂t

∂C∂x

∂2C∂x2= D - v

Viscous Fingering: When a high mobility (lower viscosity) fl uid displaces a lower mobility (higher viscosity) fl uid, a type of instability may develop known as Viscous Fingering. For such a displacement, the Mobility Ratio (see above) is high and the process may be observed in either miscible or immiscible displacements, although it occurs more readily in miscible systems. Results from an experiment are shown in Figure 43 where the dark fl uid is high viscosity and the light fl uid is low viscosity. Clearly, viscous fi ngering leads to a poorer sweep effi ciency in such fl oods.

1

PAPER NO. 28902B

HERIOT WATT UNIVERSITY

DEPARTMENT OF PETROLEUM ENGINEERING

Examination for the Degree of

Meng in Petroleum Engineering

Reservoir Simulation

Monday 17th April 1995 09.30 - 12.30

( 70% of Total marks )

NOTES FOR CANDIDATES

(1) Answer ALL the questions and try to confine your answer to the space provided on thepaper.

(2) The amount of space and the relative mark for the question will give you some idea of thedetail that is required in your answer.

(3) If you need more space in order to answer a question then continue on the back of thesame page indicating clearly (by PTO) that you have done so.

(4) The total number of marks in this examination is 262; this will be rescaled to give anappropriate weighted percentage for the exam. The marks are relative and, together withthe space available, should give an approximate guide to the level of detail required.

(5) There is a compulsory 15 minute reading time on this paper during which you must notwrite anything.

(6) You will be allocated 3 hours to complete this paper.

2

Q1. List two uses of numerical reservoir simulation for each of the following stages of fielddevelopment.

(i) At the appraisal/early field development stage

1.

(2)

2.

(2)

(ii) At a stage well beyond the maximum oil production in a large North Sea field:

1.

(2)

2.

(2)

Q2. At any stage in a reservoir development by waterflooding, the engineer may use materialbalance calculations and/or numerical reservoir simulation. Under which particular circumstances would you use each of these approaches?

(i) Material Balance?

(3)

(ii) Reservoir simulation?

(3)

Q3. Given all the problems and inaccuracies, which are known to exist in the application ofreservoir simulation, why do engineers still use it?

(2)

3

Q4

Water injector

Water injector

High k Low k

Producer

2000 mVertical scale100 m

Continuous orDiscontinuousShales ???

OWC

Reservoir X is a light oil reservoir (35º API) being developed by waterflooding. The reservoircomprises a high average permeability massive sand overlying lower average permeabilitylaminated sands. The thickness of each sand is approximately equal and there are shales at theinterface of these two sands. However, the operator is uncertain if these shales are continuous ordiscontinuous.

The following questions refer to Figure 1 above.

(i) What would the main differences be between the cases where the shales in the abovereservoir were continuous and where they were discontinuous?

(4)

(ii) Say briefly how you would go about investigating this using reservoir simulation.

(4)

4

(iii) Suppose a reservoir engineer took 10 vertical grid blocks (NZ=10) in a simulation modelof this system. Would you expect the local (kv.kh) values in each of the main reservoirsands to be similar or different? Explain your answer briefly.

Similar/different

(1)

Explain

(4)

(iv) If this reservoir well had a gas cap, then gas coning might be a problem.

What is gas coning? Draw a rough sketch.

(4)

How would you use reservoir simulation to investigate this problem?

(4)

5

(v) In which two ways would the grid used to investigate gas coning be different from thatwhich was used in the full field waterflooding simulation?

1.

(2)

2.

(2)

Q5. Two of the main numerical problems/errors that arise in reservoir simulators are due tonumerical dispersion and grid orientation.

Explain each of these terms briefly saying - what it means, its origin and how we might getround it. Draw a simple hand drawn sketch illustrating each.

(i) Numerical Dispersion: Sketch

(5)

(ii) Numerical Dispersion - Explanation?

(5)

6

(iii) Grid Orientation - sketch

(4)

(iv) Grid orientation - Explanation?

(4)

Q6. A very simple single phase pressure equation is given by Eq. 1 below.

∂∂

ÊËÁ

ˆ¯ =

∂∂

ÊËÁ

ˆ¯

Pt

Px

2

2 Eq. 1

(i) Write down how this equation is discretised in an explicit finite difference scheme - brieflyexplain your notation.

(4)

7

(ii) If an implicit finite difference scheme was used to solve Eq. 1, then a set of linearequations would arise which could be solved using either a direct or an iterative linearequation solution technique. Briefly explain each of the bold terms above:

• Implicit finite difference scheme

(4)

• Set of linear equations

(4)

• Direct linear equation solution technique

(4)

• Iterative linear equation solution technique

(4)

8

Q7. Statement: “The Equations of Two Phase Flow can be derived easily simply by usingMaterial Balance and Darcy’s Law”.

Explain this statement with reference to two phase flow - you do not need to actually derive theequations and, indeed, you may not use any equations other than Darcy’s law.

(8)

Q8. (i) Draw a schematic sketch of a single grid block of size Dx by Dy by Dz, showing theporosity f, the oil and water saturations S

o and S

w (only 2 phases present).

(2)

9

(ii) Using the sketch in part (i) above, derive expressions for (a) the volume of oil in the gridblock, V

o; (b) the mass of oil in the grid block, M

o, introducing the formation volume

factor, Bo.

(a) Vo?

(3)

(b) Mo?

(3)

(iii) Write an expression for the oil flux, Jo, saying briefly what it is, any units it might be

expressed in and explaining any terms you introduce.

(6)

10

Q9. (i) Name three ways in which a Black Oil reservoir simulation differs from aCompositional simulation model.

1.

2.

3.

(6)

(ii) Draw a simple sketch of a single grid block showing what is meant by a component and aphase.

(4)

(iii) Using the notation CIJ to denote the mass composition of component I per unit volume of

phase J (dimensions of CIJ are mass/volume), derive an expression - based on the

quantities labelled in your sketch in (ii) above - for the mass of component I in the gridblock.

(6)

11

(iv) Give one example of (a) where you would use a Black Oil model and (b) where you woulduse a Compositional simulation model.

(a) Use Black Oil model?

(3)

(b) Use Compositional model?

(3)

Q10. Figure 2: The figure below shows the basic idea of “upscaling” or “Pseudo-isation”.

"Fine" Grid Cross-Sectional Model

"Coarse" Grid Upscaled or Pseudo-ised Model

"Rock"Propts.

OIL

"Pseudo-"Propts.

With reference to Figure 2 above, answer the following:

12

(i) What is meant by “Upscaling” with reference to the modelling of say a waterflood.

(2)

(ii) What is the difference between “rock” relative permeabilities and pseudo-relativepermeabilities?

(4)

(iii) In order to perform upscaling in reservoir simulation, we need both an UpscalingMethodology and Upscaling Mathematical Techniques. Explain very briefly themeaning of the bold terms.

• Methodology

(4)

• Techniques?

(4)

13

Q11. (i) Sketch (roughly) a semi variogram for each of the following permeability models:

(a) a correlated random field with a range of 100m and a sill of 10,000 mD2; and

(b) a laminated system where the laminae are of constant width of 1cm and where the highpermeability = 2D and the low permeability = 1D. Label your sketches clearly.

(a)

(6)

(b)

(6)

(ii) What can you deduce about the standard deviation of the correlated random field in (i)(a)above.

(3)

14

(iii) Sketch the correlogram for case (i) above.

(5)

Q12. Figure 3 below shows the sketch of simple 3 layer model.

2 cm

5 cm

1 cm

k = 0.5 D

k = 2.0 D

k = 1.0 D y

x

(i) Calculate the effective permeability of the above model in the x-direction; show your working.

(5)

15

(ii) Calculate the effective permeability of the above model in the y-direction, show yourworking.

(6)

(ii) Suppose we put a very find grid (say of size 0.1 cm x 0.1 cm) on the 3 layer model in Figure 3above. If we jumbled up all the grid blocks randomly so that the new model had no discernablestructure, would the new effective permeability be: greater than that in (i) and (ii)?; less than that in (i)and (ii)?; in between these values? Explain your answer.

(6)

16

Q13. In miscible flow in a random correlated field, explain how the mixing zone grows withtime in each of the following cases (illustrate your answers with simple sketches):

(i) dispersive flow

(4)

(ii) fingering flow

(4)

(iii) channelling flow

(4)

(iv) On the same diagram below, sketch the expected type of fractional flow curve f(c) vs c) youwould expect for each type of flow.

(6)

17

Q14. In the Kyte and Berry pseudo-isation (upscaling) method, describe briefly (using adiagram) how numerical dispersion is taken into account (no detailed mathematics isrequired).

(10)

18

Q15. (i) List the main categories in the hierarchy of stratal sedimentary elements - give oneshort sentence explaining each of these.

(8)

(iii) Describe which of the above length scales of sedimentary heterogeneity are likely to havemost significance for the following reservoir flow phenomena:

* Reservoir pay-zone connectivity:

(3)

* Gravity slumping or water over-ride:

(3)* Vertical sweep efficiency:

(3)

* Residual/Remaining oil saturation

(3)

19

Q16. You have been given the following distribution of core-plug permeabilities in a particularreservoir unit which includes a higher permeability a fluvial channel sand overlying alower permeability deltaic sand:

100 400 600 800Fr

eque

ncy

Permeability (md)

With reference to Figure 4 above: (a) explain the probable reason that the permeabilitydistribution has the above form; (b) sketch the sort of permeability models (laminar, bed andformation scale) you might use for the flow simulation of this unit.

(a)

(3)

(b)

(continue on the back of this sheet if necessary) (10)

20

Q17. (i) Draw a sketch of water displacement of oil across the laminae in a water-wetlaminated system at (a) “low” flow rate (capillary dominated) and (b) “high” flow rate(viscous dominated); in this sketch show where the residual remaining oil is andgive a sentence or two of explanation.

(a)

(8)

(b)

(8)

21

(ii) Is the effective water permeability at residual (i) remaining oil saturation (across thelaminae) higher in case (a) or (b) in part (i) above? Explain.

(6)

(iii) What are the implications of the results in (i) and (ii) above for upscaling in reservoirsimulation?

(6)

Model Solutions to Examination

1

��

yyyyyyyyyyyyyyyy

Date:

1. Complete the sections above but do not seal until the examination is finished.

2. Insert in box on right the numbers of the questions attempted.

3. Start each question on a new page.

4. Rough working should be confined to left hand pages.

5. This book must be handed in entire with the top corner sealed.

6. Additional books must bear the name of the candidate, be sealed and be affixed to the first book by means of a tag provided

Subject:

INSTRUCTIONS TO CANDIDATES

8 Pages

PLEASE READ EXAMINATION REGULATIONS ON BACK COVER

No. Mk.

NAM

E:REGISTRATION N

O.:

COURSE:

YEAR:

SIGNATURE:Complete this section but do not

seal until the examination

is finished


2

Answer Notes

# => indicates one of several possible answers which are equally acceptable.

[…] => extra information good but not essential for full marks - may get bonus.


3

Q1Q1Q1Q1Q1 (i)

# 1. To perform broad scooping calculations which examine different

development options e.g. waterflooding, gas flooding etc.

# 2. To extend initial material balance calculations by examining

some other spatial factor such as well-placement or aquifer effect.

(ii)

# 1. To assess additional field management options such as infill drilling,

pressure blowdown etc.

# 2. To take the improved history match model which can be developed

after same development twice and to use this to assess various IOR

strategies e.g. gas injection, WAG or chemical flooding.

Q2Q2Q2Q2Q2

(i)

# Because of its inherent simplicity you would virtually always apply

single material balance to assess your field performance - to see if DP

decline tallies with estimated field size, sources of influx and

production.

(ii)

Would be used when a more complex development strategy requiring

spatial information is essential e.g. well placement, assessment of shale

effects, gravity segregation etc.

4

Q3Q3Q3Q3Q3

(#) Because it is the only tool we have to tackle complex reservoir

development/flow problems which extends material balance. Clearly, it

is much better than simple material balance alone.

Q4Q4Q4Q4Q4

(i)

The shale continuity strongly affects the hi/lo permeability layer

vertical communication (both pressure and fluid flow). Thus, it will

affect the effective kv/k

h (lower or zero for continuous shales) and

will strongly influence gravity slumping of water in a waterflood. In

the situation above with high k on top, some vertical communication will

help recovery.

(ii)

Set up a simple 2D cross sectional model with , say, 50 blocks in the x -

direction and 10 vertical grid blocks - 5 in each layer. Run waterflood

cases with and without shales - and some in-between cases with

transmissibility modifiers set beween Tz = 0.0 Æ 0.01 Æ0.1 Æ0.5 Æ1.0.

Compare water saturation fronts and recoveries as fraction of pv

water throughput. Result will allow us to assess the effects of the

shales in the waterflood.

(iii)

Different


5

The high perm massive sand would have a small scale kv/k

h ~1 which

would result in a similar larger scale value. In the laminated sands, the

“small” scale (say core plug scale) would have a low kv/k

h of say 0.1 to

0.01 and this would result in a correspondingly lower kv/k

h at the grid

block scale.

(iv)

Well

Gas

Oiland

Water

Oiland

Water

Gas

= perforations

Gas Coning It is the drawdown of the highly mobile (low mg) gas into

the perforations. Pattern is shown here in figure. Causes high GOR

production at a level well above the solution gas value.

6

Set up a near-well r/z geometry fine grid - possibly 50 layers and

set reservoir near-well rock properties e.g. Layering, Tz modifiers,

Rel. perms. etc.

Perform simulations to look at issues such as effect of rate, vertical

communication, gas/oil/water Rel. perms. etc…

Generally needs a fine Dr, Dz grid, often finer near the well where

most rapid changes of Sg and pressure with time occur.

(v)

1. The geometry would be different: r/z for coning and cartesian or

corner point for full field.

2. The fineness of the grid would be different. Very fine for near-

well; much coarser for full field.

# (Dimensionality too 2D vs. 3D)


7

Q5Q5Q5Q5Q5

(i)

(ii)

Numerical dispersion is the artificial spreading of saturation fronts

due to the numerical grid block structure in the simulation. It arises

because we take large grids to represent moving fronts. It can be

improved by refining the grid (globally or locally) or by using improved

numerical methods.

(iii)

P

Fluid tends toflow along (parallel)

to the grids

L

P

Wells same distanceapart in Figs A and B

LII

Fig A Fig B

I = injector ; P = producer

8

(iv)

The injected fluid tends to flow parallel with the grid from the

injector (I) to the producer (P) - see previous page. This means that

early breakthrough and poorer recoveries are seen in A then in B

above. i.e.

%00IPProducer

Fig B

Pv or Time

Actual Recovery

Fig A

Q6Q6Q6Q6Q6

(i)


9

(ii)

In this scheme the spatial term in Eq. 1 i.e. 2∂∂

P

x2 would be specified atthe new time level n+1

A set of linear equations is the following type of equation (e.g.3x3system)

a11 X

1 + a

12 X

2 + a

13 X

3 = b

1

a21

X1 + a

22 X

2 + a

23 X

3 = b

2

a31

X1 + a

32 X

2 + a

33 X

3 = b

3

where X1, X

2, X

3 are unknown - the a’s are a matrix of known

coefficients and b’s are a known right-hand side.

A direct solution method (e.g. Gaussian EliminationGaussian EliminationGaussian EliminationGaussian EliminationGaussian Elimination) is an algorithm

with a fixed number of steps which will solve these linear equations

(under certain conditions). [Typically forward elimination is applied to

get an upper triangular A* matrix and back substitution is then easily

applied to get the X solution]

In contrast, an iterative technique starts with a first estimate of the

unknown vector X (0) where the (o) denotes 0th iteration: This is then

improved by some algorithm to a better and better solution of the

original linear equations i.e. X(1) Æ X(2)… Æ X(V) until the method

converges e.g.

10

/ X(V+1) - X(V)/ < small number TOL. [Methods such as the Jacobi, LSOR,

etc. are examples of this].

Q7Q7Q7Q7Q7

mo mo

mw

In block (i, j), then material balance can be applied for each phase (e.g. oil and water) for 2-phase flow.

i, j

Mass

Accumulation of=

Amount that-

Amount that

oil over time flows in over flows out over

D t D t D t

But amount that flows in/out is given by the pressure differences

between blocks i.e.

QA k k S

P Poil i j i jro o

o i

o oij i j( , ) ( , )

. . ( )- Æ

-

ÊËÁ

ˆ¯

-( )-11

2

1m

Thus the two phase Darcy Law supplies the relation for volumetric flow

rate and pressure in the grid block. These volumetric flows can be

converted to MASS flows (x by density) and then put into the material


11

balance equation to obtain Æ a conservation equation and in pressure

equation for oil and water.

\ \ Material Balance + Darcy’s Law => 2-phase Flow Equation.

Q8Q8Q8Q8Q8

(i)

(ii)

(a) V x yo = D D D z Sof

(b)

12

(iii)

#

Q9Q9Q9Q9Q9

1. The black oil model essentially treats a phase (o,w,g) as the basic

conserved unit or “pseudo component”

2. Compositional models are based more correctly on the conservation

of components (CH4, C

10, H

2O etc.) - the black oil model simply treats

gas dissolution in oil through Rso - gas solubility

3. The compositional models incorporate a full PVT description of the

oil whereas the black oil model relies on the simple Rso type treatment.


13

(iii)

(iv)

# (a) Waterflood calculations in a low GOR - say 30∞ API - oil reservoir

with pressure maintenance.

# (b) CO2 injection in a - say 36∞ API - light oil system [Condensate

system - gas recycling etc…]

Q10Q10Q10Q10Q10

(i)

Upscaling in a waterflood essentially means getting the correct

(effective) parameters (-e.g. rel. perm.) for the larger scale grid blocks

which will reproduce a “correct” fine grid model.

(ii)

“Rock” relative permeabilities are meant to be the intrinsic

representative properties of a representative piece of reservoir rock

at the “small” (i.e. core plug) scale.

Pseudo rel. perms. are effective properties at the “larger” (usually

gridblock) scale which incorporate other effects and artefactsartefactsartefactsartefactsartefacts (e.g.

14

numerical dispersion, heterogeneity etc..) in addition to the intrinsic

“rock” rel. perms.

(iii)

MethodologyMethodologyMethodologyMethodologyMethodology

This is a geologically consistent approach to the task of upscaling. i.e.

data collection, sedimentological framework,…

[The function of the methodology is to get the geologically + fluid]

mechanically “right” answer.

Techniques?Techniques?Techniques?Techniques?Techniques?

These refer to the actual mathematical algorithm to go from a fine

grid Æ coarse grid. E.g. Kyte and Berry, Stone’s method, two phase

tensors etc…

[N.B. This just needs to reproduce the fine grid result - even if it is

WRONG - at the coarse scale]


15

Q11Q11Q11Q11Q11

(i)

(ii)

It takes a lag distance of about the “range” to see the field variability

(standard dev. - i.e. ~ 100mD) of the field.

16

(iii)

lag

1

Q12Q12Q12Q12Q12

(i)

(ii)

The effective permeability is clearly the harmonic (thickness -

weighted) average as follows:


17

(iii)

The keff

in the randomised model would be between the two answers

in (i) and (ii) above (the answer in (ii) being the lower).

e.g. Strictly in a randomised distribution of permeability the average

value tends to the geometric average (kg) in 2D

kg - is less than the arithmetic (along layer) answer.

kg - is greater than the harmonic (across layer) answer.

Q13Q13Q13Q13Q13

(i)

Note - we take the same contour values (c= 0.1, 0.5, 0.8) in all sketches

below.

18

(ii)

(iii)

(iv)


19

Q14 Q14 Q14 Q14 Q14 With no maths

Consider only ACTUAL flows of Qw and Q

o across this interface

interface only. Thus, if the fine grid water (say) flows have not

reached the coarse grid interface

then Qw = 0. => set k

rw= 0

When there is oil or water flow e.g. water flow.

20

Q15Q15Q15Q15Q15(i)

Lamina Æ The simplest unit within which we can assume (almost)

homogeneous k. (length mm Æ m)

Lamina set Æ A collection of the above (cm Æ m) e.g. core.

hi k

lo k

Bedform Æ How lamina sets are joined together geometrically to

form 3D beds.


21

2m

~50cm

Tabular cross-bedding

Bottom sets

or climbing ripples

Para-sequence/sequence-stacks of bedforms

Eroded/? top setse.g.

(ii)

• Para sequence - sequence scale

• At parasequence - sequence - also bed form influence

• Para sequence - bed form

• Lamina set - bed form

Q16Q16Q16Q16Q16

(a) There is a double peak - the bimodality probably arises from the

lower perm plugs from deltaic sands, and the higher permeability plugs

from the fluvial channel.

22

(b)

A

A

B

B

Tightly laminateddeltaic sands

Crossbeddesfluvial channel-stacked crossbeds

2 Scale pseudo-isation - inclined cross bed pseudo - bedform pseudo

laminated sandpseudo

Q17Q17Q17Q17Q17

(i) (a)

hi lo hi lo hi lo

Sw

CAPILLARY DOMINATED

Spontaneous water inhibition into the LOW k laminae occurs in Pc-dominated flow. This traps oil in the HIGH k laminae behind the front where it is well above "residual" but it can't move because the Rel. Perm. to oil in the low k water-filled laminae is so low.

HIGH "remaining"oil in hi k

Slow Flow

Water flow directionHigh water Sw in LOW perms in awater-wet system


23

(b)

hi lo hi lo hi lo

Sw

VISCOUS DOMINATEDWATERFRONT

"Fast" Flowof water

Water flow directionHigh water Sw

LOW perm in awater-cut system

Note at Viscous dominated conditions a water front goes throughwhich reduces the oil in all layers to its local "residual" level.Recovery of oil is better in this case since it is not "stranded" bydownstream capillary imbibition.

(ii)

It is higher in case (a) for the reasons already explained.

[I give a slightly over-detailed answer to part (a) and (b) above].

(iii)

The central implications are twofold.

(a) The two phase pseudo relative perms. are highly anisotropic for

such laminar systems. Along and across layer water displacement in

laminar system gives widely different pseudos.

24

Along

Across

(b) The levels of “remaining” oil can be vastly different in laminar

systems which, in simulation/upscaling, moves the pseudo rel. perm. end

points. (see above).


25

1

Course:- 28117

Class:- 28912

HERIOT WATT UNIVERSITYDEPARTMENT OF PETROLEUM ENGINEERING

Examination for the Degree ofMeng in Petroleum Engineering


Tuesday 20th April 199909.30 - 13.30

( 100% of Total marks )

NOTES FOR CANDIDATES

1. This is a Closed Book Examination.

2. 15 minutes reading time is provided from 09.15 - 09.30.

3. Examination Papers will be marked anonymously. See separate instructions forcompletion of Script Book front covers and attachment of loose pages. Do not write yourname on any loose pages which are submitted as part of your answer.

4. This Paper consists of 1 Section.

5. Attempt all Questions.

6. Marks for Questions and parts are indicated in brackets

7. This Examination represents 70% of the Class assessment.

8 State clearly any assumptions used and intermediate calculations made in numericalquestions. No marks can be given for an incorrect answer if the method of calculation isnot presented. Be sure to allocate time appropriately.

2

3

Q.1: Consider the following statement which is made referring to a reservoir development planfor a field which has been in production for some time:

“A reservoir engineer should always apply Material Balance calculations and should usually -but not always - use Numerical Reservoir Simulation”.

(i) Why should you always perform some sort of material Material Balance calculations ?

..............................................................................................................................

..............................................................................................................................

..............................................................................................................................

........................................................................................................... (4)

(ii) What is the main disadvantage of using material balance calculations in reservoirdevelopment?

..............................................................................................................................

........................................................................................................... (2)

(iii) In the above context, explain when you would use Reservoir Simulation and when you maynot use it. Give an example of each case.

When you would use Reservoir Simulation + Example:

..............................................................................................................................

..............................................................................................................................

..............................................................................................................................

........................................................................................................... (4)

When you may not use Reservoir Simulation + Example:

..............................................................................................................................

..............................................................................................................................

..............................................................................................................................

........................................................................................................... (4)

4

Q.2: Various types of 2D and 3D grids are used in reservoir simulation calculations. Describewhat you think the best type of grid is for performing calculations on each of the reservoirprocesses described below and say why.

Reservoir processes:(i) Modelling of likely gas and water coning and its effect on (vertical) well productivity in alight oil reservoir with a gas cap and an underlying aquifer;

(ii) Simulating a large number of options in an injector/producer well pair in a gas injectionscheme where the objective is to look at the effects of formation heterogeneity on gas - oildisplacement and to develop some pseudo relative permeabilities to use in a full field model;

(iii) Carrying out an appraisal of an entire flank of a complex faulted field which has severalinjector and producer wells.

(i) Which grid?............................................................................................................................

.................................................................................................................................................... (4)

Why?...........................................................................................................................................

.....................................................................................................................................................

.....................................................................................................................................................

..................................................................................................................................................... (4)

(ii) Which grid?............................................................................................................................

..................................................................................................................................................... (4)

Why?...........................................................................................................................................

.....................................................................................................................................................

.....................................................................................................................................................

..................................................................................................................................................... (4)

(iii) Which grid?..........................................................................................................................

.................................................................................................................................................... (4)

Why?..........................................................................................................................................

....................................................................................................................................................

....................................................................................................................................................

.................................................................................................................................................... (4)

5

Q.3: Numerical dispersion and grid orientation are two of the main numerical problems thatoccur in reservoir simulation. Explain in your own words, with the help of a simple sketch,the meaning of each of these terms:

(i) Numerical dispersion ? Sketch:

(5)

Numerical dispersion ? Explanation:

......................................................................................................................................................

......................................................................................................................................................

......................................................................................................................................................

......................................................................................................................................................

...................................................................................................................................................... (5)

(ii) Grid orientation ? Sketch:

(5)

Grid orientation ? Explanation:

......................................................................................................................................................

......................................................................................................................................................

......................................................................................................................................................

......................................................................................................................................................

...................................................................................................................................................... (5)

6

Q4. Figure 1 below shows the results of a series of 6 waterflooding and gas floodingcalculations (labelled A - F) in a 2D vertical cross-sectional numerical simulation model.Results are plotted as “Oil Recovery at 1PV Injection” vs. 1/NZ , where NZ is the numberof vertical grid blocks in the simulation. Assume (a) that the number of grid blocks in thex-direction (NX) is sufficiently large and is constant in all calculations; and (b) that theaxes of the graph are “quantitative”.

Figure 1

Oil recoveryat 1PV injection

40%

60%

Inverse number of grid blocks (1/NZ) -->

No. vertical gridblocks = NZ

2D cross-section

0.50.1 0.2 0.3 0.4

Key gas injection waterflood

50%

A

B

C

D

E

F

Answer the following questions:

(i) How many vertical grid blocks (NZ) were used in case F? ......... (3)

(ii) Do the simulated waterflood and gas flooding calculations become more “optimistic” or“pessimistic” as we take more vertical grid blocks in the calculation?

......................................................................................................................................................

......................................................................................................................................................

...................................................................................................................................................... (4)

7

Q4. (continued)(iii) Extrapolate each of the calculations to NZ --> ∞ for both the waterflood and the gas floodon Figure 1. Estimate the % recovery for each and the incremental recovery of the gas floodcompared with the waterflood. Comment on the implications of your result for carrying out agas flooding project in this reservoir.

Estimations...........................................................................................................

......................................................................................................................................................

......................................................................................................................................................

......................................................................................................................................................

......................................................................................................................................................

...................................................................................................................................................... (6)

Comment:

......................................................................................................................................................

......................................................................................................................................................

...................................................................................................................................................... (4)

————————————————— End of Q.4 ———————————————

Q5. on next page

8

Q. 5 Figure 2 below shows a “control volume” - block i - for single-phase compressible flow in1D. The quantities q

i+1/2 and q

i-1/2 are the volumetric flow rates of the fluid at the

boundaries of block i. All grid blocks are the same size in the x-direction (∆x) and thecross-sectional area is constant, A = ∆y.∆z (where ∆y and ∆z are the block sizes in the y-and z-directions). The density and porosity are denoted by symbols - ρ and φ respectively.

Figure 2

x

i i+1i-1

qi-1/2

∆x

Area= A= ∆y.∆z

qi-1/2

With reference to the above figure,

(i) Write a clear mathematical expression for the change in mass in block i over a time step ∆tdue to flow:

Change in mass due to flow over time step ∆t =

......................................................................................................................................................

...................................................................................................................................................... (6)

(ii) Write a clear mathematical expression for the change in mass in block i from the beginningof the time step to the end (i.e. the accumulation):

Difference in mass in block i over time step ∆t =

......................................................................................................................................................

...................................................................................................................................................... (6)

(iii) Considering the above two expressions, what equation can you now write from materialbalance ?

......................................................................................................................................................

...................................................................................................................................................... (6)

9

Q.5 (continued)(iv) Are there any assumptions in the equation you have just written in part (iii) above? Brieflyexplain.

......................................................................................................................................................

......................................................................................................................................................

...................................................................................................................................................... (4)

(v) Now use the single-phase Darcy Law in the equation you wrote in (iii) above and show how -by taking Limits ∆x, ∆t --> 0 - you obtain the pressure equation for single-phase compressibleflow: Show the steps you take.

(8)

(vi) If you have written down the answer to part (v) above correctly, then you should havewritten down a non-linear partial differential equation (PDE). What does “non-linear” mean inthis context and explain in physical terms what the main problem is with this sort of equation.

Non-linear?...................................................................................................................................

......................................................................................................................................................

...................................................................................................................................................... (4)

The problem?...............................................................................................................................

......................................................................................................................................................

...................................................................................................................................................... (4)

10

Q.6: (i) From the four assumptions listed below, show clearly how Equation 1 arises from thenon-linear equation you derived in Q.5 part (v) above. Indicate clearly where you use eachof the assumptions in your derivation.

κ∂∂

= ∂∂

Pt

Px

2

2 Equation 1

The term κ is a constant (Greek kappa - not permeability, k). Other quantities which may beused are C

f - the fluid compressibility, ρ - the density; k - the permeability; µ - the fluid viscosity;

φ - the porosity.

Assumptions: 1. The rock is incompressible.2. Permeability (k) and viscosity (µ) are constant.

3. The fluid has a constant compressibility, CPf =

1ρ

∂ρ∂ .

4. Pressure gradients are small - hence ∂∂Px

≈

2

0

(i) Answer:

(12)(continue on the back of this page if necessary indicating that you have done so)

11

Q.6 (continued)(ii) From your answer to part (i) above, write down what constant κ is in terms of the otherconstants.

κ = .............................................................................................................................................. (4)

(iii) Using the notation in Figure 3 below, apply finite differences to Equation 1 above and definethe discretised spatial derivative at the new time level, (n+1) (i.e. an implicit method). Show each stepin your working and show clearly how this leads to a sparse set of linear equations.

Figure 3

Time level

n

n+1

Space --->

i-1 i i+1

Pin

Pin+1

∆ tAll ∆ x and∆ t fixed.

∆x

(12)(continue on the back of this page if necessary indicating that you have done so)

12

Q.6 (continued)(iv) We can write down the set of linear equations that arises from applying finite differences tothe flow equations as follows:

A.x = b Equation 2

where A is a matrix, x is the vector of unknowns and vector b is known. Write out Equation 2explicitly for three equations and rearrange these to show how a simple iterative scheme can beformulated. Say very briefly how this is solved. Give one advantage and one disadvantage ofan iterative scheme.

(continue on the back of this page, if necessary, indicating you have done so) (10)

Advantage?...................................................................................... (2)

Disadvantage?................................................................................. (2)

13

Q.7: The oil flux , Jo, into and out of a grid block is shown in Figure 4 below. Other quantities

are denoted as follows: Oil saturation So; porosity, φ; Formation volume factor, B

o; Oil

density at standard conditions, ρosc

; Darcy velocity of oil, vo (similar quantities apply to the

water phase).

Figure 4

∆x

∆z

∆y

Oil Flux Jo

x x + ∆x

Jo

(i) Write an expression for the oil flux, Jo, giving any possible units.

(6)

(ii) The concentration of oil, Co, is defined as the mass of oil per unit volume of reservoir. Write

an expression for Co in terms of the quantities defined above.

(6)

14

Q.7 (continued)(iii) Prove that, in 1D two-phase flow, then for the oil phase:

(∂Jo/∂x) = - (∂C

o/∂t) Equation 3

showing your working clearly.

(10)

(continue on the back of this page, if necessary, indicating you have done so)

15

Q.7 (continued)

(vi) State the two-phase Darcy’s law for oil using (∂Po/∂x) for the pressure gradient and k

ro for the

relative permeability, and substitute this into Equation 3 above and derive the oil conservationequation.

• Two-phase Darcy Law for the oil phase (in terms of Darcy velocity, vo)

(4)

• Oil conservation equation:

(8)————————————————————End of Q.7 ——————————————

16

Q.8 In three-phase flow (oil, water and gas), we can define a gas flux, Jg, and a gas concentration,

Cg, in exactly the same way as was defined for oil in Q. 7 above.

(i) Explain physically why the gas flux and the gas concentration are more complicated than thecorresponding quantities for the flow of oil or water.

.........................................................................................................................................................

.........................................................................................................................................................

...................................................................................................................................................... (4)

(ii) Using Rso and R

º to denote the gas solubility factors, derive expressions for J

g and C

g showing

your working.

(12)


17

Q. 8 (continued)(iii) Use your expressions for J

g and C

g in the conservation Equation 3 (for gas) to write down the

first step in obtaining the conservation equation for gas.

(6)————————————————————End of Q.8 ———————————————

Q.9: Explain what history matching is in a reservoir simulation of a field saying briefly how it isdone and what can go wrong.

• History matching? How is it done? What can go wrong?

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................(10)(continue on the back of this page, if necessary, indicating you have done so)

18

Q.10 (i) Write down the formulae for the arithmetic (ka), harmonic (k

h) and geometric (k

g) averages

of a permeability field with permeabilities k1, k

2, .... k

M. The number of data points you have = M.

ka =

kh =

kg =

(10)

(ii) State how you would use these averages for calculating the effective permeabilities in thehorizontal and vertical directions in the models in Figures 5(a) and 5(b) below.

Figure 5(a) (b)

• For Figure 5(a): Horizontal keff

& Vertical keff

:

(8)

19

Q.10 (continued)• For Figure 5(b): Horizontal k

eff & Vertical k

eff :

(8)

(iii) Calculate the effective permeability for flow across the laminae in Figure 6 below. Show yourworking.

Figure 6

10 m

D, 1

cm

100

mD

, 2 c

m

20 m

D, 1

cm

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................(6)

————————————————————End of Q.10 ——————————————

20

Q.11 The diagram in Figure 7 below shows a grid consisting of 2 coarse grid blocks, with 7 x 3 fineblocks in each of these coarse blocks.

Figure 7

j= 1

2

3

∆x

∆z

i = 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Suppose you are calculating the pseudo relative permeabilities for the left coarse block using theKyte and Berry method. Assume that you have all the necessary information (saturations, pressures,flows etc.) from a fine-scale (3x14) simulation.

(i) Show clearly on Figure 7 which part of the grid you would use for calculating the followingquantities and give a brief sentence of explanation:

• the average water saturation (3)

...........................................................................................................................................................

...........................................................................................................................................................(4)

• the average pressure gradient (3)

...........................................................................................................................................................

...........................................................................................................................................................(4)

• the total flows of oil and water (3)

...........................................................................................................................................................

...................................................................................................................................................... (4)

(ii) What is the formula for the average water saturation?

Sw = (5)

21

Q. 11 (continued)

(iii) What is the weighting for the average pressure? Give a brief sentence of explanation.

(5)

..........................................................................................................................................................

........................................................................................................................................................ (3)

(iv) What is the formula for the total flow of water?

q w = (5)

————————————————————End of Q.11 ——————————————

22

Q.12 (i) By means of a simple sketch, show how a cubic packing of spherical grains is arranged.

Sketch:

(4)

(ii) Use the sketch to help you calculate the specific surface of the sample per unit volume of solid,S

s (in m2/m3).

Specific surface - working:

Specific surface per unit volume of solid, Ss = ..................... m2/m3

(6)

(iii) If the grain radius is taken to be 10µm, determine the porosity (φ) of the sample

Porosity working:

Sample porosity (φ) = ......... (6)

23

Q.12 (continued)(iv) If the grain radius was 100µm instead of 10µm, what would the porosity (φ) of the sample nowbe?

Porosity = .................................................................................................................................... (3)

(v) What do the results of parts (iii) and (iv) above suggest concerning the porosity of cubic spherepacks?

Ans............................................................................................................................................. (2)

(vi) Write down the Carman-Kozeny equation in terms of the grain diameter (D), porosity (φ), andthe specific surface per unit volume of solid, S

s

Carman-Kozeny equation:

k=

(6)

(vii) Taking the grain radius to be 10 µm and the tortuosity of the sample to be T=1, calculate anapproximate permeability in Darcies for a cubic packing of spheres (NB Use the porosity found inQ.12 part (iii) in this calculation) .

Working:

Permeability = ................. Darcies. (1 Darcy = 1x10-12 m2) (6)

24

Q.13 (i) Describe the meaning of the term “contact angle” and draw a rough sketch to illustrateyour answer

Contact angle?.........................................................................................................................................................................................................................................................................................................................................................................................................................................................

(4)

Sketch

(4)

(ii) If an oil/water meniscus is at equilibrium in a cylindrical capillary tube, what is the equationthat relates the capillary pressure, P

c, to the tube radius R and the contact angle θ?

Pc=

(5)

What is this reduced form of the equation called?

..................................................................................................................................................... (3)

25

Q. 13 (continued)(iii) Oil is introduced at the inlet face of the water-filled pore network shown in Figure 8 on thenext page. The numbers on Figure 8 refer to pore radii (in microns, µm). The pores are taken to becapillary tubes and are water-wet, the contact angle is θ = 60o in every pore and the oil/waterinterfacial tension is 80 mN/m. The capillary pressure of the system is gradually increased and oilbegins to invade the network.

Shade in the pores that become oil-filled at each of the 4 capillary pressure values Pc1, P

c2, P

c3

and Pc4 in Figure 8 (NB 14.7 psi = 105 Newtons/m2).

Show any working out below:

26

Q. 13 (continued)

Figure 8: Shade in the pores that become oil-filled at each of the 4 capillary pressure values Pc1,

Pc 2

,P

c3 and P

c4 below (NB 14.7 psi = 105 Newtons/m2).

Oil Water

5

20

12

15

10

4

2

3

8

11

3

1

12

Oil Water

5

20

12

15

10

4

2

3

8

11

3

1

12

Oil Water

5

20

12

15

10

4

2

3

8

11

3

1

12

Oil Water

5

20

12

15

10

4

2

3

8

11

3

1

12

Pc1= 0.6 psi

Pc2= 1.2 psi

Pc3= 3.0 psi

Pc4= 10.0 psi

(20)

27

Q.14 (i) Explain the differences between a drainage flood and an imbibition flood at the pore-scale, paying particular attention to the roles played by pore size, film-flow and accessibility tothe inlet (sketch each displacement in the boxes provided below to illustrate your answer).

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

.............................................................................. (10)


ImbibitionSketch

(4)

DrainageSketch

(4)

28

Q. 14 (continued)(ii) The strongly water-wet network in Figure 9 below is initially completely filled with oil and thecapillary pressure is so high that no water can currently imbibe. If the capillary pressure is slowlydecreased, so that water can invade via film-flow and snap-off, how is the residual oil distributed atthe end of imbibition (i.e. when P

c=0)? Shade in the residual oil using the network template

provided in Figure 10. The numbers on the bonds again refer to the tube radii in microns (µm).Note this network is deliberately different from that in Figure 8. Also, oil cannot enter the waterreservoir due to the presence of a water-wet membrane.

Figure 9

5

20

12

15

10

4

2

3

8

11

3

1

12

Initial distribution of oil

Water Oil

Figure 10

5

20

12

15

10

4

2

3

8

11

3

1

12

Answer: Template for final oil distribution (shade in appropriately)

Water Oil

(8)

————————————————————End of Q.14 ——————————————

29

Q15. You have been supplied with the table of mercury injection data below.

Table 1Mercury

saturation

Pc (air/mercury)

(psia)

Oil saturation Pc(oi l/water)

(psia)

0 2 0

0.2 4 0.2

0.4 5 0.4

0.5 6 0.5

0.6 7 0.6

0.8 20 0.8

0.9 60 0.9

(i) Sketch the air/mercury capillary pressure curve using the axes provided

(6)

(ii) Write down the equation used to re-scale mercury injection data to oil/water systems and use itto complete Table 1 (assume the following values for interfacial tensions and contact angles:σ

mercury/air = 360x10-3 N/m, σ

oil/water = 60x10-3 N/m, θ

mercury/air = θ

oil/water = 0o

Equation:

(6)

Now complete Table 1 at the top of this page (8)

30

Q. 15 (continued)(iii) Plot the oil/water capillary pressure curve using the axes provided below

Sketch.

(5)

(iv) Write down the equation that determines the Leverett J-function from capillary pressure data-i.e. complete the following equation:

J(Sw) = P

c(S

w) x .......................................... (5)

(Q. 15 is continued on the next page)

31

Q. 15 (continued)(v) Using the capillary pressure data shown below in Table 2, calculate an appropriate J-functionand complete the table below : assume the values k = 100 mD, φ= 0.1, interfacial tension, σ =10x10-3 N/m, and contact angle, θ = 0o. Choose any suitable units but label your sketch clearly.Sketch the J-function below.

Table 2Complete the table

Water Saturat ion Pc (psia) J(Sw)1 0

0.8 20.6 40.5 60.4 100.2 20

(6)

Write down the form of the J-function here:

J(Sw)=

(4)

Sketch the J-function here:

(6)

32

Q.16 (i) Name the two most popular tests used to determine the wettability of a reservoir rock.

The .................................................................. Test (2)

The .................................................................. Test (2)

(ii) Give the three “rules of thumb” that can often be used to distinguish between water-wet andoil-wet relative permeability curves.

Rule 1 ................................................................................................................................................

.......................................................................................................................................................(4)

Rule 2 ................................................................................................................................................

.......................................................................................................................................................(4)

Rule 3 ................................................................................................................................................

.......................................................................................................................................................(4)

(iii) Capillary pressure curves in Figure 11 below have been measured on two different core samples.The two plots are shown below. Use your knowledge of wettability variations at the pore scale toanswer the following:

Figure 11

Pc Pc

Sw Sw

(A) (B)

(iv) Which sample is probably the more water-wet? Explain your choice.

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

.......................................................................................................................................................(6)

33

Q.16 (continued)(v) Give a physical explanation for the negative leg shown in the capillary pressure curve in Figure 11(A).

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................

...........................................................................................................................................................(6)

(End of examination paper)

Total number of marks possible = 462