----------------------- Page 1----------------------MATHEMATICAL MODEL FOR DARCY FORCHHEIMER FLOW WITHAPPLICATIONS TO WELL PERFORMANCE ANALYSISByABIODUN MATTHEW AMAO, B.Sc.A THESISINPETROLEUM ENGINEERINGSubmitted to the Graduate Facultyof Texas Tech University inPartial Fulfillment ofthe Requirements forthe Degree ofMASTER OF SCIENCEINPETROLEUM ENGINEERINGApprovedAkif IbragimovChairperson of the CommitteeShameem SiddiquiCo-Chair of the CommitteeEugenio AulisaLloyd HeinzeAcceptedJohn BorrelliDean of the Graduate SchoolAugust, 2007----------------------- Page 2----------------------Texas Tech University, Abiodun Matthew Amao, August 2007ACKNOWLEDGEMENTSThis research was conducted at Texas Tech University under the supervision ofDr. Akif Ibragimov and Dr. Shameem Siddiqui. I like to express my sincere thanks

to Dr.Akif Ibragimov and Dr. Eugene Aulisa who introduced this concept to me and supportedme with the mathematical framework for the thesis. Dr. Shameem Siddiqui and Mr.Joseph McInerney were very helpful with the laboratory and experimental aspect of thethesis. My sincere gratitude goes to the Chair of the Petroleum Engineering Department,Dr Lloyd Heinze for his leadership and administrative prowess.I am indebted to all members of staff and colleagues who contributed in one wayor the other to the success of my academic pursuit at Texas Tech University.I deeply appreciate the moral support of my family back in Nigeria, myuncleJohn Oyedeji, Nengi Harry and all loved ones and friends back home.I appreciate the friendship and support of friends and members of my church inLubbock, International Christian Fellowship.Finally and most reverently, I thank the Lord for His mercy, grace andblessingswhich are too numerous for words.

ii----------------------- Page 3----------------------Texas Tech University, Abiodun Matthew Amao, August 2007TABLE OF CONTENTSACKNOWLEDGEMENTSii

ABSTRACTviLIST OF TABLESviiLIST OF FIGURESixLIST OF ABBREVIATIONSxiiCHAPTERI.

INTRODUCTION AND BACKGROUND11.1

Background3

1.2

Porous Media and Equations of Flow6

1.3

Darcys Law: Assumptions and Limitations7

1.4

Non-Darcy Flow; Darcy-Forchheimer Flow Equation9

1.5

Flow Regimes in Porous Media12

1.6

Significance of Thesis and Organization14

II

LITERATURE REVIEW172.1

Non-Darcy Flow in the Reservoir19

2.2

Flow in Fractures23

2.3

Completions, Gravel Packs and Perforations25

2.4

Beta Factor , its Measurement and Correlations26

2.5

Non-Darcy Flow Modeling30

iii----------------------- Page 4----------------------Texas Tech University, Abiodun Matthew Amao, August 2007III

PROBLEM STATEMENT343.1

ion3.2

Importance of Accurate Reservoir Pressure Predict35Limitations of Current Techniques36

3.3

Laboratory Experiments on Non-Darcy flow in Cores37

3.4.

Problem Statement42

IV

SOLUTION STATEMENT444.1

Proposed Solution44

4.2

Derivation of the Mathematical Model44

4.3

Description of the Simulator47

4.4

Numerical Computation and Algorithm48

4.5

Laboratory Measurement of Beta Factor50

V.

RESULTS OF NUMERICAL COMPUTATIONS555.1

Horizontal Well in a Rectangular Reservoir55

5.2

Centered Circular Well in a Rectangular Reservoir69

5.3

Off-centered Circular Well in a Rectangular Reser72

5.4

Centered Circular Well in a Square Reservoir75

voir

5.5

Off-centered Circular well in a Square Reservoir78

5.6

Concentric Well in a Circular Reservoir81

iv----------------------- Page 5----------------------Texas Tech University, Abiodun Matthew Amao, August 2007VI

ANALYSIS AND DISCUSSION OF RESULTS846.1

Analysis and Discussion of Experimental Results84

6.2

Analysis and Discussion of Computational Results96

VII.

CONCLUSIONS AND RECOMMENDATIONS1067.1

Conclusions106

7.2

Recommendations107

REFERENCES108

APPENDICES115A. RESULTS OF LABORATORY MEASUREMENTS OF ABSOLUTEPERMEABILITY115B. ALGORITHM FOR SELECTION OF THE RIGHT BETA FACTORCORRELATION136

C. EXPERIMENTAL SET UP AND EQUIPMENT USED IN THELABORATORY138D. VITA141

v----------------------- Page 6----------------------Texas Tech University, Abiodun Matthew Amao, August 2007ABSTRACTWell performance and productivity evaluation is a fundamental role of petroleumengineers and this is done at different phases of petroleum production; from thereservoirto the well bore through the tubulars and ultimately to the stock tank. This task requiresphysical and mathematical models that adequately characterize oil and gas flow at thesedifferent phases of petroleum production.This thesis reviews different scenarios where the effects of non-linearity in floware apparent in petroleum and gas reservoirs and cannot be neglected any more.Laboratory experiments were carried out on core samples to show non-linearity inflow,which confirms deviation from the traditional Darcy law, used in reservoir flowmodeling.Historically non-Darcy flow has only been reckoned with in high flow rate gaswells, in which it has been treated as a rate dependent skin factor and has been assumed

to act only in the vicinity of the well-bore, while neglecting the reservoir. This workseeks to show the inherent errors due to the negligence of this phenomenon, which isfundamental to the calculation of the productivity index of the well. Using themodifiednon-linear Darcy law as the equation of motion to model filtration in porous media, thisnew model is compared to the conventional Darcy law. The proposed method deliversrobust framework to model non-linear flow in the reservoir.The result of this project will equip reservoir engineers with a robusttechnique toanalyze well performance; this approach will provide better evaluation tool forselectingwells for remedial operations such as work-over or stimulation.vi----------------------- Page 7----------------------Texas Tech University, Abiodun MatthewAmao, August 2007LIST OF TABLES3.1 Result of non-Darcy flow experiment on core #13373.2 Result of non-Darcy flow experiment on core #9393.3 Result of non-Darcy flow experiment on core #26414.1 Porosity, physical properties and Lithology of core samples used524.2 Porosity ranking and cores used for permeability measurements534.3 Porosity and permeability of cores samples used in beta factor experiment545.1 Productivity index at different drain hole lengths575.2 Productivity index @ L=5000cm at different rates and beta values59

5.3 Productivity index @ L=10,000cm at different rates and beta values615.4 Productivity index @ L=20,000cm at different rates and beta values635.5 Productivity index @ L=30,000cm at different rates and beta values655.6 Productivity index @ L=40,000cm at different rates and beta values675.7 Productivity index at different rates and beta values for Geometry 5.2705.8 Productivity index at different rates and beta values for Geometry 5.3735.9 Productivity index at different rates and beta values for Geometry 5.4765.10 Productivity index at different rates and beta values for Geometry 5.5795.11 Productivity index at different rates and beta values for Geometry 5.6806.1: Beta factor correlations used for analysis846.2: Calculated beta values using the nine correlations85A.1: Experimental results of permeability measurement on core #1115A.2: Experimental results of permeability measurement on core #3118A.3: Experimental results of permeability measurement on core #6120vii----------------------- Page 8----------------------Texas Tech University, Abiodun Matthew Amao, August 2007A.4: Experimental results of permeability measurement on core #9122A.5: Experimental results of permeability measurement on core #10124A.6: Experimental results of permeability measurement on core #13126A.7: Experimental results of permeability measurement on core #22

128A.8: Experimental results of permeability measurement on core #23130A.9: Experimental results of permeability measurement on core #25132A.10: Experimental results of permeability measurement on core #26134viii----------------------- Page 9----------------------Texas Tech University, Abiodun MatthewAmao, August 2007LIST OF FIGURES1.1 Flow regimes in porous media after Basak (1977)143.1 Experimental result of non-linearity in flow through core #13383.2 Experimental result of non-linearity in flow through core #9403.3 Experimental result of non-linearity in flow though core #26424.1 Flow chart of numerical computation494.2 Experimental setup for permeability and beta factor experiments504.3 Procedure for laboratory measurement of beta factor515.1 Geometry of the horizontal drain in a rectangular reservoir565.2 Plot of productivity index at different drain hole lengths585.3 Productivity index vs. rate @ L=5000 cm605.4 Productivity index vs. rate @ L=10000 cm625.5 Productivity index vs. rate @ L=20000 cm645.6 Productivity index vs. rate @ L=30000 cm66

5.7 Productivity index vs. rate @ L=40000 cm685.8 Circular well in a rectangular reservoir (Geometry 5.2)695.9 Productivity index plot for Geometry 5.2715.10 Off-centered circular well in a rectangular reservoir (Geometry 5.3)725.11 Productivity index plot for Geometry 5.3745.12 Circular well in a square shaped reservoir (Geometry 5.4)755.13 Productivity index plot for Geometry 5.4775.14 Off-centered circular well in a square reservoir (Geometry 5.5)785.15 Productivity index plot for Geometry 5.580ix----------------------- Page 10----------------------Texas Tech University, Abiodun MatthewAmao, August 20075.16 Circular well in a circular reservoir (Geometry 5.6)815.17 Productivity index plot for Geometry 5.6836.1: Calculated beta factors for core #10, using the correlations866.2: Calculated beta factors for core #9, using the correlations876.3: Calculated beta factors for core #1, using the correlations886.4: Calculated beta factors for core #6, using the correlations896.5: Calculated beta factors for core #3, using the correlations906.6: Calculated beta factors for core #25, using the correlations916.7: Calculated beta factors for core #13, using the correlations

926.8: Calculated beta factors for core #23, using the correlations936.9: Calculated beta factors for core #22, using the correlations946.10: Calculated beta factors for core #26, using the correlations956.11: Productivity Index versus length for different rates at =0976.12: Productivity Index versus length for different rates at =2.4986.13: Productivity Index versus length for different rates at =24996.14: Productivity Index versus length for different rates at =2401006.15: Comparison of Productivity Index for all Geometries used at = 01026.16: Comparison of Productivity Index for all Geometries used at = 2.41036.17: Comparison of Productivity Index for all Geometries used at = 241046.18: Comparison of Productivity Index for all Geometries used at = 240105A.1: Darcys law plot for core #1116A.2: Klinkenberg correction plot for core #1117x----------------------- Page 11----------------------Texas Tech University, Abiodun Matthew Amao, August 2007A.3: Darcys law plot for core #3118A.4: Klinkenberg correction plot for core #3119A.5: Darcys law plot for core #6120A.6: Klinkenberg correction plot for core #6121

A.7: Darcys law plot for core #9122A.8: Klinkenberg correction plot for core #9123A.9: Darcys law plot for core #10124A.10: Klinkenberg correction plot for core #10125A.11: Darcys law plot for core #13126A.12: Klinkenberg correction plot for core #13127A.13: Darcys law plot for core #22128A.14: Klinkenberg correction plot for core #22129A.15: Darcys law plot for core #23130A.16: Klinkenberg correction plot for core #23131A.17: Darcys law plot for core #25132A.18: Klinkenberg correction plot for core #25133A.19: Darcys law plot for core #26134A.20: Klinkenberg correction plot for core #26135B.1: Beta Factor Correlation Selection Chart137C.1: Gas Permeameter, Hassler core holder and bubble flow tube138C.2: Helium Porosimeter139C.3: The core samples used for the experiments140

xi

----------------------- Page 12----------------------Texas Tech University, Abiodun Matthew Amao, August 2007LIST OF ABBREVIATIONSSymbolDefinitionAnal Area

Cross-sectio

Bon volume factor

Oil formatio

Bn volume factorg

Gas formatio

dn diameter

Average grai

Dow coefficient

Non-Darcy fl

F

Flux

FNDux

Non-Darcy Fl

huid head

height of fl

hickness

Reservoir th

J

Productivityindex

K

Permeability

Lre/ Sand bed

Length of Co

Mr weight

Gas Molecula

P

Pressure

Prvoir pressureR

Average rese

Pwfpressure

Well flowing

qate

Production r

N REber

Reynolds num

rdainage radius

Reservoir dr

rndary radiuse

External bou

rdiusw

Well bore ra

xii----------------------- Page 13----------------------Texas Tech University, Abiodun Matthew Amao, August 2007SSkin factorSTotal SkinttTimeTTemperaturevFlow velocityx, y, zRectangular coordinatesZGas compressibility factor

Greek LetterFluid densityAlpha

Inertial factorViscosityPorosityTortuosity

xiii----------------------- Page 14----------------------Texas Tech University, Abiodun Matthew Amao, August 2007Subscripto

Oil

g

Gas

w

Water

scard conditions

Stand

fure

Fract

xiv----------------------- Page 15----------------------CHAPTER IINTRODUCTION AND BACKGROUNDThe analysis and prediction of reservoir and well performance requires diverse

information which a reservoir or a production engineer must have before he/she canadequately analyze reservoir performance or predict future production under variousproduction mechanisms, the key to which is a consistent and representative mathematicalmodel of the physical parameters governing flow in the reservoir.Several techniques by which reservoir parameters can be acquired have beendevised. These include core analysis, well logging and pressure transient testing/analysis;of these techniques, pressure transient analysis gives the most representative informationon the reservoir at a scale consistent with the size of the reservoir.Pressure transient testing is simply generating and measuring pressure variationwith time in wells after a characteristic disturbance has been generated in the well;analysis of the generated data leads to an estimation of rock, fluid, well and reservoirproperties which are required in well performance engineering.Information obtained from transient testing include well-bore volume, skin,damage/improvement, reservoir pressure, permeability, porosity, reserves, reservoir andfluid discontinuities which are key input in reservoir performance analysis, wellimprovement schemes, economic analysis and production forecast.Historically, in oil field practice the productive capacity of producingwells isgenerally evaluated using the productivity index (PI), defined as the rate of productionper unit pressure drop. It has the symbol J , and it is expressed mathematicallyas:----------------------- Page 16----------------------Texas Tech University, Abiodun Matthew Amao, August 2007qJ =

(1.1)P

PR

wf

Where q = Production rateP = Average reservoir pressureRPwf = Well flowing pressureAnd based on Darcy law, the productivity index is given by;q

k hav

J =

=(1.2)

P

PR

wf141.2

ln

re B + S rw

3

4

Where,kav = Average permeabilityS = Skin factorThe productivity index J for different reservoir geometry, based on theshapefactor is given as;k hqJ =P

0.0078

av

=(1.3)P wf

A 10.06 13B ln + S 2 2 4 Cr Aw

Where,C = Shape factorAA = Drainage areaThe productivity index has been traditionally calculated based on the fundamentalassumption of the validity of Darcys law in porous media.

2----------------------- Page 17----------------------Texas Tech University, Abiodun Matthew Amao, August 2007However, Darcys law breaks down under conditions of high velocity flowwhichis proven to exist in gas wells, high permeability reservoirs, fractured reservoirs(naturally and hydraulically fractured) and in perforations, especially near thewell bore.This work seeks to review the dynamics of non-Darcy flow and how it affects theproductivity index calculation and well performance prediction in different reservoirgeometry and scenarios.

1.1 BackgroundThe physics of fluid flow in different media and conduits is a well researched areain engineering with groundbreaking works by pioneer workers in this field ofengineering. Equations describing flows in media such as cylindrical pipes, rectangularconduits, and other forms and shapes of conduits have been developed analytically overthe years.The three fundamental principles governing flow in any media and uponwhichthe development of these flow equations are based are:(a) Law of conservation of mass or the continuity equation(b) Equation of state of the fluid(c) Law governing the dynamics of fluid flow or Newtons law

3----------------------- Page 18----------------------Texas Tech University, Abiodun Matthew Amao, August 2007

Mathematical expression andstatement of these laws are given below:(a) Law of conservation ofmass or the continuity equationThis law states that the net excess of mass flux, per unit time into or out of anyinfinitesimal volume element in the fluid system is exactly equal to the change per unittime of the fluid density in that element multiplied by the free volume of the element,stated mathematically as:d(v )((

dd

v

v=

y(1.4)

)

) +

+

.( =) v

x

zdx

dydz

dt

(b) Equation of StateThis is the equation that describes the fluid and its thermodynamic flow propertiesas it relates to pressure,temperature and density. It is stated simply as;(

f,

P,

)

T

0

=(1.5)

(c) Law governing the dynamics of fluid flow (Newtons Law)This law imposeson the velocity distribution in every flow system the

requirement of a dynamicalequilibrium between the inertial forces and the viscous forcesand those due to external body forces and the internal distribution of fluid pressures. Thislaw takes into account allthe forces acting on the fluid as it flows in the medium, theforces acting on an elemental fluid particle and their equations are;

4----------------------- Page 19----------------------Texas Tech University, Abiodun Matthew Amao, August 2007(i)

Press

ure gradients in the coordinates of flowdpdp

dp,

,dxdy

dz(ii)

Exter

nal body forces, such as gravity in the direction of flowF,

F

F

,x

y

z(iii)s opposing motion or viscous forces, due to internal resistance of the

Forcefluid

to flow. An expression for viscous flow is given by:21

d

2

1

d

2

1

d

,3

+ vydx

,

+ vx

+ vz3

dy

3

dzwhere,

2d 2

d 2

d 2

+ +

anddx2

dy2

dz2

dvx

dv

dv

ydx

= = + + .v(from the continuity equation)

zdy

dzThe flow e

quation is obtained by equating the sum of these three forces statedabove to the product of mass and acceleration of the volume element of the fluid,therefore for an elemental fluid particle, the acceleration is given by the total timederivative of the velocity given by,Ddy d

dz d+

+

d+

d

d= +

v

+v

dt dy

dt dz

dt

x dx

dx d

dt

dt d

+

vDt

x

d

d

y dy

z dzCombining these par

ameters gives the Navier Stokes equation in three dimensionsDv2

1

dp

dx= + + +

Fvx(1.6a)

xDt3

dx

dx

5----------------------- Page 20----------------------Texas Tech University, Ab

iodun Matthew Amao, August 2007Dvy

dpF

2

1

d

v

= + + +

(1.6b)Dt

y

y

dy

Dvz

3

dp

2

dy

1

d

3

dz

= + + +F

v z

z

Dt

(1.6c)dz

The three laws and equations stated above are mathematically and scientificallysufficient to predict all the parameters of the flow of a viscous fluid flowing through amedium of any shape, size or geometry.The particular solution of the partial differential equations stated above for agiven medium is only possible when the boundaries of such a medium are clearlydefined. That is, the fluid system and the detailed physical conditions that serve as theinitial conditions of the system must be known before a solution can be obtained for anyflow medium or geometry.

1.2 Porous Media and Equations ofFlowA porous medium can be defined as a solid body which contains void spaces orpores that are distributed randomly; without any conceivablepattern throughout thestructure of the solid body. Extremely small voids are called molecular interstices andvery large ones are called caverns or vugs. Pores (intergranular and intercrystalline) areintermediate between caverns and molecular interstices.Fluid flow can only take place in the inter-connectedpore space of the porous

media; this is called the effective pore space.6----------------------- Page 21----------------------Texas Tech University, Abiodun Matthew Amao, August 2007Petroleum reservoirs are porous media and the storage and flow of hydrocarbonstakes place in these pore spaces which serve as conduit to the flow of oil, gasand waterduring production or the depletion of a reservoir. Some peculiarities of the porousmedia encountered in petroleum reservoirs are:(a) There is no geometry or geometrical quantity that can characterize or describethe system of pores in any porous body.(b) The pore walls are always irregularly converging or diverging and arehighlyirregular in any cross-section.(c) Visualizing pores as cylindrical tubes is not consistent with any pore systemknown in nature.These inherent and attendant characteristics of a porous medium makes itgrosslyimpossible to solve the system of partial differential equations (1.4 ), (1.5) and (1.6)describing the general fluid flow phenomena stated earlier.Literature is replete with several simplifying assumptions made by earlierresearchers to relate the pores in porous media to known shapes or geometry forwhichanalytical or numerical solution has been gotten, but none of these rightly solves theporous media problem.

1.3 Darcys Law: Assumptions and LimitationsHenri Darcy, a French civil engineer, in his 1856 publication laid the re

alfoundation of the quantitative theory of the flow of homogenous fluids through porous7----------------------- Page 22----------------------Texas Tech University, Abiodun Matthew Amao, August 2007media. As a civil engineer, he was interested in the flow characteristics of sand filtersused to filter public water in the city of Dijon in France.The result of his classic experiments, globally known as Darcys law, is thusstated: The rate of flow Q of water through the filter bed is directly proportional to thearea A of the sand and to the difference h in the height between the fluid heads at theinlet and outlet of the bed, and inversely proportional to the thickness L of the bed.This can be stated mathematically as:CAQ =

h

(1.7)Lwhere C is a property characteristic of the sand or porous media.Darcys law represents a linear relationship between the flow rate Q and the headh(pressure gradient)

.L

The constant of proportionality C in the original Darcy equation has beenkexpressed ase permeability

, where is the viscosity of the fluid and k is called th

of the porous medium. Permeability is a property of the structure of the porousmediaand it is entirely independent of the nature of the fluid. It uniquely sums up the

geometric properties of the porous media such as porosity, shape of the grains,size ofthe grains and the degree of cementation. The permeability k is considered tocompletely and uniquely characterize the dynamic properties of a porous media withrespect to flow of fluids though it.Hence, Darcys law is stated as:8----------------------- Page 23----------------------Texas Tech University, Abiodun MatthewAmao, August 2007k dpv =(1.8) dl

And more generally as:kA dpq =(1.9) dxDarcys empirical equation is a statistical average of classical hydrodynamicequation over the minute and detailed variation occurring in the individual pores; itgives a simplified macroscopic representation.Inherent in the development of the Darcy flow model are the following assumptions;a)

Darcys law assumes laminar or viscous flow (creep velocity); it do

es notinvolve the inertia term (the fluid density). This implies that the inertia oracceleration forces in the fluid are being neglected when comparedto theclassical Navier-Stokes equations.b) Darcys law assumes that in a porous medium a large surface area isexposedto fluid flow, hence the viscous resistance will greatly exceed ac

celerationforces in the fluid unless turbulence sets in.

1.4 Non-Darcy Flow; Darcy-Forchheimer Flow EquationDarcys empirical flow model represents a simple linear relationship betweenflow rate and pressure drop in a porous media; any deviation from the Darcy flowscenario is termed non-Darcy flow.

eadings

31Physical causes for these deviations are grouped under the following h;9

----------------------- Page 24----------------------Texas Tech University, Abiodun MatthewAmao, August 2007a)

High velocity flow effects.

b) Molecular effects.c)

Ionic effects.

d) Non-Newtonian fluids phenomena.However, in petroleum engineering, the most common phenomenon is the highflow rate effect. High flow rate beyond the assumed laminar flow regime can occur in thefollowing scenarios in petroleum reservoirs.a)

Near the well bore (Perforations)

b) Hydraulically fractured wellsc)

Gas reservoirs

d)

Condensates reservoirs (Low viscosity crude reservoirs)

e)

High flow potential wells

f)

Naturally fractured reservoirs

g)

Gravel packs

It is therefore imperative for reservoir engineers to develop a better

flow modelthat is adequately representative and uniquely characterizes the physical parameters andvariables in these flow scenarios.In 1901, Philippe Forchheimer, a Dutch man, while flowing gas thoroughcoalbeds discovered that the relationship between flow rate and potential gradient is nonlinear at sufficiently high velocity, and that this non-linearity increases withflow rate. Heinitially attributed this non-linear increase to turbulence in the fluid flow (it is now knownthat this non-linearity is due to inertial effects in the porous media), which he determined10----------------------- Page 25----------------------Texas Tech University, AbiodunMatthew Amao, August 200726to be proportional to avlity. Cornel and Katz

, with a being a constant of proportiona

gave a value of to a, where (beta) is called the inertial factor and is tdensityof the fluid flowing through the medium.The additional pressure drop due to inertial losses is primarily due to theacceleration and deceleration effects of the fluid as it travels through the tortuous flowpath of the porous media. The total pressure drop is thus given by Forchheimer empiricalflow model stated traditionally as;dp= v + vdx

2(1.11)

k

This can also be written in vector notation as:r

rr

(1.12)v

+v v =P Where = ,kThe Forchheimer equation assumes that Darcys law is still valid

, but that anadditional term must be added to account for the increased pressure drop. Hence thisequation will be called the Darcy-Forchheimer flow model in this thesis.Equation (1.11) is based on fitting an empirical equation through experimental data.However, Forchheimer based on these data set later propose a third order equationgiven by:dp=av +bv

2

3

+cv(1.13)dx

where a, b and c are constants as in equations (1.11) and (1.13) above.

11----------------------- Page 26----------------------Texas Tech University, Abiodun Matthew Amao, August 2007Another flow model that has been proposed for flow in porousmedia is the powerlaw model, given by:dp

n=av(1.15)

dxwhere n has a value between 1 and 2In vector notation, it is stated as:C

v

v

P n

n1 r=(1.16)

However, of these three models the most widely used is givenby equation (1.11)and it will form the basis of analysis in this project to characterizehigh velocity nonDarcy flows in porous media.

1.5 Flow Regimes in Porous MediaAnalogous to flow in pipes and conduits, several researchershave also tried todefine a flow regime in porous media to distinguish flow regimes and topredict the onsetof one or the termination of another. Typically for flow in pipes and conduits, theReynolds number is used to delineate flow regimes. A Reynolds number less than 2100implies laminar flow, while a greater number implies turbulent flow. Inporous mediahowever, there is no clear limit or a magic number that defines this transition. The nonlinearity experienced in non-Darcy flow is not a result of turbulence but inertia effects asstated earlier, hence non-Darcy flow is known to occur in porous mediaat a much more12----------------------- Page 27----------------------Texas Tech University, Abiodun Matthew Amao, August 2007lower Reynolds number, and it is not initiated by a change in flow regime. The Reynoldsnumber in porous media is given by;vdN Re =(1.17)where d is average grain diameter of the grains in the porous media. However fora mediawith non-Darcy flow (e.g. a fracture) the Reynolds number is given by;

vkN Re =(1.18)This is just another Reynolds number with the characteristic length defined by k.In the literature, depending on the flow velocity and the nature of theporousmedia different flow patterns have been observed. However four major regimes wereproposed by Dybbs and Edwards (using laser anemometry and visualization technique).These four regimes are;a)ere the

Darcy or laminar flow where the flow is dominated by viscous forces, hpressure gradient varies strictly linearly with the flow velocity. The

Reynoldsnumber at this point is less than 1.b) At increasing Reynolds number, a transition zone is observed leading toflowdominated by inertia effects. This begins in the range Re=1~10. This laminarinertia flow dominated region persists up to and Re of ~150.c)

An unsteady laminar flow regime for Re =150 ~ 300 is characterized byoccurrence of wake oscillations and development of vortices in the flow

profile.d) A highly unsteady and chaotic flow regime for Re > 300, it resembles turbulentflow in pipes and is dominated by eddies and high head losses.13----------------------- Page 28----------------------Texas Tech University, Abiodun Matthew Amao, August 2007However there is large variation in the limiting Reynolds number for thesetransition zones as published in the literature, therefore one cannot be too categoricalabout limits and transition zones as it relates to the Reynolds number in porous

media.Figure 1.1 below is a diagrammatic representation of the flow regimesin a porous49media as proposed by Basak

.

Pre-Darcy Zone

Darcy Zone

Post-Darcy Zone

LaminarTurbulentForchheimerPre-LaminarNo Flow

Figure 1.1: Flow Regimes in Porous Media after Basak (1977)

1.6 Significance of Thesis and OrganizationThe results and knowledge gained from this thesis will be useful in adequatelyevaluating production performance of wells and aid reservoir engineers in modelingreservoir flow with more robust equations. Selection of candidate wells for well

14----------------------- Page 29----------------------Texas Tech University, Abiodun Matthew Amao, August 2007engineering routines will be more objective and representative of actual scenario in thereservoir. The findings from this thesis will further illuminate known discrepancies inwell test analysis and help to ratify a fundamental source of uncertainty in well test

models.This thesis is organized into seven chapters; the contents of each chapter aresummarized.Introduction and background; this chapter contains a brief introductionto thefundamental principles of fluid flow in porous media, with a review of governingequations of flow in porous media as it relates to Darcy and non-Darcy flows.Literature review; this is an assessment of current industry practice andmethodology used to handle non-Darcy flow in different scenarios in the petroleumindustry with a review of non-Darcy flow modeling in the literature.Problem statement; a categorical expression of the problem this thesisseeks tosolve, with the motivation and importance of this solution to the petroleum industry.Solution statement; this is a procedural statement of the development of aproposed solution to the stated problem and why this approach is significantly differentfrom previous approaches. It also gives a statement of the results expected using thisprocedure.Results; a catalogue of results obtained during laboratory experiment on coresamples and numerical simulations of various reservoirs and well geometries.Discussion and analysis of results; the results obtained are compared with currentindustry practices and discussed.15----------------------- Page 30----------------------Texas Tech University, Abiodun Matthew Amao, August 2007Conclusions; the final chapter summarizes the thesis and presents the conclusions

drawn.

16----------------------- Page 31----------------------Texas Tech University, Abiodun Matthew Amao, August 2007CHAPTER IILITERATURE REVIEWIn the early days of the petroleum industry it was noted that the pressure dropmeasured in the vicinity of the wellbore was greater than the pressure drop computed

using industry-wide modeling equations36. This excessive pressure drop was explained byassuming a decrease in permeability (formation alteration) due to formation damage inthe vicinity of the wellbore. The capacity of a well to produce is generally accepted to bedirectly proportional to the pressure drop in the reservoir. Hurst and Van Everdingen36 inthe 1950s introduced a dimensionless term called the skin factor which was usedtoexplain this phenomenon36. The skin factor (S) was originally designed to give anumerical value to the additional resistance assumed to be concentrated around thewellbore resulting from drilling and completion techniques employed or the productionpractices used. This ultimately leads to an additional pressure drop, this pressure drop iscalled the skin effect. The magnitude of the skin effect determines the productivecapacity of a well. This has also been used in well performance evaluation and remedialoperations.Over the years, the skin factor has been broken down into several components. Anexpression for the total skin (S) is given below:S = S + S + Sc

p

+ S+ S + So(2.1)dGA

Where,S= skinSc= completion skin due to partial penetration17----------------------- Page 32----------------------Texas Tech University, Abiodun Matthew Amao, August 2007Sp= perforation skin

Sd= skin due to damage around the well boreSG= gravel-pack skinSA= outer boundary geometry skinSo= slanted well skinThe additional pressure drop due to high velocity flow is also expressed as anequivalent skin, Dq; where q is the flow rate and D is a composite of the following highvelocity flow terms;D = D

+ D + DR

d

+ D(2.2)dp

G

WhereDR= reservoir high velocity flow term beyond the well bore areaDd= damaged zone high velocity flow termDdp= high velocity flow term in the region surrounding the perforationsDG= high velocity flow term in a gravel packed perforationq = flow rateAssuming all the other skin sources are summed up in S, therefore, for the caseof highvelocity flows, the total skin factor will be given by;St = S + Dq(2.3)Where;St = Total skinDq = rate dependent skin factorD = Non-Darcy flow coefficient18----------------------- Page 33----------------------Texas Tech University, Abiodun Matthew Amao, August 2007It is obvious that the value of the rate dependent skin (Dq) will notbe a constant,

in comparison to the mechanical skin, as it will depend on the flow rate, in a directproportionality. This will subsequently vary the value of the total skin St .As can be seen from the sources of skin enumerated above, the petroleum industryhas known the inadequacy of Darcys law to adequately predict the pressure loss athighflow rate; however, this skin factor has been assumed to be concentrated in thevicinity ofthe wellbore i.e. at the sandface or across the completion, the effect of non-Darcy flow inthe reservoir has been neglected and assumed to be negligible.The treatments of non-Darcy flows will be reviewed under the scenariowherethese effects come into play in reservoir engineering.

2.1 Non-Darcy Flow in the ReservoirNon-Darcy flow occurs in petroleum reservoirs that have high conductivity toflow. Initially it was assumed that this phenomenon was only relevant to gas wells, butfield observations and analysis show that it relevant to oil wells as well. Thiswas proven10by Fetkovitch during a comprehensive field study of 40 oil wells.As narrated above, non-Darcy flow has been treated as a rate dependentskinfactor by the inclusion of the term Dq as an additional source of pressure loss inthevicinity of the wellbore. The various techniques for evaluating this parameter arereviewed below.

19

----------------------- Page 34----------------------Texas Tech University, Abiodun Matthew Amao, August 20072.1.1 Multi-rate TestsMulti-rate tests are traditionally used to evaluate the deliverability of a gas or oilwell, the additional pressure drop due to non-Darcy effect is calculated from theHoupeurt (back-pressure) analytical equation and from the empirical equation proposedby Rawlins and Schellhardt in 1936. These tests are listed below;(i) Flow after flow test(ii) Isochronal test(iii) Modified isochronal test

2.1.1.1 Flow-After-Flow TestsThis is also called the gas back pressure of four point test, it is conducted byproducing the well at a series of different stabilized (pseudosteady state) flow rates andmeasuring the stabilized bottom hole flowing pressure at the sand face. Each flow rate isestablished in succession, often conducted with a sequence of increasing flow rates. Amajor limitation of the testmust reach a stabilization period,

procedure is that the well

especially in low-permeability formations that take longer to reach stabilization.Schellhardt and Rawlins of the USBM developed an empirical equation for analyzingback-pressure data based on field data analysis. They proposeda relationship whichapplicable only at low pressures is given byq

C =P(

2P )

2

n(2.4)

f

s

Where,C= Stabilized performance coefficient20----------------------- Page 35----------------------Texas TechUniversity, Abiodun Matthew Amao, August 20072

2

P

P

qn = inverse slope of log-log plot of () versusf

s

The theoretical value of n ranges from 0.5, which indicates non-Darcy flow regime, to 1.0indicating a flow regime governed by Darcys lawA much more consistent analytical equation developed from the gas diffusivityequation was proposed by Houpeurt which is stated as;2Ps)

P

Aq = Bq

2

2

+

(Gas well(2.5)

R

wf

g

2P

AqP

g

2=B

2+

(Oil well

s)

(2.6)R

wf o

o

Where,ln

A

= re0.75

+

0 Bo

S

t 7.x08 10K h

r w o o BB =xk h37.08 10

3o

Do

=A

zTgln4

r e0.75

+

St

xk h7.03 10

rg

zTB =g7.03 104xk h

w

Dg2

PA Cartesian plot of (q gives a plot with intercept A and slope B,

P

2

R

wf

) against

qfrom which the value of D, can be calculated knowing all other variables.

21----------------------- Page 36----------------------Texas Tech University, Abiodun Matthew Amao, August 20072.1.1.2 Isochronal TestsThis technique was proposed by Jones, Blount and Glaze19. This test wasdesigned to shorten the stabilization time required for the flow after a flow test. This longtime is usually impractical in some cases, especially in low-permeability reservoirs. It isconducted by alternating producing the well, then shutting the well in and allowing it tobuildup to the average reservoir pressure before the beginning of the next flowperiod.Pressures are measured at several time increments during each flow period. Thetime period in which the pressures are monitored is the same relative to the stating timeof each flow period. The same method of analysis is used to analyze the data toobtain

values for D.

2.1.1.3 Modified Isochronal TestsThis technique was proposed in a paper by Brar and Aziz. It is a modification ofthe isochronal test aimed at shortening the test times required for the well tobuild up tothe average reservoir pressure in the drainage area of the well. It is conductedlike anisochronal test, except that the shut in periods are of equal duration and the flow periodsare of equal duration. The length of the shut-in period usually equals or exceeds the flowperiods.It is known to be less accurate than the isochronal test, due to thisshort timeperiods allowed for pressure build up. The data analysis is the same as the previous testtypes.

22----------------------- Page 37----------------------Texas Tech University, Abiodun MatthewAmao, August 20072.1.2 Single Well Test TechniquesThe use of a single well test to estimate the non-Darcy skin factor has been34proposed by several researchers. These include Camacho et al, Warren, Spivey etal,Kim and Kang21 . They proposed new methods for using single well tests to obtaintherate dependent skin factor, based on the algorithms they developed.

2.1.3 Correlations

Ramey proposed an equation for calculating the non-Darcy flow coefficient ifmulti-test data are not available. The expression was obtained by integrating theForchheimer equation for the drainage radius rd to the well bore rw . However,heconfirmed that the result may be in error of about 100%, based on a comparison withmulti-rate tests. The expression is given as;2.715 10

15x

Mp

k

sc

D =(2.7)Thrsc wwhere the variables have the usual notations.

2.2 Flow in FracturesThe occurrence of non-Darcy flow phenomenon in fractures is well documentedin the literature. Early workers have come to understand the importance of thisphenomenon as it affects the productivity of fractures. Fractures can either benatural orinduced e.g. hydraulic fractures. The two distinct flow regimes observed duringwell testsin fractured reservoirs point to the fact that the flow regime in the matrix isdifferent from23----------------------- Page 38----------------------Texas Tech University, Abiodun MatthewAmao, August 2007the flow regime in the fracture, although this has been thought to affect only the high ratewells.Hydraulic fracturing is a widely used completion method in the tight gasformations all over the world. Several hydraulic fracturing jobs are implemented

annually. However, the performances of these fractures are highly dependent on nonDarcy flow effects in the fracture. Several ongoing studies are looking into howtomaximize fracture design and mitigate the non-Darcy effect in fractures.

2.2.1 Hydraulic FracturesIn hydraulic fracture stimulation of wells, the wells productive capability andoverall reserve recovery is impacted by non-Darcy flow as it causes a reductionin thepropped half length to a lower effective half length. Fracture design engineershavehistorically neglected this phenomenon assuming that it only impacts high velocity wells.According to Vincent et al.37, ignoring the non-Darcy effects while designingfractures will lead to inaccurate production forecasts, suboptimal fracture design andselection of inappropriate proppant type. They opined that fluid velocities in real fractureare approximately 1000 times greater than laboratory measurements; hence laboratorymeasured proppant permeability values are not really suitable when designing fractures.Miskimins et al.26 in their investigation of flow rates at which non-Darcy flowinfluences retained fracture permeability discovered that its effect is significant across awide spectrum of flow rates from as low as 50-100 MCFD, and these decrease can rangefrom 5% at a flow rate of 50 MCFD to 30% at 400 MCFD under a given set of24----------------------- Page 39----------------------Texas Tech University, Abiodun Matthew Amao, August 2007

conditions. Presently in fracture design non-Darcy flow is integrated by accurateselection of a proppant type based on laboratory tests and field observation withparticular emphasis on the beta factor of the proppant to be used.To optimize fracture design, Lopez-Hernandez et al.24 proposed a betafactormethod to calculate the effective fracture permeabilityfkeff . This parameter isgiven by;kffkeff =(2.8)

k 1+f

vgg

This expression was derived by combining the Darcy and non-Darcy flow equationsin afracture and solving forfkeff , which determines the actual pressure drop in thefracture.Another fracture design criterion is to minimize the pressure loss dueto the inertialosses by minimizing the v2 term in the traditional Darcy-Forchheimer equation. Thiscan be achieved by selecting a proppant with an optimal beta factor.The beta factor may be more important than the reference permeabilitywhenselecting proppant for a fracturing job. Hence it is imperative to know the betafactor ofthe proppant to be used in the design, as they are not usually reported in the industry.

2.3 Completions, Gravel Packs and PerforationsSeveral workers have investigated non-Darcy flows in completions andperforations. It was observed that large pressure drops in perforated completions occurmostly in the convergence zones and the in perforation tunnel, especially in high rate oiland gas wells. Nguyen29 experimentally studied non-Darcy flow in perforations. He

25----------------------- Page 40----------------------Texas Tech University, Abiodun MatthewAmao, August 2007discovered that non-Darcy flow in perforations is a function of perforation geometry, andpermeability of the gravel. In his experiments, he used water and air as the flowing fluidand came to the conclusion that the relationship between pressure drop and flowrate isnon-linear. Therefore, a simplistic analysis of the flow using Darcys law will overpredict the productivity and cases have been found where the productivity has been overpredicted

by as much as 100%.In well performance engineering of gravel packed completions, it is imp

ortant todelineate the pressure drop due to mechanical skin or rate dependent skin (non-Darcyflow) so that the right remedial action can be taken to improve the productivityof thewell.

2.4 Beta Factor , its Measurement and CorrelationsThe beta factor , which is a constant of proportionality in the traditional DarcyForchheimer equation, was first proposed by Cornel and Katz6. It is known by severalnames which include; non-Darcy flow coefficient, inertial flow coefficient and theturbulence factor. However, in these thesis we will adopt the non-Darcy flow coefficient.It is widely agreed that is a property of the porous media; it is a strong function of thetortuosity of the flow path and it is usually determined from laboratory measurements andmulti-rate well tests.

The derived expression for the beta factor falls under two broad categories;empirical and theoretical models. The theoretical models are further divided into paralleland serial models.26----------------------- Page 41----------------------Texas Tech University, Abiodun Matthew Amao, August 2007In the parallel model, the porous medium is assumed to be made up of straightcapillary bundles of uniform diameter. According to Li and Engler22, based on the workof Ergun et al., and Polubarinova-kochina, an expression for the Beta factor fora parallelmodel is given by;c =(2.9)0.5 1.5KWhere c is a constantIn the serial type model, the pore space is serially lined up; capillaries of differentpore types are aligned in series. Li et al.22 also proposed an expression for the Beta factorfor a series model based on the work of Scheidegger, the beta factor is given as; =

c (2.10)K

Where c is a constant related to pore size distributionThere are several empirical correlations in the literature used to predict the betafactor. These expressions differ due to the varied experimental procedure, porous mediaand fluids used for the experiments. However, it is consistently shown that permeability,

porosity and tortuosity are the main parameters on which the beta factor depends. Also,some correlations have been developed for multiphase flows, hence these correlations arefunction of saturation as well.

2.4.1. Permeability Defined Beta Factors19Jones

conducted experiments on 355 sandstone and 29 limestone core

s (vuggy,crystalline, fine grained sandstone) and came up with a correlation given by27----------------------- Page 42----------------------Texas Tech University, Abiodun MatthewAmao, August 20076.15 10x = 1.55

10(2.11)

KWhere K is in md and in 1/ftPascal et al, based on mathematical analysis of data from Multirate wells inhydraulically fractured reservoirs, proposed a correlation given by,4.8 10 x = 1.176

12(2.12)

KWhere K is in md and is in 1/m.Cooke based on his experiments in using brines, reservoir oils and gases inpropped fractures, predicted the non-Darcy coefficient as, =bK

(2.13)

Where a and b are constants determined by experiments based on proppant type.

2.4.2. Correlations Based on Permeability and Porosity

Eguns empirical equation based on data found in the literature and experiments,proposed the correlation given as,8 1/ 21/ 2 3/ 2(2.14) =ab(10K )

Where a=1.75, b=150, K in Darcy and in 1/cm.Janicek and Katz, for natural porous media proposed to use the followingequations:8

5 / 4 3 / 4 (2.15)1.82=x10 KWhere K is in md and is in 1/cm.28----------------------- Page 43----------------------Texas Tech University, Abiodun MatthewAmao, August 2007Geertsma based on his experiments on consolidated and unconsolidatedsandstones, dolomites and limestone and a review of other works, he proposed anempirical correlation given by:0.005 =(2.16)0.5 5.5Kwhere K is in cm2 and in cm-1

2.4.3. Correlations Based on Permeability, Porosity and TortuosityLiu et al further worked on the data used by Geertsma, Cornell and Katz, Evansand Evans and Whitey, and by considering the effect of tortuosity they got a bettercorrelation given as,

8x

8.91 10 =(2.17)KWhere is in ft-1 and K in mdOthers include, Thauvin et al., they proposed a correlation given by,1.55 10

4x =0.98 0.K

3.3529(2.18)

Where is in cm-1 and K in DarcyThis is not an exhaustive listing, there are several other correlationsproposed inthe literature. In choosing a correlation to use in predicting the non-Darcy coefficient, Liet al.22 proposed the following guidelines.(see Appendix B)(a) Determine the lithology of the formation (e.g. from well logs)29----------------------- Page 44----------------------Texas Tech University, Abiodun MatthewAmao, August 2007(b) Determine what parameters are known or can be found, use the correlationthat has as many known parameters as possible.(c) Determine the pore geometry of the formation and the relativity offlowdirection to pore channels.

2.5 Non-Darcy Flow ModelingFluid flow in porous media in the petroleum industry has been modeled by theDarcy flow equation. The diffusivity equation has been widely used in well testmodels,reservoir simulation models and all other petroleum engineering models to simulate fluid

flow in the reservoir. One important use of these models is to predict reservoirpressureand other reservoir parameters that are required for well performance evaluationandprediction. Muskat27 was the first to utilize Darcys law in deriving fluid flowequationsin oil and gas reservoirs for different flow patterns and reservoir geometries.This hasserved the petroleum industry for a long while. However recent research and furtherinsight into non-Darcy flow phenomenon in the reservoir and scenario where it occurs isnecessitating a new look into this historical trend.Numerical modeling of non-Darcy flows began in the 1960s; some of the pioneerworkers include Smith, Swift et al., who investigated the effects of gas flow onwelltesting. Researchers in recent times are looking at newer and better ways of modelingfluid flow in porous media while integrating the Forchheimer equation for non-Darcyflow. Thus they are developing a new diffusivity equation that can be used in reservoir30----------------------- Page 45----------------------Texas Tech University, Abiodun Matthew Amao, August 2007simulators and other numerical models so that more accurate and better predictive modelscan be obtained.Belhaj et al.5 developed a new diffusivity equation thatwas used to model nonDarcy flow in the reservoir. They used a finite difference modeling scheme, based on theCrank-Nicholson and Barakat-Clark numerical modeling methods, while comparing bothDarcy and non-Darcy flows. They derived a new expression for the diffusivity equation

based on the Darcy-Forchheimer equation in two dimensions stated as;2

2

P

P

+c

= v +

P

P

v 2 +2

2

P

+

x

y

K

t

x

Based on the results of their numerical simulations, they opined that the Forchheimermodel gave more realistic result for all ranges of pressure gradients, flow rates,permeabilities, porosities, viscosity and fluid density.33Su

of Saudi Aramco, in his publication detailed how n

on-Darcy flow modelingcan be integrated into a reservoir simulator, especially for multiphase flow modeling. Hemodeled both the rate dependent skin factor in the reservoir and also at the well boretreating the two differently. He took the non-Darcy considerationinto account, both inthe cell to cell flux and in the vicinity of the well bore. His model also proposed theDarcy-Forchheimer equation for each phase flowing in the reservoir; his phase basednon-Darcy flow equation is given as

2dp =

j j q

+

j

qj

(2.19)dxkK

Arj

A 31

----------------------- Page 46----------------------Texas Tech University, Abiodun Matthew Amao, August 2007j denotes the phase, K is the relative permeability. He used a cel

y

l-to-cell nonWhere

r

Darcy flow resistance flux factor, FND to multiply the Darcy flow flux term, stated asFlux non-Darcy = FND * Flux Darcy(2.20)He gave an approximate expression for the rate dependent skin factor by the expression, kKr j

jDj =

,(2.21)

2h rj wSu35 applied his model to both oil and gas well, based on the resultof hisnumerical simulations he opined that Darcy-Forchheimer can be applied to a multiphasesystem, that non-Darcy flow in occurring in the entire reservoir can be handled in asimulator and that this model can be easily integrated with a full blown numericalsimulator.Jamiolahmady et al.17, when modeling flow in a crushed perforated rock, theydeveloped a mathematical model based on the Darcy-Forchheimer flow. From theequation in vector form they developed the following expressions V=P

V V +(2.22)k

Where the gradient operator V is the absolute value of the velocity,From which they obtained an expression for V given ask PV =1+k V(2.23)

The continuity equation for radial cylindrical coordinate system given as,

32----------------------- Page 47----------------------Texas Tech University, Abiodun MatthewAmao, August 20071

V( )+ z(2.24)rVr rz

. = Vr

=0

Is solved to obtain an expression for V given as,

+ +1

1

2k P

4

V =(2.25) 2k

The negative root is discarded, while the expression (2.25) is substituted in equation(2.24). This gives k. 2 (2.26) 1+1 4+

=0

2r

P2

k P

The above expression was solved based on the finite element method using theFemlab (COMSOL Multiphysics) mathematical modeling software. They opined thattheir model shows the limitations of the current models used in well completionengineering.

33----------------------- Page 48----------------------Texas Tech University, Abiodun Matthew Amao, August 2007CHAPTER IIIPROBLEM STATEMENTThe productivity index of a well is a powerful tool for well evaluation. It is theproduction rate divided by the drawdown. The productivity index, as an evaluation tool isonly valid when the well is flowing in a pseudo-steady state (PSS) regime. Untilthepressure transient period during a well test is passed and a steady state pressuredistribution is assumed in the well, the productivity index will not approximatea constant28with any physical significance

.

The productivity index for an ideal well remains constant, even if thewell28production rate and the reservoir pressure changes during the life of the well. A changein the productivity index of a well over its life is an indication of an anomaly, which maysuggest the presence of permeability barriers or impedance (e.g scales, asphaltenes, sandproduction and any other skin effect) to fluid flow in the reservoir. The productivity of awell is a direct function of the pressure drop in the reservoir. Hence it is imperative to

accurately delineate and evaluate the pressure drop and know the causes of suchpressuredrop in a well. This is the key goal of well performance engineering; evaluatingandcalculating the pressure drop, accurately knowing the cause of the pressure dropanddesigning a remedial action or proffering a solution to mitigate or remove the cause of thepressure drop thus increasing the productivity of the well.Therefore, in evaluating performance or non performance and in rectifying anywell problem, the source of the problem must first be identified, and then the rightsolution can be proffered to fix the problem. Based on the foregoing, it is obvious that a34----------------------- Page 49----------------------Texas Tech University, Abiodun Matthew Amao, August 2007blanket description of all well problems under the Skin umbrella does not really suffice;to adequately resolve any well problem, its source must be known. This is one ofthemain challenges of this thesis; to show how poor fluid flow modeling can affectpressurepredictions and resultant effect on the calculated well productivity index.

3.1 Importance of Accurate Reservoir Pressure PredictionThe pressure profile in the reservoir is very important to reservoir and productionengineers. The production mechanism in petroleum reservoir are driven by pressure,hence knowledge of the pressure profile is essentially an indication of the producibility ofthe reservoir. Knowledge of the reservoir pressure is important for the following reasons;a)

It gives an indication of the production mechanism of the well

b) It shows the productive capacity of the wellc)

Knowing the pressure will help determine what additional equipmen

t isrequired to lift the reservoir fluid to surface.d) It is required for reservoir management and planning.e)f)ies and for

Pressure profile help in determining new well locationsPressure profile is a source of information for reservoir properthydraulic connectivity.

Well tests and pressure surveys are usually conducted on wells to getone or someof the above information based on the pressure data obtained from the well tests.

35----------------------- Page 50----------------------Texas Tech University, Abiodun Matthew Amao, August 20073.2 Limitations of Current TechniquesA review of current industry practices as it relates to high flow ratewells wasdone in chapter 2 of this thesis. From the review it is obvious that using the historicalDarcys law to model fluid flow in high flow rate reservoir is not adequate. The nonDarcy flow problem in petroleum engineering still requires further research, until morerobust equations and models can be developed to solve this problem.Although the industry over the years has introduced a fudge factor alsocalledthe skin factor assumed to be applicable to a region of impaired permeability inthevicinity of the well bore. This has not adequately help to narrow down the problem to its

root cause and has brought in lots of uncertainties. This may explain why some remedialjobs or work-over operations have not been successful. This is simply because theproblem was never rightly diagnosed and hence, the solution applied is not applicable.A great leap in well performance engineering will occur when well or reservoirproblems are rightly diagnosed using the right models and tools, so that the proffered orrecommended solution will adequately fix the well problem at hand. The ability to rightlycalculate the individual components of the composite skin factor will help in takingcorrective measures to reduce its detrimental effect and thereby enhance the wellsproductivity. Until a problem is known, it may never have a solution or it can be rightlysaid that a problem known is half solved.

36----------------------- Page 51----------------------Texas Tech University, Abiodun MatthewAmao, August 20073.3 Laboratory Experiments on Non-Darcy flow in CoresThe following results were obtained on core samples used in the Core Laboratory(Corelab) of the Department of Petroleum Engineering Texas Tech University, to verifythe certainty of non-Darcy flows at high pressure/flow rate. The experiments wereconducted on core samples that represented different reservoir types- sandstonesandcarbonates (limestone and dolomite). The experimental results for three core samples

(#13, #26 and #9) are presented in tables 3.1, 3.2 and 3.3 respectively. Figures3.1, 3.2and 3.3 are the graphical plot showing non-linearity in flow.Table 3.1: Result of non-Darcy flow experiment on Core #13Core ID: #13Length: 6.1 cm07 mmHg = 13.15 psia

Ambient Pressure

680.

Diameter: 3.745 cm

Temperature

74 F

Viscosity of N 2

0.01

2Area: 11.015 cm7584 cpP (psi )

PPL

(atm)in

P

(atm)

Q(cc/sec)

out

K

md

Q/A

g

100472

1.57650.1115

0.8961

0.5204

7.4483

0.

201035

2.25690.2231

0.8961

1.1403

8.1596

0.

301546

2.93730.3346

0.8961

1.7032

8.1253

0.

402023

3.61770.4462

0.8961

2.2286

7.9737

0.

502589

4.29810.5577

0.8961

2.8517

8.1624

0.

603055

4.97850.6692

0.8961

3.3649

8.0263

0.

703536

5.65890.7808

0.8961

3.8949

7.9631

0.

804119

6.33930.8923

0.8961

4.5368

8.1161

0.

904548

7.01971.0039

0.8961

5.0092

7.9656

0.

1004923

7.70011.1154

0.8961

5.4225

7.7605

0.

1105283

8.38051.2270

0.8961

5.8194

7.5714

0.

1205744

9.06091.3385

0.8961

6.3269

7.5457

0.

130

9.7413

0.8961

6.5053

7.1617

0.

5906

1.4500

37----------------------- Page 52----------------------Texas Tech University, Abiodun Matthew Amao, August 2007Core#13: Non-Darcy Plot1.61.41.2)mc/mta(

1.0

0.8

L/PD

0.60.40.20.00.00.5

0.10.6

0.2

0.3

0.4

0.7Q/A (cm/s)

Figure 3.1: Experimental result of non-linearity in flow through Core #13

38----------------------- Page 53----------------------Texas Tech University, Abiodun MatthewAmao, August 2007Table 3.2: Result of non-Darcy flow experiment on Core #9Core ID: #9Length: 3.55 cm3 mmHg = 13.15 psia

Ambient Pressure

680.0

Diameter: 3.72 cm

Temperature

76 F

2

Viscosity of N

0.01

7584 cpArea: 10.869 cmP (psi )

2P

atm)

in (PL

P

(atm)

Q(cc/sec)

out

K

md

Q/A

g

1084

1.57520.0756

0.8948

0.8218

0.8097

6.9

2059

2.25560.1420

0.8948

1.5436

0.6348

6.5

3088

2.93600.2140

0.8948

2.3256

0.5221

6.5

4003

3.61640.2903

0.8948

3.1546

0.4433

6.7

5025

4.29680.3640

0.8948

3.9564

0.3852

6.7

6003

4.97720.4354

0.8948

4.7323

0.3406

6.7

7089

5.65760.4918

0.8948

5.3447

0.3052

6.4

8029

6.33800.5741

0.8948

6.2402

0.2765

6.6

9081

7.01840.6315

0.8948

6.8634

0.2527

6.4

10088

7.69880.6699

0.8948

7.2812

0.2327

6.1

39----------------------- Page 54----------------------Texas Tech University, Abiodun Matthew Amao, August 2007Core#9: Non-Darcy Plot2.52.0)mc

1.5

/mta(L/

1.0PD0.50.00.00.6

0.10.7

0.2

0.3

0.4

0.8Q/A (cm/s)

Figure 3.2: Experimental result of non-linearity in flow through Core #9

0.5

40----------------------- Page 55----------------------Texas Tech University, Abiodun Matthew Amao, August 2007Table 3.3: Result of non-Darcy flow experiment on Core #26Core ID: #26Length: 4.145 cm3 mmHg = 13.15 psia

Ambient Pressure

680.0

Diameter: 3.75 cm

Temperature

76 F

Viscosity of N

0.017

2584 cpArea: 11.04466 cmP (psi )

2

P

(atm)

P

(atm)

Q(cc/sec)

K

md

Q/A

PLin

out

g

3792

1.09890.0492

0.8948

7.5020

244.19

0.6

4703

1.16690.0657

0.8948

9.6123

234.66

0.8

5836

1.23500.0821

0.8948

10.8631

212.16

0.9

6805

1.30300.0985

0.8948

13.0384

212.20

1.1

7634

1.37110.1149

0.8948

13.9540

194.66

1.2

8666

1.43910.1313

0.8948

15.0940

184.24

1.3

9430

1.50710.1477

0.8948

15.9370

172.92

1.4

10309

1.57520.1641

0.8948

16.9085

165.11

1.5

11561

1.64320.1806

0.8948

18.2907

162.37

1.6

12514

1.71130.1970

0.8948

19.3436

157.41

1.7

13953

1.77930.2134

0.8948

19.8288

148.95

1.7

14959

1.84730.2298

0.8948

20.9396

146.06

1.8

41----------------------- Page 56----------------------Texas Tech University, Abiodun Matthew Amao, August 2007Core#26: Non-Darcy Plot0.250.20)mc 0.15/mta(L/PD

0.10

0.050.000.0.6

1.8

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1

2.0Q/A (cm/s)

Figure 3.3: Plot of Experimental result of non-linearity in flow through Core #26

3.4 Problem StatementThe buildup of the thesis up till now as been to lay the foundation of flow inporous media, describe the peculiarities of Darcy and non-Darcy flows, review currentindustry practice and show there inadequacies. This has been a gradual crescendoto thepetroleum engineering problems this thesis seeks to investigate and proffer a solution to;

these problems are summarized in the following statements. The inadequacy of Darcyslaw to model fluid flow in reservoirs with high velocity flow profiles and the resultanterror it propagates in well performance analysis.42----------------------- Page 57----------------------Texas Tech University, Abiodun Matthew Amao, August 2007The traditional use of the rate-dependent skin factor to account forthe additionalpressure loss due to high velocity flows, neglects pressure losses in the reservoir, since itonly assumes that the losses are important in the vicinity of the well bore, research hasshown that this is not the case especially in fractured reservoirs.There is no proven method of knowing flow regimes in the reservoir;thusobfuscating the judgment of a well analyst in flow modeling.

43----------------------- Page 58----------------------Texas Tech University, Abiodun MatthewAmao, August 2007CHAPTER IVSOLUTION STATEMENT4.1 Proposed SolutionThe previous chapters has adequately shown the importance and gravity of thenon-Darcy flow phenomena, and highlighted the scenario where this phenomenon occursin the prospect of oil and gas. The obvious limitations of the Darcys law as a flowmodeling equation for these scenario is evident.The proposed solution is to integrate the Darcy-Forchheimer equation into theflow modeling equation for non-linear (high velocity flows), and use the developedequation to model fluid flow in the reservoir, especially for non-linear flows.Theproductivity index of the well is then calculated using this model, with the objective thata more representative well productivity will be obtained in these scenarios.

4.2 Derivation of the Mathematical ModelIn chapter 1, the three fundamental equations required to model fluid flow in anymedia were stated as:a)

Continuity equation (Law of conservation of mass)

b) Equation of statec)

Equation of motion/dynamics (Flow Equation)

The derivation of the non-linear mathematical flow equation is given below:

The continuity equation, assuming constant porosity is given by,

44----------------------- Page 59----------------------Texas Tech University, Abiodun Matthew Amao, August 2007 div(

r+)v0 =(4.1)

t =div (vt

r)

r= div v v((4.2)t

r)

1

From product rule, P = P

t

tP

= P x

x

Substituting these expressions in equation (4.2) above, P

r P

( )div v

= PtSimplifying

r v

P

x

,

1 P r= divv v P((4.3)t

r)

1

Equation (3) above is the final form of the continuity equation used.The equation of flow is the Darcy-Forchheimer equation givenby:

dp= v + vdxk

2

And in vector form as, let = , then the expression becomeskrrr+v v =P

v

r

+P

+v

rrv=v (4.4)

0

The equation of state is given by the expression;45----------------------- Page 60----------------------Texas Tech University, Abiodun Matthew Amao, August 2007

1

(4.5)

=1( P )P 01 = eWhere(

0

is the compressibility)

Equations (3), (4) and (5) arethe three governing equations to be used in the derivation ofthe mathematical framework forthe model.The vector velocity rv (x ,t )cannot be uniquely represented as a function of the pressuregradient P , we assume an approximation given by;v

v

v v

v

r r( f , ,P=) P = (1 2 =) 3

Correspondingly,v

f = P

P

( )

Substituting these in the Darc

y-Forchheimer equation, equation (4) above, +

(f

P

( )+ )PP

(1+P

(

( )fP

(f

.P f

+) P (

(f

( =)P

0

P

))P )=P 0

2This is a form of a quadraticequation, therefore solving forf

( P ) , and taking only thepositive root as the valid sol

ution to the equation, this results in +4

+

2

P( ) =f

P

2Pthe denominator by (

Multiplying the numerator and)

2

+ +

4

P

, results in

f

P( ) =

2(4.6)2+ + P

46----------------------- Page 61----------------------Texas Tech University, Abiodun MatthewAmao, August 2007rEquation (6) above is a solution of the velocity vector v of the Darcy-Forchheimerequation.

4

The continuity equation for slightly compressible fluid from equation (4.3), isgiven by

P =divv((4.7)t

r)

1r v P

) is negligible,

For slightly compressible fluids, the term (Substituting Darcy-Forchheimer parameters into equation (4.7), results inP

( P= )P

div f(t

) (4.8)

This is the form of the partial differential equation (PDE) that is used to model the nonlinear Darcy-Forchheimer flow in porous media.In developing this model, the following assumptions have been made:a.

Pressure independent rock and fluid properties

b.

Homogenous and isotropic porous medium with uniform thickness

c.

Negligible gravity forces

4.3 Description of the SimulatorThe software used in solving the PDE above is called COMSOL Multiphysics. Ita commercial package used in solving systems of partial differential equations (PDE),typically seen in scientific and engineering problems. The solution of the PDE is basedon the finite element method (FEM) scheme for solving PDEs. The software runs thefinite element analysis with adaptive meshing and error control using a varietyofnumerical solvers.47----------------------- Page 62----------------------Texas Tech University, Abiodun MatthewAmao, August 2007

In COMSOL Multiphysics, PDEs can be described in three ways;a)

Coefficient form: Suitable for linear or nearly linear models

b)

General form: Suitable for nonlinear models

c)

Weak form: For PDEs on boundaries, edges or points or for models w

ithmixed space and time derivatives.The coefficient form of PDE model was used for solving the Darcy-Forchheimernonlinear model, in this thesis.

4.4 Numerical Computation and AlgorithmThe Darcy-Forchheimer model was applied to different reservoir geometrytoevaluate the productivity indexes of these reservoirs. A comparison is made between thecases when Darcys law is used versus when the Darcy-Forchheimer model was used tomodel flow in the reservoir. The reservoir geometry used were obtained from reservoirgeometries for which shape factors have been obtained for pseudo-steady stateproductivity index calculation as stated in chapter. The flow chart in figure 4.1 is adiagrammatic representation of the steps used in solving the model, using COMSOLMultiphysics.48----------------------- Page 63----------------------Texas Tech University, Abiodun Matthew Amao, August2007COMSOLMultiphysicsInitializeDefine form of PDEDraw Reservoir andWell GeometryDefine Boundary

Ente

r ModelingConditions and Initialuation andValues of Parametersvoir Domain

EqReserInput Values of Constantsand ParametersInput Solve ParametersSelect Solver TypeDefine Grid Size(Initialize or Refine GridMesh Size)Read off Output DataPressureProductivity Index (PI)

NO

Is Output:same?YESGenerate Plot of OutputData in EXCELEnd ofRoutine

Figure 4.1: Flow Chart of Numerical Computation49----------------------- Page 64----------------------Texas Tech University, Abiodun MatthewAmao, August 20074.5 Laboratory Measurement of Beta FactorThe Laboratory measurement of the Beta factor was done by first measuring theabsolute permeability of the core samples used in the experiments then increasing thepressure drop across the cores at an ever increasing pressure differential whilemeasuringthe flow rate. The experimental set up is shown diagrammatically in figure 4.2 below.A linear version of the Forchheimer equation was then used to calculate thecoefficient of inertial resistance, beta. (This procedure is described by Dake8

in his book,Fundamentals of Reservoir Engineering, page 259).

Figure 4.2: Experimental setup for permeability and factor measurementsThe experimental procedure used is presented diagrammatically flow in figure 4.3below.50----------------------- Page 65----------------------Texas Tech University, Abiodun Matthew Amao, August 2007StartPrepare core samples formeasurementMeasure porosity of core samplesusing helium porosimeterNo

Sor

t cores intoAre porosities insame range?

grou

ps accordingto porositiesYesMeasure gas permeability (K ) usinggnitrogen gas at low pressures (flow rate)Use Klinkenberg correction to obtainabsolute permeability (KL)Apply increasing pressure differentials across coresample and record flow rateObtain beta factor from

1

dP

=vDarcy-Forchheimer equation

+dx2 kv

Plot beta as a function ofabsolute K on a Log-Log graph

Express beta as a function ofC =absolute permeability Kk EndFigure 4.3: Procedure for Laboratory Measurement of51

Factor

----------------------- Page 66----------------------Texas Tech University, Abiodun Matthew Amao, August 2007The absolute permeability of the cores was obtained by first measuringgaspermeability using nitrogen gas, and then applying the Klinkenberg correction toobtainthe absolute permeability of the core samples.Initially 26 core samples were sampled for the experiments, but after measuringthe core porosities, it was decided to carry out permeability measurement only on tencore samples sorted based on their porosities and initial permeability tests. Table 4.1below is the spreadsheet used for the porosity calculations. Porosity was measured usingthe Helium porosimeter.Table 4.1: Porosity, physical properties and Lithology of core samples usedCore IDDiameterLengthBulk VolumeLithology#(cm)(cm)(cc)PorositySandstone13.7203.465037.6600.1829Sandstone23.7203.650039.6710.0909Sandstone33.7003.610038.8150.1730Sandstone43.7403.965043.5590.1420Sandstone53.7203.440037.3880.1699Sandstone63.7203.300035.8670.1812Sandstone73.7203.400036.9530.1247Sandstone83.7203.945042.877

0.1246Sandstone90.1838Sandstone100.1850Sandstone110.1017Sandstone120.0756Sandstone130.1377Sandstone140.1323Sandstone150.1030Sandstone160.1050Sandstone170.0812Carbonate180.0629Carbonate190.1402Carbonate200.0166Carbonate210.1114Carbonate220.1340Carbonate230.1368Carbonate240.0819Carbonate250.1457Carbonate260.0992

3.720

3.5500

38.584

3.725

3.2800

35.745

3.700

5.0800

54.621

3.700

5.5950

60.158

3.745

6.1000

67.193

3.740

5.1500

56.577

3.745

3.9400

43.400

3.745

5.6400

62.126

3.745

6.2700

69.065

3.755

6.2000

68.660

3.740

5.1000

56.028

3.745

3.2300

35.579

3.800

5.7700

65.438

3.750

4.9400

54.561

3.780

5.4400

61.048

3.750

5.0000

55.223

3.770

4.4250

49.395

3.750

4.1450

45.780

52----------------------- Page 67----------------------Texas Tech University, Abiodun MatthewAmao, August 2007The core samples were ranked based on their porosities and an initial permeabilitymeasurement done on the core samples to select the cores that were used in the finalanalysis. The core selection is given the table 4.2 below.Table 4.2: Porosity ranking and cores used for permeability measurementsLithologyCore #PorosityCommentsSandstone100.1850Sandstone90.1838Sandstone10.1829Sandstone60.1812

SandstoneSandstoneCarbonateSandstoneCarbonateSandstoneCarbonateCarbonateSandstoneSandstoneSandstoneCarbonateSandstoneSandstoneSandstoneCarbonate

35251913232214782116151126

0.17300.16990.14570.14200.14020.13770.13680.13400.13230.12470.12460.11140.10500.10300.10170.0992

Highly Frac

22417121820

0.09090.08190.08120.07560.06290.0166

Fractured

4

turedSandstoneCarbonateSandstoneSandstoneCarbonateCarbonate

Core #26 was selected because it is highly fractured and it will serve as a goodcandidateto investigate non-Darcy flow in fractured reservoir.The absolute permeability of the core samples is given in table 4.3 below; the results andanalysis of the laboratory measurements are given in appendix A.

53----------------------- Page 68----------------------Texas Tech University, Abiodun Matthew Amao, August 2007Table 4.3: Porosity and Permeability of Core samples used in factor experimentCore ID

Porosity

Permeabi

lity (md)10

0.1850

5.36

9

0.1838

6.18

1

0.1829

5.04

6

0.1812

1.77

252086

863

0.1730

3.89

25

0.1457

2.18

13

0.1377

7.58

23

0.1368

3.26

22

0.1340

0.84

26

0.0992

160.

445183894939

54----------------------- Page 69----------------------Texas Tech University, Abiodun MatthewAmao, August 2007CHAPTER VRESULTS OF NUMERICAL COMPUTATIONS

The results of numerical computations using COMSOL Multiphysics, is presentedin this chapter. Different reservoir geometry and well configurations were usedin thecomputations. The dimensions of the reservoir and the well are given for each ofthegeometry used in the computation.

5.1 Horizontal Well in a Rectangular ReservoirThe first geometry used in the numerical computation is a horizontal drain-hole ina rectangular reservoir. Figure 5.1 shows the location of the horizontal drain-hole relativeto the boundaries of the reservoir, as shown it is located in the center of thereservoir. Thedimensions used for the computation are stated below.Dimensions: Length = 800 metersWidth = 400 metersWell radius = 15 cm (6 inches)

55----------------------- Page 70----------------------Texas Tech University, Abiodun Matthew Amao, August 2007

Figure 5.1: Geometry of the Horizontal Drain in a Rectangular reservoir (Geometry 5.1)

56----------------------- Page 71----------------------Texas Tech University, Abiodun Matthew Amao, August 2007The results of the numerical computations of geometry 5.1 are given intable 5.1.It is the result of the variation of the calculated productivity index of the reservoirgeometry as length of the horizontal drain-hole and factor are varied for the geometry.Table 5.1: Productivity Index at different drain-hole lengthsProductivity Index at Different Beta ValuesLength =240005000

=0

=24

=240

10000.00237

0.23639

0.21403

0.11814

10000

20000.00265

0.31863

0.28256

0.14406

15000

30000.00298

0.40047

0.34989

0.16974

20000

40000.00336

0.48950

0.42256

0.19768

25000

50000.00382

0.59029

0.50444

0.22939

30000

60000.00436

0.70670

0.59888

0.26623

35000

70000.00500

0.84233

0.70906

0.30960

40000

80000.00577

1.00012

0.83789

0.36082

45000

90000.00668

1.18190

0.98746

0.42102

50000

100000.00774

1.38673

1.15807

0.49071

0.021690.024600.027860.031610.035990.041170.047340.054700.063410.07354

Q

=2400

55000

110000.00895

1.60952

1.34669

0.56924

60000

120000.01026

1.83912

1.54509

0.65409

65000

130000.01159

2.05735

1.73806

0.73962

70000

140000.01279

2.24032

1.90292

0.81515

0.085020.097500.110230.12164

57----------------------- Page 72----------------------Texas Tech University, Abiodun Matthew Amao, August 2007Figure 5.2 is a graphical representation of the results of the numericalcomputation; it shows the variation of the productivity index of the horizontaldrain withvariation in length at different factor values in the reservoir.

Productivity Index (P.I) vs Length of Horizontal Drainage2.52.0PI(Beta=0)PI(Beta=24)

)xednIyt

PI(Beta=240)PI(Beta=2400)PI(Beta=24000)1.5

ivitcudorP(

1.0

I.P0.5

50000

0.0500010000 15000 2000055000 60000 65000 70000

25000

30000 35000

40000

45000

Length (cm)

Figure 5.2: Plot of Productivity Index at different drain-hole lengths

58----------------------- Page 73----------------------Texas Tech University, AbiodunMatthew Amao, August 2007Table 5.2 shows the productivity index of the horizontal drain-hole at a constantlength of 5000cm while varying flow rate and factor values.Table 5.2: Productivity index @ L = 5000cm at different rates and valuesL = 5000 cmProductivity Index at Different Beta ValuesQ

=0 =240

41000

0.2364

=2.40.2338

=2

0.2140

0.1181

20000.1960

0.23640.0790

0.2314

30000.1810

0.23640.0594

0.2290

40000.1682

0.23640.0476

0.2267

50000.1570

0.23640.0397

0.2245

60000.1473

0.23640.0340

0.2223

70000.1387

0.23640.0298

0.2201

80000.1311

0.23640.0265

0.2180

90000.1243

0.23640.0238

0.2160

100000.1181

0.23640.0217

0.2140

59----------------------- Page 74----------------------Texas Tech University, Abiodun Matthew Amao, August 2007Figure 5.3 is the graphical representation of the results in table 5.2

; it shows thetrend of productivity index with flow rate at a constant drain-hole length of 5,000 cm.

Productivity Index vs Rate @ L=5000cm0.250.20xednIytivitcudorP

0.15Beta = 0Beta= 2.4Beta = 24Beta =2400.10

0.05

0

0.0010008000

20009000

300010000

4000

5000

6000

700

Rate (Q)Figure 5.3: Productivity index versus rate @ L=5000 cm

60----------------------- Page 75----------------------Texas Tech University, AbiodunMatthew Amao, August 2007

Table 5.3 shows the productivity index of the horizontal drain-hole at a constantlength of 10,000cm while varying flow rate and factor values.

Table 5.3: Productivity index @ L = 10,000cm at different rates and valuesL = 10,000 cmProductivity Index at Different Beta ValuesQ

=0 =240

4

=2.4

10000.2992

0.31860.1975

0.3165

20000.2825

0.31860.1440

0.3144

30000.2679

0.31860.1134

0.3124

40000.2548

0.31860.0935

0.3104

50000.2430

0.31860.0796

0.3084

60000.2322

0.31860.0692

0.3065

70000.2224

0.31860.0613

0.3046

80000.2135

0.31860.0550

0.3028

90000.2052

0.31860.0498

0.3009

100000.1975

0.31860.0456

0.2992

=2

61----------------------- Page 76----------------------Texas Tech University, Abiodun Matthew Amao, August 2007Figure 5.4 is the graphical representation of the results in table 5.3; it shows thetrend of productivity index with flow rate at varying beta factor values for a constantdrain-hole length of 10,000 cm.

Productivity Index vs Rate @ L=10000cm0.350.30x 0.25ednIy 0.20tivitc0.15udorP 0.10

Beta=0Beta=2.4Beta=24Beta=240

0.050.0010008000

9000

2000300010000

4000

5000Rate (Q)

6000

Figure 5.4: Productivity index versus rate @ L = 10,000 cm

7000

62----------------------- Page 77----------------------Texas Tech University, AbiodunMatthew Amao, August 2007Table 5.4 shows the productivity index of the horizontal drain-hole at a constantlength of 20,000cm while varying flow rate and factor values.Table 5.4: Productivity index @ L = 20,000cm at different rates and valuesL = 20,000 cmProductivity Index at Different Beta ValuesQ

=0 =240

4

=2.4

10000.4700

0.48950.3539

0.4874

20000.4527

0.48950.2797

0.4854

30000.4370

0.48950.2314

0.4833

40000.4225

0.48950.1977

0.4814

50000.4091

0.48950.1724

0.4794

60000.3966

0.48950.1529

0.4775

70000.3849

0.48950.1374

0.4756

80000.3740

0.48950.1247

0.4737

90000.3637

0.48950.1142

0.4719

100000.3539

0.48950.1053

0.4700

=2

63----------------------- Page 78----------------------Texas Tech University, Abiodun Matthew Amao, August 2007Figure 5.5 is the graphical representation of the results in table 5.4; it shows thetrend of productivity index with flow rate at a constant drain-hole length of 20,000 cm.

Productivity Index vs Rate @ L=200000.60.5xe 0.4dnIytiv 0.3itcudo

Beta = 0

r 0.2P

Beta= 2.4Beta = 24Beta =2400.10.010008000

20009000

300010000

4000

5000

6000

7000

Rate (Q)

Figure 5.5: Productivity index versus rate @ L = 20,000 cm

64----------------------- Page 79----------------------T