SPE-52080-STU

8/18/2019 SPE-52080-STU

http://slidepdf.com/reader/full/spe-52080-stu 1/16

Copyright 1998, Society of Petroleum Engineers Inc.

This paper was prepared for presentation at the 1998 SPE European Petroleum Conferenceheld in The Hague, The Netherlands, 20–22 October 1998.

This paper was selected for presentation by an SPE Program Committee following review ofinformation contained in an abstract submitted by the author(s). Contents of the paper, aspresented, have not been reviewed by the Society of Petroleum Engineers and are subject tocorrection by the author(s). The material, as presented, does not necessarily reflect anyposition of the Society of Petroleum Engineers, its officers, or members. Papers presented atSPE meetings are subject to publication review by Editorial Committees of the Society ofPetroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paperfor commercial purposes without the written consent of the Society of Petroleum Engineers isprohibited. Permission to reproduce in print is restricted to an abstract of not more than 300words; illustrations may not be copied. The abstract must contain conspicuousacknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O.Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

AbstractThis paper gives an overview over the different aspects of

transient pressure behaviour in fractured reservoirs.

Basically fractured reservoirs can be divided into twodifferent groups. The first consists of reservoirs of single

porosity which have been fractured artificially, the secondgroup consists of reservoirs which contain a network of

natural fractures and which have been additionally fractured

artificially (double porosity fractured reservoir). Further it is

possible to make a difference based on the radial dimensionsand the boundary conditions of a reservoir. Therefore the

systems of infinite and finite reservoirs will also be discussed.

These four cases will be presented separately. Thedifferent flow regimes will be discussed and also the

parameters which influence the pressure behaviour.

Further the interpretation of the pressure tests will be

discussed by using the type curves which correspond with the

results of the flow regimes description.Finally a short brief of the simulation of the two main

different fractured reservoirs on a numerical reservoir

simulator will be given.

IntroductionTo enhance the productivity of wells which were damaged or

were completed in tight formations, hydraulic fracturing(called "HF" in the following text) is a common and often

used technique. Hydraulic fracturing as well as acidicing have

been used with great success in many reservoirs with lownatural permeability. It was realised very early that a HF

influences the flow behaviour of a well extremely, so that

common pressure test analyses techniques may cause errors.

Thus the transient flow behaviour of fractured reservoirs became an important topic.

The different analyses of the pressure tests do not only

describe the status quo of the flow behaviour after the

treatment but describe also the form of the HF. Theinterpretation of the tests is very important for the production

strategy and even for the design of a new HF. This paper will

discuss the following topics:

The different flow regimes in a HF as well as their

relation to each other over time. The mathematical description of the infinite and finite

system.

The transient flow behaviour of a single porosity fracturedreservoir and a double porosity fractured reservoir.

The effects of storage and fracture conductivity on the

pressure behaviour. Interpretation of pressure curves based on the system of

type curve matching.

The simulation of wells in fractured reservoirs.Therefore this paper will present an overlook over the

transient pressure behaviour of fractured wells, both in single

porosity and in double porosity reservoirs as well as the

simulation of such systems.

The transient pressure behaviour is described by thesolution of the differential equation which describes each flow

regime.

To get a correct solution the correct initial and boundaryconditions are needed. So the initial and boundary conditions

of the different systems will be described at the beginning and

also the different assumptions which have been made to

simplify the system. The infinite reservoir will be discussed before the finite system. Also the behaviour in a single

porosity fractured reservoir will be discussed first. The results

will be transferred with all changes to the double porosityfractured reservoir. Fig. 1 illustrates the configuration of a

vertical HF cutting through a reservoir.

Single Porosity Reservoirs1,2,3,4

A single porosity fractured reservoir is defined as a reservoir

with a constant permeability apart from the HF. Basically two

different systems exist. The first system is an infinite reservoirwhich is infinite in its radial-dimension. The second one is the

finite reservoir. So each system is defined by two components.One is the reservoirs dimension, infinite or finite and the other

is the HF. 1,2,4

SPE 52080-STU

Transient Pressure Behaviour in fractured ReservoirsPatrick W. Pattay, Mining University of Leoben

8/18/2019 SPE-52080-STU


2 Patrick W. Pattay 52080

The system of single porosity fractured reservoir is described

by the following initial and boundary conditions:1,2

The Reservoir: An isotropic, homogeneous, horizontal,

infinite or finite, slab reservoir is bounded by upper andlower impermeable plates. The reservoir has a uniform

thickness h, permeability k and porosity φ. All these

parameters are independent from the pressure.

The Fluid: The reservoir contains a slightly compressible

fluid of a compressibility c and viscosity μ. The HF: The fluid is produced through a vertically

fractured well intersected by a fully penetrating, finite

conductivity fracture of half length xf , width wf ,

permeability k f , conductivity k f bf and porosity φf .All these parameters are constant along the whole HF and over

time. The inflow into the well occurs only through the HF. A

system which is described by these parameters is shown in

Fig. 1.In addition because of the fact that the width of the HF is

very small compared with the reservoir a linear flow inside the

HF can be assumed2. Gravity forces will be neglected, all

parameters of the reservoir and the HF are independent from

pressure and pressure gradients are small in general, but it is

not possible to neglect them. 1,2 Under all these conditions, the phenomenon is described

by the diffusivity equation in two dimensions. Based on theseassumptions the flow in the HF is described by the following

equation2.

( )∂

∂

μ φ μ ∂

∂

2

2

pf

x k f

q x t

w h

f cft

k f

pf

t+ ⋅

⋅ =

⋅ ⋅⋅

, ..........................(1)

The general solution for the well bore pressure pwf is given by

equation (2) which is based on Darcy's Law:

( )k h pi pwf

q B pwD tDx

f fD CfDf

⋅ ⋅ −

⋅ ⋅ ⋅ =

⎛ ⎝ ⎜

⎞ ⎠⎟

α μ η, , ...............(2)

In this equation pwD represents the dimensionless pressure

drop as a function of the dimensionless time, the

dimensionless fracture hydraulic diffusivity ηfD and thedimensionless fracture storage capacity CfDf .

The dimensionless time1 is defined as:

tDxf

k t

ct xf

= ⋅ ⋅

⋅ ⋅ ⋅

α

φ μ 2 ....................................................(3)

The dimensionless fracture hydraulic diffusivity1 is a quotient

of the relative permeability of the HF, the porosity and

compressibility in the reservoir and in the HF:

η φ

φfD

k f ct

k f cft

= ⋅ ⋅⋅ ⋅

........................ .................. ................ (4)

The dimensionless storage capacity1 is a quotient which takes

the dimension of the HF and the porosity and compressibility

of the HF and the formation in consideration:

CfDf

bf f cft

xf ct=

⋅ ⋅

⋅ ⋅ ⋅

φ

π ϕ .....................................................(5)

The behaviour of the pressure depends on two parameters:

The dimensionless time tD and the dimensionless fractureconductivity

1. The dimensionless fracture conductivity is a

quotient which takes the relation of the permeability inside the

HF and inside the reservoir as well as the geometry of the HF

in consideration:

( )k f bf D

k f bf

k xf

CfDf fD= ⋅

⋅ = ⋅ ⋅π η .............................(6)

Single Porosity Infinite Reservoir1,2,3. In general a

distinction must be made between a reservoir of a low and ahigh fracture conductivity, that means (k f bf )D < 300 (Case 1)

or (k f bf )D > 300 (Case 2). For these two cases two basicallydifferent curves will occur.

1

Initially there is a Fracture Linear Flow characterised by a

half slope straight line; after a Transition Flow Period, the

system may or may not exhibit a Bilinear Flow Period,

indicated by a one forth slope straight line. As time increases aFormation Linear Flow might develop. In all cases, the systemreaches, while it istabilises, a Pseudo Radial Flow.

1

Fracture Linear Flow. This behaviour occurs at verysmall values of the dimensionless time but it is exhibited in all

cases. During this time the main flow into the well bore comes

from the expansion of the system within the HF as shown inFig. 2. This is essentially a Linear Flow along the wings of the

HF

( ) pwD

k f bf D

fD tDxf

= ⋅ ⋅2

π η ...............................(7)

This equation gives a half slope straight line on a log-log plot

of pwD versus t Dxf or Δ p versus Δt. This regime is usually

masked by storage effects1. The approximate end of this

period is given by equation (8).

( )tDx

f

k f b

f D

fD

=⋅0 01

2

2

.

η.................................................(8)

Bilinear Flow Period. It is called Bilinear Flow because

two linear flow regimes occur at the same time. On one hand

there is a linear, incompressible flow inside the HF, on the

other hand there is a linear but compressible flow inside the

formation.

1,2

The Bilinear Flow exists as long as most of the fluid

entering the well bore comes from the formation near by thewell as shown in Fig. 3. and as long as the HF tips have not

yet affected the well behaviour.

8/18/2019 SPE-52080-STU


52080 Transient Pressure Behaviour in Fractured Reservoirs 3

( ) pwD

k f bf D

tDxf

= ⋅2 45

4.

.........................................(9)

This equation implies that a log-log plot of pwD versus t Dx f or

Δ p versus Δt gives a one forth slope straight linecorresponding to the forth root of the dimensionless time. The

duration of this period depends on the storage capacity and the

conductivity of the HF. So in a system with a high storagecapacity the Bilinear Flow Period may not exist at all becauseof the reason that the HF tip effects occur to early

1,2. The

approximate end of this regime is:

( )t

k bD

f f D

= −⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

−

45525

4

..

.............................................(10)

Formation Linear Flow. During this period the flow is

linear through the formation into the wings of the HF asshown in Fig. 4. This flow regime occurs clearly at reservoirs

with a high fracture conductivity, that means (k f bf ) D > 300. Ina HF with such a high conductivity the flow through the tips

of the wings of the HF gets more and more important. For a

high fracture conductivity the flux density through the lastthird of the wings of the HF is two thirds of the complete

flux.2

( )( )

pwDk f bf D

tDxf

= ⋅138.

......................................(11)

Corresponding to the square root of the dimensionless time

this equation implies also that a log-log plot of function results

a straight line with a slope of one half. The period from the

bilinear to the Formation Linear Flow behaviour will be a

convex connection. This period is called the Transition Zone.

This flow regime ends in general for a dimensionless time of0,016.

1

Pseudo Radial Flow. This flow regime is very similar to

the radial flow which commonly occurs in a reservoir. It

occurs as soon the reservoir has stabilised and behaves like acommon reservoir.

Even if the well is not a round hole any more as shown in

Fig. 5 because of the reason that the dimensions of the HF aresmall compared with the reservoir it is possible to use the term

of a Pseudo Radial Flow3.

( ) p tk t

c xwD D

f

= ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⎛ ⎝ ⎜ ⎞

⎠⎟

+

⎛

⎝

⎜⎜

⎜⎜⎜

⎞

⎠

⎟⎟

⎟⎟⎟

1

2

0 000264

2

0807092

ln.

.

Φ μ

....................(12)

The Skin Factor offers the possibility to transfer thefractured well into ideal round well with a larger radius than

the original well had. For a fractured well the skin is in most

cases negative.

Interpretations 1,2,3 . In pressure testing the system of typecurve matching is used since many years. The log-log graph

has been used commonly as a diagnostic tool to detect

different flow regimes in a transient pressure test. This systemoffers the possibility to interpret pressure test data and to

correlate them with the different flow regimes, as shown inFig. 6.

The Fracture Linear Flow shows in the log-log graph the

typical half slope straight line which makes it possible to

diagnose. For a separate analysis a graph for the dimensionless pressure drop over the square root of dimensionless time

would be more useful. This slope depends on the fracture

characteristics exempting the fracture half length xf . 1

The Bilinear Flow Period shows a one fourth slope straightline which is characteristic for this flow regime. General well

bore storage effects are causing deviations from the one fourthslope characteristic as shown in Fig. 7. The influx of the

storage capacity occurs for about three logarithmic cycles andit can also mask the hole flow regime.

1,2

An important feature of this graph is that after the BilinearFlow, where the graph is a straight line, the graph becomes a

curve either convex or concave depending on the fracture

conductivity, which depends also on storage effects.For a dimensionless fracture conductivity higher then 1.6 it

is concave and for a dimensionless fracture conductivity loweror equal 1.6 it is convex, as mentioned in the beginning of this

chapter and shown in Fig. 6.1

But it is important to realise that inside these two areasfore some parts of the type curves the shape of these type

curves is similar for different values of dimensionless fracture

conductivity. For more detailed interpretations a graph for the

dimensionless pressure drop over the forth root ofdimensionless time is very useful and offers better

interpretation possibilities.

It is important to notice that for large values of theconductivity, if the fractures tip effects are felt at the well bore

to early, the Bilinear Flow Regime dose not occur. The end of

the Bilinear Flow Regime depends on the fractureconductivity.

The Formation Linear Flow shows a one half slope straight

line. But for all values of (k f bf )D the behaviour of the Bilinearand the Formation Linear Flow is given by one curve.

The higher the fracture conductivity the longer the Linear

Flow Period1. Also in this regime for a dimensionless fracture

conductivity greater then 300 the curves are the same. The beginning of the Formation Linear Flow appears when thefracture tip effects are felt at the wellbore. That means that for

all the cases where the Bilinear Flow does not end in this

moment because of a fracture conductivity higher then 300,there is a Transition Zone between the Bilinear and the

Formation Linear Flow1,2

. This transition period has a

8/18/2019 SPE-52080-STU



characteristic shape. And if all pressure data fall on the

transition period of the curve, type curve matching is the only

analysis method available1.

The Pseudo Radial Flow shows a typical logarithmically

curve which rises for a constant value3. This result is well

known from all numerical and analytical solutions of radialflow behaviours. The influence of the HF is taken into

consideration with the changed skin as mentioned before andthe effective radius which is about half of the HF half length xf

for (k f bf ) D > 300. The skin can be expressed as a function of(k f b f ) D and xf /r w. For small values of dimensionless time the

different curves differ corresponding to the different values of

the dimensionless fracture conductive (k f bf ) D for an infinitesystem. For larger time values all curves become more and

more on line as shown in Fig. 8.

Interpretation of Data 1,2,3 . The analysis of the different

flow regimes offers the possibility to evaluate different parameters of the reservoir and the HF:

Bilinear Flow offers the possibility to calculate the half length

of the HF:

( )x

k b t

c k f

f f ebf

t

≥ ⋅ ⋅ ⋅

⋅ ⋅ ⋅

102

4α

φ μ ..........................................(13)

Bilinear Flow offers the possibility to calculate the storagecapacity of the HF:

( )

( )

( )[ ]( )[ ]

Cq B t

p

F p

F pw

M

M

wD M

wf M

= ⋅ ⋅ ⋅ ⋅ ⋅

⋅2 1

2

π α μ

Δ..........................(14)

Transient Period between bilinear and Formation Linear Flowoffers the possibility to calculate the fracture conductivity of

the HF:

( ) ( )k b xk b

xf f f

f f

f

= ⋅⎛

⎝ ⎜

⎞

⎠⎟ .................................................(15)

Pseudo Radial Flow offers the possibility to calculate the

reservoir permeability:

( ) ( )k

q B

h p p

M

wD M=

⋅ ⋅ ⋅⋅

α μ

Δ.............................................(16)

Pseudo Radial Flow offers the possibility to calculate the

effective wellbore radius:

( )

( )r

k t

c tw

M

t Dr Mw

'

'=

⋅

⋅ ⋅

α

φ μ .................................................(17)

Pseudo Radial Flow offers the possibility to calculate the Skin

factor:

Skinq B

c k h p

r

r w=

⋅ ⋅⋅ ⋅ ⋅

− ⎛

⎝ ⎜

⎞

⎠⎟

μΔ

ln ..........................................(18)

Single Porosity Finite Reservoir1,2,3,4

. In the first part the

flow regimes of an infinite fractured reservoir were described.This results will now be adapted to a fractured finite reservoir.

The area of the influx of the HF is about 4 times x f4. That

means as long as the radius of the whole reservoir is greater

than 4 times the half length of the HF the influx of the HFdoes not reach the borders of the reservoir so that it is possible

to call this an infinite reservoir2. The definition for a finite

reservoir is that the outer boundary radius of the reservoir issmaller than the influx zone of the HF. A finite fractured

reservoir is defined by its dimensions:

r

x

e

f

≤ 4 ...........................................................................(19)

But it is also possible to describe a finite fractured reservoir as

a reservoir of a finite conductivity. A low conductivity isdefined, as (k f bf ) is equal or smaller then 0.1. For such a

system it is possible to define not only the influence a of the

HF but also the effective wellbore radius depending on the

parameters of the HF. It can be observed that the r xw f

'/ is

directly proportional to the fracture conductivity, the effective

well bore radius ratio r xw f

' / varies linearly with the parameter

a=πkxf /2(k f bf ) in a log-log graph for large values of a, with a

slope of minus one.4

( )r

k b

k w

f f ' .= ⋅0 2807 .....................................................(20)

Referring to this equation it is also possible to calculate acritical fracture length beyond which there is no increase inwell productivity for practical purposes.

( ) ( )

xk b

k f

critical

f f = ⋅10 ..............................................(21)

Also in finite system a fracture Linear Flow is characterised

by a half slope straight line. After a transition period thesystem will exhibit a Bilinear Flow Period, indicated by a one

forth slope straight line. As time increase a Transition Zone

will occur, and like in all cases, the system reaches, while itstabilises, the Pseudo Radial Flow.

Fracture Linear Flow. As this behaviour occurs at very

small values of the dimensionless time it is also exhibited inthis case. It has the same behaviour with the same

mathematical description as described before.1

( ) p

k btwD

f f D

fD Dxf = ⋅ ⋅

2π η .....................................(22)

8/18/2019 SPE-52080-STU



This equation has a straight line with a slope of one half for

the dimensionless pressure versus the dimensionless time in a

log-log plot. The end of this flow regime occurs with the

following equation:

( )t

k bD

f f D

≤ −⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

−

45525

4

.. .........................................................(23)

Bilinear Flow Period. The Bilinear Flow which does not

always occur in the infinite system occurs for the finite systemdue to low conductivity. This is the main difference between

the two systems, however the explanation and the equation are

the same, so that on one hand there is a linear. incompressible

flow inside the HF and on the other hand there is a linear but

compressible flow inside the formation. The Bilinear Flowexists as long as most of the fluid entering the well bore comes

from the formation near by the well.1,2,4

( ) p

k btwD

f f D

Dxf = ⋅

2 454

..............................................(24)

Due to the previous formula of the Bilinear Flow the result is astraight line with a slope of one fourth. The duration of this

period depends on the storage capacity and the conductivity of

the HF. The end of this flow regime is given by equation 25:

( )tD

k f b

f D= ⋅0 0023

2. ......................................................(25)

Transition Zone. Between the Bilinear Flow and the

finally Pseudo Radial Flow there is a Transition Zone which isnot described by a separate equation, but it is important to

notice that this Transition Zone takes about five log cycles.3

The end of this Transition Zone is shown below:

tDr w' = 2000 ................................................................(26)

Pseudo Radial Flow. The Pseudo Radial Flow occurs afterthe end of Transition Zone. The equation for the Pseudo

Radial Flow will be basically the same as in the infinite

system.3

( ) p tk t

cx

wD D

f

= ⋅ ⋅ ⋅

⋅ ⋅ ⋅⎛ ⎝ ⎜

⎞ ⎠⎟

+

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

1

2

0 000264

2

0807092

ln.

.

Φ μ

....................(27)

It is the Skin Factor which offers the possibility to transfer the

fractured well into a ideal round well with a larger radius thanthe original well had. So the effective Wellbore radius can be

transferred into the Skin

Interpretations 1,2,3,4 . The finite system’s main difference

to the infinite system is that the Bilinear Flow occurs always,

and that there is no separated Matrix Linear Flow but a long

term Transition Zone between the Bilinear and the PseudoRadial Flow as shown in Fig. 9.

The Fracture Linear Flow shows also in the log-log graphthe typical half slope straight line which makes it possible to

diagnose it.

The Bilinear Flow Period shows a one fourth slopestraight line which is characteristic for this flow regime.

General well bore storage effects cause deviations from the

one fourth slope characteristic. The influx of the fractureconductivity occurs for about three logarithmic cycles.

Basically the conductivity lies between two limiting cases,

(k f bf )D =0.1 and (k f bf )D =300. The curves for the different

values merge from the curve representing the case of

(k f bf )=0.1.Depending on the fracture conductivity the flow is taking a

different way into the HF. The lower the fracture conductivityis the nearer at the well bore the flow into the HF is taking

place. For example for a (k f bf )D=0.1 99% of the inflow entersthe HF within the first third of the HF.

Corresponding to this fact the graph becomes as mentioned before a curve either convex or concave depending on the

fracture conductivity, which depends also on the storage

effects. For a dimensionless fracture conductivity higher then1.6 it is concave and for a dimensionless fracture conductivity

lower or equal 1.6 it is convex, so that for the finite

conductivity system only a convex curve exists.4

The Transition Zone between the Bilinear and the PseudoRadial Flow Regime is not described in a separate equation.

The curve itself is more influenced by the pseudo radial

behaviour than by the linear behaviour of the Bilinear Flow. In

general it occurs for about five dimensionless time log cycles.Therefor it is possible to interpret this flow regime.

The Pseudo Steady Flow shows also in this case it’stypical logarithmically curve which goes for a constant value.

The whole transient pressure analysis is given by a single

curve.3

q c k h p

Br

r Skine

w

0 = ⋅ ⋅ ⋅

⋅ ⎛

⎝ ⎜

⎞

⎠⎟ +

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Δ

μ ln

..........................................(28)

Interpretation of Data 1,2,3,4 . The analysis of the different

flow regimes offers the possibility to evaluate different parameters of the reservoir and the HF:

8/18/2019 SPE-52080-STU



Bilinear Flow offers the possibility to calculate the effective

well bore radius:

( )

( )r

k t

c tw

M

t Dr w

' = ⋅

⋅ ⋅

β

φ μ ...................................................(29)

Transient Period between Bilinear and Formation Pseudo

Radial Flow Behaviour: offers the possibility to calculate the

fracture conductivity of the HF:

( )k bk r

f f w=

⋅ '

.0 2807 ..........................................................(30)

Pseudo Radial Flow offers the possibility to calculate the skin

factor:

Skinq B

c k h p

r

r

e

w

= ⋅ ⋅⋅ ⋅ ⋅

− ⎛

⎝ ⎜

⎞

⎠⎟

μΔ

ln .........................................(31)

If data exhibits only the bilinear flow behaviour, the analysis

can be achieved by the fact that the slope is inversely

proportional to k b f f . And for the Pseudo Radial Flow period the slope is inversely proportional to kh of the

formation.4

Double Porosity Reservoirs 5,6,7

In recent years the interest for the evaluation of fractured wells

producing in double porosity reservoirs has been growing. The problem was first solved by assuming a uniform flux from the

reservoir into the HF, then assuming that the fracture was of

infinite conductivity. Normally it is unusual to fracture a

reservoir of double porosity but it is done in the meantime.

The double porosity model is a limit of the double permeability model which occurs in layered reservoirs, and so

the interpretation of the pressure tests of double porosity

reservoirs has a wide range of use. 5,6 A double porosity reservoir is defined as a reservoir with a

matrix of a constant permeability and natural fractures in this

matrix of a different permeability and the HF. Basically also

in a double porosity reservoir the two different systems of aninfinite and a finite reservoir exist. In general two different

concepts for the description of the different flow regimes

exist. The first one which is based on the pseudo steady stateflow model of Warren and Root and the second one which is

based on the transient matrix flow model of de Swaan and

Kaseml. In this paper only the second flow model will be

discussed.5

The system of double porosity reservoirs is described bythe following initial and boundary conditions:

5

The Reservoir: An isotropic, naturally fractured,horizontal, infinite or finite, slab reservoir is bounded byan upper and lower impermeable boundary. The reservoir

has a uniform thickness h and a permeability k and a

porosity φ for the matrix as well as for the naturally

fractured network. All these parameters are independent

of the pressure.

The Fluid: The reservoir contains a slightly compressible

fluid of a compressibility c and viscosity μ and both

properties are constant all over the reservoir and time. The Fractures: The fluid is produced through the naturally

fractured network and finally through a vertically

fractured well intersected by a fully penetrating, finite

conductive fracture of half length of xf width wf , permeability k f , porosity φf , interporosity flow

parameter hf , the storage capacity C and the relation of

the storage capacities ω.

ω =+C

C C

f

f m

.............................................................(32)

All these parameters are constant along the whole HF and overtime. The inflow to the well happens only through the HF.

A system which is described by these parameters is shownin Fig. 10. In addition gravity forces will be neglected, all

parameters of the reservoir and the HF are independent from pressure and pressure gradients are small in general. Under all

these conditions, the phenomenon is described by the

diffusivity equation in two dimensions. Due to the fact that the

width of the HF is very small compared with the reservoir a

Linear Flow inside the HF can be assumed. In this section thedimensionless pressure pwf will be represented as a function

of the dimensionless time tD, the dimensionless fracture

hydraulic diffusivity ηfD, the dimensionless fracture storage

capacity CfDf , the interporosity flow λ, the fracture

conductivity (k f bf ) and the Skin effect. 5,6

Infinite Double Porosity Reservoir 5,6,7. As mentioned before

the reservoir is represented by a fracture network and by

matrix blocks. It is assumed that the characteristics of both,the fracture network and the matrix blocks, do not vary

through the reservoir. The flow from the reservoir to the HF

occurs through the fractured network and finally through theHF only.

There are three basic flow regimes for a double porosity

reservoir existing, which can be separated by their timecomponent and the different influx of the dimensionless

fracture conductivity and storage capacity. Strictly speaking

there is no analytical solution for the finite conductivity

problem so only assumptions will be shown.

( )q k

p pmm f

* = ⋅

⋅ −γ

μ..................................................(33)

q* is the interporosity flow rate from the matrix to the

naturally fractured network, γ is the interporosity flow shape

factor which is inversely proportional to the matrix block size.For the discussion of the infinite double porosity system it

is necessary to define the interporosity flow factor λ whichshows the potential of the matrix to react to certain pressure

changes:

8/18/2019 SPE-52080-STU



λ = ⋅

⋅ ⋅

152

2k

r k xm

m f

f ............................................................(34)

Fractured Storage Dominated Flow Period. When a wellis started to produce, the fluid will be produced from the HF

and the natural fractured network, creating a pressure

difference between the natural fractures and the matrix.

Because of this pressure difference the flow is feeded by the

matrix .5,6 Thus this flow regime is influenced mainly by the storage

capacity of the HF itself, from the standpoint of the time cycle

it can be compared to the fracture Linear Flow from the single porosity fractured reservoir. This flow regime occurs for small

values of time. The flow is produced in the first moment due

to the expansion of the HF and then due to the expansion of

the fracture network; that means that the contribution of thematrix is negligible.

From the moment when the influence of the natural

fractured network occurs there is a second behaviour.5

Early Time:

( ) p

k b

t

wDf f D

Dxf =⎛ ⎝ ⎜ ⎞

⎠⎟ ⋅ ⋅

⋅π

ωΓ 54

2

4 .............................(35)

Intermediate Time:

( ) p

t

k bwD

Dx

f f D

f = +⋅ω

π3

.........................................(36)

These two equations will produce a one forth slope straight

line in the beginning and then a one half slope straight line ina double logarithmic plot of the dimensionless pressure over

the dimensionless time corresponding to the roots of the

dimensionless time.

Transient Flow Period . After the first influence of thefractured network there is a transient Flow especially for very

small CfDf if the flow in the matrix is essentially linear and the

well production is caused by the expansion of the matrix.

Under these conditions three different flows occur:5

Bilinear Flow:

( ) pwD

k f bf DAfD meD

tDxf

=⎛ ⎝ ⎜

⎞ ⎠⎟

⋅ ⋅ ⋅ ⋅⋅

π

ηΓ9

82 4 8

8 .....(37)

Pseudolinear Flow:

( )

pt

Ak b

wDDf

fD meD f f D

= ⋅

⎛ ⎝ ⎜

⎞ ⎠⎟ ⋅ ⋅ ⋅

+⋅

π

η

π4

5

4 23

Γ

...................(38)

Pseudo Steady State Matrix Flow Model:

( )

( )( )

p

k b

k bk b

w D

t

f f D

f f D

t

f f D

=

⋅ ⋅⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⋅ ⋅⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

π λ

λ

co th 2

2

2

1

4

2

2

1

4

.........................................(39)

For this last stage the well bore pressure pwD is independent oftime. This equation represents the limiting value of pressureduring the transition between the Fracture Storage Dominated

Floe Period and Total System Dominated Flow Period5.

These equations will produce a one eighth slope straight line

in the beginning, then a one half slope straight line and then a

constant corresponding to the roots of the dimensionless time.

Total System Compressibility Dominated Flow Period.

For large values of time the whole system dominates the flow behaviour.

Bilinear Flow:

( )

p

k b

twD

f f D

Dxf =

⎛ ⎝ ⎜ ⎞ ⎠⎟ ⋅ ⋅

⋅π

Γ5

42

4 ............................ (40)

Pseudolinear Flow:

( ) p t

k bwD Df

f f D

= +⋅

π3

...........................................(41)

These two equations will produce a one forth slope straightline in the beginning and then a one half slope straight line

corresponding to the roots of the dimensionless time.

Note that these equations are applicable for both thetransient and the pseudo steady state matrix flow model.

5

Interpretations. A double porosity fractured reservoir can

exhibit up to 15 different flow regimes from which the 7 mostcharacteristic were discussed in the previous chapter. From the

analysis of the different flow regimes and the mathematicaldescription of the behaviour it is possible to draw several

conclusions:

In general the system will offer a single curve which

depends on the different factors as the storage capacity, the

conductivity, the geometry and λ as shown in Fig. 11.

The Fracture Storage Dominated Flow Period showscorresponding to the roots of the dimensionless time a one

forth slope straight line in the beginning and a one half slopestraight line in the later time period. It should be noticed that

for the early time the result differs depending on the value of

CfDf . For CfDf equals one it has the same result like in thesingle porosity fractured reservoir during the Bilinear FlowRegime, equation 9.

The result is a single curve for the bilinear and the

pseudolinear flow. It is obvious that the curves for different

8/18/2019 SPE-52080-STU



values of CfDf are displaced towards smaller values of the

dimensionless time.

A fractured well in a double porosity reservoir behaves at

early time like in a homogeneous reservoir with the fracturednetwork properties. As a result the pressure behaviour might

exhibit the bilinear and the Pseudolinear Flow Period as

shown in Fig. 12. But for a large storage capacity the typical behaviour of the fractured well is hidden by the unit slope.

5,6

It should also be mentioned that the Transition Zone between the bilinear and the Pseudolinear Flow last about one

logarithmic cycle.5

The Transient Flow Period shows corresponding to the

roots of the dimensionless time a one eighth slope straight linein the beginning , a one forth slope straight line in the middle

and a constant because of the independence of the

dimensionless time in the late time period. This last equation

represents the limiting value of pressure during the transition

between the fracture network dominated and the Total SystemDominated Flow Period.

5

In the case of the transient flow the curves are primary a

function of the parameter β.

β λω

=⋅3

...........................................................................(42)

The beginning of the transient flow period is a function of λ

only. The higher λ is the higher the ability of the matrix tofeed the natural fractures. Corresponding to this the Transition

Zone will occur earlier. The tip of the curve will be influenced

by the value of ω. A type curve can be generated for a given

λ and for ω equals zero. For ω equals zero the flow shows an

asymptotic behaviour for any value of the interporosity flow

parameter ηmeD as shown in Fig. 13. 6

The end of the type curve depends only on β. This

approximation is strictly valid when infinite acting radial flowhas been reached before the start of the Transition Zone.

6

Total System Compressibility Dominated Flow Periodshows corresponding to the roots of the dimensionless time a

one forth slope straight line in the beginning and then a one

half slope straight line. This behaviour results from the fact

that there is no Pseudo Radial Flow into the well but a LinearFlow from the matrix into the naturally fractured network. So

for this system there is the possibility to use the idea of the

Pseudo Linear Flow in this stage.

Interpretation of Data 5,6 . The analysis of the different

curves offers also the possibility to calculate some reservoir

and HF parameters. For this system a new classification ofregimes and stages will be made to achieve this goal. In this

chapter three different systems will be discussed.

Bilinear Flow Analysis focuses on the early time of the

Fracture Storage Dominated Flow (equation 35) and the

Bilinear Flow of the Transient Flow Period (equation 37).

That means it takes one forth and one eighth slope straight

lines in consideration and offers the possibility to calculate the

conductivity of the HF:

( )( )

Cm

mfDf

bf

bf

= ⎡

⎣⎢⎢

⎤

⎦⎥⎥

2

1

4

..........................................................(43)

It is important to realise that the result is more valid if the data

fall on the Bilinear Flow Region.5

Trilinear Flow Analysis focuses on the Bilinear Flow Region

when most of the expansion is provided by the matrix underLinear Flow conditions. That is, in this type of flow the

moment when the Matrix Linear Flow is superimposed to the

Bilinear Flow in the HF. That means it takes the one eighthslope straight lines (equation 37) in consideration and offers

the possibility to calculate the area and permeability of the HF

but the conductivity and the formation bulk permeability must

be known:

A k m

mfb mc

tlf bf

bf tlf

⋅ = ⋅

⋅

δ

δ0

0

..............................................(44)

Transient Flow Period Analysis focuses on the Transition

Zone and offers the possibility to calculate the permeability

and the HF half length in combination:

( )k x

V c

q B

h mf

t f

⋅ = ⋅⋅ ⋅

⋅ ⋅

⋅⎛ ⎝ ⎜

⎞ ⎠⎟

22

1652. μ

φ..............................(45)

Pseudolinear Flow Analysis focuses on the Total System

Compressibility Dominated Flow Period. That means it takesthe one a half slope straight lines (equation 41) in

consideration and offers the possibility to calculate the HF

storage capacity which can be estimated from the slope:

( )

( )

Cm

mfDf

bf

bf

= ⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

2

1

2

.........................................................(46)

And it offers also the possibility to calculate the Skin:

Skinx

r

f

w

= −⋅

⎛

⎝ ⎜

⎞

⎠⎟ln

2........................................................(47)

It should be pointed out that this kind of analyses applies to

both matrix flow models.

Finite Double porosity fractured Reservoir 1,5,7. The

definition of a finite reservoir is the same as before:

r

x

e

f

≤ 4 ...........................................................................(48)

The finite double porosity fractured reservoir has the same parameters as the infinite system except of its dimensionsAlso the influencing components are the same like in the

infinite system: fracture conductivity k f bf , storage capacity

CfDf and the diffusivity ηfD.

8/18/2019 SPE-52080-STU



The finite system has also, correlating to the time and to the

different flow behaviours in the matrix and the natural

fractured network, three different flow regimes:

Fractured Storage Dominated Flow Period. As in the

infinite system this flow regime occurs in the very beginning

of the production. This flow results from the expansion of thefractured network. At this early stage the contribution of the

matrix to the flow is negligible.

( ) p

k btwD

f f fD Dxf

= ⋅ ⋅ ⋅2

η π ......................................(49)

A log-log dimensionless pressure versus dimensionless time

yields a straight line with a slope of one half corresponding to

the root of the dimensionless time.The end of this flow regime which occurs only for very

short time can be calculated by the same equation as the end

time of the Fracture Linear Flow1:

( )tDx

f

k f b

f D

fD

=⋅0 01

2

2

.

η...............................................(50)

Bilinear Flow Period. The Bilinear Flow Period can be

compared with the Bilinear Flow Period from the single

porosity Reservoir. The difference between these two flows

occurs due to the structure of the natural fracture. Thisdifference is taken into consideration by a factor which is

called the dimensionless reservoir conductivity R CD.

( ) p

k b R

t wD

f f D

CD

Dx f =

⋅⋅

2 454

....................................(51)

This equation produces one forth slope straight line

corresponding to the forth root of the dimensionless time. In a

plot of the dimensionless pressure drop versus the fourth rootof the dimensionless time the slope depends on the fracture

capacity and the formation permeability.

The end of this flow regime is given by equation 52 butadditionally a Transition Zone will occur for about two

logarithmic cycles. 7

( )t

k bD

f f D

= −⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

−

4 552 5

4

.. ...............................................(52)

Total System Compressibility Dominated Flow Period.

For the finite system the flow behaviour will also stabilise

after a certain time, and the flow conditions of the transferfrom the matrix to the fracture is pseudolinear. 7

( ) p

k b R twD

fD mD

f f mD CD fD

Dxf =

⋅

⋅ + ⋅ ⋅ ⋅

3

3 2

η η

η η....................(53)

This equation produces a straight line in a log-log graph.

Interpretation 5,7 . Storage Dominated Flow Period shows

corresponding to the root of the dimensionless time a one half

slope straight line. The flow is characterised by an early time

linear half slope straight line. This is valid only when the well bore storage is negligible

7 and the early time is not obscured

by an unit slope. For large storage the typical early behaviour

of the fractured well is hidden by a unit slope of one half andnot one forth. The reason for this is that a finite system is

much more reactive so that the difference between theexpansion of the Hf and the natural fractured network is not

that clear. It should be noticed that for the whole time the

result differs depending on the value of ω as shown in Fig. 14.

This very early behaviour is so short that it will be hidden in

most of the cases by the Transient or Bilinear Flow Regime.5,7

The Bilinear Flow Regime dominates the early time flow

behaviour so that the Storage Dominated Flow Period will be

hidden totally. Corresponding to the single porosity fracturedsystem the Bilinear Flow occurs in the finite system more

frequently than in the infinite system. The Bilinear Flow

depends on the well bore storage as shown in Fig. 14, it also

shows very clearly that the Bilinear Flow occurs more clearly

in a finite conductivity fracture.

It should also be mentioned that the Transition Zone between the bilinear and the pseudoradial flow last up to three

logarithmic cycles.

The Total System Compressibility Dominated FlowPeriod shows a straight line because of the fact that there is no

Pseudo Radial Flow as mentioned before.

Interpretation of Data. The analysis of the different

curves offers also the possibility to calculate several reservoir

and HF parameters:

The Bilinear Flow Period offers the possibility to calculate the

fracture conductivity in a plot of the dimensionless pressure

drop versus the forth root of time:

( )

k bq

m h R c k

f f

bf cD t

= ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

4411

21

4

2

. α μ

φ μ

.........................(54)

and the fracture half length:

xc

t

t R f

t

M

MD cD

= ⋅ ⋅⎡

⎣⎢⎢

⎤

⎦⎥⎥

0 0002637 12

1

2. κμφ

Δ.................................(55)

Simulation of Transient Flow in Fractured Reservoirs2,8,9,10

The simulation of a HF is a very important part of the holesimulation of a reservoir which contains a HF. Basically twodifferent approaches exist for the simulation of the Transient

Behaviour in a fractured reservoir. The first system describes a

naturally homogeneous artificially fractured reservoir asshown in Fig. 15, the second describes a double porosity

8/18/2019 SPE-52080-STU



fractured reservoir as shown in Fig. 16. In the first approach

the reservoir has a single porosity with a HF connected to the

well. In the second approach the reservoir is simulated as a

reservoir build out of blocks of different permeability asshown in Fig. 16.

10

The Simulation Technique. The simulation system is

attempted to elucidate the nature of the flow in the naturalrock. The distinguished feature of such a reservoir is that it

consists of matrix possessing high storage capacity but low permeability and fractures of a low storage capacity and a high

permeability.8,9,10

There are several aspects which influence the simulation ofa reservoir and which have to be taken into account when the

simulation of the reservoir is set up.

The Grid. The reservoir will be divided in several blocks. The

size of the different blocks depends on the chosen grid.Basically it is important to realise, that the well is represented

by single blocks, which have exactly the radius of the well.The whole reservoir is simulated by single blocks which

represent the different qualities of the rock in the particularspace and so the distribution for each parameter all over the

reservoir. Corresponding to this, the blocks have to follow the boundaries of the HF as shown in Fig. 15. This effect

influences the form of the grid. For a double porosity system

the grid is obviously much more complicated than for single porosity reservoir were only the HF as a cut through the

reservoir has to be simulated.8,9

For the simulation of a single porosity reservoir the basic

grid around the well should be chosen as shown in Fig. 15.

The HF should be divided into maximum 20 blocks per halflength. Any finer grid for the HF will not increase the quality,

but the runtime of the simulation. It is also possible to define

the matrix on both sides of the HF as an area of smaller grid. 2

For the simulation of a double porosity reservoir it isdifficult to give a general rule. The grid should be chosen

corresponding to the influence of the fractured network so that

the model has a smaller grid parallel to the fracture networkwhere the main flow occurs.

The Flow. It is assumed that only the HF produces into thewell. The natural fractures are the only path of fluid flow from

one grid block to the next towards the HF except from the

feeding process, which occurs from the matrix into the

fracture8,9

. The driving force for this flow behaviour is thePressure difference between the matrix and the fracture.

Basically there are two different systems to describe this flow.

On one side there is the pseudosteady state interporosity flow

model and on the other side there is the transient interporosityflow model. In this paper only the second system will be

discussed10

. The interporosity flow rate is proportional to the

pressure gradient on the surface of the matrix blocks, equation

56.

q A k

V

p

xmaf

m

m

m

sf

= ⋅

⋅ ⋅

⎛

⎝ ⎜

⎞

⎠⎟

μ

∂

∂..............................................(56)

Basically the flow depends on the Transmissibility.

( )Tk k

Bx y zrel~ , ,

⋅ ⋅⋅

φμ

...................................................(57)

The flow from one matrix block to the next will be

simulated as usual influenced by the matrix-matrix

transmissibility. This flow feeds, as mentioned before, thefracture.

The flow from the matrix into the fracture depends on the

block-pair transmissibility which is suitable to linear flow. If

the fracture is of a finite conductivity, its blocks will be treated

as ordinary blocks for the calculation of the transmissibility,

but if it is a fracture of an infinite conductivity, the matrixfracture flow will be governed by the half block

transmissibility of the matrix blocks. This is true as in an

infinite conductiv fracture the transmissibility of the fracturewould be practically infinite.

The flow through the fracture is similar to the flow through

the matrix. The flow is governed by the frac-fractransmissibility, which is calculated by using a linear flow

inside the fracture and the fracture permeability. In some cases

of very high permeability as in the case of an infiniteconductivity fracture it is possible to neglect any pressure drop

across the fracture.

The flow from the HF into the well bore is governed by the

fracture half-block transmissibility with the well bore block.

This half-block transmissibility is calculated using the fracture permeability and dimensions similar to the flow from the

matrix into the fracture, or it is set large enough to neglect

pressure drop from the HF into the well bore.In general the HF can be considered infinite, if the

dimensionless fracture conductivity (k f bf )D greater then 300.

These rules apply for the case of a HF in a Single PorositySystem. It is important to realise that the different flows are

taken into consideration, when the grid for the simulation is

chosen.

The Timesteps. There is no fixed rule for the timesteps, but it

is useful to have about 10 steps per logarithmic cycle2. The

choice of the timestep is always a question of the runtime, the

whole simulation should take, and how exact the simulation

should be.2

The main difference between the simulation of a single

porosity and a double porosity reservoir is the girding as it isobvious from the Fig. 15 and 16. The Simulation of a double

porosity fractured reservoir will always be more complicated

and will always take more time than the simulation of single porosity fractured reservoir.

The Data Base. For a good simulation a data base is very

important and influences directly the result and the quality ofthe simulation. The data base is given by some pre fracturing

8/18/2019 SPE-52080-STU



and some post fracturing pressure tests. But also the whole

production history of the well is important. But specially the

grid refinement in sensitive areas has a major impact on the

simulation quality and the run time.2,10

For the simulationitself it is useful to follow certain standard parameters

2,8:

Fracture Porosity:

φ f FracVolume

TotalVolume= ........................................................(58)

Matrix Porosity:

φ f Matrix Po reVo lume

TotalVolume= .............................................(59)

The HF permeability is set equivalent to the matrix

permeability by applying the following equation:

k k

fee

f

=φ

........................................................................(60)

k e is the tested effective permeability.

The shape of the HF has not such a strong influence on

the result10

The numerical system which is used to solve the equation

system of in and out flow of all cells depends on theSimulator used, but basically it is possible to add a lot ofadditional parameters like the skin factor or wellbore storage

effects.

Interpretation. The quantitative and qualitative analysis of

the test results based on the type curve matching were shown.

Basically the interpretation can be done using type curvematching but also numerical. In the numerical simulation the

analysis is especially used for the post fracture performance.The analysis is an iterative item where the start values for

certain parameters are taken from the type curve matching.

The main parameters are:

The HF half length: xf

The HF conductivity: ηDf The HF storage capacity: Cfdf

With this data the first simulation is made the parameters will

be changed for the following simulations until an optimum fitwith the pressure test data is reached.

After this procedure the exact behaviour and geometry of

the HF is defined. Based on this simulation of the HF it is possible to simulate a long term behaviour of the reservoir or

to simulate the behaviour of the reservoir with some more HF.

In general the change of the parameters is smaller than 1%.

That shows that the numerical analysis has its advantage more

in a complete reservoir simulation than in a parametercalculation.

Conclusions

1. A single porosity infinite system has four different flowregimes. The first is the Fracture Linear Flow, then comes the

Bilinear Flow, afterwards the Formation Linear Flow and

finally the Pseudo Radial Flow.2. A single porosity finite system has four different flow

regimes. The first is the Fracture Linear Flow, then comes the

Bilinear Flow and then after a Transition Zone also the finite

system ends up in the Pseudo Radial Flow. The TransitionZone between the Bilinear and the Pseudo Radial Flow occurs

for about five dimensionless logarithmic cycles.3. A Double porosity infinite system has three main flow

regimes. The first is the Fracture Storage Dominated Flow,

then comes the Transient Flow and in the end the TotalSystem Compressibility Dominated Flow occurs. The

Transition Zone between the Bilinear and the PseudolinearFlow takes about one dimensionless logarithmic cycle.

4. A double porosity finite system has three main flowregimes as the infinite system. The first is the Fracture Storage

Dominated Flow, then comes the Bilinear Flow and in the end

the Total System Compressibility Dominated Flow occurs.5. It is important to realise that the Bilinear Flow occurs

more clearly in the finite systems.

6. There is no Pseudo Radial Flow in a double porosity

reservoir but a Pseudo Linear Flow from the matrix into the

natural fractured network.7. In general the well bore storage and the fracture

conductivity have a major impact on the results. Curves varydepending on the different values of these parameters and

some flow regimes can be hidden by these variationscompletely.

8. For the simulation of fractured reservoirs two different principles exist. For a single porosity fractured Reservoir only

the HF as a volume of a high permeability has to be

implemented in the model. For a double porosity fractured theHF and the natural fractured network have to be implemented

by the Dual Porosity Model.

9. A Hf should be splitted up into 20 blocks per half

length.

10. Fore the simulation 10 time steps per logarithmic cyclesshould be simulated

8/18/2019 SPE-52080-STU



Nomenclature: A = Relative capacity parameter[ ]

B = Formation volume factor [bbl/STB]

b = Width

c = Compressibility [1/psi]

C = Storage capacity [cuft/psi]

F = Correlating parameters for the wellbore storage

h = Formation thickens [ft]

k =Permeability [dc]k f b f . = Fracture conductivity

M = A point

m = Slope of log-log straight line

p = Pressure [psi]

q = Flow rate [STB/D]

q* = Interporosity flow rate

RCD = Dimensionless Reservoir Conductivity

r = Radius

r´ = Effective Radius

S = Skin factor

T = Transmissibility

t = time [h]

w = width of the HF [ft]

x f = Fracture half lengthα = Unit conversion factors

β,λ = Interporosity Flow parameters

γ = Interporosity Flow Shape Factor

η = Hydraulic diffusely [md-psi/cp]

μ = Viscosity [cp]

φ = Porosity

ω = Ratio of storativities

Subscripts: D = Dimensionless

e = effective

f = fractured

fbf = Formation Bilinear Flow

lf = Linear Flowi = initial

t =total

w = Wellbore

xf =Based on xf

Special Functions:

( )erf x e x

= ⋅ −∫2 2

0π δμ μ

60 ( )− − =−∞

∫ E xe

i

μ

μ δμ

0

Γ = Gammafunction

Acknowledgement::The author thanks the Institute for Reservoir Engineering onthe Mining University of Leoben and especial MSc. Ahmed

Hasan, for all kinds of help they gave to him.

References:1. Cinco-Ley H. :”Transient Pressure Analysis for Fractured

Wells”, SPE 7490 (Sep. 1981) 1749-1766.

2. Cinco-Ley H., F. Samaniego V., N. Dominiguez A.:”Transient Pressure Behaviour for a Well With a Finite-

Conductivity Vertical Fracture” SPE 6014 (Aug 1981)

253-264.3. Gringarten A.C., Ramey JR. H.J., Raghavan R.

:”Unsteady-State Pressure Distribution Created by a WellWith a Single Infinite-Conductivity Vertical Fracture”

SPE 4051(Aug 1974) 347-371.4. Cinco-Ley H., Ramey Jr. H.J., Samaniego F., Rodriguez

F.: “Behaviour of Wells With Low-Conductivity Vertical

Fractures” SPE 16776 (Sep 1987) 363-3715. Cinco-Ley H. Meng, H.-Z.: “Pressure Transient Analysis

of Wells With finite Conductivity Vertical Fractures in

Double Porosity Reservoirs” SPE 18172 (Oct 1988) 645-

660.

6. Houzé O. P., Horne R.N., Ramey Jr. H.J.: “PressureTransient Response of an Infinite-Conductivity Vertical

Fracture in a Reservoir with Double-Porosity Behaviour”SPE 12778 (Sep 1988) 510-518.

7. Okoye C.U., Oshinuga A.D., Ghalambor A.: “PressureTransient Behaviour of Vertical Fractured Wells in a

Closed Squared Multilayered Reservoir” SPE 20136 (Mar1990) 417-425.

8. Gilman J.R:, Kazemi H.: “Improvements in Simulation of

Naturally Fractured Reservoirs” SPE 10511 (Feb 1982)293-316.

9. Nanba T.: ”Numerical Simulation of Pressure Transients

in Naturally Fractured Reservoirs With Unsteady-State

Matrix-to-Fracture Flow” SPE 22719 (Oct 1991) 593-

602.10. Haddad S.H., Sonrexa K.: “Simulation of Artificially

Fractured Wells With Single Well Models” SPE 21338

(Nov 1991) 137-146.

8/18/2019 SPE-52080-STU


52080

ppen ix

Flow Regime

Transient Pressure Behaviour

in ractured eservoirs

Infinite sinqel porosity fractured reservoir

Equation Begining

End

13

Fracture linear Flow:

Bilinear Flow Period:

Formation Linear Flow

Pseudo Radial Flow:

I

I I

I I

264·

k .

t

1

P o to

=

In ) 8 7 91

2 l

4>·IJ·C·

X i J

o

O.OI. (krb

)

I - D

- ,

lf

1725

I c

I =p (krbr) h ;c,k

tv

0.016

Finite sinqel porosity fractured reservoir

Fracture linear flow:

2

w = k

J;r· fl ·(D./

r r n

o

Bilinear Flow Period:

Transition Zone

Pseudo Radial Flow:

t

J r

=

Infinite double porosity fractured reservoir

Early Time:

o

Fractured Storage

Dominated Flow

Period:

Intermediet Time:

8/18/2019 SPE-52080-STU


14

Patrick Pattay

Infinite double porosity fractured reservoir

52080

Flow Regime

Equation

Bilinear Flow

Pseudolinear Flow

1r.f;

r

w

= r 3

b

. 2 ~ A f D · . r r

Pseudo Steady State

Matrix Flow

I

krb

r

: v 2 { ~ : r ) ~ r

Begining

1

End

0 01

Transient Flow Period:

Bilinear Flow:

Total system

Compressibility

dominated Flow

Period:

Pseudolinear Flow:

P o =

t 3. k

b

r r

0

Finite double porosity fractured reservoir

Fractured Storage

Dominated Flow

Period

Bilinear Flow

Period:

TotaI System

Compressibility

dominated Flow

Period:

o

O OI k

l

bl

=

h

2 log cycles later

8/18/2019 SPE-52080-STU


52080

Transient Pressure Behaviour in Fractured Reservoirs

ppendixB

\

Fracture

Well

/

1 C SE

P

Fig. 7: Storage effects on the Bil inear Flow

Fig.

5:

Pseudo Radial Flow

Fig. 6: Log-log graph

for

a fractured reservoir:

Case I : (ktb,)o

=

0.1; Case II: (ktb,)

0

=500

h

FR TURE

1IIIrr

······· r·····li···IJ

I I I I I I II I

I i

I

i I I

1

I

I

I

Fracture

IMPERME BLE

BOUNDARIES

Fig. 3: Bilinear Flow

Fig. 1: Finite conductivityHF in an infinite reservoir

Fig. 2 :Fracture Linear Flow

I

I

i

.

,

I

I

I

I

i ,

I

i

I

I I

I

i

} ~ = = : §

Well

g : = = : = = ~

/

1 1

•

p

tt

1

I

1

I

I

i

I

' I

II

..

Fig.

4:

Formation Linear Flow

Fig. 8: Type curve fo r a fractured reservoir

8/18/2019 SPE-52080-STU


16

Patrick W. Pattay

520

111-

2

10 -

6

10

5

10-<4

10

3

Ig -

2

IQ I IQ

Q

to.,

ktbt>t

Q

{ j3

'0 '

Q

10

Cl

J

0.

1

10

•

10·

1O

10

10

Fig. 9: Type curve of low conductivity HF

Fig. 13: Asymptotic behaviour during the Transient Flow

b

.

r

Fig. 10: Double porosity fractured reservoir

h

-:I:,

::1 :- ::0·7 ,

, :i·

:- ,

r:

o

•

T

.....

, :;· ......

_,r:r :r' . . .d...... I

...

OIM£NSIONLESS TINE.l

Oaf

f

- - - - - - /

.

I

I

I

I

I

I

I

I

I

I

I

I

I

7

I \

ij /

L \JL

.ti,.._ '

\ __ t; MO EL LO K

lDEAUZED .\ ATRIX

L

TO '

VIE

•

.

, .

.

.

,

-)

Fig. 15: Dual Porosity System

Fig 14: Double Porosity Fractured Finite Reservoir

10

£

[

1

1

0

'0

10

t

C

1 0-

1

i

10

-

0

10

-

10

I

10

0

10

10

lfil

2

1\1

6

I t S 1 1-<4

lfil

3

IB -

2

111-

1

le e Hal Hl

2

1 1

3

to., krbrlb

Fig. 12: Fracture dominated flow period

Fig. 11: Uniform Flux in a double porosity fractured reservoir

Fig. 16: Grid for the Simulation

of a HF

Documents

SPE-52080-STU