60b7d52d42d5a716d4

DOI 10.1007/s11242-005-1398-xTransport in Porous Media (2006) 64: 5171 Springer 2006

Effects of Fracture Boundary Conditions onMatrix-fracture Transfer Shape Factor

HASSAN HASSANZADEH and MEHRAN POOLADI-DARVISHDepartment of Chemical & Petroleum Engineering, University of Calgary, 2500 UniversityDrive NW, Calgary, AB, Canada T2N 1N4

(Received: 5 May 2005; in nal form: 26 July 2005)

Abstract. The matrix-fracture transfer shape factor is one of the important parameters inmodeling naturally fractured reservoirs. Four decades after Warren and Root (1963, SPEJ,245255.) introduced the double porosity concept and suggested a relation for it, thisparameter is still not completely understood. Even for a single-phase ow problem, inves-tigators report different shape factors. This study shows that for a single-phase ow in aparticular matrix block, the shape factor that Warren and Root dened is not unique anddepends on the pressure in the fracture and how it changes with time. We use the Laplacedomain analytical solutions of the diffusivity equation for different geometries and differ-ent boundary conditions to show that the shape factor depends on the fracture pressurechange with time. In particular, by imposing a constant fracture pressure as it is typicallydone, one obtains the shape factor that Lim and Aziz (1995, J. Petrolean Sci. Eng. 13,169.) calculated. However, other shape factors, similar to those reported in other studiesare obtained, when other boundary conditions are chosen. Although, the time variabilityof the boundary conditions can be accounted for by the Duhamels theorem, in practiceusing large time-steps in numerical simulations can potentially introduce large errors insimulation results. However, numerical simulation models make use of a stepwise approx-imation of this theorem. It is shown in this paper that this approximation could lead tolarge errors in matrix-fracture transfer rate if large time-steps are chosen.

Key words: shape factor, fractured reservoirs, matrix block, matrix-fracture boundary con-dition.

Nomenclaturea decline constant 1/[T ]A matrix surface area [L2]cm total matrix compressibility [LT 2/M]f time domain functionF laplace domain functionhm matrix block thickness [L]k porous media permeability [L2]I0 modied Bessel function of rst kind

Author for correspondence. e-mail: [email protected]

52 HASSAN HASSANZADEH AND MEHRAN POOLADI-DARVISH

I1 modied Bessel function of rst kindmatrix block thickness [L]for slab L2 =h2m

L for cylinder L2 =R2mfor sphere L3 =4/3R3m

pf pressure [M/LT 2]pm pressure [M/LT 2]pm average matrix pressure [M/LT

2]pD dimensionless matrix pressurePm matrix pressure in Laplace domainPm average matrix pressure in Laplace domainQm matrix-fracture exchange term [M/L3T ]r block radius [L]Rm outer radius of cylindrical or spherical matrix block [L]s laplace variable with respect to tDs laplace variable with respect to tt time [T ]tD dimensionless timeVm matrix bulk volume [L3]x distance from origin [L]

Greek letters dimensionless decline constant uid viscosity [M/LT ] porosity euler constant 0.5772156 uid density [M/L3] shape factor constant [1/L2]m matrix hydraulic diffusivity, L2/T

Subscriptsf fracturem matrixt total

1. Introduction

Based on Barenblatt et al. (1960) work on uid ow in fracturedmedia,Warrenand Root (1963) introduced the double porosity concept into petroleum engi-neering. In their model, a fractured medium comprises two overlapping media:matrix and fracture. Matrix has a low permeability and a high uid storage. Incontrast, fracture has a high permeability and a low storage.

In one form of double porosity models, matrix blocks act as a source orsink for a fracture system, where the uid transfer between a matrix blockand fracture system is proportional to the difference between fracture pres-sure and average matrix block pressure as given by the following equation:

Qm = km

(pm pf ), (1)

EFFECTS OF FRACTURE BOUNDARY CONDITIONS 53

where Qm is the rate of the uid transfer between the matrix and frac-ture, km the matrix permeability, the uid viscosity, the uid density,pm represents the average matrix block pressure, pf is the fracture pressure,and is called matrix-fracture transfer shape factor and has dimensions ofL2. The model based on this assumption is called a pseudo-steady state(PSS) transfer model. The matrix-fracture transfer term, Qm is related tothe matrix pressure by the following relationship:

Qm =mcm(

pm

t

), (2)

where m is the matrix porosity and cm is the matrix compressibility. Thetraditional double porosity model of Warren and Root uses Equation (1) tomodel the uid transfer between the matrix and the fracture in a naturallyfractured reservoir.

This type of model does not account for the pressure transient withinthe matrix. There is another model for uid exchange between the matrixand fracture, which accounts for the pressure transient in the matrix blockby solving the following equation

2pm = 1m

(pm

t

), (3)

where m =km/cm is the matrix hydraulic diffusivity. This type of modelis called a transient transfer model and does not use the matrix-frac-ture transfer shape factor. In the transient model, the uid transfer ratebetween the matrix and fracture is proportional to the pressure gradient atthe matrix block surface as given by Nanba (1991)

Qm = AkmVm

pm|matrix face , (4)

where A and Vm are the matrix block surface area and matrix block vol-ume, respectively. Petroleum engineering literature shows that the shapefactor remains a controversial topic. A large body of research in thearea of naturally fractured reservoirs simulation is devoted to representingan accurate matrix fracture exchange term (Kazemi et al., 1976; Thomaset al., 1983; Kazemi and Gilman, 1993; Lim and Aziz, 1995; Quintard andWhittaker, 1996; Noetinger and Estebenet, 1998; Bourbiaux et al., 1999;Coats, 1999; Noetinger et al., 2000; Penuela et al., 2002a,b; Sadra et al.,2002). In spite of extensive research on fractured reservoir modeling, nocritical improvements have been made during the last two decades. Even forthe single-phase ow problem, various investigators derive different shapefactors. The shape factor is usually derived from a simple mechanism ofpressure diffusion with constant fracture pressure as a boundary condition,


whereas the physical exchange mechanism in fractured reservoirs is morecomplex.

This paper investigates the boundary condition dependency of shapefactor, which is not reported in the earlier works. To investigate the effectof boundary conditions on the shape factor, the diffusivity equation issolved analytically in Laplace domain for different depletion regimes inthe fracture. Constant ux at a matrix-fracture interface, exponential, andlinear pressure depletion schemes are investigated. Results show that theshape factor depends on the fracture pressure and how it changes withtime.

The following sections present, rst, a brief review of previous researchworks on shape factor. The subsequent section describes the methodol-ogy used in this study. The third section presents solutions of the diffusiv-ity equation for different boundary conditions and geometries. The studysresults are then presented, leading to the discussion and conclusions.

2. Previous Works

In the following section, we describe briey the relevant literature dealingwith the single-phase shape factor. Barenblatt et al. (1960) introduced theclassic dual porosity concept in the early 1960s. Warren and Root (1963)applied this concept to reservoir engineering, principally for well testingapplications. They used a geometrical approach to derive the shape factorsfor one, two, and three sets of orthogonal fractures.

In another study, Kazemi et al. (1976) introduced the shape factor indouble porosity simulators. They obtained shape factors by discretizationof pressure equation for single-phase ow using the standard seven pointnite difference. Since then, this shape factor formulation has been usedin standard reservoir simulators. Thomas et al. (1983) presented anotherexpression for the shape factor that was validated by multiphase ownumerical simulations. Table I gives a summary of shape factors obtainedby these and other authors.

Based on the solution of the diffusivity equation with constant pressureat the matrix boundary, Coats (1989) derived the PSS shape factor con-stants of 12, 28.45, and 49.58 for one-, two-, and three-dimensional trans-fer cases (See Table I). Kazemi and Gilman (1993) used the rst term seriesapproximation of the analytical solution of the three-dimensional diffusiv-ity equation and derived the shape factor. They used a step change bound-ary condition in their derivation. Similarly, Lim and Aziz (1995) providedthe analytical shape factor constants for different matrix block geometries.They used the approximate solution of the single-phase diffusivity equa-tion with a step change boundary condition at the matrix-fracture inter-face. They validated their results with ne grid simulations as well as dual


Table I. Summary of the shape factor constants L2 found in literature

Investigator(s) Approach Fluid ow 1D 2D 3D Transient/PSS

Warren and Root Geometrical Single phase 12 32 60 PSS(1963)Kazemi et al. Numerical Single phase 4 8 12 PSS(1976)Thomas et al. Numerical Two phase 25 Transient(1983)Coats (1999) Analytical Single phase 12 28.45 49.58 PSSCoats (1999) Numerical Two phase 8 16 24 PSSKazemi and Gilman Analytical Single phase 29.61 Transient(1993)Lim and Aziz Analytical Single phase 9.87 19.74 29.61 Transient(1994)Quintard and Averaging Single phase 12 28.4 49.6Whitaker (1996)Bourbiaux et al. Numerical Single phase 20 PSS(1999)Noetinger et al. Random walk Single phase 11.5 27.1 (2000)Penuela et al. Numerical Two phase 9.87 Transient(2002)Sarda et al. (2002) Numerical Single phase 8 24 48 Transient

Pseudo-steady state transfer model.

porosity simulations. They suggested that the shape factor depends on thegeometry and physics of pressure diffusion in the matrix. Here, we willshow that the shape factor also depends on the way the pressure changesin the fracture.

Quintard and Whitaker (1996) used the volume-averaging technique toderive the shape factors for single-phase ow of slightly compressible uids.Their values are exactly the values obtained by Coats using the analyticalsolution of the diffusivity equation under the PSS assumption and constantpressure at the matrix boundaries. Bourbiaux et al. (1999) derived a shapefactor for two-dimensional matrix-fracture transfer based on a single-phasene grid simulation and PSS assumption. To evaluate the shape factor, theyperformed a ne grid simulation on a square matrix block under constantfracture pressure at the matrix boundaries. Using the results of the ne gridsimulation, the shape factor is back calculated. They derived a shape factor


that has a transient behavior and converges to a constant value of 20.1 atlate time.

Noetinger et al. (2000) presented single-phase ow shape factors calcu-lated based on the continuous time random walk technique (CTRW). Theycompared the CTRW derived shape factors with a numerical simulationand found a good agreement. Penuela et al. (2002 a, b) developed a time-dependent shape factor for slab-shape matrix block that converges to theLim and Aziz shape factor at late time based on the analytical solution ofthe diffusivity equation with constant fracture pressure as a boundary con-dition. They implemented the shape factor in a double porosity simulatorto perform the ow simulations.

Another effort to derive the single-phase shape factor has been pre-sented by Sarda et al. (2002). They used numerical simulations of a sin-gle-phase uid ow on discrete fracture models to derive the shape-factorconstant. In their model, shape factor was spatially dependent on the localproperties of the matrix blocks and the surrounding fractures.

While a few investigators have modeled the transient behavior formatrix-fracture shape factor (Bourbiaux et al., 1999; Penuela et al., 2002a, b), the derivation of the conventional shape factor for matrix-fracturetransfer does not account for the pressure transient in the matrix blocksand the pressure regime in the fracture. Furthermore, none of the matrix-fracture shape factor obtained in previous papers allows for a continuouspressure change in the facture. In this paper, we account for both pressuretransients in the matrix and pressure change in the fracture by using theLaplace domain analytical solution of diffusivity equation for the matrixblock, subject to various pressure regimes in the fracture. The followingsection describes the method we used to derive the single-phase shape fac-tor for matrix-fracture transfer.

3. Methodology

Warren and Root (1963) presented a relationship for dual porosity sys-tems that relates the matrix-fracture exchange term to uid mobility, poten-tial difference, and shape factor as given by Equation (1). To incorporatethe effects of the pressure transient in the matrix into the shape factor,the diffusivity equation for the matrix block is solved and then an averagevalue of the block pressure over the block volume can be introduced intoEquation (1). The matrix-fracture exchange term is related to the rate ofmass accumulation in the matrix block and can be shown by Equation (2).Combining Equations (1) and (2) leads to the denition of the single-phaseshape factor:

=(

pm

t

)1

m(pm pf ). (5)


Determination of pm in Equation (5) requires solution of the matrix pres-sure subject to the appropriate initial and boundary conditions. The gov-erning partial differential equation that describes the single-phase pressurediffusion in a matrix block is given by Equation (3). This equation can besolved by the Laplace transform method with an arbitrary boundary condi-tion. The Laplace domain solution for matrix pressure Pm(x, s ), can thenbe integrated to obtain the average matrix block pressure in the Laplacedomain Pm(s ), as given below:

Pm = 1Vm

Vm

Pm dVm, (6)

where s is the Laplace variable with respect to t . The parameter Pm isthe average matrix pressure in the Laplace domain and can be inverted tothe time domain by an appropriate Laplace inversion method. To obtainthe shape factor from Equation (5), one also needs the time derivative ofthe average matrix block pressure. Accordingly, from the Laplace transformtheorem

{pm

t

}= s Pm pm(0). (7)

Substituting all terms in Equation (5) gives the matrix-fracture transfershape factor.

=

1 {s Pm pm(0)}

m{

1

(Pm

)pf } . (8)Equation (8) suggests that calculation of shape factor can be performed bynding an expression for average matrix pressure in Laplace space, Pm(s ).In the following section, Equation (8) is used to obtain the shape factorfor different geometries, subject to different boundary conditions in thefracture.

4. Solutions

The solution of the diffusivity equation that leads to the derivation of theshape factor for various boundary conditions and different matrix blockgeometries is presented in this part. This is demonstrated here for a par-ticular case. The solutions for other cases are presented later.

Consider a slab shape matrix block of thickness hm, with initial pressurepi sandwiched by two parallel planes of fractures with pressure pf where

In this paper, capital letters are used for variables in Laplace domain, and the

sign is used to denote average values.


in general pf can be a function of time. The governing partial differentialequation and its associated initial and boundary conditions are given by

2pm

x2D= pm

tD, (9)

pm(xD, tD)=0, tD =0, 0xD 1,pm(xD, tD)=pf , tD >0, xD =1,pm(xD, tD)

xD=0, tD >0, xD =0,

where

p=p(xD, tD)pi.The pressure pi is the initial matrix block pressure. The parameters xD andtD are dimensionless length and time respectively and are dened as fol-lows:

xD =x/lc and tD =mt/ l2c , (10)where lc is the matrix block characteristic length. For a slab shaped matrixblock we consider half of the matrix-block thickness as the characteristiclength. The characteristic lengths of cylindrical and spherical matrix blocksare considered to be equal to the block radius.

To investigate the effect of fracture pressure on shape factor, pm isobtained for different boundary conditions using the Laplace transformmethod. In this study, we are considering step change, exponential, and lin-ear pressure depletion as well as constant ux boundary cases. In addition tothe slab geometry, the diffusivity equation is also solved for cylindrical andspherical geometry subject to the mentioned boundary condition. Functionsdescribing the pressure regime in the fracture are given in Table II.

Equation (9) with a constant boundary condition pf can be solved tond the pressure distribution in the matrix block. The Laplace base solu-tion is given by the following equation (Ozisik, 1980)

Pm = 1m

(hm

2

)2pf cosh(xD

s)

s cosh

s, (11)

where s is the Laplace variable with respect to tD. Equation (11) can beintegrated over the block volume to nd the average block pressure givenby:

Pm = 1m

(hm

2

)2pf tanh

s

s

s. (12)


Table II. Matrix pressure and its average in slab shape for different boundary conditions

Matrix and Pm Pmfracture BC

Constant 1m

(hm2

)2 [ pis

+ cosh(xD

s)

s cosh

s

]1

m

(hm2

)2 [ pis

+ (pfpi) tanh

s

s

s

]pressure pfLinear 1

m

(hm2

)2 [ pis

pis2

cosh(xD

s)

cosh(

s)

]1

m

(hm2

)2 [ pis

pis2

s tanh

s]

decline, pf =pi(1t)1/tExponential, 1

m

(hm2

)2 [ pis

pi(1s 1

s+) ( cosh(xDs)

cosh(

s)

)]1

m

(hm2

)2 [ pis

pis

s

(1 s

s+)

pf =piet tanh

s]

Constant 1m

(hm2

)2 [ pis

Qmh2m4kmcosh(xDD

s)

s

s sinh

s

]1

m

(hm2

)2 [ pis

(

h2m

4km

)Qms2

]ux, Qm

Before we substitute for Pm in Equation (8), we need to distinguishbetween the Laplace transform parameters when transform is taken withrespect to t and when taken with respect to tD called s and s, respectively(Sabet, 1991). By noting that the Laplace parameters s and s are relatedby s = (m/l2c ) s, one can write Equation (8) as

h2m =41{sPm}{

1{Pm}pf} . (13)

Equation (12) can be incorporated in Equation (13) to obtain

h2m =

1

{tanh

s

s

}{

mh2m

141{

tanh

s

s

s

}} . (14)

The product group h2m is dimensionless and will be called shape factor.For cylindrical and spherical matrix blocks, hm will be replaced by R. Thetransfer-shape factor for a slab matrix with step change in fracture pressureis obtained using Equation (14) and by inverting its Laplace functions intothe real-time domain using an appropriate Laplace inversion algorithm. Inthis work we have used the Stehfest algorithm (Stehfest, 1979).

Similarly, we have obtained the shape factor when the fracture pressureis a continuous function of time. The solutions are given in Table II. Expo-nential and linear functions have been considered. The shape factors forother matrix geometries are calculated using a similar approach. Tables IIIand IV show the results for cylindrical and spherical blocks, respectively(Hassanzadeh, 2002). The analogy between cylindrical (spherical) blocks,


Table III. Matrix pressure and its average in cylindrical shape for different boundaryconditions

Matrix and fractureBC

Pm Pm

Constant pressure pfR2mm

[pis

+ (pfpi)s

I0(r

s)

I1(Ro

s)

]R2mm

[pis

+ (pfpi)s

s

2I0(

s)

I1(

s)

]Linear decline, pf =pi(1t)1/t

R2mm

[pis

pis2

I0(rD

s)

I1(

s)

]R2mm

[pis

2pis2

s

I0(

s)

I1(

s)

]

Exponential,pf =piet

R2mm

[pis

pi(1s 1

s+)

I0(rD

s)

I1(

s)

]R2mm

[pis

pis

(1 s

s+)

2s

I0(

s)

I1(

s)

]

Constant ux, QmR2mm

[pis

(

R2mQm2km

)I0(rD

s)

s

sI1(

s)

]R2mm

[pis

(

R2m

2km

)Qms2

]

Table IV. Matrix pressure and its average in spherical matrix block for different boundaryconditions

Matrix and Pm Pmfracture BC

Constant R2m

m

[pis

+ (pfpi)s

1rD

sinh(rD

s)

sinh

s

]R2mm

[pis

+ (pfpi)s

3s

{coth

s 1

s

}]pressure pfLinear R

2m

m

[pis

pis2

s

1rD

sinh(rD

s)

sinh

s

]R2mm

[pis

pis2

3s

(coth

s 1

s

)]decline, pf =pi(1t)1/tExponential, R

2m

m

[pis

pis

(1s 1

s+)

1rD

sinh(rD

s)

sinh

s

]R2mm

[pis

3pis

s

(1 s

s+)

pf =piet(coth

s 1

s

)]

Constant R2m

m

[pis

R2mQm3km3 sinh(rD

s)

s(cosh

s 1

Rmsinh

s)]

R2mm

[pis

(

R2m

3km

)Qms2

]

ux, Qm

with matrix blocks intersected by two (three) sets of perpendicular fracturesis presented under discussions.

5. Results

In the following we present the values of shape factor for blocks of differ-ent geometries exposed to fractures with different depletion schemes. Theresults are presented in the form of a dimensionless group (L2) as a func-tion of dimensionless time. Following Lim and Aziz (1995), L is fracturespacing that results in the equivalent volume of a cylinder and or a sphere


of radius R for two and three sets of intersected fractures, respectively.Exact denition of L is given in the Nomenclature.

5.1. slab blocks

Figure 1 shows the shape factor for a slab block exposed to different deple-tion schemes in the fracture. The conventional boundary condition that isnormally used in literature is a step change for the fracture pressure. Oursolutions with this boundary condition are shown in Figure 1, and indicatea stabilized value of 9.87 at large dimensionless times. Lim and Aziz (1995)have obtained the same stabilized value.

Another boundary condition that is used is the exponential decline forfracture pressure with an exponent equals to a. We used different val-ues from 0.0001 to 1 for the exponent. The results in Figure 1 show thatfor high values of a, the shape factor starts with a transient period andthen converges to the shape factor value for the constant fracture pressureboundary condition. For very small values of a, the shape factor starts athigher values and converges to a constant value of 12. The above resultsshow that not only the transient values of shape factor depends on frac-ture pressure regime but also the late time values of shape factor depends

Figure 1. Shape factor constant for slab shape matrix block subject to differentboundary conditions.


on how pressure changes in the fracture. It is found that a range of shapefactors between 9.87 and 12 can be obtained by varying the exponent a.The exponential function with large exponent values and constant pressuredepletion regimes lead to a value of 9.87 while the exponential functionwith smaller exponents, linear, and constant ux depletion regimes give alate time value of 12. The dimensionless time for stabilization of the valueof shape factor depends on the depletion scheme in the fracture. It is notedthat for a linear decline, the shape-factor constant is not a function ofdepletion rate a.

The low and high stabilized values of shape factor are roughly 20%apart. The practical signicance of this is that the fracture-matrix uidexchange rates as calculated from Equation (14) and by the two shape-factor constants would be apart roughly by the same magnitude. This isfurther investigated in the following section.

5.2. cylindrical blocks

Similar to the shape factors obtained for a slab, we obtained two differ-ent values for shape factor both of which have been reported previously.Figure 2 shows the results for a cylindrical matrix block. For a constantfracture pressure boundary condition, the shape factor shows a transientperiod and then converges to a constant value of 18.2. Lim and Aziz(1995) reported a similar value. For an exponential pressure decline curve,results show that a smaller a gives a higher shape factor during transientperiod, such that for a very small value of a, the shape factor stabilizedat a value of 25.13, which is about 40% larger that the value of 18.2. Theresults for the linear pressure decline are not function of depletion rate andthe stabilized value is similar to the exponential decline with very smallvalues of a. Applying the constant ux boundary condition results in astabilized value of 25.13, similar to the linear decline boundary condition.Results reveal that the time to stabilization is different for different fractureboundary condition.

5.3. spherical blocks

Figure 3 gives the results for the spherical matrix block. For a constantpressure at fracture, the shape factor starts at large values and then declinesto a constant value of 25.65. Lim and Aziz (1995) reported a similar valuefor such a boundary condition. The shape factor calculated for the expo-nential decline with a large value of a is higher than the shape factor forthe constant pressure case at early time. However, they converge to thesame value at late time. As a decreases, the shape factor shows larger val-ues at early time and longer transient period. For very small values of a,


Figure 2. Shape factor constant for cylindrical shape matrix block subject to differ-ent boundary conditions.

Figure 3. Shape factor constant for spherical shape matrix block subject to differentboundary conditions.


Table V. Shape factor constant for different geometry matrix block subject to differentboundary conditions

Type of boundary condition Shape factor constants, L2

Slab Cylindrical Spherical

Constant fracture pressure 9.87 18.2 25.65Exponential, a =1 9.87 18.2 25.65Exponential, a =0.0001 12 25.13 39Linear, all a 12 25.5 39Constant ux 12 25.13 39

the shape factor converges to a constant value of 39. The linear pressuredecline boundary condition gives a similar shape factor as the exponentialcase with a small value of a. Applying the constant ux boundary condi-tion results in a stabilized value of 39 similar to the linear decline bound-ary condition.

6. Signicance of the Results

In summary, it is found that depending on the boundary condition twostabilized shape factor constant can be obtained for each geometry, whichcould be 2040% apart. Results also reveal that the time to stabilizationdepends on the boundary condition imposed on the matrix block. Further-more, it was shown in Figures 13 that the different stabilized values canbe obtained by applying an exponential boundary condition with variousa exponents, where large values of the exponent give the smaller stabilizedvalue of the shape-factor constant. These stabilized shape factor constantsfor different cases are presented in Table V.

The stabilized value of the shape-factor constant is usually used in thetraditional double porosity model. This can cause the following two typesof errors: (1) the matrix-fracture transfer would be underestimated at earlytimes, because Figures 13 show that at early times, the actual value ofshape-factor constant is larger than the stabilized value. (2) Depending onthe stabilized value of the shape factor chosen, the calculated value couldbe signicantly different even at late times.

The essential question is what is the magnitude of the error if one uses thestabilized value of shape factor instead of its transient value? We investigatedthe error introduced in the single-phase matrix fracture mass exchange termif one uses the stabilized value instead of the appropriate transient value.Here, we dene the relative errors as the difference between the ow ratesas calculated using the transient shape-factor and the stabilized-shape factor,


Figure 4. Relative error for a slab shape matrix block subject to different boundaryconditions.

divided by the rate using the transient-shape factor. A positive value wouldindicate an underestimation of the transfer term by using the stabilized val-ues. The results in Figures 46 illustrate that using the stabilized value of theshape-factor constant at early time results a signicant underestimation ofthe rate of uid transfer from matrix into fracture. For example, for a slabshape matrix block if one uses the stabilized value of shape factor, the calcu-lated rate of mass transfer from matrix block into fracture at dimensionlesstime of 0.04 would be 3050% lower than the actual transient case. The rel-ative error dies down for cases where the larger stabilized shape factor wasused. On the other hand, using the lower value of the stabilized shape-factorleads to an overestimation of the matrix-fracture transfer term late times. Thedifference, which is of the order of 2060% depending on the matrix geom-etry, does not approach zero. While, depending on the boundary condition,the magnitude of the transfer rate may be small, nevertheless the error wouldremain if one uses the lower values of the stabilized shape factor.

7. Discussion

Lim and Aziz (1995) approximated the pressure diffusion in a matrix blocksurrounded by two and three sets of perpendicular fractures by solving


Figure 5. Relative error for a cylindrical shape matrix block subject to differentboundary conditions.

Figure 6. Relative error for a spherical shape matrix block subject to differentboundary conditions.


the diffusivity equation in cylindrical and spherical geometry, respectively.According to Lim and Aziz, for two sets of perpendicular fractures withboth sets of fractures having the spacing L, the equivalent radius of thecylinder is the radius that would give the same volume as a bar with aL L square-shaped cross section. For spherical matrix blocks, Lim andAziz used an equivalent radius of the sphere that yields the same volume asa cube with length L on all sides. Here, we have used a similar approxima-tion to derive the shape factor for such matrix and fracture congurations.

Determining an accurate matrix-fracture transfer shape has been thesubject of several petroleum engineering studies. Previous studies of shapefactor are based on constant fracture pressure at the matrix boundaries.This work investigates the effects of other boundary conditions on thematrix-fracture transfer. Our results show that in addition to the matrixshape, the shape factor constant also depends on the pressure in the frac-ture and how it changes with time. The results presented in this paper indi-cate that one can obtain the different values of shape factor reported in theliterature by choosing different boundary conditions.

However, the time variability of boundary conditions can be accountedfor by using the Duhamels theorem. In the following it is shown that usinglarge time-steps can strongly affect the uid exchange between matrix andfracture. For instance, in theory a linear pressure decline in the fracture canbe modeled by an innite number of innitesimal step changes and appli-cation of the Duhamels theorem to calculate the pressure and uid efux.We consider a case where the matrix pressure can be approximated byits average value. Using Equations (1) and (2) assuming the matrix blockbehaves as a lumped system, the pressure change in the matrix block canbe described by the following ordinary differential equation

dpm

dt+ pm =pf (15)

where =km/mcm (16)We solve the above problem using Duhamels theorem and its stepwiseapproximation and compare the solutions. Solution for this ODE for con-stant and linear (pf =pi(1tD), 1/tD) fracture pressure are given bythe following equations, respectively.

pm(tD)=pf +(pi pf

)e(

24

)tD (constant fracture pressure) (17)

pm(tD)=pi{(1tD)+ 4

2

(1 e

24 tD

)}(linear fracture pressure)

(18)


As mentioned the average matrix pressure and uid efux for a linearfracture pressure can be obtained by superposition of the constant frac-ture pressure solutions. However, the accuracy of the superposition solutiondepends on the number of steps used. To demonstrate the effect of numberof steps on the average matrix pressure and uid efux, result of the linearpressure decline in the fracture and the superposition solutions of the con-stant fracture pressure with different number of steps are compared in Fig-ure 7. The dimensionless efux in this gure (7a) represents the uid efuxdivided by its stabilized value under the linear fracture decline. Resultsshow that using superposition of large pressure steps underestimates theuid efux signicantly. The average matrix pressure and the uid efux fora linear fracture pressure can be obtained by superposition of the constantpressure solution only if small pressure steps are used. Therefore, in prac-tice using large time-steps in numerical simulations can potentially intro-duce large errors in simulation results.

We found a shape factor constant of 9.87 and 12 for a slab shape matrixblock. Warren and Root (1963) and Lim and Aziz (1995) reported valuesof 12 and 9.87, respectively. For the two-dimensional case, we found val-ues of 18.2 and 25.13. A shape factor in this range was reported beforeby different authors such as Kazemi and Gillman (1993), Lim and Aziz(1995), Bourbiaux et al. (1999), and Sadra et al. (2002). For the three-dimensional cases, we derived values of roughly 25 and 39. Thomas et al.(1983), Kazemi and Gillman (1993), and Lim and Aziz (1995) reportedshape factor values of 25, 29.61, and 29.61, respectively, for three-dimen-sional transfer cases.

(a) (b)

Figure 7. Dimensionless uid efux (a) and dimensionless matrix pressure (b) for alinear pressure decline in fracture as a function of dimensionless time comparedwith the superposition solutions of the constant fracture pressure steps. Solutionsare compared for different numbers of steps.


The focus of this paper was on single-phase ow between matrix andfracture. Under multi-phase ow conditions, capillary and gravity forcescould become important and the saturation functions lead to nonlinear-ity of the ow equations. It is expected that under these conditions, thetransfer shape-factor depends on many other factors and using a constantvalue, may lead to signicant errors. This may explain (but not justify) thecommon practice where the shape factor is treated as a matching parame-ter (Thomas, 1983, Bourbiaux et al., 1999).

8. Conclusions

Previous studies have shown the effect of matrix shape and ow physics onthe shape-factor constant. In this study, we have investigated the effect ofthe pressure depletion regime in the fracture surrounding the matrix on thesingle-phase ow shape factor. Laplace domain analytical solutions of thediffusivity equation for various matrix block geometries and boundary con-ditions presented here have led to the following conclusions:

The matrix-fracture transfer shape factor depends on the pressureregime in the fracture and how it changes with time. Depending onthe pressure regime in the fracture a range of stabilized values can beobtained. The upper value is obtained from a slow (linear or exponen-tial) pressure depletion in the fracture and the lower bound by a fastdepletion in the fracture.

The time variability of the fracture boundary condition can beaccounted for by the superposition solution of the constant fracturepressure only through a large number of pressure steps.

The boundary condition dependency of a shape factor can be character-ized by applying an exponential-decline boundary condition with vary-ing decline exponents, where fast declines lead to a smaller value ofthe shape-factor constant. A range of shape factors can be obtained byassigning different exponents.

It is shown that using the stabilized shape factor introduces large errorsin the rate of matrix-fracture transfer by uid expansion at early andlate times.

For single-phase ow applications, using the shape factor is meaning-ful when it is derived based on an appropriate geometry, physics, andboundary conditions.

Acknowledgements

We would like to thank Jacques Hagoort (Hagoort & Assoc.) for hiscomments and discussion that greatly improved the earlier version of the


paper. After the completion of this work, we have noticed a study byChang [20] using an approach different than ours, that has led to similarresults. The nancial support for this work was provided by Alberta EnergyResearch Institute (AERI) and NSERC. This support is gratefully acknowl-edged. The rst author wishes to acknowledge the nancial support of theNational Iranian Oil Company (NIOC) during the course of this study.

References

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