A Semianalytic Solution for Flow in Finite-Conductivity Vertical Fractures by Use of Fractal Theory M. Cossio, SPE, Texas A&M University; G.J. Moridis, SPE, Lawrence Berkeley National Laboratory; and T.A. Blasingame, SPE, Texas A&M University Summary The exploitation of unconventional reservoirs complements the practice of hydraulic fracturing, and with an ever-increasing demand in energy, this practice is set to experience significant growth in the coming years. Sophisticated analytic models are needed to accurately describe fluid flow in a hydraulic fracture, and the problem has been approached from different directions in the past 3 decades—starting with the work of Gringarten et al. (1974) for an infinite-conductivity case, followed by contributions from Cinco-Ley et al. (1978), Lee and Brockenbrough (1986), Ozkan and Raghavan (1991), and Blasingame and Poe (1993) for a finite-conductivity case. This topic remains an active area of research and, for the more-complicated physical scenarios such as multiple transverse fractures in ultratight reservoirs, answers are currently being sought. Starting with the seminal work of Chang and Yortsos (1990), fractal theory has been successfully applied to pressure-transient testing, although with an emphasis on the effects of natural frac- tures in pressure/rate behavior. In this paper, we begin by per- forming a rigorous analytical and numerical study of the fractal diffusivity equation (FDE), and we show that it is more funda- mental than the classic linear and radial diffusivity equations. Thus, we combine the FDE with the trilinear flow model (Lee and Brockenbrough 1986), culminating in a new semianalytic solution for flow in a finite-conductivity vertical fracture that we name the “fractal-fracture solution (FFS).” This new solution is instantane- ous and comparable in accuracy with the Blasingame and Poe so- lution (1993). In addition, this is the first time that fractal theory is used in fluid flow in a porous medium to address a problem not related to reservoir heterogeneity. Ultimately, this project is a demonstration of the untapped potential of fractal theory; our approach is flexible, and we believe that the same methodology could be extended to different applications. One objective of this work is to develop a fast and accurate semianalytical solution for flow in a single vertical fracture that fully penetrates a homogeneous infinite-acting reservoir. This would be the first time that fractal theory is used to study a prob- lem that is not related to naturally fractured reservoirs or reservoir heterogeneity. In addition, as part of the development process, we revisit the fundamentals of fractals in reservoir engineering and show that the underlying FDE possesses some interesting qualities that have not yet been comprehensively addressed in the literature. Introduction Power-Law Distribution of Hydraulic Properties With Respect to Space. All well testing is derived, in one form or another, from the diffusivity equation, which is known in either linear or radial form. The simplest set of assumptions that can be made for the study of a reservoir system include single-phase and single-component laminar flow through a homogeneous and iso- tropic reservoir. These assumptions proved to be too simplistic to address the problems encountered. Several modifications have been proposed, including pressure-dependent permeability (Fran- quet et al. 2004), multiple interacting-media models (Warren and Root 1963; Abdassah and Ershaghi 1986; Camacho-Velazquez et al. 2005), multiphase flow (Lee and Wattenbarger 1996), or sorption effects (Clarkson et al. 2012). However, it is not possible to introduce too much nonlinearity into the diffusivity because it quickly becomes difficult to obtain an analytic solution, which ren- ders it impractical for well-testing and history-matching purposes. In this work, we explore the mathematical implications of modify- ing the porosity and permeability distributions with power-law relations, originally proposed by the introduction of fractal theory to well testing. Although research in the area of pressure-transient analysis of naturally fractured systems has made important advances, it became apparent that the models did not always give satisfactory results (Acun ˜a et al. 1995). The work of Chang and Yortsos (1990) contains basic theoretical formalism as it pertains to spe- cific petroleum-engineering applications of fractal theory. Their original contribution involved a modification of the Warren-Root model so that, instead of having a network of linearly arranged matrix “sugar cubes,” the permeable fractures embedded within the matrix would be arranged in a fractal fashion. A comprehen- sive review of fractal applications to reservoir engineering, fol- lowing the Chang and Yortsos (1990) model, may be found in Cossio (2012). Chang and Yortsos (1990) perform a control-volume deriva- tion of flow across a differential shell that contains a fractal-frac- ture network embedded in a Euclidean matrix. Their derivation culminates in the following power-law expressions for porosity and permeability, respectively: /ðrÞ¼ AV S G r Dd ð1Þ kðrÞ¼ AV S m G r Ddh ð2Þ where D is the mass-fractal dimension (dimensionless); h is the conductivity index (dimensionless); d is the Euclidean-embedding dimension (dimensionless); A is the site-density parameter (m –D ); V s is the volume per site (m 3 ); G is the geometry factor (m 3–d ); m is the fracture-network parameter ðm hþ2 Þ; r is the radius from the center of the wellbore (m); / is the porosity (fraction); and k is the permeability (m 2 ). The term d, always an integer, indicates the Euclidean dimen- sion of the matrix in which the fractures are embedded. The term D indicates, roughly speaking, the fractal dimension of the frac- tal-fracture network embedded in the Euclidean matrix. The h characterizes the diffusion process. h ¼ 0 indicates when the diffu- sion process is a classical random walk (i.e,. an ideal diffusion process). Higher values of h indicate that the capillary paths in the fractal network are highly tortuous [e.g., hindered diffusion (Acun ˜a et al. 1995)]. Consequently, the authors never consider a negative value of h, because, physically, it means that there is “enhanced diffusion.” Without some external output, such as me- chanical stirring to create turbulence, it is probably not possible to .......................... ........................ Copyright V C 2013 Society of Petroleum Engineers This paper (SPE 153715) was accepted for presentation at the SPE Latin American and Caribbean Petroleum Engineering Conference, Mexico City, Mexico, 16–18 April 2012, and revised for publication. Original manuscript received for review 27 January 2012. Revised manuscript received for review 8 August 2012. Paper peer approved 31 August 2012. February 2013 SPE Journal 83
Transcript
A Semianalytic Solution for Flow inFinite-Conductivity Vertical Fractures
by Use of Fractal TheoryM. Cossio, SPE, Texas A&M University; G.J. Moridis, SPE, Lawrence Berkeley National Laboratory; and
T.A. Blasingame, SPE, Texas A&M University
Summary
The exploitation of unconventional reservoirs complements thepractice of hydraulic fracturing, and with an ever-increasingdemand in energy, this practice is set to experience significantgrowth in the coming years. Sophisticated analytic models areneeded to accurately describe fluid flow in a hydraulic fracture,and the problem has been approached from different directions inthe past 3 decades—starting with the work of Gringarten et al.(1974) for an infinite-conductivity case, followed by contributionsfrom Cinco-Ley et al. (1978), Lee and Brockenbrough (1986),Ozkan and Raghavan (1991), and Blasingame and Poe (1993) fora finite-conductivity case. This topic remains an active area ofresearch and, for the more-complicated physical scenarios such asmultiple transverse fractures in ultratight reservoirs, answers arecurrently being sought.
Starting with the seminal work of Chang and Yortsos (1990),fractal theory has been successfully applied to pressure-transienttesting, although with an emphasis on the effects of natural frac-tures in pressure/rate behavior. In this paper, we begin by per-forming a rigorous analytical and numerical study of the fractaldiffusivity equation (FDE), and we show that it is more funda-mental than the classic linear and radial diffusivity equations.Thus, we combine the FDE with the trilinear flow model (Lee andBrockenbrough 1986), culminating in a new semianalytic solutionfor flow in a finite-conductivity vertical fracture that we name the“fractal-fracture solution (FFS).” This new solution is instantane-ous and comparable in accuracy with the Blasingame and Poe so-lution (1993). In addition, this is the first time that fractal theoryis used in fluid flow in a porous medium to address a problem notrelated to reservoir heterogeneity. Ultimately, this project is ademonstration of the untapped potential of fractal theory; ourapproach is flexible, and we believe that the same methodologycould be extended to different applications.
One objective of this work is to develop a fast and accuratesemianalytical solution for flow in a single vertical fracture thatfully penetrates a homogeneous infinite-acting reservoir. Thiswould be the first time that fractal theory is used to study a prob-lem that is not related to naturally fractured reservoirs or reservoirheterogeneity. In addition, as part of the development process, werevisit the fundamentals of fractals in reservoir engineering andshow that the underlying FDE possesses some interesting qualitiesthat have not yet been comprehensively addressed in the literature.
Introduction
Power-Law Distribution of Hydraulic Properties With
Respect to Space. All well testing is derived, in one form oranother, from the diffusivity equation, which is known in eitherlinear or radial form. The simplest set of assumptions that can bemade for the study of a reservoir system include single-phase andsingle-component laminar flow through a homogeneous and iso-
tropic reservoir. These assumptions proved to be too simplistic toaddress the problems encountered. Several modifications havebeen proposed, including pressure-dependent permeability (Fran-quet et al. 2004), multiple interacting-media models (Warren andRoot 1963; Abdassah and Ershaghi 1986; Camacho-Velazquezet al. 2005), multiphase flow (Lee and Wattenbarger 1996), orsorption effects (Clarkson et al. 2012). However, it is not possibleto introduce too much nonlinearity into the diffusivity because itquickly becomes difficult to obtain an analytic solution, which ren-ders it impractical for well-testing and history-matching purposes.In this work, we explore the mathematical implications of modify-ing the porosity and permeability distributions with power-lawrelations, originally proposed by the introduction of fractal theoryto well testing.
Although research in the area of pressure-transient analysis ofnaturally fractured systems has made important advances, itbecame apparent that the models did not always give satisfactoryresults (Acuna et al. 1995). The work of Chang and Yortsos(1990) contains basic theoretical formalism as it pertains to spe-cific petroleum-engineering applications of fractal theory. Theiroriginal contribution involved a modification of the Warren-Rootmodel so that, instead of having a network of linearly arrangedmatrix “sugar cubes,” the permeable fractures embedded withinthe matrix would be arranged in a fractal fashion. A comprehen-sive review of fractal applications to reservoir engineering, fol-lowing the Chang and Yortsos (1990) model, may be found inCossio (2012).
Chang and Yortsos (1990) perform a control-volume deriva-tion of flow across a differential shell that contains a fractal-frac-ture network embedded in a Euclidean matrix. Their derivationculminates in the following power-law expressions for porosityand permeability, respectively:
/ðrÞ ¼ AVS
G� rD�d ð1Þ
kðrÞ ¼ AVSm
G� rD�d�h ð2Þ
where D is the mass-fractal dimension (dimensionless); h is theconductivity index (dimensionless); d is the Euclidean-embeddingdimension (dimensionless); A is the site-density parameter (m–D);Vs is the volume per site (m3); G is the geometry factor (m3–d); mis the fracture-network parameter ðmhþ2Þ; r is the radius from thecenter of the wellbore (m); / is the porosity (fraction); and k isthe permeability (m2).
The term d, always an integer, indicates the Euclidean dimen-sion of the matrix in which the fractures are embedded. The termD indicates, roughly speaking, the fractal dimension of the frac-tal-fracture network embedded in the Euclidean matrix. The hcharacterizes the diffusion process. h¼ 0 indicates when the diffu-sion process is a classical random walk (i.e,. an ideal diffusionprocess). Higher values of h indicate that the capillary paths in thefractal network are highly tortuous [e.g., hindered diffusion(Acuna et al. 1995)]. Consequently, the authors never consider anegative value of h, because, physically, it means that there is“enhanced diffusion.” Without some external output, such as me-chanical stirring to create turbulence, it is probably not possible to
This paper (SPE 153715) was accepted for presentation at the SPE Latin American andCaribbean Petroleum Engineering Conference, Mexico City, Mexico, 16–18 April 2012, andrevised for publication. Original manuscript received for review 27 January 2012. Revisedmanuscript received for review 8 August 2012. Paper peer approved 31 August 2012.
February 2013 SPE Journal 83
achieve this just from the geometry of porous media. All otherfractal parameters (A, Vs, G, and m) are briefly discussed, but it isnot clear how one may obtain values for them. Their physicalmeaning may be clear only after an exhaustive foray into the math-ematics of fractal theory, which is probably beyond the sphere ofinterest of the practicing reservoir engineer.
Regardless of this possible limitation, this power-law formula-tion allowed for the development of a “fractal” radial-diffusivityequation intended to facilitate the study of reservoirs thought tocontain a network of fractures assumed to have a fractal distribu-tion. The concept of a fractal- fracture network embedded in aEuclidean matrix has been adopted successfully by other authors,such as Beier (1994), Flamenco-Lopez and Camacho-Velazquez(2003), Camacho-Velazquez et al. (2008), and Fuentes-Cruz et al.(2010). It is worth noting that Chang et al. (1990) did experimentwith different values of D (with h¼ 0) and showed that classicresults can be reproduced. However, this was not the main driveof their work and, to our knowledge, this idea has not been furtherdeveloped beyond the paragraph where it is mentioned.
Acuna et al. (1995) later presented Eqs. 1 and 2 in a differentway. They are now
/ðrÞ ¼ /0
r
r0
� �D�d
ð3Þ
kðrÞ ¼ k0
r
r0
� �D�d�h
; ð4Þ
where k0, /0, and r0 are permeability (m2), porosity (fraction),and radius (m), respectively, chosen at arbitrary points of ourchoice. In other words, the terms A, Vs, G, and m are convenientlyeliminated and condensed into the /0=rD�d
0 and k0=rD�d�h0 terms.
Study of the FDE
Analytical Considerations. Fractal Porosity/Permeability Re-lations (FPPR). In this work, we use Eqs. 3 and 4 as steppingstones for our development. For simplicity, we are going to con-sider only a flat system, thereby forcing d¼ 2. Furthermore, wework exclusively in Cartesian coordinates. Thus, our FPPR are
kðxÞ ¼ kwx
xw
� �D�2�h
ð5Þ
/ðxÞ ¼ /w
x
xw
� �D�2
; ð6Þ
where kw is the permeability at the edge of the wellbore (m2); /w
is porosity at the edge of the wellbore (fraction); xw is the dis-tance from the center of wellbore to the edge of the wellbore (m);D is the mass fractal dimension (dimensionless); h is the conduc-tivity index (Chang and Yortsos 1990) or fractal exponent (Acunaet al. 1995) (dimensionless); and x is distance from the center ofthe wellbore (m).
Motivation: Flow Equivalency. A classical analysis of fluidflow in porous media with a constant permeability and porositythroughout the reservoir results in the following 1D diffusivityequations:
@2pD
@x2D
¼ @pD
@tDð7Þ
@2pD
@r2D
þ 1
rD
@pD
@rD¼ @pD
@tD; ð8Þ
where pD is the dimensionless pressure; rD is the dimensionlesswellbore radius in radial coordinates; and xD is the dimensionlesswellbore radius in Cartesian coordinates.
Eqs. 7 and 8 are expressed in dimensionless variables and arederived, starting from the continuity equation, in Cartesian and ra-dial coordinate systems, respectively. As shown in Appendix A,we shall perform the same derivation in Cartesian coordinates,
but instead of having constant hydraulic properties, we use theFPPR (Eqs. 5 and 6) and obtain the following relationship, whichwe can call the FDE:
@2pD
@x2D
þ D� h� 2
xD
@pD
@xD¼ xh
D
@pD
@tD; ð9aÞ
where we have used the following dimensionless variables:
A cursory inspection of the FDE (Eq. 9a) yields the followinginteresting observations:� If we set fD ¼ 2; h ¼ 0g, we obtain exactly Eq. 7 from Eq.
9a. This means we are dealing with a linear reservoir with constanthydraulic properties (i.e., the classic Euclidean linear-flow case).� Similarly, if we select fD ¼ 3; h ¼ 0g, we obtain exactly
Eq. 8 from Eq. 9a. That Eq. 9a was derived in Cartesiancoordinates and Eq. 8 in radial coordinates implies that aconstant-hydraulic-properties radial flow is equivalent to a line-arly-increasing-hydraulic-properties linear flow. As this is animportant concept, we provide an illustration in Fig. 1.
This seems to suggest that, at least mathematically, the FDE(Eq. 9a) is more fundamental than Eqs. 7 or 8, and it can act as a sortof bridge between the two classic flow regimes, linear and radial.Chang and Yortsos (1990) first proposed this idea, but this observa-tion was not the main thrust of their work; no further work has beendone capitalizing on this concept. Furthermore, they propose thath> 0.
Numerical Validation. To advance our hypothesis, the FDEneeds to be validated, and numerical simulation is an ideal mecha-nism for this validation. We use a finite-volume black-oil reser-voir simulator to create a simple linear 1D reservoir with porosityand permeability that change with distance from the wellbore, asstipulated in the FPPR (Eqs. 5 and 6). For this validation, werequire a sufficiently fine space discretization (i.e., a sufficientlylarge number of cells¼ 1,000) to achieve a smooth porosity andpermeability profile. Each cell represents a subdomain with itsown (and different from all others) permeability and porosity.Furthermore, because it is not possible to have infinite reservoirsor infinite hydraulic properties in a simulator, an appropriatelydefined finite reservoir system is needed for comparison with theanalytical solution of Eq. 9a. The two possible options are either ano-flow or a constant-pressure outer boundary; we chose to usethe no-flow outer boundary option for this purpose. Eq. 10 is theanalytical solution to Eq. 9a with a no-flow outer-boundarycondition—its development is shown in Appendix B.
pDðxD;D; h; L; zÞ
¼ xaD �
Inxc
D
ffiffizp
c
� �Kn�1
Lc ffiffizp
c
� �þ In�1
Lc ffiffizp
c
� �Kn
xcD
ffiffizp
c
� �
z3=2 In�1
Lc ffiffizp
c
� �Kn�1
ffiffizp
c
� �� In�1
ffiffizp
c
� �Kn�1
Lc ffiffizp
c
� �� �� � � � � � � � � � � � � � � � � � � ð10Þ
Numerical simulations were performed for eight differentcases for which the fractal parameters are detailed in Table 1.Complete details of the simulation may be found in Cossio(2012). A sketch of the reservoir is presented in Fig. 2. There isexcellent agreement between the analytical and numerical cases,as shown in Fig. 3. Thus, we have validated numerically ourhypothesis that the FDE can model a multitude of flow regimeswithin a 1D system.
Beier (1994) applied instantaneous line-source functions, in thesame spirit as Gringarten et al. (1974), to a single vertical fracture,thus fully penetrating a reservoir that is assumed to have a perme-able fracture network that is fractally distributed. He consideredthe cases of infinite conductivity and uniform flux, and success-fully applied them to a field case. Because this is the only publica-tion, as far as we know, that applies fractal theory specifically to
hydraulic fractures, we believe that this is a research area that islargely unexplored, and the remainder of this paper is devoted tothis particular problem.
Mathematical Model. In this paper, our goal is to combine thetrilinear flow model given by Lee and Brockenbrough (1986) withfractal theory to develop a fast and accurate semianalytical
TABLE 1—PARAMETERS OF THE EIGHT CASES TO BE SIMULATED
Case
Number D h
k (md) U (%)
xD ¼1 xD ¼L xD¼1 xD¼L
1 2 0 1.0 1.0 0.1 0.1
2 2.25 0 1.0 4.47 0.1 0.447
3 2.50 0 1.0 20 0.1 2.0
4 2.75 0 1.0 89.44 0.1 8.94
5 3.0 0 1.0 400 0.1 40.0
6 3 �0.75 1.0 3.577�104 0.1 40.0
7 1.969 �0.7778 1.0 87.74 30.0 24.91
8 1.661 �0.323 1.0 0.91 5.0 0.65
Production
0 0.1 m
No flow
40.0 m
x
Fig. 2—Schematic of the 1D linear reservoir common to all cases. This reservoir has 1,000 gridblocks in the x-direction, each withits own porosity and permeability, as stipulated in the FPPR.
∂2pD
∂rD
∂pD
∂rD
∂pD
∂tD
1
rD2
+ =
∂2pD
∂xD
∂pD
∂xD
∂pD
∂tD
D – θ – 2
xD2
+ =
Classic radial diffusivity equation
A radial system withconstant {k, φ } properties
A linear system withlinearly increasing {k, φ }properties
is equivalent to
xDθ
Fractal diffusivity equationwith {D=3, θ=0}
k
φ
X
k (x ) = kw
xxw
φ (x ) = φ w
xxw
Fig. 1—Schematic of the equivalency of fractal linear flow and classic radial flow. Even though the top and bottom cases describedifferent physical scenarios, the FDE predicts that both should yield the same pressure signal at the wellbore. This has been veri-fied analytically and numerically in this work.
February 2013 SPE Journal 85
solution for the problem of a producing well with a single verticalfracture that fully penetrates an infinite-acting homogeneous res-ervoir. We first note that fast and reliable solutions for this physi-cal scenario already exist in the literature—Blasingame and Poe
(1993) provide an instantaneous “trilinear pseudoradial solution”(TPRS) that is based on a coupling of the trilinear-flow solution(which does not model radial flow) and the solution for a uni-form-flux/infinite-conductivity vertical fracture (which doesmodel pseudoradial flow). In our development of this semianalyti-cal solution, we believe that this is the first application of fractaltheory to address a flow problem that is not related to reservoirheterogeneity.
The basic premise of Lee and Brockenbrough (1986) is toidealize the flow into the hydraulic fracture by connecting thefracture to a system of sequentially connected 1D (linear) reser-voirs, where this combination of three reservoirs (including thefracture) will produce a different flow regime. Each of these linearreservoirs is called a “region,” each has a governing diffusivityequation, and each reservoir communicates with the other twothrough their common boundaries (which maintain common fluxconditions). This scenario is depicted in Fig. 4.
Mathematically, the trilinear flow system (with fractal geome-try) is described as follows.� Region 3 (formation flow):
Fig. 3—Comparison of analytical and numerical results of the FDE. Note that Case 1 is equivalent to a Euclidean linear case, andCase 5 is equivalent to a Euclidean radial case. The excellent agreement in Case 5 validates numerically the equivalency proposedin Fig. 1.
Top View
Region 2 Region 3
Region 1
Fracture
Flow direction
Y
X
bf
xf
Fig. 4—Schematic of the trilinear flow concept, as proposed byLee and Brockenbrough (1986). Because of symmetry, only aquadrant of the flow domain is considered. Region 3 flows inthe x-direction and meets Region 2 in the dashed lines at x 5 xf;Regions 2 and 3 are both formation flow. Region 2 flows in they-direction and meets Region 1 in the dashed lines at y 5 bf.Region 1 represents an idealized vertical fracture. Finally, thefracture flow of Region 1 feeds the wellbore, represented by thecircle.
The terms in these equations are defined as follows:
p1D ¼khðpi � p1Þ
q B lfor oil in Region 1 (similar expressions in
Regions 2 and 3).
And the remaining terms are given as
tD ¼k
/lctx2f
t;C1 ¼k/f cft
kf /ct;
xD ¼x
xf;FcD ¼
kf bf
k x f;
yD ¼y
bf; and a ¼ 2
FcD:
CDf ¼C
2p/cthx2f
b ¼ � pFcD
Essentially, the only difference between the original Lee andBrockenbrough (1986) model and the present work is that we havereplaced the Euclidean linear-diffusivity equations, as stipulated inthe original paper, with their fractal counterparts (i.e., Eqs. 11a,12a, and 13a). If we set D3 ¼ D2 ¼ D1 ¼ 2 and h3 ¼ h2 ¼ h1¼ 0,then we obtain the original Euclidean formulation. In Lee andBrockenbrough’s (1986) original paper, the solution gave goodperformance at early times (i.e., tD � 1)—however, because of thewell-known inability of the Lee and Brockenbrough (1986) solu-tion to model pseudoradial flow, this solution begins to fail atapproximately tD¼ 1. Our hypothesis is that we can correctly cap-ture pseudoradial flow by use of the FFS formulation that we pro-posed in the preceding. This will be an approximate/semianalyticalsolution because we must calibrate the D and h parameters, but ourexpectation is that we will capture the appropriate flow regimes inthe pressure and pressure-derivative function.
Calibration of an Approximate Solution. In the process ofcombining the FDEs into the trilinear flow solution, we haveintroduced six unknowns into the problem, (D1, D2, D3, h1, h2,and h3). At this point, it is not clear whether these unknowns areconstant, nor what variables the unknowns should be correlated
with (although an intuitive variable of correlation is the fractureconductivity). At this stage, our strategy is first to numericallyoptimize these parameters (D1, D2, D3, h1, h2, and h3) for eachindividual FcD case and then to graphically compare each parame-ter with FcD to establish whether a single-variable correlation isappropriate.
We select the Cinco-Ley and Meng (1988) solution as the“standard” against which we will correlate the fractal-based,trilinear flow solution. Their solution is given as
where Eqs. 14a and 14b imply the following system of equations:
½Aij�qfDiðzÞpwDðzÞ
� �¼ Bj
� ; ð14cÞ
where the dimensionless-pressure solution in the real (time) domainis obtained by means of numerical inversion of the pwDðzÞ solutionthat is given in the Laplace domain. We note that, as a standard, theCinco-Ley and Meng solution (1988) does provide a physically andmathematically rigorous treatment of the problem in the Laplacedomain, but it is a discretized solution that is cumbersome, is com-plex to set up, and is computationally expensive. Consequently,their solution is not well-suited to history-matching applications.
To calibrate the D and h values by use of the Cinco-Ley andMeng (1988) solution as a standard, we constructed a Fortran code(Chapman 2008) to perform the numerical optimization. This pro-gram incorporates the Levenberg-Marquardt optimization algo-rithm (More et al. 1984) coupled to the FFS (i.e., our FDEscombined with the trilinear flow solution). A flow chart of this opti-mization process is depicted in Fig. 5. Our optimization/calibrationprocess begins with an initial guess for each of the values of theunknown fractal parameters; we then compute the FFS by use ofthese values and compare the results with the Cinco-Ley and Meng(1988) solution. To define an “objective function” for optimization,we must first define the function(s) that shall be our basis—in ourcase, we consider both the pressure function and the pressure-deriv-ative function, coupled with a defined weighting of each function.For this research, we have defined our objective function (OF) as
OF ¼ 0:25
����pwD �FFS �pwD �CMS
����pwD�CMS
þ 0:75
����p0wD �FFS �p0wD �CMS
����p0wD�CMS
; ð15Þ
where the individual components in Eq. 15 are defined as� pwD�FFS is the dimensionless wellbore pressure (FFS).� pwD�CMS is the dimensionless wellbore pressure [Cinco-Ley
and Meng (1988) solution].� p0wD�FFS is the dimensionless wellbore pressure derivative
(FFS).� p0wD�CMS is the dimensionless wellbore pressure derivative
[Cinco-Ley and Meng (1988) solution].The optimization process is terminated when the Levenberg-
Marquardt algorithm determines by means of a specified tolerancethat the objective function cannot be minimized further. SeeCossio (2012) for details on the determination of the weightingcoefficients 0.25 (for dimensionless-pressure function) and 0.75(for dimensionless-pressure-derivative function).
. . . . . . . . .
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. . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . .
February 2013 SPE Journal 87
For this calibration, we have selected the Cinco-Ley and Meng(1988) solution, and we will estimate the D1, D2, D3, h1, h2, andh3 parameters by numerical optimization. As a precautionarymeasure, we have chosen to simplify the problem because theoptimization of six parameters may invite issues of nonuniquenessand thus will substantially increase computational cost. As such,we consider three different “scenarios,” in which we leave aparticular region(s) in its linear (nonfractal) form. These scenariosare listed in Table 2. In Table 3, we show the analytical solutionfor each scenario, and a detailed derivation for each case may befound in Cossio (2012). In all cases, the skin S, wellbore-storagecoefficient CDf, and fracture-storage factor C1 were set to zero.
Optimization Results. With the calibration/optimizationapproach, we were able to match the Cinco-Ley and MengSolution (1988) satisfactorily with all three scenarios (Table 2)over 13 log cycles of dimensionless time by use of the optimizer(Fig. 5). Fig. 6 shows the L1 relative-error norm of pressure (vs.the fracture conductivity) for each optimized scenario. Inspectionof Fig. 6 and Table 3 leads to the following conclusions:� Scenario 1 is by far the least accurate (most simple form/
analytical solution).� Scenario 3 is the most accurate (most complex form/analyti-
cal solution).� Scenario 2 lies between the other two scenarios in terms of
both solution complexity and accuracy.� All scenarios fail for FcD< 0.5, and this should be the lower
limit of applicability.� Scenarios 2 and 3 are essentially equivalent for practical
purposes (L1 error <0.3%).From these observations, we can pick the scenario to develop
into a closed-form solution; we propose Scenario 2 as the optimalcompromise between accuracy and solution complexity. Thederivation of the analytical solution of Scenario 2 is shown inAppendix C.
Final Form of the FFS—Correlations and Results
Parameter Correlations. As shown in Fig. 7, the optimizedfractal parameters of Scenario 2 are well-behaved when correlatedagainst the dimensionless fracture conductivity (FcD)—as such,we believe it is possible to establish univariate correlations (i.e., y
vs. x, where x¼FcD) for each fractal parameter (D2, D3, h2, andh3). To develop these correlations, we used TableCurve 2D(2012) software, which provides an extensive library of possibledata models, fitted and ranked statistically for a given regressionof y vs. x.
Our correlations for Scenario 2 were constructed as follows:� h2 ¼ f ðFcDÞ: h2 is a unique function of FcD.� D2 ¼ f ðFcDÞ: D2 is a unique function of FcD.� D3 ¼ f ðD2Þ: D3 is a defined function of D2.� h3 ¼ f ðD3Þ: h3 is a defined function of D3.There may seem to be a contradiction between correlating the
parameters for Region 2 (D2 and h2) in terms of the dimensionlessfracture conductivity (FcD) and then defining the parameters forRegion 3 (D3 and h3) in terms of the parameters for Region 2 (D3
directly and h3 indirectly). However, these definitions for D3 andh3 arose from our correlation efforts, and we believe that theremay be a physical basis for these correlations (but this is notexplored in our present work). For this work, our efforts havefocused on establishing robust and accurate correlations for thefractal parameters (in this case: D2, D3, h2, and h3), and webelieve that we have created (at the least) a practical solution forapplication of the FFS for this scenario.
The parameter correlations for Scenario 2 are provided here:� h2 ¼ f ðFcDÞ: h 2 is a unique function of FcD.
� h2 ¼A1 þ C1xþ E1x2 þ G1x3
1þ B1xþ H1x2 þ F1x3½where x ¼ log10ðFcDÞ�:
� � � � � � � � � � � � � � � � � � � ð16Þ
The following coefficients were determined by use of theTableCurve 2D (2012) software:
A1¼�0.34048432, E1¼�0.34553019,B1¼ 1.918772436, F1¼ 3.154995757,C1¼�0.26952048, G1¼�0.98791385,H1¼ 0.433916281.This correlation (Eq. 16) has a coefficient of determination
r2¼ 0.9999 (Cossio 2012).� D2 ¼ f ðFcDÞ: D2 is a unique function of FcD
� D2 ¼A2 þ C2xþ E2x2 þ G2x3
1þ B2xþ H2x2 þ F2x3;where x ¼ log10ðFcDÞ
� � � � � � � � � � � � � � � � � � � ð17Þ
Next iteration
InitialGuess
Fractal FractureSolution
Objectivefunction
Optimizer
Finish Final
Cinco-MengSolution
InitialDi
θi Di
θi
Yes
No
Is OF smallenough?
Fig. 5—Optimization process of the fractal parameters in the FFS. We begin with an initial guess of the values of the unknownfractal parameters, compute the FFS with them, compare the results with the Cinco-Ley and Meng (1988) solution, and attempt tooptimize the parameter values by minimizing an objective function. The optimization is terminated when the Levenberg-Marquardtalgorithm determines that the objective function OF cannot be minimized further.
TABLE 2—DETAILS OF SCENARIOS TO BE OPTIMIZED
Scenario Region 1 Region 2 Region 3 Unknowns To Be Optimized tD Range Considered FcD Range Considered
1 Linear Fractal Removed D2 and h2 10–6 to 107 10–1 to 104
2 Linear Fractal Fractal D2, D3, h2, and h3 10–6 to 107 10–1 to 104
3 Fractal Fractal Fractal D1, D2, D3, h1, h2, and h3 10–6 to 107 10–1 to 104
88 February 2013 SPE Journal
10–1 100 101 102 103 104 105
10–1 100 101 102 103 104 105
Dimensionless Fracture Conductivity (FcD)
Val
ue o
f Num
eric
ally
Opt
imiz
edFr
acta
l Par
amet
ers
— S
cena
rio 2
3
2
1
0
–1
–2
3
2
1
0
–1
–2
Legend:D3D2
θ3θ2
Fig. 7—Values of optimized fractal parameters vs. thedimensionless fracture conductivity (FcD). The FFS (Scenario 2)was numerically optimized to obtain values for the four fractal pa-rameters to match the Cinco-Ley and Meng (1988) solution. Notethat all curves are well-behaved and should lend themselves toapproximations by smooth, closed-form functions.
2.5
2.0
1.5
p D L
1 E
rror
Nor
m, p
erce
nt
1.0
0.5
0.010–1 100 101
Dimensionless Fracture Conductivity (FcD)
102 103 104 105
10–1 100 101 102 103 104 105
0.0
0.5
1.0
1.5
2.0
Legend:
2.5
Scenario 1Scenario 2Scenario 3
Fig. 6—The FFS was derived analytically for the three scenariosconsidered in Table 2. The solutions of Table 3 were eachnumerically optimized to match the Cinco-Ley and Meng (1988)solution. The L1 relative-error norm (%) is plotted againstfracture conductivity for each of these scenarios.
TABLE 3—ANALYTICAL SOLUTIONS FOR ALL SCENARIOS
Scenario Analytical Solution
1n2 ¼
h2 þ 3� D2
h2 þ 2, w ¼ 1
FcD2ffiffizp Kn2�1
2ffiffizp
h2þ2
�Kn2
2ffiffizp
h2þ2
� � h2n2
24
35
8<:
9=;
1=2
,(20)
pwDðD2; h2;FcD; zÞ ¼p
FcD z w tanhðwÞ.
2 n3 ¼h3 þ 3� D3
h3 þ 2, X ¼ ffiffi
zp Kn3�1
2ffiffizp
h3 þ 2
� �
Kn3
2ffiffizp
h3 þ 2
� � , (21)
n2 ¼h2 þ 3� D2
h2 þ 2,
w ¼ 1
FcD2ffiffiffiffiffiffiffiffiffiffiffiffizþ Xp Kn2�1
2ffiffiffiffiffiffiffiffiffiffiffiffizþXp
h2 þ 2
� �
Kn2
2ffiffiffiffiffiffiffiffiffiffiffiffizþ Xp
h2 þ 2
� � � h2n2
26664
37775
8>>><>>>:
9>>>=>>>;
1=2
,
pwDðD3; h3;D2; h2; FcD; zÞ ¼p
FcD z w tanh ðwÞ.
3 n3 ¼h3 þ 3� D3
h3 þ 2, X ¼ ffiffi
zp Kn3�1
2ffiffizp
h3 þ 2
� �
Kn3
2ffiffizp
h3 þ 2
� � , (22)
n2 ¼h2 þ 3� D2
h2 þ 2, w ¼ 1
FcD2ffiffiffiffiffiffiffiffiffiffiffiffizþ Xp Kn2�1
2ffiffiffiffiffiffiffiffiffiffiffiffizþXp
h2 þ 2
� �
Kn2
2ffiffiffiffiffiffiffiffiffiffiffiffizþ Xp
h2 þ 2
� � � h2n2
26664
37775
8>>><>>>:
9>>>=>>>;
1=2
,
n1 ¼h1 þ 3� D1
h1 þ 2, D ¼
In1�12cwc
� �Kn1�1
wc
� �� In1�1
wc
� �Kn1�1
2cwc
� �
In1
wc
� �Kn1�1
2cwc
� �þ In1�1
2cwc
� �Kn1
wc
� � ,
c ¼ h1 þ 2
2,
pwDðD3; h3;D2; h2; D1; h1;FcD; zÞ ¼p
FcD z wD.
February 2013 SPE Journal 89
The following coefficients were determined by use of theTableCurve 2D (2012) software:
A2¼ 1.732300052, E2¼ 4.547075660,B2¼ 0.145343775, F2¼ 1.016405890,C2¼�1.00458904, G2¼ 1.566926538,H2¼ 2.270990983.This correlation (Eq. 17) has a coefficient of determination
r2¼ 0.9989 (Cossio 2012).� D3 ¼ f ðD2Þ: D3 is a defined function of D2.This correlation is defined on the basis of the observed behav-
ior of the parameters D3 vs. D2. We considered that a quadraticform would best fit this relationship:
D3 ¼ A3ðD2Þ2 þ B3D2 þ C3: ð18Þ
The following coefficients were determined by use of theTableCurve 2D (2012) software:
A3 ¼ 1:4814;B3 ¼ �7:3109;C3 ¼ 10:03:
This correlation (Eq. 18) has a coefficient of determinationr2¼ 0.9992 (Cossio 2012).� h3 ¼ f ðD3Þ: h3 is a defined function of D3.This correlation is defined on the basis of the observed behav-
ior of the parameters h3 vs. D3. We considered that a linear formwould best fit this relationship:
h3 ¼ A4D3 þ B4: ð19Þ
The following coefficients were determined by use of theTableCurve 2D (2012) software:
A4¼ 1.2063, B4¼�3.1532.
This correlation (Eq. 19) has a coefficient of determinationr2¼ 0.9999 (Cossio 2012).
We next use our correlation relations (i.e., Eqs. 16 through 19)as components of the FFS procedure (i.e., Eq. 21 for Scenario 2,shown in Table 3). By use of this FFS for Scenario 2, we thengenerated comparator cases (pD and p0D functions) to visuallyassess the relative accuracy of our approach for this scenario.Because all the parametric correlations have very good or excellentstatistical behavior, our expectation is that this FFS formulationshould yield good correlations with the reference solution (Cinco-Ley and Meng 1988).
Results
We now present results generated by use of the “Scenario 2”closed-form FFS (i.e., Eqs. 16 through 19 and 21) compared with
the “reference” solution of Cinco-Ley and Meng (1988). In thiscomparison, we vary the dimensionless fracture conductivity(FcD) over the range of 0.6 � FcD � 104, where this range encom-passes very low conductivity (0.6) to nearly infinite conductivity(104). Our approach is to compare the dimensionless-pressurefunction (pD) and dimensionless-pressure-derivative function (p0D)separately to assess the relative accuracy of each function inisolation.
In Fig. 8 we present the pD vs. tD functions for FFS (Scenario 2)and the Cinco-Ley and Meng reference solution in log-log format(13 log cycles are shown; this is 13 orders of magnitude in tD).There appear to be only minor discrepancies at very small values oftD for the low-conductivity cases (0.6 � FcD � 1). In this “log-log”view, no other discrepancies are apparent in the solutions.
In Fig. 9 we present the p0D vs. tD functions for FFS (Scenario 2)and the Cinco-Ley and Meng reference solution in log-log format(13 log cycles are shown; this is 13 orders of magnitude in tD).There appears to be only a very minor discrepancy at extremelysmall values of tD for the FcD¼ 0.6 case. In short, in this log-logview, we observe no significant issues/discrepancies in thesolutions.
In Fig. 10 we present the L1 error norm for the pD function (for0.6 � FcD � 104) vs. the dimensionless fracture conductivity(FcD), for the FFS and the TPRS (Blasingame and Poe 1993). InFig. 10 we observe that the pD(tD) values generated by use of theFFS are generally more accurate and more stable than thoseobtained from the TPRS proposed by Blasingame and Poe (1993).We observe that the Blasingame and Poe (1993) solution (TPRS)exhibits its worst behavior in the pD function (i.e., 0.5 � L1 errornorm �2%) for the range of 102 � FcD � 104, which is actuallysomewhat unexpected because the TPRS uses the infinite-conduc-tivity vertical-fracture solution as its basis.
In Fig. 11 we present the L1 error norm for the p0D function(for 0.6 � FcD � 104) vs. the dimensionless fracture conductivity(FcD), for the FFS and the TPRS (Blasingame and Poe 1993). Incontrast to Fig. 10, the Blasingame and Poe (1993) solution,TPRS, exhibits relatively stable behavior in the L1 error norm—inparticular, 0.7 � L1 error norm �1.2%—and varies somewhatrandomly with FcD, although the highest observed errors occur forthe 102 � FcD � 104 cases.
Discussion
Applicability and Limitations of the FPPR. There are acouple of important observations to be made about the FPPR(Eqs. 5 and 6).� A cursory numerical exercise by use of typical conventional
reservoir values reveals that, if D =2 (and/or h = 0), then theFPPR describe a situation that may be physically improbable— oreven impossible. For example, if we set xw¼ 0.10 m, /w¼ 15%
Fig. 8—(Scenario 2) Log-log plot of dimensionless-pressure function vs. dimensionless time for the FFS and Cinco-Ley and Mengsolutions (1988).
90 February 2013 SPE Journal
(or 0.15 fraction), kw¼ 1.9738� 10–13 m2 (200 md), D¼ 2.5, andh¼ 0, we find that at 20 m from the wellbore, we should expect apermeability of 2.82 darcies and a porosity of 212.13%.� Although the physical meaning of parameters D and h is
generally well-understood by specialists, they remain conceptu-ally elusive because these parameters cannot be easily related totangible physical quantities. This makes these concepts difficult todigest by nonspecialized practitioners and hinders their wide-spread acceptance.� When discussing the physical meaning of these relations,
Acuna et al. (1995) are careful to issue a caveat, stating that “itmust be stressed again that they do not correspond to point values(local averages) but to the porosity and permeability of regions ofsize r [sic]. It should be also stressed that [Eqs. 3 and 4] do notimply that the conventional porosity and permeability are radiallydependent around a given well. They only suggest that in a fractalmedium, all properties of any region of size r [sic] are scale-dependent following a power law.”
For these reasons, Eqs. 3 and 4, on which the FPPR are based,have not received much attention in the literature, and to ourknowledge, no attempt has yet been made to produce new solutions
or methodologies on the basis of the idea of a radially changingporosity and permeability in the literal sense of the definition.
General Remarks on the Implications of the FDE. We vali-dated the analytical solution (Eq. 10) of the FDE (Eq. 9a) by use ofnumerical simulation (Fig. 3). It is important to note that thereis nothing “fractal”—in the strict sense of the word—about our res-ervoir simulation. Specifically, there are no chaotic processes, norandomly distributed network of fractures, and no double-porosity assumptions. This leads us to suggest that we were suc-cessful in this validation effort not because the flow is fractal perse, but rather because both the FDE and the reservoir simulatorsolve equations on the basis of the principle of conservation ofmass. In short, we have solved exactly the same problem both ana-lytically and numerically, providing a “proof of concept” for theFDE approach. Building on this observation, in this work we areeffectively divorcing ourselves from the concept of a fractal-frac-ture network embedded in a matrix—the consensus in the literaturethus far. We are not limited by the need to have a fractal-fracturenetwork in the system—by relaxing this constraint, we can exercisemore freedom in our wielding of the FDE (e.g., we allow h< 0).
3.0
2.5
2.0
1.5
1.0
0.5
0.0
3.0
Blasingame and Poe Solution2.5
2.0
1.5
1.0
0.5
0.0
p D′ L
1 E
rror
Nor
m, p
erce
nt
10–1 100 101 102 103 104 105
10–1 100 101 102 103 104 105
Dimensionless Fracture Conductivity (FcD)
Legend:
Fractal Fracture SolutionClosed form, Scenario 2
Fig. 11—(Scenario 2) L1 relative-error norms for the dimension-less-pressure-derivative solutions for the closed-form FFS andthe TPRS (Blasingame and Poe 1993) vs. the dimensionlessfracture conductivity (FcD). Reference solution was obtainedfrom Cinco-Ley and Meng (1988).
3.0
2.5
2.0
1.5
1.0
0.5
0.0
3.0
Blasingame and Poe Solution 2.5
2.0
1.5
1.0
0.5
0.0
p D L
1 E
rror
Nor
m, p
erce
nt
10–1 100 101 102 103 104 105
10–1 100 101 102 103 104 105
Dimensionless Fracture Conductivity (FcD)
Legend:
Fractal Fracture SolutionClosed form, Scenario 2
Fig. 10—(Scenario 2) L1 relative-error norms for the dimension-less-pressure solutions for the closed-form FFS and the TPRS(Blasingame and Poe 1993) vs. the dimensionless fractureconductivity (FcD). Reference solution was obtained from Cinco-Ley and Meng (1988).
Fig. 9—(Scenario 2) Log-log plot of dimensionless-pressure-derivative function vs. dimensionless time for the FFS and Cinco-Leyand Meng solutions (1988).
February 2013 SPE Journal 91
Ours is a theoretical study in which the mathematically intriguingpossibilities offered by the FDE are exploited, and it does not needto be inextricably linked to the presence of fractal fractures.
In Fig. 1, we illustrate a case in which two reservoirs in differ-ent coordinate systems with different hydraulic properties produceidentical pressure signals. Similarly, the FFS introduced in thiswork and the Cinco-Ley and Meng (1988) solution produce nearlyidentical pressure signals, even though the way the fracture ismodeled mathematically in each case is different. Thus, we havetwo examples that it is possible to create “equivalent flow sys-tems” with relative ease, thanks to the flexibility introduced byvarying k and / with distance.
Furthermore, if D¼ 2 implies linear flow, and D¼ 3 impliesradial flow, what does it mean if D is equal to, say, 2.5? Is this theequivalent of elliptical flow? If so, what does it look like? At pres-ent, we do not have answers to these questions, but we intuitivelysuspect a relationship between the fractal parameters {D,h} andthe shape of the pressure waves and/or streamlines.
Applicability and Limitations of the FFS. On the basis of ourobservations in Figs. 10 and 11, we believe that the FFS andTPRS methods are sufficiently accurate for practical applicationsfor 0.5 � FcD. For cases of even lower conductivity in which bothsolutions begin to fail, Blasingame and Poe (1993) also observedthe same behavior; they suggested that the discretization chosen forthe Cinco-Ley and Meng (1988) solution (30 elements) may poseproblems at low conductivities. We do not attempt to offer any fur-ther explanation. However, recall the physical significance ofFcD< 0.5: the formation is more than twice as conductive as the hy-draulic fracture. Considering the extremely low permeabilities ofunconventional reservoirs and the definition of dimensionless frac-ture conductivity FcD ¼ ðkf bf Þ=ðk xf Þ, we know that it is highlyimprobable that we will encounter such a situation in practice.
The Use of Other Formulations for the Distribution of
Hydraulic Properties. We chose the power-law FPPR (Eqs. 5and 6) as a starting point because they have been successfullyapplied and have a physical significance attached to them. Weknow it is possible to derive a diffusivity equation in dimension-less form, and analytical solutions are readily available (Konget al. 2009; Yun et al. 2009). However, we are not suggesting thatthis is the only possible formulation for k(x) and /(x), and we cer-tainly invite the technical audience to attempt other formulations.However, for a new formulation to be successful, we believe itshould lead to the derivation of a diffusion-process partial-differ-ential equation; this diffusion equation must be reducible todimensionless form; and analytical solutions for any combinationof boundary conditions must be readily obtainable.
Possible Future Research Directions. We showed that the FDEis capable of modeling different flow regimes, including linearand radial flow. Why do we go through the trouble of combiningwith the trilinear flow model? The Cinco-Ley and Meng (1988)solution has bilinear flow at early times, and it transitions to pseu-doradial flow at late times. Therefore, the flow regimes that arearound a fracture change with time, something that the FDE is notable to capture, in the form that we have presented it. It may bepossible to take the FDE by itself (not coupled to the trilinear flowmodel) and numerically optimize the fractal parameters {D,h}against time to match the Cinco-Ley and Meng (1988) solution.At this point, we are not considering the fractal parameters to bedynamically changing with time, but it is an intriguing possibilitythat may yield interesting results in future investigations.
Conclusions
The following conclusions can be drawn:1. We perform a rigorous analytical study of the FDE (Eq. 9a).
We revisit the FPPR (Eqs. 5 and 6) and discuss their originsand physical meaning, and we show why they have beenignored in the literature. We make a straightforward, but strik-ing observation: Depending on the value of the fractal parame-
ters D and h chosen, it is possible to obtain both the classic-lin-ear and radial-flow solutions. Chang and Yortsos (1990) brieflydiscuss this, but it is only presented as a concomitant result, and,to our knowledge, no further work has been conducted trying toexploit this property.
2. In light of this observation, a rigorous numerical study of theFDE was performed. The pressure signal predicted by theanalytical solution (Eq. 10) was successfully matched bynumerical simulation (Fig. 3). For what we believe is the firsttime, fractal theory is used in reservoir engineering to addressa problem that is not related to heterogeneous media. Webelieve that the FDE may have potential because of its flexibil-ity in describing a multitude of flow regimes.
3. We combined the FDE with the trilinear flow solution (Fig. 4),originally proposed by Lee and Brockenbrough (1986). Wecall this new solution the FFS. We replaced the three originallinear-diffusivity equations of Regions 1, 2, and 3 with theirfractal counterparts (Eqs. 11a, 12a, and 13a). In doing so, weintroduced six unknowns to the problem, the fractal parametersD1, D2, D3, h1, h2, and h3. Our goal is to match the Cinco-Leyand Meng (1988) semianalytical solution for flow in a singlevertical hydraulic fracture.
4. Because it is not obvious what values these parameters shouldhave, we coupled the FFS with the Levenberg-Marquardtnumerical optimization algorithm (Fig. 5). We compared theaccuracy of three different scenarios (Fig. 6). We chose theone with four unknowns by making a compromise of solutioncomplexity and accuracy. We found the parameters to be well-behaved (Fig. 7), and we developed relationships with respectto dimensionless fracture conductivity FcD (Eqs. 16 through19). Finally, we combined all these efforts in a closed-formFFS that has performance (Figs. 10 and 11) comparable withthe Blasingame and Poe (1993) TPRS. We do not recommendthe use of the FFS for values FcD< 0.5.
5. Because of the flexibility of the FFS observed during thecourse of these investigations, we are optimistic about the pos-sibility of extending this methodology to develop solutions forcurrently intractable problems (such as multiple transversefractures or sorption effects)—however, at this point, this ispurely speculative until such work is carried out. Table 3 con-tains all the analytical solutions if the user would want to dohis/her own matches with a different scenario. Furthermore,new solutions developed by use of our methodology will stillneed to be calibrated from a known exact solution—thus, theproblem would only be partially solved, even if promisinghistory matches and production forecasts are achieved.
Nomenclature
a ¼ parameter defined after Eq. 13dA ¼ site-density parameter, m–D
b ¼ parameter defined after Eq. 13dbf ¼ fracture width, mB ¼ formation volume factor, RB/STBct ¼ total compressibility, Pa–1
n ¼ parameter defined in Eq. C-2p ¼ reservoir pressure, Paq ¼ flow rate, m3/sr ¼ radius from center of well, mS ¼ skin factort ¼ time, seconds
Vs ¼ volume per site, m3
92 February 2013 SPE Journal
x,y ¼ space coordinates, mxf ¼ fracture half-length, mz ¼ Laplace-space variablea ¼ parameter defined in Eq. C-2c ¼ parameter defined in Eq. C-2h ¼ conductivity indexl ¼ viscosity, Pa�s [cp]/ ¼ porosity, fractionw ¼ a variable defined in Eq. C-11X ¼ parameter defined in Eq. C-5
Subscripts
D ¼ dimensionlessf ¼ related to the fracturei ¼ initial conditiono ¼ oilw ¼ at the edge of the wellbore
1,2,3 ¼ index of flow region
Acknowledgments
This work was supported by RPSEA (Contract No. 08122-45)through the Ultra-Deepwater and Unconventional Natural Gasand Other Petroleum Resources Research and Development Pro-gram as authorized by the United States Energy Policy Act of2005. In addition, we acknowledge the support of the Supercom-puting Facility at Texas A&M University.
References
Abdassah, D. and Ershaghi, I. 1986. Triple Porosity Systems for Repre-
senting Naturally Fractured Reservoirs. SPE Form Eval 1 (2):
113–127. http://dx.doi.org/10.2118/13409-PA.
Acuna, J.A., Ershaghi, I., and Yortsos, Y.C. 1995. Practical Application of
Fractal Pressure-Transient Analysis in Naturally Fractured Reservoirs.
SPE Form Eval 10 (3): 173–179. http://dx.doi.org/10.2118/24705-PA.
Beier, R.A. 1994. Pressure-Transient Model for a Vertically Fractured
Well in a Fractal Reservoir. SPE Form Eval 9 (2): 122–128. http://
dx.doi.org/10.2118/20582-PA.
Blasingame, T.A. 2010a. Petroleum Engineering 620—Fluid Flow in
In this derivation, we show the main steps for the derivation ofthe FDE in 1D form in a Cartesian coordinate system. The samederivation in complete detail may be found in Cossio (2012).
We start with the continuity equation for a slightly compressi-ble single-phase flow in a porous medium (Lee and Wattenbarger1996).
where x is distance from the center of the wellbore (m); xD isdimensionless distance; A is reservoir cross section (m2); q is pro-duction rate (m3/s); B is formation volume factor (dimensionless);l is viscosity (Pa�s); t is time (seconds); tD is dimensionless time;pD is dimensionless pressure; pi is initial reservoir pressure (Pa); pis sandface pressure (Pa); and ct is total compressibility (Pa–1).
Appendix B—Derivation of the Particular Solutionof the FDE
We will show the main steps to solve Eq. 9a for a no-flow outer-boundary condition. We will not consider wellbore storage or skinfactor here. The same derivation with complete detail may befound in Cossio (2012).
@2pD
@x2D
þ D� h� 2
xD
@pD
@xD¼ xh
D
@pD
@tD;
with the following boundary conditions:
Initial condition; pD ¼ 0 when tD ¼ 0 ðB-1aÞ
Inner-boundary condition;dpD
dxD
� �xD¼1
¼ �1 ðB-1bÞ
No-flow; outer-boundary condition;dpD
dxD
� �����xD¼L
¼ 0;
� � � � � � � � � � � � � � � � � � � ðB-1cÞ
where L is dimensionless distance to the boundary.By taking the Laplace transform of Eq. 9a and by use of B-1a,
we obtain
@2pD
@x2D
þ D� h� 2
xD
@pD
@xD¼ xh
D z pD ðB-1dÞ
According to Bowman (1958), the general solution to Eq.B-1d may be expressed as
pDðxD; zÞ ¼ xaD A In
ffiffizp
cxc
D
� �þ B Kn
ffiffizp
cxc
D
� �� �; ðB-2Þ
where a ¼ hþ 3� D
2, n ¼ hþ 3� D
hþ 2, and c ¼ hþ 2
2.
We also note the useful pressure derivative:
dpD
dxD¼ xaþc�1
D
ffiffizp
A In�1
ffiffizp
cxc
D
� �� B Kn�1
ffiffizp
cxc
D
� �� �� � � � � � � � � � � � � � � � � � � ðB-3Þ
We need to determine the constants A and B. By taking theLaplace transform of both boundary conditions (Eqs. B-1b andB-1c) and solving the following system for A and B,
limxD!1
@PD
@xD
� �¼ �1
z
limxD!L@PD
@xD
� �¼ 0
;
8>><>>: ðB-4Þ
we obtain the following particular solution:
pDðxD;D; h;L; zÞ
¼ xaD
Inxc
D
ffiffizp
c
�Kn�1
Lc ffiffizpc
�þ In�1
Lc ffiffizpc
�Kn
xcD
ffiffizp
c
�z
32 In�1
Lcffiffizp
c
�Kn�1
ffiffizp
c
�� In�1
ffiffizp
c
�Kn�1
Lcffiffizp
c
�h i� � � � � � � � � � � � � � � � � � � ðB-5Þ
We now show that, at the wellbore (xD¼ 1), this result can bereduced to the classic linear- and radial-flow cases:� If we set (xD¼ 1, D¼ 2, h¼ 0, L¼ 2), after some algebraic
This is exactly the classic result from the literature (Blasin-game 2010a) for linear flow.� Similarly, if we set (xD¼ 1, D¼ 3, h¼ 0), we obtain
pDðxD ¼ 1; D ¼ 3; h ¼ 0; L; zÞ
¼ I0ðffiffizpÞK1ðL
ffiffizpÞ þ I1ðL
ffiffizpÞK0ð
ffiffizpÞ
z3=2½I1ðLffiffizp ÞK1ð
ffiffizp Þ � I1ð
ffiffizp ÞK1ðL
ffiffizp Þ� ðB-7Þ
This is exactly the classic result from the literature (Blasin-game 2010b) for radial flow.
Appendix C: Derivation of the Analytical Solutionof the FFS, Scenario 2
A detailed account of this derivation may be found in Cossio(2012).
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94 February 2013 SPE Journal
Region 3
By taking the Laplace transform of Eq. 11a and by use of the ini-tial condition (Eq. 11b), we obtain
@2p3D
@x2D
þ D3 � h3 � 2
xD
@p3D
@xD¼ xh3
D zp3D ðC-1Þ
Just as in Appendix B, the general solution to Eq. C-1 is
p3DðxD; zÞ ¼ xa3
D A3In3
ffiffizp
c3
xc3
D
� �þ B3Kn3
ffiffizp
c3
xc3
D
� �� �� � � � � � � � � � � � � � � � � � � ðC-2Þ
In all cases (here and throughout the paper), ni ¼ hiþ3�Di
hiþ2,
ai ¼ hiþ3�Di
2, and ci ¼ hiþ2
2are constants.
We need to determine the constants A3 and B3. We take theLaplace transform of the two boundary conditions (Eqs. 11c and11d) and solve the following system for A3 and B3:
limxD!1ðp3DÞ ¼ p2D
limxD!1ðp3DÞ ¼ 0
�ðC-3Þ
We found it straightforward to solve such a system of equa-tions by use of software with symbolic math capabilities, such asMathematica (2010). We obtain the particular solution to Eq. C-2as
p3DðxD; zÞ ¼ p2Dxa3
D
Kn3
ffiffizp
c3xc3
D
�Kn3
ffiffizp
c3
� ðC-4Þ
We will also need the following result:
@p3D
@xD
����xD¼1
¼ �p2D XðzÞ; ðC-5Þ
where XðzÞ ¼Kn3�1
2ffiffizp
h3 þ 2
� �
Kn3
2ffiffizp
h3 þ 2
� � ffiffizp
.
Region 2
By taking the Laplace transform of Eq. 12a, with the initialcondition (Eq. 12b), and with Eq. C-5 substituted into Eq. 8a, weobtain
@2p2D
@y2D
þ D2 � h2 � 2
yD
@p2D
@yD¼ yh
D ðzþ XÞp2D ðC-6Þ
The general solution is
p2DðyD; zÞ ¼ ya2
D A2In2
ffiffiffiffiffiffiffiffiffiffiffiffizþ Xp
c2
yc2
D
� �þ B2 Kn2
ffiffiffiffiffiffiffiffiffiffiffiffizþ Xp
c2
yc2
D
� �� �� � � � � � � � � � � � � � � � � � � ðC-7Þ
We need to determine the constants A2 and B2. By taking theLaplace transform of the two boundary conditions (Eqs. 12c and12d) and by solving the following system for A2 and B2,
limyD!1 p2D � S@p2D
@yD
� �¼ p1D
limyD!þ1ðp2DÞ ¼ 0
;
8<: ðC-8Þ
we obtain the particular solution to Eq. C-7 as
p2DðyD; zÞ ¼p1D ya2
D Kn2
ffiffiffiffiffiffiffizþXp
c2yc2
D
�Sffiffiffiffiffiffiffiffiffiffiffiffizþ Xp
Kn2�1
ffiffiffiffiffiffiffizþXp
c2
�þ Kn2
ffiffiffiffiffiffiffizþXp
c2
� ðC-9Þ
We will also need the following result in Region 1:
@p2D
@yD
����yD¼1
¼p1D a2Kn2
ffiffiffiffiffiffiffizþXp
c2
��
ffiffiffiffiffiffiffiffiffiffiffiffizþ Xp
Kn2�1
ffiffiffiffiffiffiffizþXp
c2
�þ n2ffiffiffiffiffiffiffi
zþXp Kn2
ffiffiffiffiffiffiffizþXp
c2
�h in oSffiffiffiffiffiffiffiffiffiffiffiffizþ Xp
Kn2�1
ffiffiffiffiffiffiffizþXp
c2
�þ Kn2
ffiffiffiffiffiffiffizþXp
c2
�� � � � � � � � � � � � � � � � � � � ðC-10Þ
Region 1
By taking the Laplace transform of Eq. 13a, with setting (D1¼2,h1¼0), with Eq. 13b, and by substituting Eq. C-10 into Eq. 13a,we obtain
@2p1D
@x2D
¼ w2p1D; ðC-11Þ
with w ¼�a a2 Kn2
ffiffiffiffiffiffiffiffiffiffiffiffizþ Xp
c2
� ��ffiffiffiffiffiffiffizþXp
Kn2�1
ffiffiffiffiffiffiffiffiffiffiffiffizþ Xp
c2
� �þ
n2ffiffiffiffiffiffiffiffiffiffiffiffizþ Xp Kn2
ffiffiffiffiffiffiffiffiffiffiffiffizþ Xp
c2
� �� ��
SffiffiffiffiffiffiffizþXp
Kn2�1
ffiffiffiffiffiffiffiffiffiffiffiffizþ Xp
c2
� �þKn2
ffiffiffiffiffiffiffiffiffiffiffiffizþ Xp
c2
� � þ C1z
2664
3775
1=2
.
The general solution to Eq. C-11 is
p1D ðS; C1; xD; zÞ ¼ A1e�xDw þ B1exDw ðC-12Þ
We need to determine the constants A1 and B1. By taking theLaplace transform of the two boundary conditions (Eqs. 13c and13d) and by solving the following system for A1 and B1,
limxD!0
@p1D
@xD
� �¼ 1
zbð1� CDf pwD z2Þ
limxD!1
@p1D
@xD
� �¼ 0
8>><>>: ; ðC-13Þ
we obtain the following particular solution:
p1DðCDf ; FcD; S; C1; xD; zÞ
¼ bðCDf pwD z2 � 1Þ ðexDw þ e2w�xDwÞðe2w � 1Þ z w
� � � � � � � � � � � ðC-14Þ
If we solve for pwD in the equation p1D ðxD ¼ 0Þ ¼ pwD, weobtain the solution for the wellbore pressure for the trilinear model,
pwDðCDf ; FcD; S; C1; zÞ ¼ b
zðb CDf z� w tanh ðwÞ ;
� � � � � � � � � � � � � � � � � � � ðC-15Þ
with the same w as defined in Eq. C-11.If we set S¼C1 ¼ CDf ¼ 0, then Eq. C-15 simplifies to
pwDðFcD; zÞ ¼ pFcD z w tanh ðwÞ ; ðC-16Þ
with
w ¼ 1
FcD2ffiffiffiffiffiffiffiffiffiffiffiffizþ Xp Kn2�1
2ffiffiffiffiffiffiffizþXp
h2þ2
�Kn2
2ffiffiffiffiffiffiffizþXp
h2þ2
� � h2n2
24
35
8<:
9=;
1=2
:
This solution can then be inverted numerically from Laplacespace by use of the Gaver-Stehfest algorithm (Stehfest 1970).
Manuel Cossio is a Reservoir Engineer for Pioneer NaturalResources. His research interests include numerical analysis,fluid flow in porous media, reservoir simulation and unconven-tional reservoir systems. He holds an MEng in AeronauticalEngineering from Imperial College London (2006) and an MSin Petroleum Engineering from Texas A&M University, CollegeStation (2012).
George Moridis has been a Staff Scientist in the Earth SciencesDivision of LBNL since 1991, where he is the Deputy Program
Lead for Energy Resources, is in charge of the LBNL researchprograms on hydrates and tight/shale gas, and leads thedevelopment of the new generation of LBNL simulation codes.Moridis is a visiting professor in the Petroleum Engineering Dept.at Texas A&M University, and in the Guangzhou Center for GasHydrate Research of the Chinese Academy of Sciences; he isalso an adjunct professor in the Chemical Engineering Dept. atthe Colorado School of Mines, and in the Petroleum and Natu-ral Gas Engineering Dept. of the Middle East Technical Univer-sity, Ankara, Turkey. He holds MS and PhD degrees from TexasA&M University and BS and ME degrees in chemical engineer-ing from the National Technical University of Athens, Greece.Moridis is the author or coauthor of over 65 papers in peer-reviewed journals, more than 175 LBNL reports, paper presen-tations and book articles, and three patents. He was an SPE(Society of Petroleum Engineers) Distinguished Lecturer for2009–10, and was elected an SPE Distinguished Member in2010. He is the recipient of a 2011 Secretarial Honor Award ofthe U.S. Department of Energy. He is an Associate Editor of fourscientific journals, and a reviewer for 24 scientific publications.
Tom Blasingame is a Professor and is the holder of the RobertL. Whiting Professorship in the Department of PetroleumEngineering at Texas A&M University in College Station Texas.He holds BS, MS, and PhD degrees from Texas A&M University —all in Petroleum Engineering. In teaching and research activ-ities he focuses on petrophysics, reservoir engineering, analysis/interpretation of well performance, and technical mathemat-ics. Blasingame is a Distinguished Member of the Society ofPetroleum Engineers (2000) and a recipient of the SPE Distin-guished Service Award (2005), the SPE Uren Award (for technol-ogy contributions before age 45) (2006), the SPE Lucas Medal(SPE’s preeminent technical award) (2012), and has served asan SPE Distinguished Lecturer (2005-2006). Blasingame has pre-pared more than 100 technical articles, and he has chairednumerous technical committees and technical meetings. Bla-singame also served as Assistant Department Head (GraduatePrograms) for the Department of Petroleum Engineering atTexas A&M from 1997 to 2003, and he has been recognizedwith several teaching and service awards from Texas A&MUniversity.
