This paper was accepted for presentation at the SPE International Oil Conference and Exhibition in Mexico held in Cancun, Mexico, 31 August–2 September 2006, and revised for publication. Original manuscript received for review 8 November 2010. Revised paper received for review 31 March 2011. Paper peer approved 12 April 2011 as SPE paper 104028.
SummaryMethodology is presented and proved for determination of the best-estimate parameter values affecting the matrix/fracture-inter-face fluid transfer in naturally fractured reservoirs. Fracture/sur-face-hindered interface transfer of immiscible fluids is consid-ered between matrix blocks and surrounding natural fractures. Improved matrix/fracture-transfer models are applied on the basis of presumed matrix-block shapes. Analytical solutions and the lim-iting isotropic-matrix long-time shape factors developed for special boundary conditions are used for interpretation of typical labora-tory tests conducted using rectangular- and cylindrical-shaped rock samples. Workable equations and straight-line data-plotting schemes are developed for effective analysis and interpretation of laboratory data obtained from various-shaped oil-saturated reser-voir-rock samples immersed into brine. Applications concerning the water/air and water/decane systems in laboratory core tests are also presented. The present approach allows rapid determination of the characteristic parameters of the matrix/fracture-transfer models for various-shaped matrix blocks, which are essential for predic-tion of petroleum recovery from naturally fractured reservoirs. The methodology is verified using various experimental data, and the values of the characteristic parameters (e.g., the average diffusion-coefficient and the interface-skin-mass-transfer coefficient) are determined. The Arrhenius (1889) equation is shown to represent the temperature dependency of these parameters effectively.
IntroductionThis article provides a significant advancement over previous attempts toward an improved formulation of the matrix/fracture transfer functions by taking into account the hindered-interface transfer of immiscible fluids. This is accomplished on the basis of the analytical procedures derived from the rigorously linearized and transformed diffusion equation. A practical methodology is developed and implemented with conventional laboratory tests for determination of the effective parameters of the matrix/frac-ture-transfer functions (e.g., the average diffusion coefficient and interface-skin-mass-transfer coefficient). The proposed analytical approach is instrumental for accurate fitting of the hindered matrix/fracture-transfer model to a variety of test data and determination of its parameters conveniently.
Flow through naturally fractured petroleum reservoirs is gener-ally described by simultaneous numerical solution of the equations describing the flow of gas, oil, and brine phases in the fracture and matrix media, coupled with the matrix/fracture interface-bound-ary conditions. The overwhelming computational effort required for this purpose is usually alleviated on the basis of a fracture-flow and matrix-source/sink formulation. This requires the use of approximate analytical matrix/fracture interface-transfer func-tions derived under certain simplifying assumptions (de Swann 1978, 1990; Civan 1998; Rasmussen and Civan 1998; Civan and Rasmussen 2001). Frequently, the parameters of the transfer func-
tions are estimated by laboratory core imbibition tests or reservoir production history matching. There is a considerable amount of experimental data available in the literature, but the data-analysis and -interpretation techniques lack in accuracy and suitability for practical applications. This article presents a rigorous approach and demonstrates its validity by effective analysis and interpretation of various experimental data.
The derivation of adequate transfer functions expressing the matrix/fracture interface flow has attracted significant attention. Consequently, numerous theoretical models (Warren and Root 1963; de Swann 1978, 1990; Kazemi et al. 1976, 1992; Moench 1984; Zimmerman et al. 1993; Civan 1993, 1994a; Gupta and Civan 1994; Lim and Aziz 1995; Reis and Cil 2000) and empirical correlations (Zhang et al. 1996; Guo et al. 1998; Matejka et al. 2002; Gallego et al. 2007) have been proposed. In numerous cases, theoretical trans-fer functions have been developed using the approximate analytical solutions obtained from the simplified and/or linearized diffusion equations for presumed-shape matrix blocks subject to certain matrix/fracture interface-boundary conditions.
The fracture surface may be damaged by various mechanisms in petroleum reservoirs. Impairment of permeability by deposition of mineral matter and debris and by mechanical processes is referred to as formation damage (Civan 2007a). Buildup of a stagnant thin fluid film (stationary hydrodynamic boundary layer) over the fracture surface is referred to as the pseudodamage (Civan 2007a). The combined effects of such adverse processes may severely hin-der the exchange of the fluid phases across the interface between matrix block and fracture medium. Mostly, the previous approaches assumed constant fluid conditions (Dirichlet type) (e.g., prescribed saturation) over the surfaces of matrix blocks. Then, Duhamel’s rule has been applied to account for variable-fracture-fluid condi-tions (de Swann 1978, 1990). In contrast, Moench (1984), Wal-lach and Parlange (2000), and Rasmussen and Civan (2003, 2006) considered the fracture-skin effect (Cauchy boundary condition) caused by the formation and/or pseudodamages.
Exhaustive experimental data are available in the literature from laboratory imbibition tests using various-shaped reservoir-core samples and fluids (Zhang et al. 1996; Ma et al. 1997; Reis and Cil 2000; Xie and Morrow 2001). Laboratory measurement of the matrix/fracture-interface flow is usually carried out using small-sized rock samples (dimensions on the order of a few inches) under certain prescribed conditions suitable for convenient interpretation of experimental measurements. Laboratory tests have been conducted using rectangular-, cylindrical-, and spherical-shaped reservoir-rock samples exposed to various boundary conditions (Fig. 1).
Conventional laboratory testing of core samples is rather straightforward, carried out by immersion of oil-saturated core samples into brine and measuring the recovery of oil during water imbibition at different times (Matejka et al. 2002). Cil (1997), however, conducted similar tests with water/air and water/decane systems. Nevertheless, the interpretation of experimental measure-ments is a rather challenging task, requiring an accurate model for proper formulation of the exchange of immiscible fluids across the matrix/fracture interface. The differential equation and its bound-ary conditions describing the flow of immiscible fluids in a matrix block by capillary diffusion are highly nonlinear, and an adequate approximate analytical solution of this equation is required in order to be able to devise a reliable direct analytical interpretation
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procedure for unique determination of the best-estimate parameter values.
Therefore, the primary focus of the present study is a rigorous characterization of the exchange of immiscible fluids across the matrix/fracture interface by model-assisted analysis of laboratory core tests. Our studies provide some critical insights for proper determination of the parameters of the transfer functions used in simulation of naturally fractured reservoirs using the analytical solutions obtained from a rigorously linearized nondimensional diffusion equation (Civan and Rasmussen 2006). The approach facilitated here for proper modeling of flow through a matrix block by capillary diffusion can be referred to as the leaky-tank model (Civan 2000). Also, the transfer functions obtained for prescribed-shape matrix blocks are dependent on the shape of matrix blocks. However, because the analytical solution of the linearized diffusion equation was derived assuming a constant fracture-fluid saturation, ultimately the application of Duhamel’s rule is required to account for variable fracture-fluid saturation in reservoir simulation, as demonstrated by de Swann (1978, 1990), and Civan (1994a, 1998).
For this purpose, we elaborate upon the practical applications of importance concerning the hindered-matrix/fracture-transfer models, and the data analysis and interpretation method proposed by Civan and Rasmussen (2003, 2005, 2006). This approach is significant because it not only provides a best estimate of the average capillary-diffusion coefficient but also the value of the interface-skin-mass-transfer coefficient. These are the essential parameters required for accurate determination of the matrix/
fracture interface-transfer functions. Further, we show that the temperature dependency of these parameters can be represented adequately using the Arrhenius (1889) equation.
Civan and Rasmussen (2003, 2005, 2006) derived the special analytical solutions of the transient-state capillary-diffusion equa-tion for the various-shaped matrix blocks depicted in Fig. 1 under various boundary conditions by taking advantage of the symmetry. The limiting isotropic-matrix long-time shape factors for various-shaped matrix blocks subject to different boundary conditions are presented. These can provide convenience in interpretation of the conventional laboratory core tests. They presented the full-time solution and the short- and long-time asymptotic analytical solu-tions for 1D, 2D, and 3D Cartesian matrix blocks, circular-cylinder and annular-shaped matrix blocks, and spherical matrix blocks. These analytical matrix/fracture-transfer functions are used in the present article to analyze and interpret various experimental data obtained with different-shaped reservoir rock samples, and the characteristic parameters of the transfer functions are deter-mined. Simultaneously, these exercises confirm the presence of the hindered-interface-fluid-transfer mechanism and the validity of our approach.
FormulationBriefly, our formulation describes the isothermal capillary-diffu-sion phenomenon in a water-wet matrix block having a prescribed initial fluid condition (oil-saturated) suddenly immersed into an infinite medium having a different fluid condition (water). This causes a matrix-to-fracture oil transfer by spontaneous water imbibition. We consider a single capillary-diffusion coefficient that is dependent upon saturation, assuming that the immiscible fluid phases [e.g., oil and water (brine)], act as single pseudocomponents (fixed-composition mixtures). This is commensurate with the common approach followed by the Buckley-Leverett formulation of waterflooding of oil reservoirs. The details of the mathemati-cal modeling and analytical solutions are presented elsewhere by Civan and Rasmussen (2005) and Rasmussen and Civan (2006). In the following sections, the workable equations of this model are summarized and the results are manipulated to develop and apply the various convenient straight-line plotting schemes. These provide practical means of determining the best-estimate values of the average capillary-diffusion coefficient over the mobile-fluid-saturation range and the interface-skin-mass-transfer coefficient. The latter cannot be determined by the previous interpretation methods, assuming prescribed saturation values at the matrix-block surface.
Basic Equations of Fluid Transfer Through Matrix Blocks. The formulation is presented in general for any shape of matrix block having a volume V and surrounded by an external surface of area A exposed to the fracture fl uid. We consider an immiscible and incompressible two-phase fl uid system (e.g., oil and water). The nonwetting and wetting phases are indicated by the subscripts n and w. The symbol Sw denotes the saturation of the wetting phase, and Swi and Sni denote the immobile wetting and nonwetting-phase saturations in the matrix, respectively. Swf is the wetting-phase saturation in the fractures surrounding the matrix block, assumed constant, commensurate with the usual experimental protocol. However, the matrix/fracture-transfer function obtained by the analytical solution generated using an arbitrarily or conveniently selected constant fracture-wetting-phase saturation Swf value can be used readily by applying Duhamel’s theorem to account for the effect of the variable fracture-wetting-phase saturation Swf occur-ring in the fractures of naturally fractured reservoirs, according to Rasmussen and Civan (1998) and Civan et al. (1999).
The transport of the wetting phase inside a matrix block (Fig. 2) can be described by the following equations (Civan and Rasmussen 2005). The flow of the wetting phase in a matrix block by capillary diffusion is described by
∂( )∂
= ∇ ∇⎛⎝⎜
⎞⎠⎟
��� �w w
w c w
S
tD
KSi i
K, . . . . . . . . . . . . . . . . . . (1)
Fig. 1—Representing the matrix blocks in laboratory tests by parallepiped, cylinder and spherical shaped rock samples with different boundary conditions. (1) Table 1 and Table 4/SS3, all-faces-open parallelepiped; (2) Table 3/SS4, sides-open par-allelepiped; (3) Table 3/SS1, two-adjacent-faces-closed parallel-epiped; (4) Table 3/S15-1, sides-closed parallelepiped; (5) Table 3, all-faces-open cylinder; (6) Table 3/SA2-3 annular; (7) Table 3/S2-6, side-open cylinder; (8) Table 3/S15-1, side-closed cylin-der; (9) Table 3/BC13, one-end-open cylinder (Zhang et al. 1996; Xie and Morrow 2001; Guo et al. 1998); and (10) spherical.
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where t is time; � is porosity of matrix; � and K are the average porosity and permeability, respectively; and K is the permeability tensor of the matrix block. �w is the density of the wetting fluid. The capillary-diffusion coefficient Dc is defined as
D D S Fk p
S
Kc c w w
r n
n
c
w
= ( ) = −⎛⎝⎜
⎞⎠⎟� �
d
d. . . . . . . . . . . . . . . . . . . . (2)
The fractional-wetting-phase function Fw is given by
Hence, the capillary-diffusion coefficient is a scalar function (Civan 1994b), which depends on the average porous matrix properties (�, K ) and dimensions, the wetting-phase and nonwet-ting-phase viscosities (� �w n, ), the petrophysical properties (e.g., the relative permeability of the wetting and nonwetting phases), and the capillary pressure (k k prw rn c, , ), which are functions of the wetting-phase saturation Sw.
The initial condition inside the matrix block is given by the uniformly distributed immobile wetting-phase saturation Swi
S S tw wi= =, 0. …………..…………………………….. (4)
The symmetry condition imposed along a plane passing through the block center is
− ∇ = >DK
S tc w�K
ni i 0 0, . …………...…………… (5)
The imbibition boundary condition over the surfaces of the matrix block is
where �s = Ds /bs is defined as a phenomenological skin mass-transfer coefficient associated with the variable Sw. This relates the macromechanics phenomenological parameters to the micro-mechanics parameters. The symbols Ds and bs represent the skin diffusivity and thickness, respectively.
We apply two consecutive transformations on Eqs. 1 through 6 to obtain a linearized model that can be used for convenience in analyzing experimental data obtained with different-shaped matrix blocks, as described in the following.
First-Level Transformations. Allow S to denote the normalized wetting-phase saturation, given by
In addition, we represent the Dc function given by Eq. 2 as a product of a dimensional constant Do and a dimensionless function f(S) of the normalized saturation S as
D S D S D f Sc w c o( ) = ( ) = ( ) . . . . . . . . . . . . . . . . . . . . . . . . . (8)
We define Dm as the average value of Dc over the mobile-wet-ting-fluid-saturation range, according to
D D s s D S S D f Sm c w wS
S
c owi
ni= = ( ) = ( )−
∫ ∫( )d d d1
0
1 � � � ��S0
1
∫ , . . . . . (9)
where sw is the wetting-fluid-saturation variable of integration and �S is the normalized wetting-phase saturation variable of integra-tion. A numerical example illustrating the application of Eq. 9 is given in Appendix A.
Next, a new dependent variable �C, referred to as the pseudo-saturation here, is defined as
�CD S S
D S S
D Sc w wS
S
c w wS
S
c wwi
w
wi
ni≡
( )( )
=( )∫
∫−
d
d1
dd
d d
S
D
D S S D S S
wS
S
m
w wS
S S
wi
w
wi
w
∫
∫ ∫= ( ) = ( )� �0
. . . . . . . . . . . . . . (10)
Notice that, only as S varies between zero and unity, its transformed counterpart �C does also; that is, �C S =( ) =0 0 and�C S =( ) =1 1 .
The nondimensional distances, time, and diffusivity are defined, respectively, by
We define K as a geometric average permeability of a matrix block, given as K K K Kx y z≡ ( )1 3
, where K K Kx y z, ,and are the permeability components in the Cartesian x-, y-, and z- direction, assumed constant, and L denotes a geometric-average length for a matrix block given by L V= 1 3/ . Thus, for example, for the various matrix shapes considered in the present study (Fig. 1) it would be given by the following expressions:
Ly
Lz
x
y
z
O
Fracture Matrix
Lx
Lx/2
Lz/2
Ly/2
Skin
Transfer
bs
Fig. 2—First octant of a rectangular-parallelepiped-shaped ma-trix block and schematic surface skin hindering fluid transfer across the matrix block and the surrounding fracture.
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L L L L
L L
x y z= ( )=
1 3/(parallelepiped)
(cube)
L R= ⎛⎝⎜
⎞⎠⎟
4
33
1 3�
/
(sphere)
(cylL R Lz= ( )� 2 1 3/iinder)
, . . . . . . . . . . . . . . . . (12)
where L L Lx y z, ,and are dimensions in the Cartesian x-, y-, and z- direction, and R is the radius.
We assume constant fluid and rock properties, but anisotropic rock permeability. Consequently, substituting Eqs. 7 and 11 into Eqs. 1 and Eqs. 4 through 6 yields, respectively,
where �s is the interface-skin-mass-transfer coefficient and Kn is the permeability component in the normal unit vector direction.
Notice that we still need to express Sw in Eq. 16 in terms of �C ,as shown in the following subsection.
Second-Level Transformations. Consider the following truncated Taylor-series expansion of the integral given in Eq. 10 so that we can express the right side of Eq. 16 in terms of the transformed variable �C :
where S* is an appropriately selected value such that the truncated Taylor series closely approximates the integral given by Eq. 18, as illustrated in Appendix A by means of an example.
Consequently, eliminating Sw between Eq. 16 and Eq. 18 yields
Further, we introduce a pseudotime or new independent time-transformed variable T according to Rasmussen and Civan (2006) such that
� � �= ( )∂ ≈ ∂( , ), .X T D S T . . . . . . . . . . . . . . . . . . . . . . . . (24)
Consequently, applying Eqs. 20 through 24 to Eqs. 13 through 16, and then neglecting the residual terms arising from lineariza-tion, the transfer of the wetting phase by countercurrent nonwetting-and wetting-phase displacement inside the matrix blocks owing to the capillary-diffusion mechanism can be described adequately by the following general nondimensional linearized formulation (Rasmussen and Civan 2006):
∇ = − >C C Tnin � , on matrix/fracture interface, 0.. . . . . . . (28)
The linearized model presented here enables analytical solu-tions for various matrix-block shapes, as described in Appendix B (Civan and Rasmussen 2005).
Shape Factor. Defi ne an average dimensionless transformed-satu-ration function C over the matrix block by
For convenience, we express the rate term d dC T/ in terms of the average dimensionless transformed-saturation function C inside the matrix block relative to the dimensionless transformed-saturation function in the fracture medium (Warren and Root 1963; Kazemi et al. 1976, 1992). This can be accomplished by integrating Eq. 25 over the matrix block, assigning the value of the resulting integral term, and expressing the matrix/fracture-interface transfer as shown here:
where d is called the nondimensional shape factor and clearly depends on the shape of the matrix block over which the integra-tion is taken. Hence, when C is a known function of T, then Eq. 30 can be used as a defining equation for a time-variable shape factor d . The shape factor varies with time for short times, but it attains a constant-limit value for long times. The long-time limits of the shape factors presented in Table 1 for various configurations have been determined as described in Appendix B.
Further, integrating Eq. 30 and applying Eq. 11 yield
C T C T To
T
( ) = − ( )⎡
⎣⎢
⎤
⎦⎥∫exp d d′ ′
0
. . . . . . . . . . . . . . . . . . . . . (31)
Or
C t CL
D t t to m
t
( ) = − ( ) ( )⎡
⎣⎢
⎤
⎦⎥∫exp ,
12
0
′ ′ ′d d . . . . . . . . . . . . . . (32)
where T' and t' denote the variables of integrations.
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Recovery Factor. The volume q(t) of an incompressible fl uid accu-mulated in a matrix block of volume V in the time t is given by
q t q t t V C Tt
( ) ( ) ( ) ,= = − −⎡⎣ ⎤⎦∫ � d0
1� . . . . . . . . . . . . . . . . . . (33)
where � denotes the effective porosity excluding the volume occupied by the immobile fluids (i.e., the connate water and residual oil) present in the pore space. According to the posing of our problem, the volume of the fluid q(t) that diffused out in the time t is the volume of the fluid that is gained or recovered by the
TABLE 1—ISOTROPIC-MATRIX SURFACE-SKIN-CONDITION DEPENDENCY OF THE LIMITING LONG-TIME SHAPE FACTOR
Matrix Block Type
Lim and Aziz (1995) Present Study
Zero Skin Finite Skin
Zero Skin
β → ∞
Shape Special Cases LT
dσ LTdσ Eigen equation
oK LTdσ
Infinite-Surface Slab ( )L
Ends open and sides closed cube
2π 24 oK 2 tano oK K = β / 2π 2π
Infinite-Long Square-
Column 2( )L Ends closed and sides open cube
22π 28 oK 2 tano oK K = β / 2π 22π
Cube 3( )L All sides open 23π 212 oK 2 tano oK K = β / 2π 23π
Cube With One Side Closed _ _ 2
24 24o
oK
K∗⎡ ⎤
+⎢ ⎥⎢ ⎥⎣ ⎦
tano oK K∗ ∗ = β
2 tano oK K = β
/ 2π
/ 2π
294
π
Cube With Two Adjacent Sides Closed _ _ 2
242o
oK
K∗⎡ ⎤
+⎢ ⎥⎢ ⎥⎣ ⎦
tano oK K∗ ∗ = β
2 tano oK K = β
/ 2π
/ 2π
232
π
Cube With Three Adjacent Sides Closed _ _ 23 oK∗ tano oK K∗ ∗ = β / 2π 23
4π
Cube With Only One Side Open _ _
2oK ∗ tano oK K∗ ∗ = β / 2π 2 / 4π
Infinite-Long Cylindrical-Column
2 2
z
R L
L L
π ==
Ends closed and sides open
5.78
18.17
π =
2oKπ 1
0
( )
( )o
oo
J KK
J Kβ=π
2.405 5.78π
Cylinder With One End Closed And Sides Open
zL
R
=
π
_ 2 2o oK K∗π + 1
0
( )
( )o
oo
J KK
J Kβ=π
tano oK K∗ ∗ = β
2.405
/ 2π 2
5.78
/ 4
π
+ π
Cylinder With All Sides Open
zL
R
=
π
_ 2 24o oK K ∗π + 1
0
( )
( )o
oo
J KK
J Kβ=π
2 tano oK K∗ ∗ = β
2.405
/ 2π 2
5.78π
+ π
Cylinder With Only One End Open
zL
R
=
π
_ 2oK ∗ tano oK K∗ ∗ = β 2π 2 / 4π
Annular Cylinder With All Sides Open
/o iR Rη ≡ 2(1 )
z
o
L
R= π − η
_ 2 2
2
(1 )
4
o
o
K
K ∗
− η π
+
2
2
42
(1 )
b b aca
− + −
β=− η π
2 tano oK K∗ ∗ = β
( )
( )
( )
( )
o o
o o
o o
o o
J K
Y K
J K
Y K
η=η
,
( 0)a →
/ 2π
2 2
2
(1 ) oK− η π
+π
Sphere
3 34( )3
R Lπ = Open ( )2 3 8 34 3
25.67
π
=
2/324
3 oKπ⎛ ⎞
⎜ ⎟⎝ ⎠ 1
tano
o
K
K− = β
π 2/38/34
3⎛ ⎞ π⎜ ⎟⎝ ⎠
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external region that is the fracture medium. This is defined by the recovery factor RF(T) as
The recovery factor is a measured quantity in the laboratory. A theoretical expression for the recovery factor as a function of time can be obtained by inserting C T( ) in place of RF(T) by means of Eqs. 31 and 34 (Civan and Rasmussen 2003, 2005).
ln ( ) ln ( ) ( ) ,1−[ ] = ⎡⎣ ⎤⎦ −RF T C ToLT
dLT� � . . . . . . . . . . . . . (35)
where the superscript LT denotes the long-time behavior.Note that we obtained Eq. 35 by solving Eq. 31 using the con-
stant value of the limiting shape-factor attained as T → ∞ (Table 1). The semilog graph emphasizes the long-time behavior. A different sort of graph is more suitable to emphasize the short-time behavior. The short-time behavior is different from the early-time behavior. An examination of the functions that describe the short-time behav-iors of C T( ) suggests that plotting C T( ), and thus, RF(T) vs. T , is appropriate for capturing the essence of the small-time behavior (Civan and Rasmussen 2003).
Recovery Rate. The volume rate of fl uid recovered by the fracture medium is equal to the volume rate �q t( ) of the incompressible fl uid diffusing out of the matrix block. On the basis of Eqs. 11, 12, 31, and 33, the semilog plot of experimental data is given by (Civan and Rasmussen 2003)
ln ( ) ln ( ) ( ) ( )�q T C D LdLT
oLT
m dLT[ ] = Δ⎡⎣ ⎤⎦ − � � � � TT , . . . . . . (36)
where Δ = −S Swi wf . Alternatively, in dimensionless form, recovery rate is
ln( )
ln ( ) ( )�q T t
qCc
dLT
oLT
dL
∞
⎡
⎣⎢
⎤
⎦⎥ = Δ⎡⎣ ⎤⎦ − � � TT T( )� , . . . . . . . . (37)
in which t qc and ∞ denote the characteristic diffusion-time scale and the characteristic pore volume, given respectively by
Matrix/Fracture-Transfer Function. Substituting Eq. 31 into Eq. 33 and taking the derivative with respect to time results in
�q T VC T T To d d
T
( ) exp= − ( ) − ( )⎡
⎣⎢
⎤
⎦⎥∫� ′ ′d
0
. . . . . . . . . . . . (40)
Or
�q t VC tL
D t t to d m d
t
( ) exp= − ( ) − ( ) ( )⎡
⎣∫�
12
0
′ ′ ′d⎢⎢⎤
⎦⎥ . . . . . . . . (41)
Hence, once the analytical solution of the linearized diffusion equation has been obtained, as described in Appendix B accordingto Rasmussen and Civan (2003, 2006), then the shape factor can be calculated using Eq. 30 and can be substituted into Eq. 41, which constitutes the matrix/fracture-transfer function required for naturally-fractured-reservoir simulation. A typical example of the application of the matrix/fracture-transfer functions is given elsewhere by Civan and Rasmussen (2001).
Data Analysis and Interpretation MethodologyThe present methodology yields the average value Dm (Eq. 9) of the capillary-diffusion coefficient Dc, but not the capillary-diffusion function itself, defined by Eqs. 2 and 8. Its value is independent of the shape of the matrix block. Hence, the same value should be
obtained regardless of the shape of the core samples used in coun-tercurrent-diffusion experiments. This average value Dm of the diffu-sion coefficient obtained here is not intended to describe the binary (wetting/nonwetting) diffusion in the porous medium. Instead, it is used to quantify the interface-skin-mass-transfer coefficient �s by Eqs. 17 or 20 using the Dm value. The capillary-diffusion coefficient Dc is calculated by Eq. 2. Once the Dm value has been obtained, a reasonable approximation to the capillary-diffusion coefficient Dc may be obtained using this value, as illustrated in Appendix A.
The methodology used for the analysis of experimental data of the recovery factor and rate types is described in the following subsections. It is demonstrated that the long-time approximation offers a means for extracting the values of the average diffusion coefficient Dm , the Biot number �, and the interface-skin-mass-transfer coefficient �s from a well-defined set of data, providing the recovery factor as a function of time, especially for large times. For this purpose, the parameters given in Table 1 for different matrix-block shapes are facilitated. A discussion of the effect of the measurement errors is presented in Appendix C.
Data Analysis Using the Recovery Factor. If the semilog graph of Eq. 35 is plotted vs. the physical time, then the long-time approximation can be identifi ed with a straight-line correlation of the form
Note that a and b should both be positive, and b should have the units of per unit time. The y-axis intercept relation allows the interface-mass-transfer Biot number � to be determined; that is,
Corresponding relations can be established if base-10 logarithms are used. According to the theory, there is a limiting value of the y-axis intercept parameter a that corresponds to the case where � → ∞ (Civan and Rasmussen 2005). For example, this corre-sponds to Ko = � / 2 for the isotropic cube (Table 1), for which
a = ( ) =3 8 0 632ln .� . The interface-skin-mass-transfer coefficient �s is calculated by Eq. 17 using the Dm value calculated by Eq. 46.
Data Analysis Using the Recovery Rate. If the semilog graph of Eq. 37 is plotted vs. the physical time, then the long-time approxima-tion can be identifi ed with a straight-line correlation of the form
After � is determined, then �dLT ( ) can be determined and the
average-diffusion coefficient Dm can be obtained from the slope relation given by Eq. 46. Corresponding relations can be established if base-10 logarithms are used. The interface-skin-mass-transfer coefficient �s is calculated by Eq. 17 using the Dm value.
Applications and ResultsVarious experimental data of the recovery factor types and rate types vs. time are analyzed by the present methodology. These applications demonstrate that hindered-transfer functions should
be considered for accurate interpretation of the matrix/fracture-interface flow. Simultaneously, the best-estimate values of the average-capillary-diffusion and interface-skin-mass-transfer coef-ficients are determined using the straight-line plotting schemes developed in the preceding section. The applicability of the Arrhe-nius (1889) equation for effective correlation of the temperature dependency of these parameters is also demonstrated.
Analysis of Experimental Data Using the Recovery Factor. Cil (1997) and Reis and Cil (2000) used parallelepiped-shape fi red-Berea-sandstone matrix blocks. The blocks saturated with a non-wetting fl uid (air or decane) were fully immersed and suspended in distilled water. Those authors determined the matrix block’s average water saturation and the cumulative fl ow across the total matrix-block surface with time based on the measurements of the weight of the suspended blocks vs. time and application of Archi-medes, principle of buoyancy. The properties of the various sys-tems tested by Cil (1997) and Reis and Cil (2000) and parameters calculated in the present study are reported in Table 2.
Fig. 3 shows the least-squares linear correlation of the Cil (1997) recovery-factor-vs.-time data according to Eq. 42 for the water/air systems at 1, 23, 43, 60, and 81oC and for one water/dec-ane system. As shown in Fig. 4, when applying an Arrhenius-type equation (Arrhenius 1889), the correlations of Dm sand � with the absolute temperature are obtained as follows:
Fig. 3—Linear correlation of the Cil (1997) measured data of recovery factor (RF) vs. time (t) using the present model.
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required for capillary diffusion and interface fluid transfer (cal/mol), respectively; K( ) is the absolute temperature; Rr is the coefficient of regression; and �R (1.987 cal/mol-K) is the universal gas constant.
The preceding correlations were developed in temperature, rather than viscosity and interfacial tension (IFT), for the following reasons. First, temperature is a more-practically-measurable inde-pendent variable than viscosity and IFT. Second, correlations of temperature-dependent parameters should not be attempted using temperature-dependent variables, to avoid compounding the errors associated with their measurement (Civan 2007b, 2011).
Reis and Cil (2000) did not report the permeability and porosity of their matrix block. Therefore, we could not determine the value of the time-scaling factor from their dimensionless plots of data. Consequently, we could not determine the values of � �, ,s mDand . Thus, these values are not reported in Table 2. As can be observed from Fig. 5, their data can be more-meaningfully plotted linearly according to Eq. 42 in the semilog coordinates. Using a nondimen-sional time does not affect the straight-line plotting of their data. Note that the experimental data deviate from the straight line as the
measurement errors are amplified at later times when approaching the steady state, as explained in Appendix C.
Guo et al. (1998) tested all-faces-open oil-saturated cylindrical siltstone core samples with a brine/oil system from the Spraberry reservoir. They measured the fractional oil recovery as a function of time. Figs. 6 and 7 show the long-time linear behavior of the recovery-factor-vs.-time response at different conditions according to Eq. 42 in the semilog coordinates. The values of the core and fluid properties, and the parameters calculated, are reported in Table 3.
Zhang et al. (1996) and Xia and Morrow (2001) tested oil-saturated cylindrical-, annular-, and rectangular-shaped water-wet Berea sandstone cores under various boundary conditions. They partially coated selected surfaces of the core samples with epoxy resin and subjected them to water imbibition. They measured the fractional oil recovery as a function of time. Fig. 8 shows the long-time linear behavior of the recovery-factor-vs.-time response of various core and fluid systems used by Xia and Morrow (2001)
according to Eq. 42 in the semilog coordinates. Fig. 9 shows the similar result for the core and fluid system used by Zhang et al. (1996). The values of the core and fluid properties, and the param-eters calculated, are reported in Table 4.
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.0027 0.0032 0.0037
ln(1
03 D
iffus
ion
coef
ficie
nt, c
m2 /
s) o
r ln
(103
Ski
n m
ass-
tran
sfer
coe
ffici
ent,
cm/s
)
Reciprocal temperature, 1/K
Diffusion coefficient
Skin mass-transfer coefficient
Fig. 4—Temperature dependency of the diffusion coefficient and skin-mass-transfer coefficient for water/air-system data of Cil (1997).
–3.5
–3.0
–2.5
–2.0
–1.5
–1.0
–0.5
0.0
0 1 2 3 4 5
log
(1–R
F)
T, Dimensionless time
Fig. 5—Linear correlation of the Reis and Cil (2000) recovery-factor vs. dimensionless-time data.
–0.55
–0.50
–0.45
–0.40
–0.35
–0.30
–0.25
–0.20
–0.15
–0.10
–0.05
0.00
0 5 10 15 20
Rec
over
y Fa
ctor
, log
(1–R
F)
Time, days
SP-33SP-11SP-1SP-21SP-24SP-27HP-R4
Fig. 6—Linear correlation of the Guo et al. (1998) measured data of recovery fraction (RF) vs. time (t) using the present model.
–0.15
–0.10
–0.05
0.00
0 20 40 60 80 100 120
Rec
over
y Fa
ctor
, log
(1–R
F)
Time, days
SP-H9
SP-H8
Fig. 7—Linear correlation of the Guo et al. (1998) data of recov-ery fraction (RF) vs. time (t) using the present model.
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Analysis of Experimental Data Using the Recovery Rate. As a part of their experimental studies described previously, Reis and Cil (2000) also measured the rate of fl ow across the total matrix block surface with time. As can be observed from Fig. 10, their data can be plotted linearly according to Eq. 47 in the semilog coordinates. The values of the parameters are given in Tables 3 and 4. Again, the experimental data deviate from the straight line as the measurement errors are amplifi ed at later times when approaching the steady state. (See Appendix C for explanation).
ConclusionsThe following conclusions are offered on the basis of the present studies and results:• We demonstrated the necessity of considering a hindered matrix/
fracture interface-fluid-transfer function because the fracture-surface conditions in petroleum-bearing rocks usually contain some formation damage and/or pseudoskins that can limit the matrix/fracture-interface fluid transfer. The finite-skin effect may result from formation of a stationary fluid film over the matrix block surfaces, referred to as pseudoskin, or from an actual mechanical skin because of formation damage by various processes, including deposition of minerals and other debris and by adverse rock/fluid interactions.
• Our improved nonlinear differential model adequately repre-sents the immiscible-fluid displacement taking place inside a matrix block by capillary-diffusion by considering a fracture/sur-face-hindered interface-imbibition-boundary condition between matrix blocks and surrounding natural fractures, a time-varying shape factor characterizing the effect of the matrix-block shape, and a scalar capillary-diffusion coefficient that is dependent upon the matrix-block volume-average porosity, permeability, and the matrix-medium fluid saturation (through the dependency of the mobility ratio and capillary pressure on saturation).
• Our interpretation method is based on an analytical solution obtained after proper linearization of the nonlinear transient-state capillary-diffusion model by means of the integral transformations of both the dependent-variable saturation and the independent-variable time. Our formulation uses a capillary-diffusion coef-
ficient averaged over the mobile-wetting-fluid-saturation range, which relates to the saturation-dependent capillary-diffusion coef-ficient through the integral transformation. In contrast, most previ-ous approaches assume a constant value for the capillary-diffusion coefficient to be able to generate an analytical solution.
• The nondimensional shape factor resulting from our formulation is not a constant but is time-variable. Hence, once the analytical solution of the linearized diffusion equation has been obtained, then the time-variable shape factor can be calculated directly by our formulation.
• We show that although the shape factor varies with time for short times, it nevertheless attains a constant-limit value for long times. We take advantage of this condition to develop workable equations and semilog straight-line data-plotting schemes for effective analy-sis and interpretation of laboratory data obtained from various-shaped reservoir-rock samples initially saturated by a nonwetting fluid and later immersed in a wetting fluid. This approach leads to the uniqueness of the determined parameter values.
• We demonstrate that the semilog straight-line data-plotting approach using the constant long-time shape factors allows rapid simultaneous determination of the characteristic parameters of the matrix/fracture-transfer models for various-shaped matrix blocks, which are essential for predicting petroleum recovery from naturally fractured reservoirs (e.g., the average capillary-diffusion coefficient Dm and the interface-skin-mass-transfer coefficient �s). The values of these parameters can be extracted conveniently from the intercept and slope values of the straight-line regression of the long-time experimental data of either the recovery factor or the recovery rate because our formulation relates the intercept and slope of the semilog straight line to these coefficients.
• We show that the temperature dependency of these parameters can be correlated effectively using the Arrhenius (1889) equation.
• The values of the interface-skin-mass-transfer coefficient �s and, therefore, the Biot number � become significantly large ( )� → ∞ when the skin effect is negligible, as observed with several core samples considered in the laboratory tests analyzed in this study.
TABLE 3—PARAMETERS AND LONG-TIME BEHAVIOR OF THE RECOVERY-FACTOR-VS.-TIME RESPONSE OF CYLINDRICAL-CORE/FLUID SYSTEMS
• Previous experimental studies primarily measured the fluid recovery from matrix blocks in laboratory core tests. However, the present modeling effort suggests that experimental studies should also consider measuring one of the skin parameters, either the skin diffusivity Ds or thickness bs. Then, the remain-der can readily be determined from the ratio of ( / )�s s sD b= , referred to as the interface-skin-mass-transfer coefficient here, and obtained from the mass-transfer Biot number by the present analysis.
Nomenclature a, a' = intercept, dimensionless A = constant, dimensionless b = slope, T–1
bs = skin thickness, L Bn = Biot number, dimensionless c = constant, dimensionless C = redefi ned pseudosaturation, dimensionless �C = pseudosaturation, dimensionless C = matrix-block volume average water saturation,
dimensionless Co = initial matrix-block volume average water saturation,
dimensionless Dc = capillary-diffusion coeffi cient, L2/T Do = dimensional constant, L2/T
Dm = average capillary-diffusion coeffi cient, L2/T Dmo = pre-exponential constant, L2/T Ds = skin-diffusion coeffi cient, L2/T ED = activation energy required for capillary diffusion,
ML2/T2/mol Es = activation energy required for interface fl uid trans-
fer, ML2/T2/mol f(S) = dimensionless function of normalized saturation S,
dimensionless Fw = zero capillary pressure and gravity effect on
fractional wetting phase, fraction k = constant, dimensionless krn = relative permeability of nonwetting phase, dimen-
sionless krw = relative permeability of wetting phase, dimension-
less K = geometric average permeability, L2
K = permeability tensor, L2
Kn = permeability component in the normal-unit- vector direction, L2
K K Kx y z, , = permeability components in the Cartesian x-, y-, and z-directions, L2
L = length, L L = geometric average of side lengths of rectangular
parallepiped, L L L Lx y z, , = side lengths of rectangular parallepiped in the Car-
tesian-coordinate directions, L n = normal unit vector pc = capillary pressure, M/L–T2
q = volume of fl uid diffused out of matrix block, L3
�q = volume rate of fl uid diffused out of matrix block, L3/T q∞ = characteristic pore volume, L3
Q = transformed variable, dimensionless R = radius, L
�R = universal gas constant, 1.987 cal/(mol-K) Ri = inner radius, L Ro = outer radius, L Rr = coeffi cient of regression, dimensionless RF = recovery factor, fraction, dimensionless RFexp = experimental value of the recovery factor, fraction,
dimensionless RFtrue = true value of the recovery factor, fraction, dimen-
sionless s = integration variable, dimensionless
–0.25
–0.20
–0.15
–0.10
–0.05
0.00
0 25 50 75 100
Rec
over
y Fa
ctor
, log
(1–
RF
)
Time, days
SS3-All faces open parallelepiped
S15-1: Two ends open cylinder
S2-6: Two ends closed cylinder
SS1: Two adjacent Lx/Lz and Ly/Lz faces closed parallelepiped
SS4: Two Ly/Lz ends open parallelepiped
SA2-3-All faces open annular
Fig. 8—Linear correlation of the recovery-factor-vs.-time data of Xie and Morrow (2001).
–0.0025
–0.0020
–0.0015
–0.0010
–0.0005
0.0000
0 500 1,000 1,500
Rec
over
y Fa
ctor
, log
(1
–RF
)
Time, minutes
Fig. 9—Linear correlation of the recovery factor vs. time data of Zhang et al. (1996).
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TABLE 4—PARAMETERS AND LONG-TIME BEHAVIOR OF THE RECOVERY-FACTOR-VS.-TIME RESPONSE OF PARALLELEPIPED-CORE/ AND CYLINDRICAL-CORE/FLUID SYSTEMS
)ecnerefeR 1002( worroM dnA eiX )6991( .la te gnahZ
Fig. 10—Linear correlation of the Reis and Cil (2000) normalized imbibitions-rate-vs.-dimensionless-time data.
sw = wetting-fl uid-saturation variable of integration, frac-tion
S = normalized wetting-phase saturation, fraction �S = normalized wetting-phase-saturation variable of
integration, dimensionless S* = appropriately selected value for Taylor-series
approximation, dimensionless Sn = nonwetting-phase saturation, fraction Sni = immobile nonwetting-phase saturation, fraction Sw = wetting-phase saturation, fraction Swi = immobile wetting-phase saturation, fraction Swf = wetting-phase saturation in the fractures, fraction x y z, , = distances in the Cartesian-coordinate directions, L X Y Z, , = dimensionless distances in the Cartesian-coordinate
directions, dimensionless t = time, T t' = time variable, T tc = characteristic diffusion time scale, T T = dimensionless time T' = dimensionless-time variable V = matrix-block volume, L3
� = coeffi cient, dimensionless � = Biot’s number, dimensionless �n = Biot’s number in a normal direction, dimensionless Δ = saturation difference, Swi – Swf , fraction
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ε = percent error measured as a decimal fraction, di-mensionless
= temperature, �s = skin-mass-transfer coeffi cient, L/T �so = pre-exponential constant, L/T �n = viscosity of nonwetting phase, M/L–T �w = viscosity of wetting phase, M/L–T �w = density of the wetting fl uid, M/L3
= constant, dimensionless d = shape factor, dimensionless �d
LT ( ) = long-time limit of shape factor, dimensionless � = dimensionless time � = representative average matrix porosity, fraction
Subscripts f = fracture fl uid condition LT = long-time m = average n = nonwetting fl uid ni = irreducible nonwetting fl uid o = initial state s = skin w = wetting fl uid wi = irreducible wetting fl uid ∞ = long-distance condition
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Appendix A: Average Capillary-Diffusion Coefficient and Transformed Variable From Capillary-Diffusion FunctionRasmussen and Civan (2006) derived the following expression for the case of unit endpoint mobility ratio and linear relative perme-ability relationships, using the empirical petrophysical functions given by Donaldson et al. (1991):
D S D f S DS S
cSc o o( )( )
( )= ( ) = −
+1
1 2 , . . . . . . . . . . . . . . . . . . . (A-1)
where f(S) is an appropriate dimensionless function of the nor-malized saturation S (e.g., assumed previously) and Do and c are empirical fitting parameters. The average value of Dc is given by
D D s s Dc
cc
cm c o= = +⎛⎝⎜
⎞⎠⎟ +( ) −⎡
⎣⎢⎤⎦∫ ( ) lnd
0
1
3 2
21
2⎥⎥. . . . . . . . . (A-2)
Note the maximum value Dmax of Dc occurs at Scm =
+1
2.
Hence, Eq. A-2 also can be expressed as the following, relating the parameters c, Dm, and Dmax, which may be estimated by history matching:
D
Dc
c
cc
cm
max
ln= +( ) +⎛⎝⎜
⎞⎠⎟ +( ) −⎡
⎣⎢⎤⎦⎥
4 12
12
3 2 . . . . . . . . . . . . . . (A-3)
This approach may be readily extended for nonunit endpoint mobility (Civan et al. 1999) and more-general cases. However, determination of the capillary-diffusion function is beyond the focus of the present study.
Next, we illustrate the applicability of the Taylor-series approxi-mation given by Eq. 18 by an example. Consider the following expression of the capillary-diffusion coefficient for the unit-end-point-mobility case obtained from Eq. A-1:
where �( )c D Do m≡ . The transformed dependent variable is determined by Eq. 18
using Eq. A-4 as
Q SC
c
D S S
c
c
c
c
S
( )( )
,
( )
l
≡ =( )
= +⎛⎝⎜
⎞⎠⎟
∫� � �
� �
d0
2
1 2nn( )
( )1
2
1+ − + +
+⎡⎣⎢
⎤⎦⎥
cSS c cS
cS
. . . . . . . . . . (A-5)
The transformed variable Q approximated by the truncated Taylor series given by Eq. 18 is sufficiently close to the exact transformed variable Q given by Eq. A-5, as illustrated in Fig. A-1 as a function of S (e.g., for c = 0.5 and S* = 0.61). The difference is acceptable in view of the measurement errors involved in the experiments. This provides a justification for the transformation approach used in our method. This example is also instructive in providing useful insights into the shortcomings of the previous approaches using linear models on the basis of the assumption of a constant value for the diffusion coefficient Dc. Our approach alleviates this limitation by the convenient transformation methods presented in this article. This provides a reasonable approximation in view of the measurement errors without assuming a constant value for the capillary-diffusion coefficient Dc.
Appendix B: Analytical Product Solutions by Separation of Variables Laboratory measurement of the matrix/fracture-interface flow is usually carried out using small-sized spherical-, cylindrical-, annular, and rectangular-shaped core samples having all or selected surfaces exposed to the fracture fluid, as depicted in Fig. 1. Table 1 summarizes the results obtained by the analytical solutions of Eqs. 25 through 28 for these samples, as described in the following. The details are presented elsewhere by Civan and Rasmussen (2003, 2005).
Rectangular Parallelepiped.All Faces Open. We derive the full-, long-, and short-time solu-tions.
Full-Time Solution. We describe our analytical solution approach for the Cartesian diffusion equation. The same method is applied for other geometries. Thus, we consider a product func-tion of the form
C X Y Z T X T Y T Z T( , , , ) ( , ) ( , ) ( , )= Ξ � . . . . . . . . . . . . . (B-1)
to satisfy the diffusion equation. Substituting Function B-1 into Eq. 25 and carrying out the differentiations, we can arrange the result in the following form:
that is, if Ξ(X,T), �(Y,T), and ( , )Z T satisfy their correspond-ing 1D diffusion equations. Of course, the boundary conditions and initial conditions must be appropriate if the product solution (Eq. B-1) is to be suitable. For our case, the initial condition and the boundary conditions (Eqs. 26 through 28) are appropriate for such a product solution. The Fourier-series solution can be placed in this form (a similar situation exists when the equation is written in terms of axisymmetric cylindrical variables.) The product solution is particularly useful for obtaining the short-time approximations.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.0 0.2 0.4 0.6 0.8 1.0
Tran
sfor
med
var
iabl
e, Q
Dimensionless saturation, S
Exact
Taylor expansion
Fig. A-1—The transformed variable Q approximated by the truncated Taylor series sufficiently represents the exact trans-formed variable Q as a function of S for c = 0.5 and S* = 0.61.
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Long-time Approximation. For the long-time approximation, only the leading term in the series is retained.
Short-Time Approximation. For the short-time approximation, we start with the Laplace-transform analysis of Eqs. 25 through 28, where �Ξ( , )X s denotes the transform of Ξ( , )X T with respect to T and s is the transform variable. For short times, the Laplace-trans-form variable s is large, in which case the exponential term in the denominator can be neglected. The approximation for short times is therefore obtained and then the resulting expression can be inverted by means of Laplace-transform tables leading to the short-time solutions. We apply the solution derived in this section to generate a number of special solutions, as described in the following.
One Face Closed. Consider the 1D results determined previously. Allow the regime of the spatial variable x to be restricted to the
half-range 02
1≤ ≤x
Lx
. Then, the results now describe a slab of
thickness L Lx x∗ ≡ / 2 bounded by an infi nite impermeable plane
(closed face) at x = 0 and an infi nite permeable plane (open face) at x Lx= ∗ . Thus, the x dependency given by the previously given all-faces-open case is replaced by the new x dependency by shifting the coordinate. Then, the resulting confi guration is a parallelepiped with fi ve open surface faces and one closed surface face.
Two Adjacent Faces Closed. We modify the y equation of the all-faces-open case; therefore, the old symmetry plane at y = 0 now is to be used as a closed external surface. Then, if the functions for x and y are changed accordingly, the function for z remains unchanged and the resulting parallelepiped has four faces that are open and two adjacent faces that are closed.
Three Faces Closed. Suppose the function for z is also modifi ed in accordance with the procedure described previously for x and y. Then, the resulting parallelepiped has three faces that are open and three that are closed. The three closed faces intersect at the origin of the coordinate system.
A Portion of a Slab. Consider the 1D problem as delineated previously. If a rectangular area L Ly z is marked out on one of the bounding planes and projected on to the other, then a parallelepi-ped shape is constructed with a volume of V L L Lx y z= . The four circumferential faces between the two end planes are impermeable and closed, and the two end faces are permeable and open. Thus, a parallelepiped that has two open faces and four that are closed can be represented using the solution for the all-faces-open case. Then, from this, we can obtain the shape that is represented by one open face and fi ve closed faces.
A Portion of an Infi nite Rectangular Column. Consider the 2D problem as delineated previously. Construct a fi nite volume V L L Lx y z= between two slices taken across the column a distance Lz apart. The two end faces are impermeable and closed. Thus, we have the solution that represents a parallelepiped with two closed faces that oppose one another and four open faces that are circumferential. The surfaces in this confi guration are arranged differently than previously mentioned.
Axisymmetric Circular Cylinder Matrix BlockAll Faces Open. We consider cylindrical coordinates r, �, and z, where r is the distance measured radially from an axis of symmetry z and � is the azimuthal angle. It can be shown that a function of the form C R Z T C R T C Z TR Z( , , ) ( , ) ( , )( ) ( )= ⋅ satisfies the diffusion equation.
Solid Radial Problem. First, an eigenfunction that satisfi es the symmetry condition about the centerline is derived. Then, a func-tion of the form C R T( , ) can be obtained by the product solution. We assume that the outer surface is permeable and open.
Annular Problem. The annular cylinder is similar to the solid cylin-der analyzed previously, except that it has an inner surface of radius
Ri . Ro denotes the outer radius. In this case, the volume and charac-teristic length, respectively, are V R R L L Vc i z= −( ) ≡� 2 2 1 3and / .With this change, the normalization of the variables is the same as before. We assume that the inner surface is permeable and open, similar to the outer boundary.
Solution for the Longitudinal Problem. The problem for the longitudinal function C Z TZ( ) ( , ) is the same as the 1D problem treated previously. The solution holds for both ends of the cylinder as permeable or open. The problem is modifi ed by shifting the coordinates to allow for one end of the cylinder to be imperme-able or closed.
Circular Cylinder With Closed Sides and Open Ends. This confi guration is formed from a portion of a slab in a manner similar to that previously described for a parallelepiped. Allow a circular area �Rc
2 to be marked out on one of the planes and projected onto the other. Then, a cylindrical shape is constructed between the two planes with a volume of V R Lc z= � 2 . The cylindrical side surface is impermeable and closed, and the two end faces are permeable and open. The solution holds with L V= 1 3/ .
Circular Cylinder With Sides and One End Closed and the Other End Open. Consider the 1D problem, only modifi ed in the fashion as previously mentioned. Then, the equation in terms of Z is used, and L V R Lc z= = ( )∗1 3 2 1 3/ /
.� This describes a circular cylin-der for which the cylindrical side surface and one end are imperme-able and closed, and the other end is permeable and open.
Spherical Matrix BlockWe consider spherical coordinates r, , and �, where r is the distance measured radially from a point origin and and � are polar and azimuthal angles. The relation with the Cartesian coor-dinates is x = r sin cos �, y = r sin sin �, and z = r cos . The derivation of the diffusion equation in spherical coordinates for an anisotropic medium is carried out. Here, we specialize to the case of spherical symmetry for a sphere of radius r Rs= and volume V Rs= 4 33� / .
Appendix C: Effect of Measurement ErrorsErrors made in obtaining and/or recording the recovery-factor data are more significant at long times than for short times, as observed in the various data analyzed in this article. We can explain this issue in terms of the following.
Suppose that the error made in gathering the experimental value of the recovery factor is a certain percentage of the true value; that is,
where k and are constants. On a semilog plot, the data for the experimental value of the recovery factor would then be repre-sented by the function
ln ln lnexp1 1 1−( ) = − + − −⎛⎝⎜
⎞⎠⎟
⎡
⎣⎢
⎤
⎦⎥RF k T
e
k
T
ε . . . . . . . . . (C-3)
Therefore, the scatter in the data increases as time increases. Thus, every effort should be taken to obtain data with special accuracy at long times.
Faruk Civan is the Martin G. Miller Chair Professor in the Mewbourne School of Petroleum and Geological Engineering at the University of Oklahoma. He formerly held the Brian and Sandra O’Brien Presidential and Alumni Chair Professorships. Previously, Civan worked at the Technical University of Istanbul in Turkey. His principal research interests include fossil- and
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sustainable-energy-resources development; carbon seques-tration; unconventional gas reservoirs; reservoir and well/pipe-line hydraulics and flow assurance, and mitigation; formation- and completion-damage mitigation; oil and gas processing, transportation, and storage; multiphase transport phenomena in porous media; environmental-pollution control; and math-ematical modeling and simulation. Civan is the author of two books, Porous Media Transport Phenomena and Reservoir Formation Damage: Fundamentals, Modeling, Assessment, and Mitigation. He has published more than 275 technical articles in journals, edited books, handbooks, encyclopedias, and confer-ence proceedings, and has presented more than 100 invited seminars and/or lectures at various technical meetings, com-panies, and universities worldwide. Civan holds an advanced degree from the Technical University of Istanbul, an MS degree from the University of Texas at Austin, and a PhD degree from
the University of Oklahoma, all in chemical engineering. He has served on numerous American Institute of Chemical Engineers (AIChE) and SPE technical committees. Civan is a member of AIChE and SPE, and is a member of the editorial boards of several journals. He has received 20 honors and awards, includ-ing five distinguished lectureship awards and the 2003 SPE Distinguished Achievement Award for Petroleum Engineering Faculty. Civan can be reached at [email protected].
Maurice L. Rasmussen is the David Ross Boyd Professor of the School of Aerospace and Mechanical Engineering at the University of Oklahoma. His research interests are in fluid mechanics and applied mathematics. Rasmussen is the author of a book on hypersonic flow and coauthor of a book on applied mathematics for engineers. He holds a PhD degree in aeronautics and astronautics from Stanford University.
