Joint modeling of acoustic velocities and electrical conductivity from ...

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109, B01202, doi:10.1029/2003JB002443, 2004

Joint modeling of acoustic velocities and electrical conductivity from unified microstructure of rocks E. Kazatchenko, M. Markov, and A. Mousatov Instituto Mexicano del Petro´leo, Me´xico D.F., Me´xico Received 11 February 2003; revised 8 August 2003; accepted 24 October 2003; published 15 January 2004.

[1] A technique for the joint modeling of the electrical conductivity and acoustic

velocities in porous rocks is proposed. The technique is based on the model of twocomponent media, composed by grains, which constitute a solid frame and pores saturated by a fluid. For this model we used symmetrical effective medium approximation method that provides the simulation of the acoustic and electrical parameters for multicomponent systems with equally treated constituents. The individual element of each component as a pore or grain was approximated by an ellipsoid. The aspect ratios for grain and pore ellipsoids are introduced as a function of porosity. By applying this technique we performed the calculation of acoustic velocities and electrical conductivity for carbonate formations with a primary pore system. For such formations the experimentally determined acoustic and electrical parameters are described by the empirical regression equations for the P and S wave velocities versus porosity and Archie’s law. The ellipsoid aspect ratios were obtained by minimizing the differences of both predicted P wave velocity and conductivity with experimental data. The results obtained demonstrate that electrical conductivity is more sensitive to the pore and grain geometry than acoustic velocities. To ensure the conductivity for the low porosities, the form of pores tends to a needle shape. The simulation technique developed is the base for the petrophysical INDEX inversion of well log data to reconstruct the microstructure of porous rocks. TERMS: 5102 Physical Properties of Rocks: Acoustic properties; 5109 Physical Properties of Rocks: Magnetic and electrical properties; 5112 Physical Properties of Rocks: Microstructure; KEYWORDS: microstructure, joint modeling, electrical conductivity, acoustic velocities Citation: Kazatchenko, E., M. Markov, and A. Mousatov (2004), Joint modeling of acoustic velocities and electrical conductivity from unified microstructure of rocks, J. Geophys. Res., 109, B01202, doi:10.1029/2003JB002443.

1. Introduction [2] Carbonate formations have a complex structure of pore space defined by a primary pore system of matrix and secondary porosity related to vugs and microfractures. The pore system types and their porosity values significantly influence the determination of the basic formation characteristics including permeability prediction and reservoir reserve estimation. For such complicated heterogeneous media, the evaluation of microstructure from experimental well log data is based on an integrated analysis of various geophysical parameters. We consider the reconstructing of porous formation microstructure to be a specific type of inversion problem. Petrophysical inversion requires the development of a technique for the simulation of rock physical parameters, based upon a unified microstructure model which provides correct estimates of the acoustic velocities and electrical conductivity. However, the modeling of some physical characteristics for geological formations is performed for models designed specially for a single estimated parameter. Copyright 2004 by the American Geophysical Union. 0148-0227/04/2003JB002443$09.00

[3] The self-consistent methods developed for composite materials have a wide application for simulation of rock properties. These methods include the effective medium approaches [Bruggeman, 1935; Berryman, 1980, 1992; Norris, 1985; Berge et al., 1993;] that are based on the solution of the elastic or electrical equations for a single inclusion placed in some effective host. Introducing the effective medium takes into account inclusion interactions. Generally two basic approaches are used for the modeling of rock geophysical parameters: the effective media approximation (EMA method) and the differential effective model (DEM). The application of DEM is limited to two-component systems in which the first component corresponds to a matrix and the second as inclusions placed in this matrix. The EMA method allows the multicomponent systems when all components are treated equally with no one material distinguished as a host to be described [Berryman, 1980, 1995; Norris, 1985]. [4] To describe elastic parameters and phenomenon of acoustic wave propagation both EMA and DEM are used [O’Connell and Budiansky, 1974; Budiansky and O’Connell, 1976; Berryman, 1980; Norris, 1985; Berge et al., 1993]. However, the DEM is convenient to characterize terrigenous homogeneous formations with unimodal pore spaces placed

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KAZATCHENKO ET AL.: JOINT MODELING

in the solid matrix [Hornby et al., 1994]. The EMA method allows the elastic wave velocities for formations with multicomponent solid frame and complex pore structures to be obtained [Berryman, 1992; Mavko et al., 1998]. It was demonstrated [Berge et al., 1993] that P wave velocities predicted by the self-consistent effective medium approach fit the experimental data for synthetic sandstone (sintered glass beads). The shapes of both the beads that constitute the solid frame and the pores were approximated by spheres. [5] DEM is often applied for the simulation of electrical properties. In this case the medium is represented by high resistivity inclusions (solid grains) placed in a conductive liquid matrix [Sen et al., 1981, 1997; Mendelson and Cohen, 1982; Norris et al., 1985; Chinh, 2000]. Such a matrix guarantees the existence of electrical conductivity for small porosities. However, this approach does not allow shear wave propagation in porous rock to be described. [6] The approach to the joint calculation of acoustic and electrical properties of porous formations from the unified microstructure model was presented by Sheng [1991]. Using the combination of EMA and DEM methods, Sheng considered a porous rock to be a three-component composite consisting of a conductive fluid, cement, and solid grains of spheroidal form. The matrix is a homogeneous half-andhalf mixture of the fluid and cement. The introduced matrix ensures a nonzero shear modulus and an electrical conductivity for low porosities. The effective matrix parameters are first calculated by the EMA method. Then the DEM method is applied to find the electrical and acoustic parameters of the medium composed by the effective matrix with spheroidal solid grains. However, the grain aspect ratio was assumed to be constant for the considered porosity interval, and it was selected by adjusting the predicted resistivity according to Archie’s law. [7] The inversion of acoustic velocities to determine the pore space structure was proposed by Cheng and Tokso¨z [1979]. They presented a method to obtain the pore aspect ratio spectrum applying linearized iterative inversion of the acoustic velocities as function of pressure and saturation condition. [8] In the present work, we propose a technique for the joint simulation of acoustic and electrical parameters proceeding from a unified microstructural model of porous media. The model is a two-component medium of solid grains and pores. The grains of high resistivity form the solid frame. The pores are saturated by a conductive fluid. The elements of each component are approximated by three-axis ellipsoids with different aspect ratios which are introduced as functions of porosity. We applied the EMA method when the grains and pores are considered as equivalent components in the composite medium. Such an approach (the model and the simulation method) ensures a nonzero shear modulus and an electrical conductivity for low porosity. [9] We applied this technique to calculate the acoustic velocities and electrical conductivity for carbonate formations with a primary pore system. For such formations the experimentally determined acoustic and electrical parameters are described by the empirical regression equations for the compressional wave (VP) and shear wave (VS) velocities versus porosity and by Archie’s law, respectively. The ellipsoid aspect ratios were defined by simultaneous minimization of the difference of the predicted Pwave velocity and

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conductivity with the experimental data. The best matching of the calculated and experimental data is achieved when the pore and grain aspect ratios are porosity functions. The assumption about the variable component geometry allows the EMA approach for modeling electrical conductivity up to very low porosities (1 – 2%) to be applied.

2. Theoretical Background of Modeling Technique [10] In this section we describe the technique for joint simulation of acoustic velocities and electrical conductivity, using the same microstructure model of porous rocks. The applied model presents a homogeneous isotropic medium composed by two components: grains, which constitute a solid skeleton of high resistivity and pores saturated by a conductive fluid. For such a model we used the symmetrical EMA approach that solves the homogenization problem for multicomponent systems with equally treated constituents (i.e., not identifying any component as a solid or fluid host) [Berryman, 1980, 1992; Norris, 1985]. The individual element of each component as a pore or grain was approximated by an ellipsoid. We assumed that the aspect ratios for grain and pore ellipsoids are a function of porosity. 2.1. Elastic Moduli [11] The general EMA equations for the elastic properties of the medium composed by N components were obtained by Korringa et al. [1979], Berryman [1980, 1992], and Norris [1985]: N X

Ci ðLi L*ÞT ðiÞ ¼ 0;

ð1Þ

i¼1

where Ci is a volumetric concentration of the ith component (Ci = 1), L* is the elastic tensor of the effective medium, Li is the elastic tensor of ith component, and T (i) is Wu’s tensor [Wu, 1966]. [12] For isotropic inclusions the components of tensor Li are defined as 2 Ljklm ¼ Ki djk dlm þ mi djl dkm þ djm dkl djk dlm ; 3

where Ki and mi are the bulk and shear moduli of ith component, dij is Kronecker delta. Tensor T (i) relates the strain tensor inside individual element of ith component ei with the uniform field of stain e0 far from it: ei = T (i)e0. [13] For the medium composed by arbitrary distributed isotropic components, the system (1) can be presented as N P i¼1 N P

Ci ðKi K*ÞPi ¼ 0 ð2Þ Ci ðmi m*ÞQi ¼ 0;

i¼1

where K* and m* are bulk and shears moduli for effective medium, respectively, and ðiÞ

Pi ¼ 13 Tjjll ; ðiÞ Qi ¼ 15 Tjljl Pi ;

where repeated subscripts are summed.

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[14] Tensor T (i) is obtained from the solution of oneparticle problem for the strain of an ellipsoid placed in the infinite effective medium and affected by a given strain field that is uniform far from the inclusion. For the ellipsoidal inclusion this tensor is h i1 T ðiÞ ¼ I þ S ðiÞ ðL*Þ1 ðLi L*Þ ;

ð3Þ

where I is the fourth-order isotropic identity tensor and S (i) is Eshelby’s tensor [Eshelby, 1957]. [15] The components of Eshelby’s tensor depend on L* and the aspect ratio of the element of the ith component. The expressions for these components are given in Appendix A. 2.2. Electrical Conductivity [16] For the electrical conductivity, the EMA equations for isotropic medium with ellipsoidal components were obtained by [Sen et al., 1981; Norris et al., 1985]: N X

" Ci

i¼1

#

3 X

ðsi s*Þ

k¼1

s* þ nk ðsi s*Þ

ðiÞ

¼ 0;

ð4Þ

where si and s* are electrical conductivities of the ith component and effective medium, respectively, nk is depolarization factor [Stratton, 1941] that using the appropriate ellipsoidal coordinate system [Landau and Lifshitz, 1960] can be determined as a1 a2 a3 nk ¼ 2

Z1 0

ds

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

: 2 s þ ak s þ a21 s þ a22 s þ a23

ð5Þ

2.3. Determination of Aspect Ratios [17] Previous work [O’Connell and Budiansky, 1974; Sen et al., 1981; Mendelson and Cohen, 1982; Sheng, 1991; Xu and White, 1995] assumed that the shape of elements (i.e., aspect ratios) for each component remains the same for the used interval of porosity values. For the present model we considered that the ellipsoid aspect ratios of both solid grains and fluid saturated pores are functions of porosity. The introduction of the varied aspect ratios is based on the supposition that the microstructures of porous rocks ( pore size distribution and its connection network) for high and low porosities are different. The variations of the aspect ratios can be obtained from two classes of experimental data. The first includes data about the porosity and microstructural statistical characteristics measured on cores. The second class presents the acoustic velocities and electrical conductivity from cores or well log data. In this case the aspect ratios are determined by a joint inversion of velocities and conductivity, adjusting the calculated parameters of equations (2) and (4) with experimental data. We estimated the aspect ratios using the second class of data. To fit the calculated and experimental data we solved a nonlinear problem for minimizing of a cost function:

F a1p ; a2p ; a1g ; a2g

2 0 13 ðd Þ M VPk VPk ln sk X A5 @ ¼ min4 þ 1 ; ðd Þ ln sðdÞ VPk k¼1 k ð6Þ

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(d) where VPk, VPk are the calculated and measured P wave velocities, and sk, sk(d) are the calculated and measured electrical conductivities, respectively. The minimization procedure was performed for the four aspect ratios aip = a2/a1, a2p = a3/a1, a1g = b2/b1, and a2g = b3/b1, where aj, bl are the ellipsoid semiaxes for the pores and the grains, respectively. [18] We used the piecewise constant approximation for the aspect ratios in each porosity interval of 3%. The applied inversion algorithm is based on the quasi-Newton method with finite difference calculation of gradients [Dennis and Schnabel, 1983].

3. Modeling of Acoustic Velocities and Electrical Conductivity for Carbonate Formation [19] The technique described above was applied for calculation of the acoustic velocities and electrical conductivity for carbonate formations. We considered the carbonate formations with only a primary porosity defined by intergranular pore system. Rocks with such a microstructure can be characterized as a two component composite material. [20] The generalized regression equation for a dolomite formation saturated with water for porosities from 0.01 to 0.32 was given by Mavko et al. [1998]: VP ¼ 6:61 9:38f

ð7Þ

VS ¼ 3:58 4:72f;

ð8Þ

where VP and VS are given in units of km/s. [21] Dolomite has the following characteristics: bulk density rg = 2.84 g/cm3, bulk modulus Kg = 75.17 GPa and shear modulus mg = 36.4 GPa [Mavko et al., 1998]. Saturating water parameters are: rf = 1 g/cm3, Kf = 2.25 GPa, electrical conductivity sf = 1 m, and the ratio of the water and grain conductivities is taken to be 105. [22] The relationship of the electrical conductivity to a primary porosity for the carbonate formations that has no secondary porosity as microfractures and vugs, was obtained by Kazatchenko and Mousatov [2002] on the basis of the statistical analysis of numerous experimental data. The formation factor for this type of rock is described by Archie’s law [Archie, 1942] s* ¼ fm sf

ð9Þ

with the cementation exponent m = 2. [23] Using equations (2) – (5) for the components of corresponding tensors obtained for ellipsoids, we calculated the P wave velocity and conductivity as functions of (d) and s(d) porosity f. The VPk k required in equation (6) were provided by the generalized regression equations for dolomite (equation (7)) and Archie’s law (equation (9)). The minimization of a fitting error (equation (6)) allows determination of the pore and grain geometry (i.e., values of the aspect ratios) over the porosity range of 2– 31%. These were then compared to the theoretical S wave velocities for the best pore and grain geometry with the regression equation (8) to further confirm the validity of the model structure obtained.

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of accuracy (Figures 1a and 2a). The relative error for the P wave velocity is less than 0.05%, and for the conductivity does not exceed 5% for porosities over the range of 4– 31% (Figures 1b and 2b). [26] The EMA using spheroids with nonvariable aspect ratios was rarely applied to model electrical properties, because it predicted vanishing conductivity (percolation threshold) at relatively high porosity values. Allowing variation of the pore and grain aspect ratios moves the percolation threshold to lower porosities near 2% in agreement with measured data for carbonate collections. [27] The calculated S wave velocities coincide with experimental equation both for constant and for variable aspect ratios (Figure 3a). The maximum relative errors

Figure 1. Results of joint inversion for electrical conductivity. (a) Theoretical and experimental relationships electrical conductivity versus porosity: squares, Archie’s law s*/sf = f2 ; curves, calculated electrical conductivities. (b) Relative error between predicted conductivity and Archie’s law. Curve 1, constant aspect ratios for pores and grains; curve 2, variable aspect ratios for pores and constant aspect ratios for grains; and curve 3, variable aspect ratios for pores and grains. [24] First, we assumed constant aspect ratios of pore and grain elements on the all porosity interval (2 – 31%). For this situation, the best coincidence of theoretical and experimental data is achieved when the aspect ratios of grains and pores are a1g = 0.3048, a2g = 0.0903 and a1p = 0.0803, a2p = 0.0252 (Figures 1a and 2a). The matching of the acoustic data is satisfactory, with a relative error P = (VPtheor VPexp)/VPexp less than 2% (Figure 2b). However, the conductivity relative error s = (stheor sexp)/sexp is rather high and corresponds to 30% in the porosity interval 5 – 10% (Figure 1b). For porosities lower than 5% the predicted conductivity is reduced considerably and tends to the percolation threshold. [25] To better fit the modeling and experimental data for electrical conductivity we make the axis ratios for pores and grains depend on the porosity. The variation of the aspect ratios allows the predicted and experimental data in the porosity interval 3 – 31% to be adjusted with a high degree

Figure 2. Results of joint inversion for P wave velocity. (a) Theoretical and experimental relationships P wave velocity versus porosity: squares, regression equation for saturated dolomite VP = 6.6 9.38f; curves, calculated VP. (b) Relative error between predicted P wave velocity and regression equation. Curve 1, constant aspect ratios for pores and grains and variable aspect ratios for pores and constant aspect ratios for grains; and curve 2, variable aspect ratios for pores and grains.

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tendency from porosity. The scatter of the aspect ratios and cross sections for the grains (Figures 6 and 7) indicates that the velocities and conductivity have lower sensitivity to the grain geometry than to the pore shape. The predicted variations of forms for skeleton grains and saturated pores correspond to the recently reported experimental data for carbonate formations on pore microstructure changes depending on porosity [Song et al., 2002]. [29] Taking into account the nonregular change of the grain section area and good agreement of the velocityporosity relations with experimental data for both constant and variable aspect ratios we fixed the grain form and performed the minimization just for variable pore aspect ratios. The grain aspect ratios were chosen equal to the one obtained at the constant pore and grain geometry (a1g = 0.3048 and a2g = 0.0903). The elastic velocities in this case coincided with the calculated ones for the constant pore and

Figure 3. Results of joint inversion for S wave velocity. (a) Theoretical and experimental relationships S wave velocity versus porosity: squares, regression equation for saturated dolomite VS = 3.58 4.72f; curves, calculated VS. (b) Relative error between predicted S wave velocity and regression equation. Curve 1, constant aspect ratios for pores and grains and variable aspect ratios for pores and constant aspect ratios for grains; and curve 2, variable aspect ratios for pores and grains. s = (VStheor VSexp)/VSexp are 2% and 3%, respectively (Figure 3b). [28] The geometry of grains and pores is presented in Figures 4 and 5. For high porosities (28 – 31%) the pores have ellipsoidal form with aspect ratios a1p = 0.300 and a2p = 0.097. When the porosity decreases from 31% to 2% the ellipsoids transform into the needle-form pores with the small aspect ratios a1p = 0.065 and a2p = 0.022 (Figure 4). This pore geometry maintains the electrical conductivity for low concentrations of conductive component. The solid grain geometry changes from the ellipsoids with a1g = 0.408, a2g = 0.269 to penny shapes characterized by a1g = 0.984, a2g = 0.098 (Figure 5). The variations of aspect ratios with porosity are presented in Figure 6. The section area normalized to the square major semiaxis for the pore ellipsoid (S*p = pa1pa2p) are reduced from 0.092 to 0.004 when the porosity decreases (Figure 7). The variation of normalized section area S*P is approximated by the polynomial curve S*p = 12.384f3 3.143f2 + 0.29f 0.002 with the correlation coefficient R2 = 0.999. The grain section area varies in the interval 0.26 – 0.36 and does not have clear

Figure 4. Pore forms as a function of the porosity values. (a) Porosity f = 3%, aspect ratios a1 = 0.065, a2 = 0.022; (b) f = 11%, a1 = 0.091, a2 = 0.032; and (c) f = 27%, a1 = 0.300, a2 = 0.097.

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Figure 6. Variation of the aspect ratios for grains (triangles) and pores (circles) as function of porosity. Curve 1, a1g; curve 2, a2g; curve 3, a1p; curve 4, a2p. The aspect ratios for the porosity 0.03, 0.11, and 0.27 correspond to the grain and pore shapes presented in Figures 4 and 5.

Figure 5. Grain forms for the different porosity values. (a) Porosity f = 3%, aspect ratios a1 = 0.984, a2 = 0.098; (b) f = 11%, a1 = 0.682, a2 = 0.134; and (c) f = 27%, a1 = 0.408, a2 = 0.269.

variable aspect rations which depend on the porosity value. The technique provides for a nonzero shear modulus as well as electrical conductivity at low porosity. [31] The simulation of acoustic velocities and electrical conductivity was carried out for saturated dolomite with primary pore system. The ellipsoid aspect ratios were obtained by minimizing the differences of both predicted P wave velocity and electrical conductivity with experimental data. The results obtained demonstrate that the technique used allows modeling of acoustic and electrical

grain aspect ratios (Figures 2 and 3). Small deviations of the theoretical VP and VS from the regression equations indicate that acoustic parameters have low sensitivity to the changing of the microstructure geometry and are basically defined by the porosity values. The electrical conductivity for the porosities 10– 31% fits the experimental data better than for the case of constant aspect ratios (relative error does not exceed 10%). In the porosity interval 2 – 10% the error increases significantly up to 40%. It means that electrical conductivity has a high sensitivity to both pore and grain geometry.

4. Conclusions [30] We have presented a technique for the joint simulation of the acoustic velocities and electrical conductivity of porous rocks. The technique is based on the unified microstructure model and the EMA symmetrical approach. The porous media are considered as a two component material composed of solid grains (solid frame) and pores saturated with conductive fluid. The pores and grains are approximated by three-axial ellipsoids with different aspect ratios. The peculiarity of the model consists in introducing the

Figure 7. Normalized cross section areas S* = pa1a2 of grains (triangles) and pores (circles). The approximation curve for pore-section area is S*p = 12.384f3 3.143f2 + 0.29f 0.002 with correlation coefficient R2 = 0.999.

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properties of porous formation with a high accuracy. The calculation performed shows that the electrical conductivity is more sensitive to the pore and grain geometry than the acoustic velocities. However, the acoustic data are required to stabilize the aspect ratio determination. [32] The estimation of the aspect ratios for experimental data allowed evaluation of the variation of microstructure for carbonate formations. With decreasing of porosity, the grain form changes from spheroid to penny shape and the pore form tends to needle shape. The relative pore section area has a strong polynomial relationship with porosity values and can be used as a characteristic of pore space structure. We consider that the simulation technique developed here is a basis for the petrophysical inversion of well log data to reconstruct the microstructure of porous rocks.

Appendix A:

Components of Eshelby’s Tensor

[33] The components of Eshelby’s tensor S (i) for the ellipsoidal inclusions are given by the series of expressions [Eshelby, 1957]. When the ellipsoid semiaxes satisfy the inequality a1 > a2 > a3, these expressions are

S1212 ¼

S1111 ¼

3 1 2n a2 J11 þ J1 ; 8pð1 nÞ 1 8pð1 nÞ

S1122 ¼

1 1 2n a2 J12 þ J1 ; 8pð1 nÞ 2 8pð1 nÞ

ðA1Þ

2

1 1 2n ðJ1 þ J2 Þ a þ a22 J12 þ 16pð1 nÞ 1 16pð1 nÞ

where n is Poisson’s coefficient of effective medium, Jj, Jik integrals given by 4pa1 a2 a3 J1 ¼

1=2 f F ðq; K Þ E ðq; K Þg 2 a1 a22 a21 a23 4pa1 a2 a3 J2 ¼

1=2 a22 a23 a21 a23

( )

1=2 a2 a21 a23 E ðq; K Þ a1 a3

J1 þ J2 þ J3 ¼ 4p

and 3J11 þ J12 þ J13 ¼

4p a21

3a21 J11 þ a22 J12 þ a23 J13 ¼ 3J1

J12

( 1=2 ) a21 a23 q ¼ arcsin a21

a21 a22 a21 a23

We used Carlson’s algorithm [Carlson, 1979] to calculate the elliptical integrals F(q, K ) and E(q, K ). The components of the Wu’s tensor T (i) [Wu, 1966] were calculated numerically using equations (A1).

Notation Ci concentration of the ith component. Li, L* elastic tensors of ith component and effective medium. Ljklm elastic tensor of isotropic component. dij Kronecker delta; dij = 1 if i = j and dij = 0 if i 6¼ j. Ki, K* bulk moduli of ith component and effective medium. mi, m* shear moduli of ith component and effective medium. T (i) Wu’s tensor. ei strain inside inclusion. e0 uniform field of stain far from inclusion. I fourth-order isotropic identity tensor. S (i) Eshelby’s tensor. n Poisson’s coefficient of effective medium. F(q,K ), E(q,K ) elliptic integrals of the first and second kind. si, s* electrical conductivity of ith component and effective medium. nk depolarization factor. a1, a2, a3 ellipsoid semiaxis for pores. b1, b2, b3 ellipsoid semiaxis for grains. f porosity. (d) calculated and measured P wave velocity. VPk, VPk VS S wave velocity. ) calculated and measured electrical consk, s(d k ductivities. sf conductivity of saturating fluid. rg, Kg, mg density, bulk and shear moduli of solid grains. rf, Kf density and bulk modulus of saturating fluid. a1g, a2g grain aspect ratios. a1p, a2p pore aspect ratios. s, P, S relative error of the fitting calculated parameters with experimental data for electrical conductivity, P and S wave velocities. S* normalized section area to the square major semiaxis of ellipsoids. R correlation coefficient. [34] Acknowledgments. The authors are grateful to express gratitude to the Mexican Petroleum Institute, where in the framework of the Research Program ‘‘Naturally Fractured Reservoirs’’ this study was fulfilled.

J2 J1 ¼ 2 a1 a22

In these expressions, F(q,K ) and E(q,K ) are the elliptic integrals of the first and second kind where

K¼

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1=2 :

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Mendelson, K. S., and M. N. Cohen (1982), The effect of grain anisotropy on the electrical properties of isotropic sedimentary rocks, Geophysics, 47, 257 – 263. Norris, A. N. (1985), A differential scheme for the effective moduli of composites, Mech. Mater., 4, 1 – 16. Norris, A. N., P. Sheng, and A. J. Callegari (1985), Effective-medium theories for two-phase dielectric media, J. Appl. Phys., 57, 1990 – 1996. O’Connell, R. J., and B. Budiansky (1974), Seismic velocities in dry and saturated cracked solids, J. Geophys. Res., 79, 5412 – 5426. Sen, P., C. Scala, and M. H. Cohen (1981), A self-similar model for sedimentary rocks with application to the dielectric constant of fused glass beard, Geophysics, 46, 781 – 796. Sen, P., W. E. Kenyon, H. Takezaki, and M. J. Petricola (1997), Formation factor of carbonate rocks with microporosity: Model calculations, J. Pet. Sci. Eng., 17, 345 – 352. Sheng, P. (1991), Consistent modeling of the electrical and elastic properties of sedimentary rocks, Geophysics, 56, 1236 – 1243. Song, Y.-Q., N. V. Lisitza, D. F. Allen, and W. E. Kenyon (2002), Pore geometry and its geological evolution in carbonate rocks, Petrophysics, 43(5), 420 – 424. Stratton, J. A. (1941), Electromagnetic Theory, McGraw-Hill, New York. Wu, T. T. (1966), The effect on inclusion shape on the elastic moduli of a two-phase material, Int. J. Solids Struct., 2, 1 – 8. Xu, S., and R. White (1995), Poro-elasticity of clastic rocks: A unified model, paper V presented at 36th Annual Logging Symposium, Soc. of Prof. Well Log Anal., Houston, Tex.

E. Kazatchenko, M. Markov, and A. Mousatov, Instituto Mexicano del Petro´leo, Eje Central, La´zaro Ca´rdenas 152, C.P.07730, Me´xico, D.F., Me´xico. ([email protected]; [email protected]; [email protected])

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