Computers and Geotechnics

Transcription

1 Compters and Geotechnics 38 (2) Contents lists available at ScienceDirect Compters and Geotechnics jornal homepage: Effective medim methods and a comptational approach for estimating geomaterial properties of poros materials with randomly oriented ellipsoidal pores A.P. Svorov, A.P.S. Selvadrai Department of Civil Engineering and Applied Mechanics, McGill University, Montreal, QC, Canada H3A 2K6 article info abstract Article history: Received 2 November 2 Received in revised form 25 Febrary 2 Accepted 8 April 2 Available online May 2 Keywords: Effective elastic properties Disc-shaped voids Self-consistent estimates of elastic properties Mori Tanaka estimates of elastic properties Comptational estimates The role of effective medim approaches and the differential scheme in estimating the overall elastic modli of a poros medim with randomly oriented pores is examined. The analytical estimates of the elastic modli are compared with approximations available in the literatre that are valid for small pore aspect ratios. Accracy of the analytical estimates is frther established by performing finite element simlations. Finite element estimates are obtained for a model of a poros medim containing three families of spheroidal pores arranged in a mtally orthogonal configration. This model is regarded as a close approximation to a poros medim with randomly oriented pores. Also, experimental data available for several sandstones and granites was sed to analytically examine the inflence of the aspect ratio of pores on overall properties. This information is also sed to provide an estimate for the permeability of the poros medim. Ó 2 Elsevier Ltd. All rights reserved.. Introdction This work relates to the determination of the overall elastic modli of a poros medim. The pores of the region can be empty or flid-filled, and are treated as either discrete spherical voids or as oblate spheroidal voids with small aspect ratios. The distribtion of pore orientations is assmed to be random. On the macroscale sch a poros medim is considered to be statistically homogeneos and isotropic, i.e., there are two overall elastic modli (e.g., blk and shear modli) that characterize the elastic behavior of the material. While exact analytical expressions for the overall elastic modli of a poros material are difficlt to determine, several procedres exist for estimating the elastic modli of sch a solid. Among them are the poplar effective medim methods that inclde the selfconsistent method [5] and the Mori Tanaka method [8]. The differential scheme [7] is also a poplar method for obtaining the overall properties [29]. For flat oblate spheroidal pores with small aspect ratio (i.e., penny-shaped cracks), Kachanov s scheme is freqently sed to estimate the overall elasticity properties [6]. The works of Benson et al. [2] and Gegen and Sarot [2] have shown that Kachanov s estimates for the overall elastic modli of a cracked material provide a good fit to the experimental data for several rocks. Bary [] sed the Mori Tanaka method to estimate the elastic modli of poros cement paste. Girad et al. Corresponding athor. addresses: svora23@yahoo.com (A.P. Svorov), patrick.selvadrai@ mcgill.ca (A.P.S. Selvadrai). [8] estimated the elastic modli of rocks consisting of flattened ellipsoidal pores with an aspect ratio of /2 and spherical inclsions of mineral phases sing the Mori Tanaka method. Gresc et al. [] sed a two-step homogenization procedre to determine the overall thermal condctivity of poros rocks (i.e., the Mori Tanaka method to find the properties of clay with calcite and qartz inclsions and the method of Ponte Castaneda and Willis [2] to accont for the presence of pores in the clay matrix). The aspect ratio of the oblate spheroidal pores sed by Gresc et al. [] was /2. Girad, Gresc et al. [9,] sed the Mori Tanaka method for two stages of the homogenization procedre to determine the overall thermal condctivity of poros rocks. Gery et al. [3] estimated the elastic properties of an argillite composed of a clay matrix and spherical inclsions of qartz and calcite sing three approaches: the dilte, the self-consistent and the Mori Tanaka methods. They fond that the Mori Tanaka method was able to accrately reprodce the experimental data for the elastic modli. Roberts and Garboczi [23] sed a finite element model of a poros medim with overlapping spherical pores and overlapping oblate spheroidal pores with an aspect ratio of /4. They fond that the differential scheme accrately matched their finite element reslts for the overall Yong s modls for porosities of p to.5. The self-consistent estimates were only accrate for porosities lower than.2. Whether the differential scheme or the selfconsistent method can be sed to accrately estimate the overall elastic properties of a poros medim with the cracks of a smaller aspect ratio cannot be decided a priori X/$ - see front matter Ó 2 Elsevier Ltd. All rights reserved. doi:.6/j.compgeo.2.4.2

2 722 A.P. Svorov, A.P.S. Selvadrai / Compters and Geotechnics 38 (2) A finite element model of the poros material consisting of randomly distribted penny-shaped cracks with small aspect ratios can be sed to comptationally assess which of the estimates for the overall modli yields the most accrate reslt. The comptational approach, while formally straightforward, does not always lead to convenient implementation since the mesh shold conform to the complex geometry of the matrix-pore interfaces. Comptational experience with the se of mltiphysics finite element code COMSOL sggests that only a model having a relatively small nmber of cracks, i.e., poros media with very small vales of porosities, can be sccessflly meshed and analyzed. Even for very small porosities, the size of the representative volme element mst be sfficiently small to ensre a sccessfl generation of the reqired finite element mesh. The program COMSOL has been sccessflly sed for solving flid flow problems for the real 3D pore geometry of a granlar material obtained throgh X-ray compter assisted micro-tomography [9]. The microstrctre of the material consisted of solid grains of approximately spherical shape and a pore-network domain with a volme fraction of Estimates of overall elastic modli and their properties Consider a poros medim with randomly-distribted pores of any shape. The pores can be empty (dry) or flid-filled. (The elastic constants of the matrix are K s, G s, E s, m s, the blk modls of the flid is K f.) The porosity or volme fraction of the pores is denoted by n. For sch a poros medim some important relationships can be smmarized. Consider the constittive eqation for the isotropic stress r I in the poros medim nder ndrained conditions [4,22,2,24,25], r I ¼ K D V ap ¼ K V where V is the volmetric strain, p is the flid pressre, K D is the blk modls of the poros medim with empty pores (drained blk modls), K is the blk modls of the poros medim with flid-filled pores (ndrained blk modls), a = K D /K s is the Biot coefficient. For a poros medim nder ndrained conditions, a frther constittive relation is, a V þ n þ a n p ¼ K f K s From () and (2) we can write the Biot Gassmann relationship as, a 2 K ¼ K D þ n=k f þða nþ=k s This eqation relates the blk modls of the poros medim with flid-filled pores K to the blk modls of the poros medim with empty pores K D. Consider now the constittive eqation for the deviatoric stresses s ij in the poros skeleton. Since the flid in the pores is assmed to be inviscid, the deviatoric stress is, s ij ¼ 2G D e ij ¼ 2G e ij where G D is the shear modls of the poros medim with empty pores (drained shear modls), G is the shear modls of the poros medim with flid-filled pores (ndrained shear modls). From (4) it is clear that the shear modli of the drained and ndrained poros medim satisfy the relationship, G ¼ G D ðþ ð2þ ð3þ ð4þ ð5þ Experimental data presented by Thomsen [27] for several rocks sggests that the eqalities (3) and (5) mst be indeed satisfied. It trns ot that not all estimates for the effective modli of the poros medim satisfy the relationships (3) and (5). To demonstrate this we restrict or attention to the poros medim with ellipsoidal pores. Consider first a poros medim in which pores have the shape of oblate spheroids with a small aspect ratio, q (q is the ratio of the semi-minor axis to the semi-major axis of the spheroid). When q, these correspond to penny-shaped cracks, bt for brevity we shall refer to these pores as cracks. The cracks are assmed to be randomly distribted. Mori Tanaka estimates for the overall elastic modli of sch a poros material with empty pores are [3], K D ¼ K s þ g 6 m 2 s 9 2m s G D ¼ G s þ g 32 ð m sþð5 m s Þ ð6þ 45ð2 m s Þ where g =3n/(4pq) is the crack density, and n is the porosity. These eqations have been derived with the assmption that the aspect ratio of the oblate spheroids is mch smaller than nity, i.e., in the limit q?. Ths, they are essentially a trncated version of Mori Tanaka estimates valid only for small aspect ratios. For a poros material with penny-shaped cracks, the scheme by Kachanov can also be sed to estimate the components of the overall compliance matrix. For a random distribtion of empty cracks, components of the overall compliance matrix take the form [2], S ¼ =E s þ 32ð m2 s Þ g 3ð2 m s ÞE s 3 m s g ¼ =E s þg 32ð m2 s Þ 3m s 2 5 3ð2 m s ÞE s 3 m s g ¼ m s þg 6ð m2 s Þ 2 5 E s 45ð2 m s Þ S 22 ¼ =ð4g D Þ¼=ð4G s Þþ 32ð m2 s Þ 2g 3ð2 m s ÞE s 4 3 m s g ð7þ 2 5 S 22 ¼ S 33 ¼ m s E s þ 32ð m2 s Þ 3ð2 m s ÞE s The blk modls K D can be obtained from (7) as K D =(S + S 22 + S 33 ) /3, and the overall shear modls as G D = /4S 22. This will lead to the reslts of the Mori Tanaka method (6). Ths, for an isotropic distribtion of empty cracks, Kachanov s scheme is eqivalent to the trncated version of the Mori Tanaka method derived nder the limit q?. Consider now a poros medim with randomly oriented flidfilled cracks. Kachanov s estimate of the components of the overall compliance matrix are, S ¼ =E s þ 32ð m2 s Þ g 3ð2 m s ÞE s 3 þ w g 5 S 22 ¼ S 33 ¼ m s þ 32 m2 s w g E s 3ð2 m s ÞE s 5 S 22 ¼ =ð4g D Þ¼=ð4G s Þþ 32ð m2 s Þ 3ð2 m s ÞE s 4 where w ¼ m s df 2 ; d f ¼ 3ð 2m sþpq þ d f 4ð m 2 s Þ 2g 3 þ w g 5 K s K f Upon simplifications and sing the identities, / K D =3(S + S 22 + S 33 ), /G D =4S 22, we obtain expressions for the blk and shear modli of the poros medim with flidfilled cracks (ndrained elastic modli), as follows: K ¼ K s þ g 6 m 2 s d f 9 2m s þ d f G ¼ G s þ g 32ð m sþ 3 þ df ð2 þd f m s A 45ð2 m s Þ ð8þ ð9þ ðþ

3 A.P. Svorov, A.P.S. Selvadrai / Compters and Geotechnics 38 (2) Table Normalized blk and shear modli, K /K s, G /G s, for poros medim with randomly oriented flid-filled pores estimated by Kachanov s scheme (K), non-trncated version of Mori Tanaka method (MT), effective medim method (EM), selfconsistent method (SC), differential scheme (DIF). Porosity is.5. K /K s Aspect ratio, q K MT EM SC DIF / / / / G /G s / / / / Table 2 Normalized blk and shear modli, K D /K s, G D /G s, for poros medim with randomly oriented empty pores estimated by Kachanov s scheme (K), non-trncated version of Mori Tanaka method (MT), effective medim method (EM), self-consistent method (SC), differential scheme (DIF). Porosity is.5. K D /K s Aspect ratio K MT EM SC DIF / / / / G D /G s / / / / If the blk modls of the flid K f?, d f? and () redces to the reslts for the poros medim with empty cracks given by (6). We observe, however, that as predicted by Kachanov s theory, the shear modls of the poros medim with flid-filled cracks G is not eqal to G D. The Kachanov s estimates for the overall blk modli K D and K satisfy Biot Gassmann relationship (3). To jstify this statement, note that from (6 ) and the definition of Biot coefficient, 3 ð 2m 4 sþ= m 2 s ¼ 4 gð aþ=a, where g =3n/(4pq) is the crack 3 density. Then, from definition of d f in (9) it follows that d f = n(k s / K f )( a)/a. Using this expression for the d f in ( ) it can be shown that the Biot Gassmann relationship is satisfied. The Mori Tanaka estimates also satisfy relationship (3) for any shape of the randomly-distribted pores, i.e., with any aspect ratio. The jstification for this statement is given in Appendix A (see also Tables and 2). Ths, the eqivalence of the estimates of the overall drained blk modls K D, proven previosly, implies a similar eqivalence of the estimates of the ndrained blk modls K. 3. Nmerical reslts and modelling Table and Fig. give the estimates of the normalized overall blk and shear modli for the poros medim with flid-filled cracks and spherical cavities (i.e., K /K s, G /G s ). The estimates are obtained by Kachanov s scheme (K), varios effective medim methods inclding the self-consistent method (SC) and the differential scheme (DIF). The Mori Tanaka (MT) estimates are not trncated (i.e., they can be sed for any aspect ratio). The estimates of Fig.. Normalized overall modli for poros medim nder ndrained conditions estimated by non-trncated version of Mori Tanaka method (thick solid line), Kachanov s scheme (thin solid line), self-consistent method (dash-dot line) and differential scheme (dashed line). effective medim method (EM) are obtained by choosing the elastic modli of the comparison (reference) medim as ( n) K s, ( n)g s. Porosity n =.5, K s = 25 GPa, m s =.3, K f = 2.2 GPa. The reslts given in Table and Fig. indicate that Kachanov s estimates for the overall modli of the poros medim with pennyshaped cracks are the closest to the Mori Tanaka estimates and converge to the Mori Tanaka estimates faster than to any other estimate as the aspect ratio q?. Table 2 and Fig. 2 give the estimates of the normalized overall blk and shear modli of the poros medim with empty, or dry, cracks and spheres, i.e., K D /K s, G D /G s. The estimates are obtained by Kachanov s scheme (K), the non-trncated version of the Mori Tanaka method (MT), the self-consistent method (SC), effective medim method, Bdiansky and O Connell method [5], and the differential scheme. The estimates of effective medim method (EM) are obtained by choosing the elastic modli of the comparison (reference) medim as ( n)k s,( n) G s. The porosity is set to.5. Fig. 3 compares the difference between the shear modli G and its Biot-consistent conterpart G BC ¼ G D. The smallest difference between the shear modli G and G D is obtained from Kachanov s

4 724 A.P. Svorov, A.P.S. Selvadrai / Compters and Geotechnics 38 (2) Fig. 2. Normalized overall modli for poros medim nder drained conditions estimated by non-trncated version of Mori Tanaka method (thick solid line), Kachanov s scheme (thin solid line), self-consistent method (thick dash-dot line), Bdiansky and O Connell method (thin dash-dot line), and differential scheme (dashed line). scheme and the Mori Tanaka method, and the largest difference occrs in the self-consistent method. However, for the poros medim with spherical pores and disks the relationship G = G D is satisfied by all estimates. The same conclsion can be drawn from Tables and 2. Fig. 3 also shows the difference between the blk modli K and its Biot-consistent conterpart K BC. The latter is evalated from Eq. (3) sing the estimated vale of the blk modls K D. For all pore aspect ratios, the Kachanov s and Mori Tanaka estimates for the overall blk modls satisfy Biot Gassmann relationship (3). The estimates obtained by effective medim method also satisfy Biot Gassmann relationship (3) as it can be inferred from Tables and 2 (not shown in the figre). However, the self-consistent and differential scheme estimates do not. For the poros medim with spherical pores and disks, Biot Gassmann relationship (3) is satisfied by all estimates. Table 3 shows experimentally determined properties of several sandstones and granites docmented by Zimmerman [3]. Also, it shows their porosities and Biot coefficients a. Using the Mori Tanaka method, the aspect ratio of the pores was back-calclated Fig. 3. Normalized difference between ndrained modli estimated by nontrncated version of Mori Tanaka method, Kachanov s scheme, self-consistent method or differential scheme and the respective Biot-consistent modli obtained from the eqalities (3) and (5). Table 3 Poroelastic modli for varios sandstones and granites as given by Zimmerman [3]. Aspect ratio q of penny-shaped pores (cracks) is evalated by Mori Tanaka method G D (GPa) K D (GPa) a n q g ¼ 3n 4pq Berea sandstone / Weber sandstone / Ohio sandstone / Pecos sandstone / Boise sandstone / Rhr sandstone / Tennessee marble / Charcoal granite / Westerly granite / by sing the optimization toolbox available in MatLab TM. The objective fnction was the difference between the elastic modli of the drained material determined sing the Mori Tanaka method

5 A.P. Svorov, A.P.S. Selvadrai / Compters and Geotechnics 38 (2) and the 2 experimental vales of the modli, i.e., K MT D Kexp D þ G MT 2. D Gexp D The inpt for MatLab bilt-in optimization fnction fminimax was the set of experimentally measred elastic modli K exp D, Gexp D, and the Biot coefficient aexp. The reslt obtained was the shear modls of the matrix material G s and aspect ratio of the pores q. The optimal soltion corresponded to the zero vale of the objective fnction. From Table 3 it can be seen that the aspect ratio of pores (cracks) is smaller for rocks with smaller porosities. This is to be expected becase only in the presence of pores with small aspect ratios can a rock have a large Biot coefficient a (e.g.,.65, when the porosity is very small). In addition, the pores with very small aspect ratios have a low percolation threshold and their presence can explain non-zero vales of permeability even when the porosity is very small. It shold also be noted that the crack densities g =3n/(4pq) for sandstones listed in Table 3 range from.5 to bt the crack densities for granites and marble are in the range.5.2. To simplify comptations, the rock was assmed to have pores of a single shape, i.e., a distribtion of pore shapes was not considered. The aspect ratio presented in Table 3 shold, therefore, be treated as the average aspect ratio. The presence of pores with large aspect ratios, i.e., spherical pores, is still possible in the rock types examined, bt their volme fraction mst be limited and the presence of a certain volme of flat pores with a small aspect ratio is necessary. Using information on the average aspect ratio of pores, other qantities sch as the crack density, permeability and the local stress fields at the crack tip can be estimated. We now present the reslts of the finite element comptations. The comptational code COMSOL was sed to create several models of the poros medim with orthogonally oriented bt otherwise randomly distribted pores, with the possibility of their intersection. In each model all pores have the same size and aspect ratio. The aspect ratios of the pores are (spherical), / and /3. The modelled region has a cboidal shape with sides five times larger than the diameter of the spheroidal pores. The ability of the COMSOL code to create sch a model and generate the finite element mesh depends on the size of the model, the volme fraction of the pores, their aspect ratio and their orientation. For spherical pores, the COMSOL code was able to create a model of the poros medim with porosities as high as.226, bt for pores with an aspect ratio of / the maximm porosity that cold be modelled was only.36. The error messages generated by the code while creating a mesh prevent the ser from modelling a material region with higher vales of porosities when the pores have a small aspect ratio. This is regarded as a limitation of the code. A typical model sed in comptations is shown in Figs. 4 and 5. The aspect ratio of pores is /, the volme fraction of pores is.36. To simplify the model, instead of randomly oriented pores, only three families of mtally orthogonal cracks (pores) were modelled. It is important to note that the reslting system has a cbic material symmetry. If L is a 6 6 stiffness matrix for this system, then it is fond that L 44 <(L L 2 )/2 whereas for the isotropic case L 44 =(L L 2 )/2 = G. Bt for small porosities and sfficiently large aspect ratio of pores, L 44 (L L 2 )/2 and the system can be considered approximately isotropic. For the model shown in Figs. 4 and 5 there are 32 pores oriented parallel to the x y plane, 29 pores parallel to the y z plane, and 28 pores parallel to the x z plane. Fig. 6 shows the Biot coefficient a = K D /K s estimated by the Mori Tanaka method (solid line) and the differential scheme (dashed line) as a fnction of porosity n. The reslts of the finite element comptations for the poros material with randomly oriented overlapping pores are indicated with stars and open circles. The elastic properties of the material of the solid phase are taken as K s = 25 GPa, m s =.3. Fig. 4. The finite element model of a cboidal region of a poros material with the oblate spheroidal pores of aspect ratio /. The volme fraction of pores is.36. The Biot coefficient evalated sing the comptational scheme is shown for two sitations where the external bondary conditions correspond to either niform stress or niform strain. De to the relatively small size of the finite element model employed, the Biot coefficient depends on the type of the bondary conditions sed. When the stress bondary condition is applied, the RVE response is too compliant and an pper bond for the Biot coefficient is obtained, i.e., the overall elastic modls K D trns ot to be small. When the strain bondary condition is applied, the RVE response is too stiff and the lower bond for the Biot coefficient is obtained, i.e., the overall elastic modls K D is large. The correct soltion shold lie between these finite element (FE) bonds.

6 726 A.P. Svorov, A.P.S. Selvadrai / Compters and Geotechnics 38 (2) Biot coefficient α = ( K D /K s ) aspect ratio /3 aspect ratio / aspect ratio.3.2 Mori Tanaka method differential scheme. FEM niform stress FEM niform strain porosity n Fig. 6. The Biot coefficient for a poros material with randomly distribted overlapping pores estimated by the Mori Tanaka method, the differential scheme and a finite element approach..9 aspect ratio /3 Shear coefficient ( G D /G s ) aspect ratio /.4 aspect ratio.3.2 Mori Tanaka method differential scheme. FEM niform stress FEM niform strain porosity n Fig. 7. Shear coefficient G D /G s for a poros material with randomly distribted overlapping pores estimated by the Mori Tanaka method, the differential scheme and a finite element approach. Fig. 5. The side views of the finite element model of a cboidal region of a poros material with the oblate spheroidal pores of aspect ratio /. The volme fraction of pores is.36. Fig. 6 shows that for RVE with spherical pores, both Mori Tanaka and differential schemes provide the estimates for the Biot coefficient that lie within the finite element (FE) bonds althogh the Mori Tanaka estimate is almost coincident with the lower bond. For RVE with the pores of aspect ratio /, the Mori Tanaka and differential scheme estimates are still close to each other bt the differential scheme tends to slightly nderestimate the blk modls K D becase it lies slightly above the pper FE bond on the Biot coefficient. Fig. 7 shows the shear coefficient defined as G D /G s estimated by the Mori Tanaka method (solid line) and the differential scheme (dashed line). The comptational reslts are denoted by the stars and circles. The shear coefficient evalated sing the comptational approach is shown for both niform shear stress and niform shear strain bondary conditions. The shear coefficient is again dependent on the type of the bondary conditions applied. The imposed niform shear stress condition reslts in a smaller overall shear modls G D, and a larger shear coefficient than that for the imposed shear strain condition. Fig. 7 shows that for the spherical pores, the Mori Tanaka and differential scheme estimates for the shear coefficient are close to each other and both tend to lie closer to the lower finite element bond on the shear coefficient. For the pores with aspect ratio /, the estimates obtained from the Mori Tanaka and differential scheme are close only for very small range of porosities, and for these porosities the Mori Tanaka estimate falls slightly below the lower comptationally derived bond for the shear coefficient. The small vales of porosities sed in these comptations, however, limit validity of these conclsions.

7 A.P. Svorov, A.P.S. Selvadrai / Compters and Geotechnics 38 (2) New Biot-consistent estimates of elastic modli Let K, G be the estimates of the blk and shear modli of the poros medim (drained or ndrained). Let s denote the elastic modli of the comparison medim by K and G. Recall that in the Mori Tanaka method K = K s and G = G s, where K s, G s are the elastic modli of the solid phase. The Mori Tanaka estimates K MT, G MT for two-phase medim play an important role in that they coincide with the pper Hashin Shtrikman bond [4], i.e., any meaningfl estimates mst satisfy the ineqalities, 6 K 6 K MT ; 6 G 6 G MT ðþ The eqalities () are satisfied by the effective medim methods as long as the elastic modli of the comparison medim satisfy the following ineqalities [28,6]: 6 K 6 maxfk s ; K f g; 6 G 6 G s ð2þ For the drained poros medim the flid blk modls K f =, for the ndrained one it is not zero bt sally mch smaller than K s. Clearly there are infinite nmber of K, G that satisfy (2) and, therefore, infinite nmber of the estimates can be generated that satisfy the bonds (). Or goal is, however, to select the estimates that are Biot-consistent, i.e., satisfy both relationships (3) and (5). The reslts presented in Tables and 2, namely those of Mori Tanaka method and EM method, sggest that if comparison medim is chosen to be identical for drained and ndrained poros media, then the Biot Gassmann relationship (3) is indeed satisfied. However, the eqality of the shear modli (5) is satisfied only if the pore shape is spherical or disk-shaped, i.e., the pore aspect ratio is either one or zero. This sggests that the new Biot-consistent estimates K BC, G BC for the poros medim can be defined as follows. Select the properties of the comparison medim for the drained material sch that the ineqalities (2) are satisfied with K f =. Given the properties of the comparison material, obtain the estimates of the blk and shear modli for the drained poros medim, i.e., K D, G D. Let K, G be the estimates of the elastic modli for the ndrained poros medim obtained sing the same comparison material. These estimates satisfy the respective Hashin Shtrikman bonds, i.e., K 6 K MT, G 6 G MT as long as K f 6 K s which is sally the case. The estimates K D, K are Biot-consistent since they satisfy (3) bt, in general, G D G. Now define the new Biot-consistent estimates for the ndrained poros medim simply as, K BC a 2 ¼ K D þ ¼ K n=k f þða ; nþ=k s a ¼ K D K s G BC ¼ G D ð3þ The estimate of the blk modls K BC in (3) satisfies the Hashin Shtrikman bond () simply becase the estimate K satisfies this bond. Frthermore, G BC satisfies the Hashin Shtrikman bond since G BC ¼ G D 6 G MT D 6 GMT. Ths, the new estimates defined in (3) are Biot-consistent and satisfy the Hashin Shtrikman bonds. It shold be noted that if m D is the Poisson s ratio for the drained poros medim with elastic modli K D, G D, then the Poisson s ratio for the ndrained system with Biot-consistent elastic modli K BC G BC satisfies m D 6 m BC 6 :5. The proposed estimates can be considered a generalization of the reslts obtained by Thomsen [27] and Endres [7]. Thomsen [27] modified the self-consistent estimate, namely, the expression derived by Bdiansky and O Connell [5] in sch a way that the new estimate is Biot-consistent, i.e., both eqalities (3) and (5) are satisfied. To achieve this, the properties of the comparison medim were assmed dependent on the flid pressre conditions, i.e., whether the poros medim is drained or ndrained. Endres [7] sed essentially Mori Tanaka method (called in his paper by EIAS, method) and modified the concentration factor for the shear strain so that the new estimate satisfies the eqality for the shear modli (5). 5. Estimate for permeability It is natral to ask whether the effective medim approaches proposed here can be extended to provide estimates for the permeability of a poros medim, which is an important property of a flid-satrated mltiphase media [25]. The permeability of a poros rock containing cracks can be expressed as [2], k ¼ 2 5 fw2 qg ð4þ where q and g are the aspect ratio and crack density as before, f is the percolation factor fraction of connected pores, and w is the average crack apertre. In the stdies by Benson et al. [2] the crack apertre ranges from.2 lm tolm. According to the model of [] the percolation factor is eqal to zero for small crack densities satisfying p 2 g/4 < /3 and approximately for large crack densities satisfying p 2 g/4 >. For the range /3 < p 2 g/4 < the percolation factor can be approximated by the expression [], f ¼ 9 p g 2 ð5þ 3 Using definition of the crack density, Eq. (4) can be simplified to, k ¼ n p fw2 ð6þ where n is the porosity. Table 4 compares the measred vales of permeability [3] and the vales compted sing (6). The average crack apertre was assmed to be eqal to.5 lm for all rocks. It can be seen from Table 4 that for sandstones the agreement between measred and estimated vales of the permeability is acceptable at least when the porosity n is sfficiently small. This is perhaps explained by the narrower distribtion of pore shapes in the low porosity rocks. When the porosity n is large (e.g.,.26), the estimate for the permeability may be inaccrate bt can be corrected by choosing the crack apertre to be correspondingly larger than.5 lm. For charcoal granite and marble, the estimated permeability is zero for the two rocks de to the fact that the percolation factor f = for small vales of crack densities, g =.625,.879. It appears that more refined theories are needed to estimate the transport properties of very low permeability rocks. The estimate for the permeability of Westerly Granite with a crack density of g =.97 can also be corrected by making the crack apertre smaller than.5 lm. When the distribtion of the pore space is amorphos, the poros medim has a spatial heterogeneity that reqires alternative Table 4 Permeability (m 2 ) for varios sandstones and granites as given by Zimmerman [3] and estimated from Eq. (6). The crack apertre is.5 lm. n Measred Estimated Berea sandstone.9.87e 3.5E 5 Weber sandstone E E 6 Ohio sandstone E 5.5E 5 Pecos sandstone E 6.59E 5 Boise sandstone E 3 2.7E 5 Rhr sandstone.2.97e 6.59E 6 Tennessee marble E 2 Charcoal granite E 2 Westerly granite. 3.95E 9 4.7E 8

8 728 A.P. Svorov, A.P.S. Selvadrai / Compters and Geotechnics 38 (2) approaches for the estimation of effective properties. In a recent stdy, Selvadrai and Selvadrai [26] investigated the effective permeability of a poros medim with a spatial heterogeneity in the isotropic measre of the point-wise permeability. These athors have demonstrated throgh experimental and comptational modelling of the heterogeneity that a geometric mean provides an accrate estimate for the effective isotropic permeability of a poros medim with spatial heterogeneity of permeability, which has a lognormal distribtion. 6. Conclsions In this stdy analytical estimates of the overall elastic modli of a poros medim with randomly oriented flat oblate spheroidal pores were obtained sing effective medim methods and Kachanov s scheme. It was shown that Kachanov s scheme is eqivalent to the trncated version of the Mori Tanaka method when the aspect ratio of the pores is very small. It was also demonstrated that estimates obtained from the Mori Tanaka method satisfy the Biot Gassmann Eq. (3) for any shape of pores, bt the estimates obtained by the self-consistent method or differential scheme satisfy the Biot Gassmann eqation only for spherical and disk-shaped pores. The analytical estimates for the elastic modli were compared with the reslts obtained sing a comptational approach applied to a model of a poros medim with overlapping spherical pores and one containing three families of overlapping oblate spheroidal pores of identical size arranged in a mtally orthogonal configration. The comptational code COMSOL was sed to obtain nmerical reslts. The ability of the finite element program COM- SOL to create sch a model and generate the finite element mesh depends on the volme fraction of pores and their aspect ratio. In general, the smaller the aspect ratio of the pores, the lower the maximm porosity that can be modelled by sing the COMSOL code is. The analytical estimates for the overall blk modls agree closely with the blk modls obtained sing the comptational approach. The analytical estimates for the overall shear modls tend to be closer to the comptational estimates obtained sing the niform shear strain bondary conditions applied to cboidal RVE. However, in order to provide more definitive conclsions, higher porosities need to be modelled by alternative comptational strategies. The pore aspect ratios for several sandstones and granites were estimated sing the Mori Tanaka method and available experimental reslts. In general, it was fond that the aspect ratio of pores is smaller for rocks with smaller porosities. The crack density of sandstones was fond to be higher than the crack density of granites. Using information on the aspect ratio of pores, the estimates for the permeability were also obtained and compared with the experimental vales. The agreement between the two sets of reslts was fond acceptable for all sandstones with small porosities, n <.6, and even for certain sandstones with porosities as high as n =.9,.2. Acknowledgement The work described in this paper was spported by an NSERC Discovery Grant awarded to A.P.S. Selvadrai. Appendix A The effective medim methods for estimating the overall elastic modli of the two-phase heterogeneos solids are reviewed in this Appendix. All effective medim methods are formally eqivalent [6]. They differ only in the choice of the comparison (reference) medim. For example, in the Mori Tanaka method the stiffness of the comparison medim L is eqal to that of the matrix material L s, and in the self-consistent method L = L, where L is the overall stiffness matrix of the poros medim that is soght. In all effective medim methods, each inclsion or a pore is envisioned to be embedded within a large volme of the comparison medim. Ths, the interaction problem between mltiple inclsions is replaced by a mch simpler problem of a single inclsion in a homogeneos comparison medim sbjected to a niform strain or stress field. If a niform strain field is applied to the comparison medim, the strain field with the inclsion can be fond as, p ¼ T p ða:þ where T p is the strain concentration factor for the inclsion (or pore). It can be fond as, T p ¼ðI PðL L p ÞÞ ða:2þ where P is the Hill s polarization tensor [28], L p is the stiffness matrix of the inclsion. The matrix P depends only on the properties of the comparison medim, i.e., L and the shape of the inclsion. For an isotropic comparison medim, the forth-order tensor P is given by Ponte Castaneda and Willis [2]. We recast their reslts in a matrix form, i.e., by treating P as a 6 6 matrix rather than the tensor. Sppose that the inclsion is an oblate spheroid that has the minor semi-axis directed along the x -axis of the Cartesian coordinate system (x, x 2, x 3 ). In this coordinate system the 6 6 matrix P can be fond as, n l l l k þ m =4 k m =4 l k m =4 k þ m =4 P ¼ ða:3þ m p C A p where n, l, k, m, p are the constants [2] defined in terms of the aspect ratio of the spheroid q and the elastic modli K, G of the comparison medim, k ¼ 7h 2q2 4q 2 h G þ 3K h 2q 2 þ 2q 2 h 8ð q 2 ÞG ð4g þ 3K Þ l ¼ ðg þ 3K Þ h 2q 2 þ 2q 2 h 4ð q 2 ÞG ð4g þ 3K Þ n ¼ 6 5h 8q2 þ 8q 2 h G þ 3K h 2q 2 þ 2q 2 h 2ð q 2 ÞG ð4g þ 3K Þ m ¼ 5h 2q2 2q 2 h G þ 3K 3h 2q 2 6ð q 2 ÞG ð4g þ 3K Þ 24 3h 2q2 G þ 3K 2 3h þ 2q 2 3q 2 h p ¼ 8ð q 2 ÞG ð4g þ 3K Þ where for oblate spheroids q <, h i q cos ðqþ qð q 2 Þ =2 h ¼ ð q 2 Þ 3=2 and for prolate spheroids q >, h i q cosh ðqþþqðq 2 Þ =2 h ¼ ðq 2 Þ 3=2 ða:4þ The concentration factor for the matrix phase T s can be defined as,

9 A.P. Svorov, A.P.S. Selvadrai / Compters and Geotechnics 38 (2) T s ¼ðI PðL L s ÞÞ ða:5þ where L s is the stiffness of the matrix phase, and P is an isotropic matrix which can be fond from the eqality, ft p g¼fði PðL L p ÞÞ g¼ði PðL L p ÞÞ ða:6þ Here doble brackets signify averaging over all possible orientations of the pores. Ths, to determine P the concentration factor for the inclsions or pores (A.2) needs to be evalated and averaged over all orientations. Note that the formla (A.5), which gives the concentration factor for the matrix phase, is rather new since the se of arbitrary comparison material to estimate the properties of a poros medim with pores of arbitrary shape has not received sfficient attention. For a random distribtion of pores, the components of the 6 6 concentration factor {T} can be expressed in terms of components of the matrix T as, ftg ¼ 3 5 ðt þ T 22 þ T 33 Þþ 5 ðt 2 þ T 2 þ T 3 þ T 3 þ T 23 þ T 32 Þ þ 2 5 ðt 44 þ T 55 þ T 66 Þ ftg 2 ¼ 5 ðt þ T 22 þ T 33 Þþ 2 5 ðt 2 þ T 2 þ T 3 þ T 3 þ T 23 þ T 32 Þ 5 ðt 44 þ T 55 þ T 66 Þ ftg 44 ¼ 2 5 ðt þ T 22 þ T 33 Þ 5 ðt 2 þ T 2 þ T 3 þ T 3 þ T 23 þ T 32 Þ þ 5 ðt 44 þ T 55 þ T 66 Þ The matrix T can be fond in the local coordinate system of an inclsion (pore) according to (A.2). The strain in the inclsion can also be expressed in terms of the overall strain applied to the poros medim itself, p ¼ A p ða:7þ where A p is the total strain concentration factor for the inclsions (pores). This factor can be obtained in the form, A p ¼fT p gðnft p gþð nþt s Þ ða:8þ Eq. (A.7) ths gives the orientation averaged strain in the inclsions. The overall stiffness matrix of the poros medim can be fond as, L ¼ L s þ nðl p L s ÞA p ða:9þ If the properties of a drained poros medim are soght, the stiffness matrix of the pores is zero L p =. On the other hand, for ndrained conditions, L p is a reglar stiffness matrix that is obtained by setting the shear modls G f of the flid in the pore to zero. Formal eqivalence of all effective medim methods allows s to find the self-consistent estimate iteratively. Starting with some comparison medim L ðþ, possibly different from the solid phase L s, we generate, sing (A.5), (A.6) and (A.9), the estimate of the overall modli L (). At the next iteration we se this estimate as the new comparison medim L, i.e., L ð2þ ¼ L ðþ. Every time, the overall stiffness matrix obtained at a previos iteration serves as a comparison medim at the sbseqent iteration, i.e., L ðiþþ ¼ L ðiþ. Iterations are terminated when the difference between the stiffness of the comparison medim L ðiþ and the overall stiffness matrix L(i) is small. Now we prove that the Mori Tanaka estimates of the blk modls satisfy the Biot Gassmann eqation. For an arbitrary matrix A we define the volmetric part of the matrix as ½AŠ V ¼ ða 3 þ A 22 þ A 33 þ A 2 þ A 2 þ A 3 þ A 3 þ A 23 þ A 32 Þ. First of all note that from (A.9) the blk modls of the ndrained medim K can be represented as, K ¼ K s þ na o f ðk f K s Þ ða:þ where A o f is the volmetric strain concentration factor nder ndrained conditions. For the drained poros medim, K D ¼ K s þ na o f ð K sþ ða:þ where A o f is the volmetric part of the strain concentration factor nder drained conditions. If the Biot Gassmann Eq. (3) is satisfied, then it can be shown that, a A o f A f ¼ ¼ n þða nþk f =K s þða o ða:2þ f ÞK f =K s where A o f ¼ a=n, and a =( K D /K s ) is the Biot coefficient. Then from (A.2),! ¼ A f A o þ K f f A o ða:3þ K f s In the Mori Tanaka method, the volmetric concentration factors A f, A o f can be fond from (A.8) in terms of partial volmetric concentration factors as follows: A o f ¼ðn þð nþðt o f Þ Þ ; A f ¼ðn þð nþðt f Þ Þ ða:4þ where T f and T o f are partial volmetric concentration factors for the ndrained and drained poros media, respectively. They are defined as volmetric parts of the averaged matrices, i.e., T f =[{T p }] V, T o f ¼½fT o p gš V. Partial volmetric concentration factors can be expressed from (A.2) as, T f ¼ð P V K s þ P V K f Þ T o f ¼ð P V K s Þ ða:5þ where P V is a certain factor related to the Hill s polarization tensor P. P V depends on the shape of the pore and the elastic modli of the matrix material, i.e., solid phase, bt not on the modli of the pores. P V is niqe both for T f and T o f, and can be expressed in terms of To f ÞK as P V ¼ð T o f s. It is important to note that (A.4) is valid only for the Mori Tanaka method, bt the formla (A.5) is valid for any choice of comparison material once K s is replaced with the blk modls of that material. Now after sbstittion of (A.5) into (A.4), the eqality (A.3) or (A.2) can be proved. Conseqently, the Biot Gassmann Eq. (3) is satisfied. References [] Bary B. Simplified copled chemo-mechanical modeling of cement pastes behavior sbjected to combined leaching and external slfate attack. Int J Nmer Anal Meth Geomech 28;32: [2] Benson P, Schbnel A, Vincigerra S, Trovato C, Meredith P, Yong RP. Modeling the permeability evoltion of microcracked rocks from elastic wave velocity inversion at elevated isostatic pressre. J Geophys Res 26;. B422. [3] Benveniste Y. A new approach to the application of Mori Tanaka s theory in composite materials. Mech Mater 987;6: [4] Biot MA. General theory of three-dimensional consolidation. J Appl Phys 94;2: [5] Bdiansky B, O Connell RJ. Elastic modli of a cracked solid. Int J Solids Strct 976;2:8 97. [6] Dvorak G, Srinivas MV. New estimates of overall properties of heterogeneos solids. J Mech Phys Solids 999;47: [7] Endres A. Geometrical models for poroelastic behavior. Geophys J Int 997;28: [8] Girad A, Gresc C, Do DP, Homand F, Kondo D. Effective thermal condctivity of transversely isotropic media with arbitrary oriented ellipsoidal inhomogeneities. Int J Solids Strct 27;44:8 33.

10 73 A.P. Svorov, A.P.S. Selvadrai / Compters and Geotechnics 38 (2) [9] Girad A, Hynh QV, Hoxna D, Kondo D. Effective properties of transversely isotropic rock-like composites with arbitrarily oriented ellipsoidal inclsions. Mech Mater 27;39:6 24. [] Gresc C, Girad A, Homand F, Kondo D, Do DP. Effective thermal condctivity of partially satrated poros rocks. Int J Solids Strct 27;44: [] Gegen Y, Dienes J. Transport properties of rocks from statistics and percolation. Math Geol 989;2(): 3. [2] Gegen Y, Sarot J. Crack-indced anisotropy in crstal rocks: predicted dry and flid-satrated Thomsen s parameters. Phys Earth Planet Interiors 29;72:6 24. [3] Gery AAC, Cormery F, Shao JF, Kondo D. A comparative micromechanical analysis of the effective properties of a geomaterial: effect of mineralogical compositions. Compt Geotech 2;37: [4] Hashin Z, Shtrikman S. A variational approach to the theory of the elastic behavior of mltiphase materials. J Mech Phys Solids 963;: [5] Hill R. Self-consistent mechanics of composite materials. J Mech Phys Solids 965;3: [6] Kachanov M. Elastic solids with many cracks and related problems. Adv Appl Mech 994;3: [7] McLaglin R. A stdy of the differential scheme for composite materials. Int J Eng Sci 977;5: [8] Mori T, Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclsions. Acta Metall 973;2:57 4. [9] Narsilio GA, Bzzi O, Fitys S, Yn TS, Smith DW. Upscaling of Navier Stokes eqations in poros media: theoretical, nmerical and experimental approach. Compt Geotech 29;36:2 6. [2] Norris A. On the correspondence between poroelasticity and thermoelasticity. J Appl Phys 992;7(3):38 4. [2] Ponte Castaneda P, Willis JR. The effect of spatial distribtion on the effective behavior of composite materials and cracked media. J Mech Phys Solids 995;43(2):99 5. [22] Rice JR, Cleary MP. Some basic stress-diffsion soltions for flid-satrated elastic poros media with compressible constitents. Rev Geophys Space Phys 976;4: [23] Roberts AP, Garboczi EJ. Comptation of the linear elastic properties of random poros material with a wide variety of microstrctre. Proc Roy Soc A 22;458: [24] Selvadrai APS, editor. Mechanics of poroelastic media. The Netherlands: Klwer Academic Pblishers; 996. [25] Selvadrai APS. The analytical methods in geomechanics. Appl Mech Rev 27;6:87 6. [26] Selvadrai APS, Selvadrai PA. Srface permeability tests: experiments and modelling for estimating effective permeability. Proc Roy Soc Math Phys Sci Ser 2;466: [27] Thomsen L. Biot-consistent elastic modli of poros rocks: low-freqency limit. Geophysics 985;5: [28] Walpole LJ. On the overall elastic modli of composite materials. J Mech Phys Solids 969;7: [29] Zimmerman RW. Elastic modli of a solid containing spherical inclsions. Mech Mater 99;2:7 24. [3] Zimmerman RW. Micromechanics of poroelastic rocks. In: Markov K, Preziosi L, editors. Heterogeneos media: micromechanics modeling methods and simlations; 2. p. 4 7.