New Renormalization Schemes for Conductivity Upscaling in Heterogeneous Media

Transcription

1 Transp Porous Med () 85: DOI.7/s New Renormalization Schemes for Conductivity Upscaling in Heterogeneous Media M. R. Karim K. Krabbenhoft Received: 8 February / Accepted: April / Published online: 3 April Springer Science+Business Media B.V. Abstract Two new renormalization schemes for conductivity upscaling in heterogeneous media are presented. The schemes follow previous ones by performing the renormalization over square cells of size d with d being the dimensionality. Contrasting with previous schemes, the two-dimensional scheme makes use of the exact block-conductivity. On the basis of the structure of the exact two-dimensional block-conductivity, an analogous threedimensional scheme is proposed. The new schemes are tested on a number of benchmark problems and are shown to be significantly more accurate than existing schemes. Keywords Renormalization Upscaling Heterogeneous media Homogenization Introduction A fundamental problem in many fields of science and engineering is that of representing a heterogeneous conductivity field by a single effective conductivity that characterizes the conditions on some suitable macroscopic length scale. This process is known variously as upscaling or homogenization and common technological applications include the determination of effective hydraulic conductivities in petroleum and water resources engineering and the determination of effective thermal conductivities, molecular diffusivities, and dielectric constants for natural and man-made composites. Common to these problems is the assumption that the transport is governed by a gradient type law on the microscale (the smallest length scale considered) as well as on the macroscale (the largest length scale considered). That is, the physical laws governing the transport are assumed to be scale invariant. The assumption that the physics is invariant across the range of length scales considered is a most convenient one which allows for straightforward application of a large number of standard homogenization procedures. One such procedure is known as renormalization. Originally conceived in the field of quantum electrodynamics, the procedure was first applied M. R. Karim K. Krabbenhoft (B) Centre for Geotechnical and Materials Modelling, University of Newcastle, Callaghan, NSW, Australia kristian.krabbenhoft@gmail.com

2 678 M. R. Karim, K. Krabbenhoft.98 k e k e k e k e k e.3 k e =.5 Fig. Example of block renormalization starting from a 3 3 grid. The problem is the classic random checkerboard with equal phase fractions. The conductivities are k = and k =, giving an effective conductivity of k e = k k =. Also shown in the figures are the Wiener bounds (the harmonic and arithmetic means) to the problem of hydraulic conductivity upscaling by King (989). The basic idea, which is illustrated in two spatial dimensions in Fig., is the following. Given a square domain consisting of n cells, each with a unique conductivity, a partial upscaling is performed by replacing each block in the grid by a single cell with an appropriate representative block-conductivity. This gives a grid consisting of n cells, and the procedure is then repeated recursively until only one macro-cell remains. The conductivity of this cell is the sought effective conductivity. Starting from this basic idea, a large number of different strategies have been developed (see e.g.,green and Patterson 7; Lunati et al. ; Renard and de Marsily 997; Renard et al. ). For the type of procedure described above, the most basic issue is that of determining the representative conductivity of a four-phase block. A fundamental issue is here what boundary conditions should be applied to the block. Strictly speaking, the concept of effective conductivity of random heterogeneous media applies only to volumes of an infinite extent. For finite-size volumes, for example blocks, the effective conductivity will depend on the boundary conditions. In classic renormalization procedures, the boundary conditions are usually of the mixed type. In the two-dimensional case, no-flow Neumann boundary conditions are enforced on two opposite edges while the Dirichlet boundary conditions are enforced on the other two edges (see Fig. ). Similarly, in three dimensions, Dirichlet conditions are applied on two opposite faces while Neumann boundary conditions are enforced on all other faces. In ensemble averaging procedures, pure Dirichlet and Neumann boundary conditions are known to result in upper and lower bounds, respectively, on the effective conductivity while mixed boundary conditions of the type shown in Fig. result

3 New Renormalization Schemes 679 Fig. Four-phase checkerboard with mixed boundary conditions for calculating the block-conductivity in the x-direction y k 3 k 4 No-flow b. c. P P k k x No-flow b. c. in an effective conductivity that falls in between these two bounds Karim and Krabbenhoft ; Ostoja-Starzewski and Schulte 996). Considering the approximate nature of renormalization, the use of mixed boundary conditions thus appears to be quite reasonable. In the following, all results on block-conductivities assume mixed boundary conditions of the kind shown in Fig.. In the context of hydraulic conductivity upscaling, renormalization was first considered by King (989) who devised a finite-difference type solution to the calculation of blockconductivities. In the two-dimensional case this leads to a closed-form expression for the block-conductivity. For the three-dimensional case similar arguments can be used although the equivalent closed-form expression is somewhat more problematic to derive and rather expensive to compute (Green and Patterson 7). The immediate question that arises concerns the accuracy of the block-conductivity solution. Surprisingly, although the relevant two-dimensional solution has been available for some years (Craster and Obnosov ; Milton ; Mortola and Steffe 985), it has not to our knowledge been employed in renormalization schemes. The first aim of the present paper is therefore to use this solution in a conventional renormalization scheme as described above. Second, inspired by the structure of the exact two-dimensional block-conductivity solution, we propose an approximate expression for the block-conductivity of a three-dimensional block. The new schemes are validated against both analytical and numerical (finite element/difference type) solutions. Means For later reference, a number of concepts related to mean values are briefly summarized in the following. Let k = (k, k,...,k n ) be an array of n non-negative conductivities. The arithmetic mean is then given by μ a (k) = n n i= k i = k + k + +k n n = k a () The harmonic mean is given by μ h (k) = ( n ) n = k i= i n ( k + k + + k n ) =[μ a (k )] = k h ()

4 68 M. R. Karim, K. Krabbenhoft where k is to be understood as an element-by-element operation. As is well-known, these means provide rigorous upper and lower bounds on the effective conductivity: k h k e k a (3) In addition to the arithmetic and harmonic means, the geometric mean is also useful. This is given by ( n ) n μ g (k) = k i = (k k k n ) n = exp[μa (ln k)] =k g (4) i= It can be shown that the arithmetic, harmonic, and geometric means are related by Furthermore, it is straightforward to show that k h k g k a (5) k h μ g (k a, k h ) k a (6) Thus, the geometric mean of the harmonic and arithmetic means are bounded by these means themselves. This is a useful property that will be utilized later on. 3 Two-Dimensional Block-Conductivity A crucial feature of the two-dimensional renormalization scheme of King (989) is the calculation of block-conductivities as described in the Introduction. King originally proposed that the block-conductivity of a block involving four different conductivities (see Fig. ) be estimated on the basis of a finite difference approximation as: k e,x = 4(k + k 3 )(k + k 4 )A (k + k + k 3 + k 4 )A + B (7) where A = k k (k 3 + k 4 ) + k 3 k 4 (k + k ) B = 3(k + k )(k 3 + k 4 )(k + k 3 )(k + k 4 ) (8) The block-conductivity in the y-direction is obtained by interchanging k and k 3. As noted by Yeo and Zimmerman (), this approximation is potentially rather inaccurate. Indeed, for the case of a two-phase checkerboard (k = k 4 and k = k 3 ), we have 8r( + r) k e /k = ( + 3r)(3 + r), r = k /k (9) where k e = k e,y = k e, while the exact solution, due to Dykhne (97), is given by k e /k = r () Thus, for large r, the King approximation implies a finite relative block-conductivity of k e /k.67 which is in evident contrast to the exact solution.

5 New Renormalization Schemes 68 k 3 k 4 k 3 k 4 k 3 k 4 y k k k k k k x (a) (b) Fig. 3 Four-phase checkerboard and splittings (a)and(b) for flow in the x-direction 3. Exact Solution for Four-Phase Checkerboard Two approximate solutions to the exact block-conductivity of the four-phase checkerboard can be obtained by the considering the splittings shown in Fig. 3. For splitting (a), the blockconductivity is given by k L,x = μ a [μ h (k, k ), μ h (k 3, k 4 )]= k k (k 3 + k 4 ) + k 3 k 4 (k + k ) (k + k )(k 3 + k 4 ) () while for splitting (b) we have k U,x = μ h [μ a (k, k 3 ), μ a (k, k 4 )]= (k + k 3 )(k + k 4 ) k + k + k 3 + k 4 () It may be shown that () and() constitute, respectively, lower and upper bounds on the exact block-conductivity, hence subscripts L and U. These bounds are in a certain sense similar to conventional harmonic and arithmetic bounds and it could therefore be expected that a more accurate estimate would be obtained by the geometric mean of these two bounds. In fact, as first conjectured by Mortola and Steffe (985) and later demonstrated rigorously by Milton ()and by Craster and Obnosov (), this estimate of the block-conductivity constitutes the exact solution. It may be expressed as where [ ] ( ) k e,x = (k L,x k U,x ) (k + k 3 )(k + k 4 ) I3 = (3) (k + k )(k 3 + k 4 ) I I = k + k + k 3 + k 4 I 3 = k k (k 3 + k 4 ) + k 3 k 4 (k + k ) (4) Again, the block-conductivity in the y-direction is obtained by interchanging k and k 3. In doing so, we note that I and I 3 are coordinate invariant. 4 Three-Dimensional Block-Conductivity Inspired by the structure of the exact two-dimensional solution discussed in the previous section, we now construct a three-dimensional approximation along the same lines. The splittings analogous to the two-dimensional case are shown in Fig. 4. For configuration (a),

6 68 M. R. Karim, K. Krabbenhoft k 7 k 8 k 7 k 8 k 7 k 8 k 3 k 4 k 6 k 3 k 4 k 6 k 3 k 4 k 6 y k k k k k k z (a) (b) x Fig. 4 Three-dimensional eight-phase checkerboard and splittings (a)and(b) for flow in the x-direction the block-conductivity is given by where k L,x = μ a [μ h (k, k ), μ h (k 3, k 4 ), μ h (k 5, k 6 ), μ h (k 7, k 8 )] = k k S + k 3 k 4 S 34 + k 5 k 6 S 56 + k 7 k 8 S 78 (k + k )(k 3 + k 4 )(k 5 + k 6 )(k 7 + k 8 ) S ij = (k + k )(k 3 + k 4 )(k 5 + k 6 )(k 7 + k 8 ) (6) k i + k j For configuration (b) we have k U,x = μ h [μ a (k, k 3, k 5, k 7 ), μ a (k, k 4, k 6, k 8 )] = (k + k 3 + k 5 + k 7 )(k + k 4 + k 6 + k 8 ) (7) k + k + k 3 + k 4 + k 5 + k 6 + k 7 + k 8 Finally, the block-conductivity is taken as the geometric mean of the upper and lower bounds: [ ] ( ) k e,x = (k L,x k U,x ) (k + k 3 + k 5 + k 7 )(k + k 4 + k 6 + k 8 ) J5 = (8) (k + k )(k 3 + k 4 )(k 5 + k 6 )(k 7 + k 8 ) I where I = k + k + k 3 + k 4 + k 5 + k 6 + k 7 + k 8 (9) J 5 = k k S + k 3 k 4 S 34 + k 5 k 6 S 56 + k 7 k 8 S 78 with S ij given by (6). We note the similarity in structure to the two-dimensional solution (3). Alternatively, the block-conductivity can be expressed as ( ) ( ) ( ) ( ) k e,x = k + k + k3 + k4 + k5 + k6 + k7 + k8 (k + k 3 + k 5 + k 7 ) + (k + k 4 + k 6 + k 8 ) This form is much better suited for numerical implementation, first because it involves significantly fewer floating point operations than (8) and second, because all intermediate results are well scaled. The above expressions coincide with this exact two-dimensional solution for all cases where the three-dimensional checkerboard reduces to a two-dimensional geometry, e.g., for (5) ()

7 New Renormalization Schemes 683 (a).3 New scheme, eqn. (8) (b).5 Green & Patterson (7) scheme Frequency.. Frequency Error (%) 4 4 Error (%) Fig. 5 Error distributions for new approximate 3D block-conductivity formula (8)(a) and that of Green and Patterson (7)(b) k i = k i+4, i =,, 3, 4. However, we can immediately rule out the possibility that () is exact. Indeed, for the 3D analog of the D two-phase board discussed previously, i.e., k 4 = k 6 = k 7 = k and k 3 = k 5 = k 8 = k, the exact block-conductivity for k /k is given by k e = k k (Keller 987). In contrast, the approximation ()predictsk e = k k regardless of the ratio k /k. 4. Evaluation of 3D Block-Conductivity Approximation Besides the Keller problem, we are unaware of 3D problems with known analytical solutions by which the analytical expression (8) may be evaluated. Consequently, the block-conductivity of a large number of eight-phase checkerboards were determined numerically and the results compared to the analytical expression. The finite element method was used and in each case, a cube of the type shown in Fig. 4 was discretized using 64 3 = 6, 44 four-node elements with equal side lengths. The results for, random boards, with conductivities ranging between and, are shown in Fig. 5a. Also shown for comparison in Fig. 5b is the error in the formula proposed by Green and Patterson (7) which is based on the D King formula (7). It is seen that the new 3D approximation is significantly more accurate. Indeed, the mean absolute error is only around % versus approximately % for the Green and Patterson formula. Moreover, whereas the probability of an error in excess of 5% is only % with the new formula, it is 69% with the Green and Patterson formula. For errors in excess of, 5, and % the probabilities are 45, 8, and 8% for the Green and Patterson formula versus only.,.4, and.% with the new scheme. 5Results In the following, the new renormalization schemes are tested on a number of common examples. Where possible, we compare the results obtained with results from the literature and with the results of renormalizations using the block-conductivity of King (in D) and Green (in 3D). 5. Random Checkerboards The random checkerboard constitutes a severe test example for analytical and numerical homogenization techniques. A board consisting of n n squares of conductivities k and k in proportions φ and φ is considered. In the case of equal phase fractions, φ = φ =,

8 684 M. R. Karim, K. Krabbenhoft D random checkerboard, k / k = D random checkerboard, k / k =.8.8 k e / k.6.4 k e / k Legend: φ φ Present renormalization, King renormalization, Torquato et al. (999), Fig. 6 Effective conductivity of two-dimensional random checkerboards 4 point bounds the exact effective conductivity is given by k e = k k. For other phase fractions, no known analytical solutions exist. However, for the cases where k /k = and k /k = respectively, Torquato et al. (999) have provided results from Brownian motion simulations that are believed to be rather accurate. Furthermore, the effective conductivity is bounded by the four-point bounds (see Milton ; Torquato ): k L k e k U () where with k L = k φ φ (k k ) k +y, k U = k φ φ (k k ) k +y () y = k (k + k ζ ), k + k ζ y = k (k + k ζ ), k + k ζ (3) k =φ k + φ k, k =φ k + φ k, (4) k ζ = ζ k + ζ k, k ζ = ζ k + ζ k, (5) φ = φ, ζ = ζ (6) For the D random checkerboard, the microstructural parameter ζ is given by ζ = φ (7) For the three-dimensional counterpart of the D random checkerboard, three-point bounds that depend only on the phase fractions and the microstructural parameter ζ have been established as (see Milton, ; Torquato,, for details): k L = + ( + φ )β (φ ζ φ )β k, k U = k φ φ (k k ) + φ β (φ ζ + φ ) k + k ζ (8) where β = k k, k k ζ = φ (9) and k, k, and k ζ are given by (4) (5).

9 New Renormalization Schemes D random checkerboard, k / k = 3D random checkerboard, k / k =.8 k e / k.6.4 k e / k Legend: Present renormalization, φ φ Green & Patterson renormalization, Fig. 7 Effective conductivity of three-dimensional random checkerboards 3 point bounds The results of the two-dimensional checkerboard, for k /k = and k /k =, are shown in Fig. 6. Also shown in the figures are the four-point upper and lower bounds as well as the results obtained by Torquato et al. (999) using a first-passage random walk algorithm and the results of renormalization using the block-conductivity formula (7) ofking (989). We here see that the results of the new renormalization scheme match those of Torquato et al. (999) quite closely while the results of the King renormalization underestimate the effective conductivity. Indeed, whereas the present results consistently fall within the four-point bounds, the results of the King renormalization fall outside these bounds for the lower phase contrast of k /k =. As noted by Torquato et al. (999), the effective conductivity has a tendency to be closer to the four-point lower bound for phase fractions φ below the site percolation threshold, φ c.597, and closer to the upper bound above this threshold. The results for the three-dimensional random checkerboard are shown in Fig. 7 together with the three-point bounds and the results of renormalization using the block-conductivity formula proposed by Green and Patterson (7). The effective conductivity here displays a behavior similar to that of the two-dimensional case, i.e., a marked increase with increasing phase fraction φ that becomes more pronounced as the phase contrast k /k increases. Also, the deviation between the present renormalization and that of Green and Patterson (7) follows that of the two-dimensional case with the maximum deviation being in excess of 5% (for φ =.3 andk /k = ). Finally, in order to examine the capabilities of the various renormalization schemes in capturing the percolation properties of the two- and three-dimensional random checkerboards, we determined the effective conductivity as function of the phase fraction for the case where k /k tends to infinity (in practice, we used k /k = ). The results of these calculations are shown in Fig. 8. We here see that all the schemes considered capture the site percolation threshold quite well. However, the effective conductivities above the percolation threshold differ considerably, especially for the three-dimensional case. 5. Log-Normal Media The hydraulic conductivity of natural geological formations is often assumed to vary spatially according to the log-normal distribution: [ ] P(k; μ ln,σ ln ) = exp (ln k μ ln) kσ ln π σln (3)

10 686 M. R. Karim, K. Krabbenhoft k e / k D random checkerboard, k / k = Present renormalization King renormalization 4 point upper bound φ c =.597 k e / k D random checkerboard, k / k = Present renormalization Green & Patterson renormalization 3 point upper bound φ c = φ φ Fig. 8 Effective conductivity of two- and three-dimensional random checkerboards with infinite phase contrast where k = k(x, y, z) is the spatially variable hydraulic conductivity, μ ln = μ a (ln k) is the arithmetic mean of its natural logarithm, and σ ln = σ(ln k) is the standard deviation of ln k. Over the years, isotropic log-normal media have been the subject of numerous studies (see Sanchez-Vila et al. 6, for a recent review) and a number of analytical results are available. First, for the two-dimensional case, the effective conductivity is given by the geometric mean: k e = k g = exp[μ a (ln k)] =exp(μ ln ) (3) This result, which is attributed to Matheron (967), does not hold in three dimensions where the effective conductivity does depend on the standard deviation of the hydraulic conductivity. This dependence has been conjectured to take the form (Matheron 967) k e /k g = exp [ ( d )σ ln ] (3) where d is the dimensionality. The one- and two-dimensional effective conductivities are covered by this conjecture and numerical results indicate that the three-dimensional effective conductivity follows the formula quite closely up to at least σln = 7(Dykaar and Kitanidis 99; Neuman and Orr 993). Furthermore, using the method of perturbations and assuming small σln, Gutjahr et al. (978) found: k e /k g = + ( d ) σ ln (33) This expression constitutes the first-order Taylor approximation (in σln )to(3). Later, again using small perturbations, Dagan (993) extended this result to k e /k g = + ( d ) σ ln + ( d ) σ 4 ln (34) which is the second-order Taylor approximation to (3), thus further strengthening the conjecture. However, shortly thereafter, De Wit (995) found the next term in the series to obtain the highest order approximation so far: k e /k g = + ( d ) σ ln + ( d ) [ σ 4 ln + 6 ( d ) 3 ] + ε σln 6 (35) The quantity ε is equal to zero for d =, while, for d = 3, De Wit found ε =.4, thus effectively disproving the Matheron conjecture.

11 New Renormalization Schemes 687 k e / k g..5 D lognormal medium Exact Present renormalization King renormalization k e / k g D lognormal medium Matheron Numerical (FE) conjecture Eqn. (36) De Wit approximation Present renormalization σ ln Green & Patterson renormalization σ ln Fig. 9 Effective conductivity of two- and three-dimensional lognormal media We first consider the two-dimensional case using the exact block-conductivity and that of King (989). The results of the analyses are shown in Fig. 9. We here see that while new scheme verifies the exact solution, use of the King block-conductivity formula leads to a normalized effective conductivity k e /k g that decreases with σ ln, in evident contrast to the exact solution. The results of the three-dimensional case are shown in Fig. 9. Also shown are the results obtained using the scheme of Green and Patterson (7) as well as the effective conductivities predicted by the Matheron conjecture (3) and the De Wit approximation (35). Furthermore, a number of numerical results are shown. These latter results were obtained by generating cubes. Initially, each voxel (volume pixel) in these cubes was represented by a single finite element and the effective conductivity was then extracted using standard computational homogenization procedures (e.g., Jiang et al. ). It was, however, found that this approach tended to overestimate the computed conductivities quite significantly. Consequently, the additional elements were added by subdividing the cubes systematically to eventually end up with 56 3 meshes. For low to moderate values of σ ln this degree of refinement appears to be adequate while, for large values of σ ln, the computed effective conductivities may still be somewhat overestimated. Furthermore, the representivity of the original 6 3 cubes is questionable, i.e., a larger initial volume would be desirable which, however, would imply a lower degree of accuracy in the finite element analysis for a fixed level of subdivision. These issues with representivity and finite element accuracy are reflected in the results which tend to become somewhat oscillatory for larger values of σ ln. Nevertheless, the results do reveal a number interesting facts. First, numerically computed effective conductivities begin to depart significantly from the Matheron conjecture and the De Wit approximation rather early on, at around σln =.5. As the De Wit approximation possesses sixth-order accuracy, this is somewhat surprising. On the other hand, there appears to be excellent agreement between the numerical results and those obtained using the new renormalization scheme, at least up to around σln = 5. Above this value the numerical results are too inaccurate to draw any firm conclusions, but bearing in mind that the effective conductivity converges from above as the mesh is refined, a similarly good agreement could well be expected for larger values of σ ln. Finally, and somewhat ironically, while the numerical results appear follow the Matheron conjecture quite well for σln <.5, the accuracy of the new renormalization scheme is not particularly good in this range. Bearing this in mind, we propose the following approximation: k e /k g = + 6 σ ln σ 4 ln + 39 σ 6 ln (36)

12 688 M. R. Karim, K. Krabbenhoft Swiss-cheese model Inverse Swiss-cheese model Fig. Swiss-cheese and inverse Swiss-cheese models with a sphere volume fraction of φ =.6 (thetwo assemblies are the inverse of each other) This expression fits the numerical results well for small values of σ ln while it gradually tends to the renormalization predictions for larger values of σ ln. 5.3 Swiss-Cheese Model We finally consider the so-called Swiss-cheese model which consists of an assembly of overlapping spheres (see Fig. ). This model is frequently used in the study of transport properties of porous geological media such as sandstone (Ewing and Hunt 6; Klemm et al. ; Torquato ). Spheres of identical volume are here distributed at random and identified as the solid phase while the surrounding material (the matrix ) is identified as the pore space. Alternatively, in the inverse Swiss-cheese model, the spheres play the role of the pore space while the matrix is thought of as the solid skeleton. In the following, we refer the matrix as the -phase and the sphere phase as -phase. The percolation properties of the Swiss-cheese model are as follows. The matrix phase is connected for volume fractions φ.3 while the sphere phase is connected for φ.9. Consequently, there is a rather large range,.3 φ.7 (.9 φ.97), where the material is bicontinuous, i.e., both phases are connected. In the following, results for various Swiss-cheese models are presented. In each case, a unit cube containing 5 3 voxels was generated and approximate, digitized, spheres of diameter.5 were inserted at random on this lattice. The results for phase contrasts of k /k = and k /k = 5 are shown in Fig.. In all cases, the renormalization results fall between the rigorous three-point upper and lower bounds. They are further in reasonable agreement with the results of Kim and Torquato (99) who used a first-passage simulation technique similar to one of Torquato et al. (999). However, the agreement is not as good as in the case of the random checkerboards considered in Sect. 5.. Furthermore, and quite surprisingly, the results of the new renormalization scheme do not differ greatly from those of the Green and Patterson scheme. This trend is also observed in Fig. where the phase contrast approaches infinity. Such a phase contrast corresponds to the situation encountered in diffusion through porous media of the type shown in Fig.. For the Swiss-cheese model both renormalization schemes thus underestimate the effective conductivity (or diffusivity) at least compared to the results of Kim and Torquato (99). Also, for both the Swiss-cheese and the inverse

13 New Renormalization Schemes Swiss-cheese model, k / k = Swiss-cheese model, k / k = 5.8 k e / k.6.4 k e / k Porosity, φ Legend: Present renormalization, King renormalization, Kim & Torquato (99), Fig. Effective conductivity of Swiss-cheese models with finite phase contrasts Porosity, φ 3 point bounds. Swiss-cheese model, k / k =. Inverse Swiss-cheese model, k / k =.8.8 k e / k.6.4 k e / k Porosity, φ Legend: Present renormalization, King renormalization, Kim & Torquato (99), Porosity, φ 3 point upper bound Fig. Effective conductivity of Swiss-cheese and inverse Swiss-cheese models with infinite phase contrast Swiss-cheese model, the percolation thresholds (φ c =.3 and φ c =.9) are overestimated somewhat. The Swiss-cheese model differs from the examples previously considered in that the correlation length, depending on the volume fractions of the two phases, is of an appreciable magnitude. For such media, renormalization schemes can be expected to be less accurate due to their local nature as is indeed confirmed by the examples considered. 6 Conclusions Two new renormalization schemes for conductivity upscaling in heterogeneous media have been presented. The schemes follow previous ones, the only difference being that new blockconductivity expressions are used. In the two-dimensional case, the exact block-conductivity is used while the three-dimensional scheme makes use of an approximation inspired by the structure of the exact D solution. Examples show that the performance of the new schemes is significantly better than existing ones, especially for media with limited internal correlation lengths.

14 69 M. R. Karim, K. Krabbenhoft References Craster, R.V., Obnosov, Y.V.: Four-phase checkerboard composites. SIAM J. Appl. Math. 6, () Dagan, G.: Higher-order correction of effective permeability of heterogeneous isotropic formations of lognormal conductivity distribution. Transp. Porous Media, 79 9 (993) De Wit, A.: Correlation structure dependence of the effective permeability of heterogeneous porous media. Phys. Fluids 7, (995) Dykaar, B.B., Kitanidis, P.K.: Determination of the effective hydraulic conductivity for heterogeneous porous media using a numerical spectral approach. Results. Water Resour. Res. 8, (99) Dykhne, A.M.: Conductivity of a two-dimensional two-phase system. Sov. Phys. JETP 3, (97) Ewing, R.P., Hunt, A.G.: Dependence of the electrical conductivity on saturation in real porous media. Vadose Zone J. 5, (6) Green, C.P., Patterson, L.: Analytical three-dimensional renormalization for calculating effective permeabilities. Transp. Porous Media 68, (7) Gutjahr, A.L., Gelhar, L.W., Bakr, A.A., McMillan, J.R.: Stochastic analysis of spatial variability in subsurface flow : Evaluation and application. Water Resour. Res. 4, (978) Jiang, M., Jasiuk, I., Ostoja-Starzewski, M.: Apparent thermal conductivity of periodic two-dimensional composites. Comput. Mater. Sci. 5, () Karim, M.R., Krabbenhoft, K.: Extraction of effective cement paste diffusivities from X-ray microtomography scans. Transp. Porous Media (, in press). doi:.7/s y Keller, J.B.: Effective conductivity of periodic composites composed of two very unequal conductors. J. Math. Phys. 8, 56 5 (987) Kim, I.C., Torquato, S.: Effective conductivity of suspensions of overlapping spheres. J. Appl. Phys. 7, (99) King, P.R.: The use of renormalization for calculating effective permeability. Transp. Porous Media 4, (989) Klemm, A., Kimmich, R., Weber, M.: Flow through percolation clusters: NMR velocity mapping and numerical simulation study. Phys. Rev. E 63(454) () Lunati, I. et al.: A numerical comparison between two upscaling techniques: non-local inverse based scaling and simplified renormalization. Adv. Water Resour. 4, () Matheron, G.: Elements pour une theorie des milieux poreux. Masson et Cie, Paris (967) Milton, G.W.: Proof of a conjecture on the conductivity of checkerboards. J. Math. Phys. 4, () Milton, G.W.: The Theory of Composites. Cambridge University Press, Cambridge () Mortola, S., Steffe, S.: A two-dimensional homogenization problem. Atti Della Accademia Nazionale Dei Lincei, Serie VIII 73(3), 77 8 (985) Neuman, S.P., Orr, S.: Prediction of steady state flow in nonuniform geologic media by conditional moments: Exact nonlocal formalism, effective conductivities, and weak approximation. Water Resour. Res. 9, (993) Ostoja-Starzewski, M., Schulte, J.: Bounding of effective thermal conductivities of multiscale materials by essential and natural boundary conditions. Phys. Rev. B 54, (996) Renard, P., de Marsily, G.: Calculating equivalent permeability: a review. Adv. Water Resour., (997) Renard, P. et al.: A fast algorithm for the estimation of the equivalent hydraulic conductivity of heterogeneous media. Water Resour. Res. 36, () Sanchez-Vila, X., Guadagnini, A., Carrera, J.: Representative hydraulic conductivities in saturated groundwater flow. Rev. Geophys. 44, 46 (6) Torquato, S.: Random Heterogeneous Materials. Springer, New York () Torquato, S., Kim, I.C., Cule, D.: Effective conductivity, dielectric constant, and diffusion coefficient of digitized composite media via first-passage-time equations. J. Appl. Phys. 85, (999) Yeo, I.W., Zimmerman, R.W.: Accurary of renormalization method for computing effective conductivities of heterogeneous media. Transp. Porous Media 45, 9 38 ()