Corresponding author: A. Fiori, Dipartimento di Scienza dell'Ingegneria Civile, Faculty of Engineering, Università di Roma Tre, via Volterra 62, I-00146 Rome, Italy. (email@example.com)
 We consider three-dimensional uniform flow in heterogeneous porous media characterized by a spatially variable hydraulic conductivity K(x); the latter is considered as a stationary random function of lognormal distribution (mean ln KG and variance of Y = ln K) and of finite integral scale IY. We investigate the effective conductivity Kef of such formation by adopting a structural model of cubical inclusions that tessellate the space. The domain is large compared to IY. The dependence of Kef/KG upon is determined by numerical simulations that are performed in parallel employing 2.5 × 109 cells. This technical note focuses on numerical issues in media with strong heterogeneity: domain discretization and the averaging scheme for the intercell conductivity. The effective conductivity is also derived using the self-consistent approximation.
 Numerical simulations of flow and transport through strongly heterogeneous porous formations are becoming more and more popular. They are needed for several reasons. For example, they serve as numerical laboratories in exploring physical transport phenomena that occur in such formations and in testing flow and transport theories. Most of the available numerical codes are developed for 2-D flow and transport. However, number of numerical codes capable of modeling 3-D flow and transport is increasing. Our interest is in 3-D uniform flow through heterogeneous aquifer formations, where the hydraulic conductivity K(x) is spatially variable. Because of its uncertainty and scarcity of measurements, K is modeled as a space random function. Analysis of field data pertaining to many aquifers led to a univariate normal probability density function (pdf) f(Y) of Y = ln K [e.g., Freeze, 1975; Rubin, 2002] of mean and variance . Furthermore, stationarity is a common assumption, unless a trend can be clearly identified; stationarity of K has been identified in several sites [see, e.g., Rubin, 2002], including the recent analysis of the Columbus site by Bohling et al. . We consider here this type of K structure.
 Application of numerical models to highly heterogeneous formations is hampered by several problems, including, but not limited to, ergodicity, domain size, discretization, and selection of intercell conductivities. In the present technical note, we explore some of these problems by analyzing the effective conductivity Kef of highly heterogeneous porous formations. Determination of Kef in groundwater is one of several engineering problems where heterogeneous media are replaced by homogeneous ones of constant, effective, properties. This topic is of wide interest, as underscored by the large number of works addressing it, which were published in the hydrological literature (see, e.g., Sanchez-Vila et al.  and the reviews by Renard and de Marsily  and Neuman and Di Federico ).
 We study here a solution for the effective conductivity Kef of an isotropic aquifer. While the solution for weak heterogeneity, applying to is well known, we have recently carried out systematic numerical investigations for highly heterogeneous 3-D formations, by adopting a structural model made by blocks of different and independent K (multi-indicator model (MIM)). We have solved the problem for a medium made up from spherical inclusions of uniform radii arranged in the space at the highest possible volume fraction, while the voids among them are filled by a homogeneous matrix of conductivity K0 = Kef [Jankovic et al., 2003]. This is an idealization that allowed us to solve the problem with high accuracy for as large as 10, by expanding the flow variables in a series of spherical harmonics.
 The salient question is whether Kef computed using this structural model applies to other structural models, such as the one made of cubes of uniform side length that fully tessellate the space. While other structural models need to be examined in addition to the cube-based one, the present extension is significant because of the absence of homogeneous background and the presence of nonsmooth boundaries that delineate zones of constant K. The development of a solution for the cube-based model is hampered by numerical difficulties caused by jumps in K and a lack of an analytic solution for flow past an isolated cube in uniform flow, which is necessary to derive an approximate analytic solution for Kef. The numerical solution is achieved here using the finite difference method, which is more general, but whose accuracy is not warranted beforehand.
 It is emphasized that in spite of the seemingly artificial character of the setup, it represents a random medium of lognormal univariate pdf (KG, ) and of linear covariance of finite integral scale IY. Since field data, even in the case when aquifers were sampled exhaustively (see, e.g., the analysis of the MADE (Macro Dispersion Experiment) aquifer measurements carried out by Bohling et al. ), allow us to identify these parameters solely, the simplified structure can be regarded as representative. Furthermore, it submits the numerical procedure to a severe test because of the jump of K values at block interfaces. Therefore, the accuracy criteria are conservative with regard to smoother fields, which may be more realistic for many aquifers.
 Thus, the main issue addressed here, regarding the accuracy of the numerical method, depends on the grid size and internode averaging scheme, which is submitted to a severe test for the blocks of highly contrasting conductivity. To the best of our knowledge, no such investigation was conducted in the past for the domains of the size considered here. The second contribution of this technical note is development of a self-consistent approximation (SCA) for the cube-based model and a comparison to numerical simulations. Finally, we show that Kef may depend on the higher-order statistical characterization of the hydraulic conductivity structure.
 We remark that the widely employed multi-Gaussian model is different from the above one, for the same first- and second-order moments of Y; the differences manifest at higher-order and multipoint moments of Y. Alternatives to the multi-Gaussian assumption are available, like indicator fields [Wen and Gomez-Hernandez, 1998]. However, the scarcity of data prevents, in practice, the identification of such moments, and therefore, it is of both theoretical and practical interest to evaluate the impact of the structure on Kef for the given second-order moments solely.
2. General Mathematical Statement
 With Ω the flow domain, steady flow in porous media is governed by Darcy's law, , and continuity, , in which V is the specific discharge and H is the pressure head. Combining these two equations yields
 The flow domain Ω is selected as a rectangular block of length L and square cross section A of side B, delimited in a Cartesian system by 0 < x < L, , . The log conductivity Y = ln K is a second-order stationary random space function with a normal pdf, defined by the mean , variance , and two-point covariance (r = x − y) of integral scale . We assume that .
 The conditions on the boundary are as follows:
i.e., a column of impervious lateral boundaries with flow driven by a constant head gradient J = (H0 − HL)/L.
 Under ergodicity assumption , the effective conductivity is defined by , where .
 Unlike the 1-D and 2-D cases, no exact solution is available for the 3-D flow case. A common approximation is the first-order solution in which does not depend on the pdf of Y and on its covariances. Generalization of 1-D, 2-D, and the 3-D first-order result has been proposed by Landau and Lifshitz  and Matheron :
where D is the space dimension. This conjecture is supposedly valid for finite .
3. Numerical Simulations
 As mentioned in section 1, we model the Y field as an ensemble of M adjacent cubical blocks of side 2R of constant, but different, K(j) (j = 1, 2, …, M), of volume ω(j), and centroid location (j = 1, 2, …, M) such that the total domain volume is Ω = AL = 8R3M. The elements are placed on a cubical grid with nodes at distance 2R in the x, y, z directions. The structure is completely defined by assuming (i) that Y(j) is a random normal variable and (ii) that Y(j) and Y(k) are uncorrelated for any , i.e., properties of different inclusions are independent.
 The various covariances can be determined from these definitions. Thus, it is easy to ascertain that the two-point autocorrelation ρY is given by the product of three linear covariance functions [see Dagan, 1989, p. 170] with the integral scale IY = R. These are different from the ones pertaining to spherical inclusions of radius R [see, e.g., Jankovic et al., 2003] for which the covariance is spherical and IY = 3R/4.
 The problem has to be solved for Y(j), selected as a normal vector of independent random values. Once this is achieved, the spatial mean specific discharge is determined from = and the effective conductivity by .
 Numerical solution of the steady flow problem is obtained using a node-centered finite difference method. Continuity equation is hence written for each node using heads of neighboring nodes. In order to allow for a large number of nodes (just over 2.5 × 109), parallel processing was used for both storage and computations. Layer-based domain decomposition is combined with a conjugate gradient solver to allow for storage of the large system matrix (only seven nonzero entries exist in each matrix row) and to speed up computations. With domain decomposition, each processor needs to store only a small portion of the large sparse system matrix (only nonzero matrix entries are stored) and various vectors, including solution vector that contains unknown heads at grid nodes. Communications between processors are limited to nodes at boundary between domains covered by two processors that share the boundary. These communications must be carried for each iteration.
 The conjugate gradient method is implemented with a diagonal preconditioner. While other preconditioners would likely improved the convergence speed, a simple diagonal one is selected for ease of implementation in parallel. Full convergence has been achieved for all cases that were simulated. Matrix multiplication, the core component of conjugate gradient method, is implemented in parallel to increase computation speed. Code correctness was successfully checked against analytic-element-based code for spheroidal inclusions [Suribhatla et al., 2011] for several settings of densely packed elements.
 The flow domain is made of a grid of M = 17 × 17 × 17 = 4913 blocks; B = L = 34IY. The domain size was selected such as to ensure ergodicity of the head and specific discharge fields for any value of ; the issue will be recalled in the sequel. Eighty finite difference cells are used in each direction for each block. The number of cells hence equals 1360 in each direction. The total number of nodes in the domain is 13613 = 2.52 × 109. The blocks' log conductivity follows a normal distribution of zero mean and variance . A unit head gradient J = 1 drives the flow. The solution was obtained on 17 eight-processor nodes at the supercomputer cluster housed at the Center for Computational Research, State University of New York at Buffalo. Convergence was monitored using maximum mass imbalance at each node every 10th iteration, by computing after 20,000 and 25,000 iterations for and by visual inspection of head contours. Computing 20,000 iterations was found to be sufficient for all cases, while low simulations needed significantly fewer iterations. Simulations have lasted up to 34 h.
 While nodes in block interiors do not require any averaging of hydraulic conductivity, nodes at boundaries of blocks require local averaging of K of neighboring blocks. A node that is placed at a box side shared by two blocks requires K averaging of two block K values to compute the components of flux tangential to the boundary. Similarly, averaging of four block K values is applied for nodes shared by four blocks. Because of constant grid sizes in three directions, simple arithmetic, geometric, and harmonic local averaging is used to examine the impact of averaging schemes. No K averaging is required for computation of normal flux across block boundaries. If cell-centered finite difference scheme is selected instead of node-centered scheme for present model, no K averaging is necessary for computation of flux tangential to the boundary. However, computation of normal flux in that case requires averaging of K values.
 Criteria for selecting the “best” intercell averaging scheme have been discussed since the pioneer work by Ababou et al. , who proposed the geometric averaging scheme. We remind that the popular code MODFLOW employs the harmonic intercell mean as default. Despite its importance, the issue does not seem to be much debated in the literature. We remark that, for a sufficiently dense discretization, in terms of cells per IY, all the averaging intercell schemes should provide the same result. However, for large the required number of cells may be very large, as discussed in the sequel.
 A buffer zone at the domain boundary, one block wide (2R = 2IY), is excluded from the analysis because of visibly reduced variability in specific discharge in that zone. The remaining part of the domain made of 15 × 15 × 15 blocks is used for the computation of the average specific discharge and . This 15 × 15 × 15 domain was found to be sufficiently large to warrant ergodicity since values for two independent realizations of with arithmetic local averaging were less than 1% different. values for two realizations of with geometric local averaging were equally similar. The driving head gradient was also determined by averaging over the core of stationarity, and it was larger than unity by 2% for .
 Despite a high resolution of 80 cells per block (40 per R = IY), the local averaging scheme was found to have measurable impacts on both node head values and for large . For and the difference in between arithmetic (that produced highest ) and harmonic local averaging (that produced smallest ) was under 6%. For , the difference was 12%. For simulations with , arithmetic and geometric local averaging have produced similar results, so simulations with harmonic local averaging were not conducted. Impact of local averaging on transport (not reported here) was found to be even more significant for . While results here will be reported up to , the last point in Figure 1 should be taken with some reservation considering this issue. Because of memory limitations, it was not possible to further increase the number of cells per block in the domain of size needed in order to ensure ergodic behavior of the head and specific discharge. In addition to Kef, the average and variance of specific discharge perturbations are recorded as a function of κ = K/Kef for comparison with the self-consistent solution and presented later in the manuscript (Figure 2).
 The derivation of Kef using SCA, presented in next section, requires solution for isolated cubes of varying K placed in a homogeneous background. This is accomplished by running 200 numerical simulations for a single cube with Y = ln (K) between Y = −10 and Y = 10, with step of 0.1. The background conductivity K0 was set to unity since the solution depends only on the relative conductivity κ. Because of symmetry across three orthogonal planes that pass through center of the inclusion, each case is solved for one eighth of the inclusion volume and associated background.
 The domain was set to 5R × 5R × 5R and solved using a grid of 500 × 500 × 500 cells with arithmetic local averaging and parallel processing with 32 processors. The average interior specific discharge in x direction is computed for each case. Results are found to be insensitive to the domain size and resolution. Considering the very large resolution (100 cells per R), this was an expected result.
 The SCA assumes that the flow problem can be solved for each inclusion ω(j) of conductivity K(j) separately in the domain Ω, by assuming that it is surrounded by a homogeneous matrix of unknown Kef. With the associated perturbation of the space averaged specific discharge over the entire volume, Kef is determined from the self-consistent requirement that the total disturbance due to all the inclusions is equal to zero. It leads at the continuous limit to the SCA general equation , where κ = K/Kef, and f is the pdf of K.
 The procedure was applied by Dagan  to K lognormal and to spherical inclusions, for which leads to the integral equation:
which can be easily solved numerically. The resulting Kef/KG, function of , was found to be in excellent agreement with the accurate numerical solution for the ensemble of dense inclusions [Jankovic et al., 2003] for .
 Because of the absence of analytic solution for isolated cube ω of conductivity K within a column Ω of conductivity Kef, the perturbation is computed numerically in terms of , where is the interior specific discharge. By averaging the head gradient , the final result is . However, unlike the case of the sphere, the specific discharge field for a cube could be determined only numerically, as described in the previous section. Nevertheless, once is determined for an array of κ values, it can be used to solve numerically for any pdf of K, particularly the lognormal one.
 The dependence of the numerically determined for the cubes and the analytical for the spheres is displayed in Figure 2 as a function of κ. It is seen that in spite of the difference in shape the two are very close for κ < 2 and uc is larger than us by 0.18 for . The impact of this finding upon the derivation of Kef is discussed in the following.
5. Results, Discussion, and Conclusions
 The main topic of this study is the numerical derivation of Kef of the multi-indicator structure of cubical inclusions of independent normal Y, for a large range of values. The main scope is to emphasize problems in the numerical solution for flow through highly heterogeneous porous formations and to elucidate few issues regarding Kef in such formations. The dependence of Kef/KG upon is represented in Figure 1 for three internode averaging schemes: arithmetic, geometric, and harmonic (for the latter, only the cases were performed). The schemes lead to very close results for , with the maximum difference of 12% for . We remark that, for structures like the multi-Gaussian one, for which the values of Y of adjacent nodes are correlated and do not undergo jumps, it is expectable that the differences between averaging scheme are even smaller.
 The impact of the intercell conductivity scheme on the results raises a warning against possible insufficient numbers of cells per integral scale employed in numerical simulations of highly heterogeneous formations, when . We remind that 2-D numerical simulations carried out in the past have typically employed a number of cells less than 10/IY, with the exception of Schwarze et al.  and Trefry et al. , who have used 50/IY and 82/IY, respectively [see also de Dreuzy et al., 2007]. Because of memory limitations, discretization is usually lower for 3-D flow simulations. To our best knowledge, the largest total number of cells adopted in a 3-D simulation was of 1.2 × 107 cells (2 orders of magnitude lower than the present study), with 4 cells per IY [Englert et al., 2006]; common values adopted in other studies are of the order of 105 cells. Insufficient discretizations are expected to have a much larger impact on transport simulations.
 The dependence of Kef/KG on as determined by the self-consistent (effective medium) approximation by solving numerically the integral equation for both cubes and spheres is also shown in Figure 1. The first striking result is that the shape has a negligible impact, and the two values are very close for . This is explained by closeness of uc and us displayed in Figure 2, the difference for large κ affecting only a small portion of the medium even for . The SCA results are very close to the numerical ones for a dense packing of spheres for as large as 10, as shown in Jankovic et al. . This is confirmed in the present numerical simulations (section 3) for . This match is explained by Figure 2 where uc and us determined using the numerically computed values of Vin for are included. For (Figure 1), the numerical solution for cubes differs from the one for spheres or the SCA for both. Whether this is due to the difference in the shapes of the autocorrelation functions for the two structures or to the numerical problems encountered at this high value is not clear at present.
 Returning to the Kef issue, we have also represented in Figure 1 the first-order approximation a well as the Landau-Matheron (LM) conjecture While the values based on the inclusion models are close to the first-order approximation, the difference with the LM relationship is quite large for , as already found by Jankovic et al. . If we assume that previous numerical solutions (section 3) validated the LM conjecture for multi-Gaussian structures, the conclusion is that for formations of lognormal distributions of identical KG, , and IY, the 3-D effective conductivity depends on the higher-order correlations of Y and on the shape of ρY(r). This result casts some doubts on the feasibility to derive Kef in an accurate manner from structural data, as the common characterization procedures do not generally permit a statistical description of K beyond the second-order one. Extension of the present highly resolved numerical simulations to multi-Gaussian structures of identical linear ρY and to anisotropic structures is planned for the future.
 The authors wish to thank the Center for Computational Research, State University of New York at Buffalo (especially Cynthia Cornelius), for assistance in running numerical simulation and for significant CPU time, which was used in conducting simulations.