## 1. Introduction

[2] 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*. [2012]. We consider here this type of *K* structure.

[3] 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 *K*_{ef} of highly heterogeneous porous formations. Determination of *K*_{ef} 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*. [1995] and the reviews by *Renard and de Marsily* [1997] and *Neuman and Di Federico* [2003]).

[4] We study here a solution for the effective conductivity *K*_{ef} 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 *K*_{0} = *K*_{ef} [*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.

[5] The salient question is whether *K*_{ef} 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 *K*_{ef}. The numerical solution is achieved here using the finite difference method, which is more general, but whose accuracy is not warranted beforehand.

[6] It is emphasized that in spite of the seemingly artificial character of the setup, it represents a random medium of lognormal univariate pdf (*K _{G}*, ) and of linear covariance of finite integral scale

*I*. 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

_{Y}*Bohling et al*. [2012]), 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.

[7] 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 *K*_{ef} may depend on the higher-order statistical characterization of the hydraulic conductivity structure.

[8] 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 *K*_{ef} for the given second-order moments solely.