Effective conductivity of an anisotropic heterogeneous medium of random conductivity distribution

"\"The paper deals with the effective conductivity tensor K(ef) of anisotropic random media subject to mean uniform flux. The hydraulic conductivity K field is modeled as a collection of spheroidal disjoint inclusions of different, isotropic and independent Y = ln K; the latter is a random variable with given distribution of variance sigma(2)(Y). Inclusions are embedded in homogeneous background of anisotropic conductivity K(0). The K field is anisotropic, characterized by the anisotropy ratio f, ratio of the vertical and horizontal integral scales of K. We derive K(ef) by accurate numerical simulations; the numerical model for anisotropic media is presented here for the first time, and it generalizes a previously developed model for isotropic formations [I. Jankovic, A. Fiori, and G. Dagan, Multiscale Model. Simul., 1 (2003), pp. 40-56]. The numerical model is capable of solving complex three-dimensional flow problems with high accuracy for any configuration of the spheroidal inclusions and arbitrary K distribution. The numerically derived K(ef) for a normal Y is compared with its prediction by (i) the self-consistent solution K(sc), (ii) the first-order approximation in sigma(2)(Y), and (iii) the exponential conjecture [L. J. Gelhar and C. L. Axness Water. Resour. Res., 19 (1983), pp. 161-180]. It is found that the self-consistent solution K(sc) is very accurate for a broad range of the values of the parameters sigma(2)(Y), f and for the densest inclusions packing. In contrast, the first-order solution strongly deviates from K(ef) for large sigma(2)(Y), as expected, and the exponential conjecture is generally unable to correctly represent the effective conductivity. The numerical solution for the potential is expressed as an infinite series of spheroidal harmonics, attached to the interior and exterior of each inclusion, which accounts for the nonlinear interaction between neighboring inclusions.\"