Effective conductivity of an anisotropic heterogeneous medium of random conductivity distribution

The paper deals with the effective conductivity tensor K_ef of anisotropic randommedia 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 σ²_y. Inclusions are embedded in homogeneous background of anisotropic conductivity K ₀. The Kfield 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 numericalmodel 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 threedimensional 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 σ²_y , and (iii) the exponential conjecture [L. J. Gelhar and C. L. AxnessWater. 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 σ²_y ,f and for the densest inclusions packing. In contrast, the first-order solution strongly deviates from K_ef for large σ²_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.