Effective Conductivity of an Anisotropic Heterogeneous Medium of Random Conductivity Distribution
R. M. Suribhatla, Igor Janković, Aldo Fiori, Antonio Zarlenga, Gédéon Dagan · Multiscale Modeling and Simulation · 2011
The paper deals with the effective conductivity tensor [Formula: see text] of anisotropic random media subject to mean uniform flux. The hydraulic conductivity [Formula: see text] field is modeled as a collection of spheroidal disjoint inclusions of different, isotropic and independent [Formula: see text]; the latter is a random variable with given distribution of variance [Formula: see text]. Inclusions are embedded in homogeneous background of anisotropic conductivity [Formula: see text]. The [Formula: see text] field is anisotropic, characterized by the anisotropy ratio [Formula: see text], ratio of the vertical and horizontal integral scales of [Formula: see text]. We derive [Formula: see text] 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 [Formula: see text] distribution. The numerically derived [Formula: see text] for a normal [Formula: see text] is compared with its prediction by (i) the self-consistent solution [Formula: see text], (ii) the first-order approximation in [Formula: see text], 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 [Formula: see text] is very accurate for a broad range of the values of the parameters [Formula: see text],[Formula: see text] and for the densest inclusions packing. In contrast, the first-order solution strongly deviates from [Formula: see text] for large [Formula: see text], 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.