Effective Conductivity of Random Multiphase 2D Media with Polydisperse Circular Inclusions

The effective conductivity Kef of porous formations of spatially variable permeability K is determined for media of random two-dimensional and isotropic structures. The medium is modeled as an ensemble of multiphase circular inclusions of different Y = lnK, characterized by a pdf f(Y ), and of different radii R (polydisperse medium), of pdf f(R|Y ), which are implanted in a matrix of K = K0. A large number of inclusions are embedded in a large circle, to allow for exchange of space and ensemble averaging. For symmetrical pdf f(Y ) = f(−Y ) and symmetrical f(R|Y ), the Matheron exact relationship Kef = KG (the geometric mean) applies. The main aim of the article is to determine the deviation of Kef from KG for symmetrical f(Y ) but nonsymmetrical f(R|Y ). This is related to recent studies on the effect on Kef of connectivity of spatial domains of different K classes. The problem is solved numerically by an accurate and efficient iterative procedure and by a novel, approximate, analytical method. The two procedures are illustrated and compared for the configuration of two phases of conductivities K1, K2, of equal volume fractions, of different radii R1 and R2, respectively, within a matrix of K0 = KG = (K1 K2) 1/2. Even for very high heterogeneity (K1/K2 = 1000) it is found that the effect of variable R is relatively modest and it manifests mainly at the largest attainable volume fraction. The simple analytical approximation, valid for moderate volume fractions, is applied to investigation of Kef for normal f(Y ), and for two values of R, for Y < 0 and Y > 0, respectively. The results are of interest for similar heterogeneous media and for other physical processes governed by linear relationships between the flux and the driving potential gradient.