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=ln K, 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.