Effective conductivity of random multiphase 2D media with polydisperse circular inclusions

The effective conductivity K _ef 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 K _ef = K _G (the geometric mean) applies. The main aim of the article is to determine the deviation of K _ef from K _G for symmetrical f(Y ) but nonsymmetrical f(R|Y ). This is related to recent studies on the effect on K _ef 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 K ₁,K ₂, of equal volume fractions, of different radii R1 and R ₂, respectively, within a matrix of K0 = K _G = (K ₁ K ₂) ^1/2. Even for very high heterogeneity (K ₁/K ₂ = 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 K _ef 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.