Bounds and estimates for transport coefficients of random and porous media with high contrasts

Abstract

Bounds on transport coefficients of random polycrystals of laminates are presented, including the well-known Hashin–Shtrikman bounds and some rigorous bounds involving two formation factors for a two-component porous medium. A class of self-consistent estimators is then formulated based on the observed analytical structure both of these bounds and also of earlier self-consistent estimates [of the coherent potential approximation (CPA) or CPA-type]. A numerical study is made, assuming first that the internal structure (i.e., the laminated grain structure) is not known, and then that it is known. The purpose of this aspect of the study is to attempt to quantify the differences in the predictions of properties of a system being modeled when such organized internal structure is present in the medium but detailed spatial correlation information may or (more commonly) may not be available. Some methods of estimating formation factors from data are also presented and then applied to a high-contrast fluid-permeability numerical simulation data set. Hashin–Shtrikman bounds are found to be very accurate estimates for low contrast heterogeneous media. But formation factor lower bounds are superior estimates for high contrast situations. Other related bounds by Bergman that interpolate between the Hashin–Shitrikman bounds and the formation factor bounds are also briefly discussed. The self-consistent estimators developed here also tend to agree better with data than either the bounds or the CPA estimates, which themselves tend to overestimate values for high-contrast conducting composites.