Conductivity Estimates of Fractal Models of Geological Media

Geological media are inherently fractal over multiple length scales, exhibiting a wide range of fractal dimensions (Formula presented.). We study the size dependent conductivity of models of geological media based on so called regular random fractals. Specifically, we consider generalizations of the classic Menger sponges where the impermeable cells can take random locations. This results, for a given random fractal generator and a given construction order, in multiple realizations of the target media, each one maintaining the same (Formula presented.). We use direct numerical simulation to estimate the conductivity, over a number of patterns, with orders (Formula presented.). The conductivity bounds and estimates for regular fractal media are, in general, not applicable to these random fractal models. The conductivity of a specific realization may differ from the ensemble average by a few percents, which suggests a lack of self-averaging; however, the self-averaging is regained if the size of the domain of interest is beyond the maximal fractality length scale. We propose a low computational cost approach for estimating the ensemble average conductivity of a given random fractal at a given order, which matches the order of magnitude predicted from more intensive calculations. Through tabulation of conductivities of generators of random fractals with small scaling factors and fitting models, we explain how these fractals can model geological media in which fractal dimensions and conductivities (or effective diffusion coefficients) are measured.