Statistics of random walks in geologically relevant conductance fields

Statistics of diffusion, modeled by random walks, such as the mean number of distinct sites visited S(t) at time t, the mean probability P_{0}(t) of being at the origin of the walk, and the mean-squared displacements 〈R^{2}(t)〉 of the random walkers have been studied extensively in the past in both regular lattices and such disordered media as percolation clusters and other fractal structures, and universal power laws for such quantities have been derived. S(t) provides insight into reaction properties of geological formations, while P_{0}(t) is directly linked with the problem of back diffusion in remediation of groundwater aquifers. In all such studies, it was assumed that the conductances of the bonds that connect nearest-neighbor sites of the lattices are equal. Motivated by the problem of transport and reaction in large-scale porous media that are characterized by a broad spatial distribution of hydraulic conductances, we demonstrate, using extensive Monte Carlo simulations, that the statistics of random walks, when the conductances are broadly distributed, depend on the structure of the distribution. Five geologically relevant conductance distributions, namely, normal, log-normal, fractional Brownian motion (FBM), log-FBM, and stable distributions, are considered and random walks in a two-dimensional model with the five distributions are simulated. The first two distributions are uncorrelated, while the last three induce long-range correlations in the values of the conductance. We show that if S(t)∼t^{p} and P_{0}(t)∼t^{-ζ}, the exponents p and ζ may depend on the conductance distribution, in which case they are neither equal to those for homogeneous lattices, nor those for percolation clusters and other fractal structures. For at least three of the conductance distributions, diffusion is anomalous, with 〈R^{2}(t)〉 not growing linearly with the time t, even in the long-time limit. In addition to being of scientific interest, and the fact that transport processes in geomedia are simulated by random walks, the dependence of such statistics on the distribution of the conductances and their deviations from the statistics of random walks in homogeneous systems, and percolation and other types of fractal structures, indicate that diffusion in highly heterogeneous porous media is anomalous, and is described by fractional partial differential equations in which the temporal and spatial fractional orders depend on the details of the conductance distribution.