Dispersion in porous media, continuous-time random walks, and percolation

A promising approach to the modeling of anomalous (non-Gaussian) dispersion in flow through heterogeneous porous media is the continuous-time random walk (CTRW) method. In such a formula on the waiting time distribution psi(t) is usually assumed to be given by psi(t) similar to t(-1-alpha), with alpha fitted to the experimental data. The exponent a is also related to the power-law growth of the mean-square displacement of the solute with the time t < R-2(t)> similar to t(zeta). Invoking percolation and using a scaling analysis, we relate alpha to the geometrical exponents of percolation (nu, beta, and beta(B)) as well as the exponents mu and e that characterize the power-law behavior of the effective conductivity and permeability of porous media near the percolation threshold. We then explain the cause of the nonuniversality of alpha in terms of the nonuniversality of mu and e in continuum systems, and in percolation models with long-range correlations, and propose bounds for it. The results are consistent with the experimental data, both at the laboratory and field scales.