Hydrodynamic dispersion near the percolation threshold: scaling and probability densities

Hydrodynamic dispersion in random network models of porous media near the percolation threshold is investigated. This is done by studying the random walk of a particle in flow through the random network. Various scaling regimes (which depend on the Peclet number which is the ratio of diffusion and convection times) are identified, and the scaling relations for the mean-square displacement of the walk, both in the direction of macroscopic flow and in directions perpendicular to the macroscopic flow, are derived and related to those of anomalous diffusion on percolation clusters. It is shown that dispersion can give rise to superdiffusion in which the mean-square displacements of the random walk grow with time faster than linearly, while the spectral dimension of such random walks can be significantly larger than two, which is the critical dimension for diffusion on fractal systems. The author proposes a new equation for the probability density of finding the random walker at a point at a given time and discusses a method by which the probability density for first passage times of the walker can be determined.