A random walk model for diffusion in the presence of high-diffusivity paths

Abstract

Abstract Diffusion in the presence of high-diffusivity paths is an important issue of current technology. In metals high-diffusivity paths are identified with dislocations, grain boundaries, free surfaces and internal microcracks. In porous media such as rocks, fissures provide a system of high-flow paths. Recently, based on a continuum approach, these phenomena have been modelled, resulting in coupled systems of partial differential equations of parabolic type for the concentrations in the bulk and in the high-diffusivity paths. This theory assumes that each point of the media is simultaneously occupied by more than one diffusion or flow path. Here a simple discrete random walk model of diffusion in a media with double diffusivity is given. The continuous version of this model gives rise to precisely the coupled system of partial differential equations obtained from the continuum approach. For the discrete model the problem of the unrestricted particle is considered. This problem corresponds to the source solutions of the continuous model and an approximate asymptotic expression is deduced from the random walk model for the total concentration. This expression illustrates clearly the departures from classical findings due to the high-diffusivity paths.