Diffusion in random networks

Abstract

The ensemble averaging technique is applied to model mass transport by diffusion in random networks. The system consists of an ensemble of random networks, where each network is made of pockets connected by tortuous channels. Inside a channel, fluid transport is assumed to be governed by the one-dimensional diffusion equation. Mass balance leads to an integro-differential equation for the pocket mass density. The so-called dual-porosity model is found to be equivalent to the leading order approximation of the integration kernel when the diffusion time scale inside the channels is small compared to the macroscopic time scale.As a test problem, we consider the one-dimensional mass diffusion in a semi-infinite domain. Because of the required time to establish the linear concentration profile inside a channel, for early times the similarity variable is xt−1/4 rather than xt−1/2 as in the traditional theory. This early time similarity can be explained by random walk theory through the network.