Percolation and conductivity in evolving disordered media

Abstract

Percolation theory and the associated conductance networks have provided deep insights into the flow and transport properties of a vast number of heterogeneous materials and media. In practically all cases, however, the conductance of the networks' bonds remains constant throughout the entire process. There are, however, many important problems in which the conductance of the bonds evolves over time and does not remain constant. Examples include clogging, dissolution and precipitation, catalytic processes in porous materials, as well as the deformation of a porous medium by applying an external pressure or stress to it that reduces the size of its pores. We introduce two percolation models to study the evolution of the conductivity of such networks. The two models are related to natural and industrial processes involving clogging, precipitation, and dissolution processes in porous media and materials. The effective conductivity of the models is shown to follow known power laws near the percolation threshold, despite radically different behavior both away from and even close to the percolation threshold. The behavior of the networks close to the percolation threshold is described by critical exponents, yielding bounds for traditional percolation exponents. We show that one of the two models belongs to the traditional universality class of percolation conductivity, while the second model yields non-universal scaling exponents.