Anomalous diffusion in two-dimensional Euclidean and prefractal geometrical models of heterogeneous porous media

Abstract

Disordered systems are known to induce anomalous diffusion. This phenomenon may be important for environmental applications such as contaminant transport and nutrient availability. However, few studies have investigated anomalous diffusion in this context. In particular, the relationship between pore space geometry and anomalous diffusion is not well understood. We report on numerical simulations of solute diffusion within the water-filled pore spaces of two-dimensional geometrical models of heterogeneous porous media. Euclidean and mass, pore, and pore-solid prefractal lattices were used to generate random pore networks with varying porosity (phi) and lacunarity (L). The objectives were to investigate the effects of phi and L on the solute random walk dimension (d(w)) and to identify which of these models best represents a natural porous medium. Solute diffusion was simulated using a stochastic cellular automaton based on the "myopic ant'' algorithm. Estimates of d(w)>> 2 occurred with increasing frequency as phi -> 0, indicating scale dependency in the standard diffusion coefficient at low porosities. The relationship between d(w) and phi for the mass and pore-solid prefractal lattices was the closest to that for natural 2-D systems (i.e., soil thin sections). The presence of large, interconnected pore spaces (L -> 1) at low porosities reduced the intensity of anomalous diffusion (d(w)-> 2). A power law relationship based on the product of d(w) and L explained >96% of the total variation in phi regardless of the type of lattice considered. The potential predictive capability of this approach for natural porous media deserves further investigation.