A Particle Tracking Model for Non-Fickian Subsurface Diffusion

In this paper, a new particle tracking technique is described which can simulate non-Fickian diffusion within porous media. The technique employs fractional Brownian motions (fBms), a generalization of regular Brownian motion. These random fractal functions allow both super- and subdiffusive particle paths to be produced and hence non-Fickian diffusion of the resulting panicle clouds can be modeled. In recent years, fBm trace functions have been used by many authors to reproduce self-affine random fields to simulate various porous media properties. In contrast, a method is detailed herein which uses self-similar spatial fBm trajectories to simulate directly non-Fickian behavior of the particle clouds. Although fractal trajectories have been previously suggested as the basis for possible methods of modeling non-Fickian diffusion, the authors believe that this paper contains the first algorithm to be presented which does not require an a priori knowledge of the end condition of the random walk and, more importantly, allows both a definable scaling exponent and (fractal) diffusion coefficient to be specified. The resulting non-Fickian diffusion using the new algorithm is illustrated and some applications are discussed. The purpose of this paper is to bring the potential usefulness of fBm trajectories in simulating non-Fickian processes within homogeneous media to the attention of numerical modelers active in the simulation of subsurface diffusive processes. The method has a particular environmental application in the simulation of the non-Fickian dispersion of groundwater contaminants through porous media.