Scale-Dependent Subsurface Dispersion: A Fractal-Based Stochastic Model

A stochastic model that can simulate non-Fickian (scale-dependent) subsurface dispersion by imposing long-range correlation structure within a particle-based transport scheme is detailed here. The model uses increments from self-similar random fractal processes. These processes [fractional Brownian motions (FBMs)] are characterized by an index H, which is related to the correlation properties of the function. Temporally correlated FBMs that mimic the faster-than-Fickian growth observed in contaminant plumes in heterogeneous subsurface porous media are chosen. It is shown how FBMs can be incorporated within traditional Langevin-based formulations of the random-walk particle tracking models to describe mass displacements. A Fokker-Planck diffusion equation with a time-dependent tensor and its solution are presented to describe FBM anomalous dispersion. A relationship of the modelʹs parameters to measurable porous media parameters is given.