Monte Carlo simulation of superdiffusion and subdiffusion in macroscopically heterogeneous media

Abstract

Monte Carlo simulations are developed to approximate one-dimensional superdiffusion and subdiffusion in macroscopically heterogeneous media with discontinuous or continuous transport parameters. For superdiffusion characterized by a space fractional (α-order) derivative model, one empirical reflection scheme is built to track particle trajectory across an interface with discontinuous dispersion coefficient D, where the reflection probability depends on both α and the ratio of D. Different from the superdiffusive case, anomalous diffusion described by a time fractional derivative model can be decomposed into a motion component and a hitting time process, where the discontinuity affects only the motion process, implying an efficient Monte Carlo simulation of decoupled continuous time random walks. The discontinuity of effective porosity n is also discussed, and results show the influence of the ratio of n on solute particle dynamics. In addition, for anomalous superdiffusion and subdiffusion in heterogeneous media with spatially continuous D and n, Langevin analysis reveals that the corresponding particle dynamics contain three independent stable Lévy noises scaled by D, the gradient of D, and the gradient of ln(n). A new implicit Eulerian finite difference method is also developed to solve the spatiotemporal fractional derivative models and then extensively cross verify the Lagrangian solutions. Further testing against one field example of mixed superdiffusion and subdiffusion reveals the applicability and flexibility of the novel Monte Carlo approach in simulating realistic plumes in macroscopically heterogeneous media with locally variable transport parameters. Copyright 2009 by the American Geophysical Union.