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Monte Carlo studies of flow and transport in fractal conductivity fields: Comparison with stochastic perturbation theory

A Monte Carlo simulation of flow and transport is employed to study tracer migration in porous media with evolving scales of heterogeneity (fractal media). Transport is studied with both conservative and reactive chemicals in media that possess physical as well as chemical heterogeneity. Linear kinetic equations are assumed to relate the sorbed phase and the aqueous phase concentrations. The fluctuating log conductivity possesses the power law spectrum of a fractional Brownian motion (fBm). Chemical heterogeneity is represented as spatially varying reaction rates that also are assumed to obey fBm statistics and may be correlated to the conductivity field. The model is based on a finite difference approximation to the flow problem and a random walk particle‐tracking approach for solving the solute transport equation. The model is used to make comparisons with the nonlocal transport equations recently developed by Deng et al. [1993], and Hu et al. [1995, 1997]. The results presented herein support these nonlocal models for a wide range of heterogeneous systems. However, the infinite integral scale associated with the fractal conductivity has a significant effect on the prediction of the nonlocal theories. This suggests that integral scale should play a role in stochastic Eulerian perturbation theories. The importance of the local‐scale dispersion depends to a great extent on the magnitude of the local dispersivities. The effect of neglecting local dispersion decreases with the decrease in local dispersivity.