Physical Pictures of Transport in Heterogeneous Media: Advection-Dispersion, Random Walk and Fractional Derivative Formulations

Details Physical Pictures of Transport in Heterogeneous Media: Advection-Dispersion, Random Walk and Fractional Derivative Formulations Physical Pictures of Transport in Heterogeneous Media: Advection-Dispersion, Random Walk and Fractional Derivative Formulations

2002-02-19

arxiv.org/abs/cond-mat/0202327v2

Physics

Condensed Matter

Statistical Mechanics

14 pages, REVTeX4, accepted to Wat. Res. Res; reference added

Scientific paper

The basic conceptual picture and theoretical basis for development of transport equations in porous media are examined. The general form of the governing equations is derived for conservative chemical transport in heterogeneous geological formations, for single realizations and for ensemble averages of the domain. The application of these transport equations is focused on accounting for the appearance of non-Fickian (anomalous) transport behavior. The general ensemble-averaged transport equation is shown to be equivalent to a continuous time random walk (CTRW) and reduces to the conventional forms of the advection-dispersion equation (ADE) under highly restrictive conditions. Fractional derivative formulations of the transport equations, both temporal and spatial, emerge as special cases of the CTRW. In particular, the use in this context of L{\'e}vy flights is critically examined. In order to determine chemical transport in field-scale situations, the CTRW approach is generalized to non-stationary systems. We outline a practical numerical scheme, similar to those used with extended geological models, to account for the often important effects of unresolved heterogeneities.

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