Diffusion theory for transport in porous media: Transition‐probability densities of diffusion processes corresponding to advection‐dispersion equations

Abstract

Local‐scale spatial averaging of pore‐scale advection‐diffusion equations in porous media leads to advection‐dispersion equations (ADEs). While often used to describe subsurface transport, ADEs may pose special problems in the context of diffusion theory. Standard diffusion theory applies only when characteristic coefficients, velocity, porosity, and dispersion tensor, are smooth functions of space. Subsurface porous‐material properties, however, naturally exhibit spatial variability. Transitions between material types are often abrupt rather than smooth, such as sand in contact with clay. In such composite porous media, characteristic coefficients in spatially averaged transport equations may be discontinuous. Although commonly called on to model transport in these cases, standard diffusion theory does not apply. Herein we develop diffusion theory for ADEs of transport in porous media. Derivation of ADEs from probabilistic assumptions yields (1) necessary conditions for convergence of diffusion processes to ADEs, even when coefficients are discontinuous, and (2) general probabilistic definitions of physical quantities, velocity, and dispersion tensor. As examples of how the new theory can be applied to theoretical and numerical problems of transport in porous media, we evaluate several random walk methods that have appeared in the water resources literature.