A Macrodispersivity for transport in arbitrary nonuniform flow fields: Asymptotic and preasymptotic results
Lunati I., Attinger S., Kinzelbach W., Water Resource Research, 38, 10, 5-1 — 5-11, 2002
We use homogenization theory to investigate the asymptotic macrodispersion in arbitrary nonuniform velocity fields, which show small-scale fluctuations. In the first part of the paper, a multiple-scale expansion analysis is performed to study transport phenomena in the asymptotic limit epsilon << 1, where epsilon represents the ratio between typical lengths of the small and large scale. In this limit the effects of small-scale velocity fluctuations on the transport behavior are described by a macrodispersive term, and our analysis provides an additional local equation that allows calculating the macrodispersive tensor. For Darcian flow fields we show that the macrodispersivity is a fourth-rank tensor. If dispersion/diffusion can be neglected, it depends only on the direction of the mean flow with respect to the principal axes of anisotropy of the medium. Hence the macrodispersivity represents a medium property. In the second part of the paper, we heuristically extend the theory to finite epsilon effects. Our results differ from those obtained in the common probabilistic approach employing ensemble averages. This demonstrates that standard ensemble averaging does not consistently account for finite scale effects: it tends to overestimate the dispersion coefficient in the single realization.