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Macrodispersivity for transport in arbitrary nonuniform flow fields: Asymptotic and preasymptotic results
Author(s) -
Lunati Ivan,
Attinger Sabine,
Kinzelbach Wolfgang
Publication year - 2002
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/2001wr001203
Subject(s) - mathematics , anisotropy , homogenization (climate) , flow (mathematics) , statistical physics , mathematical analysis , scale (ratio) , limit (mathematics) , tensor (intrinsic definition) , geometry , physics , ecology , quantum mechanics , biology , biodiversity
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 ε ≪ 1, where ε 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 ε 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.