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Physical pictures of transport in heterogeneous media: Advection‐dispersion, random‐walk, and fractional derivative formulations
Author(s) -
Berkowitz Brian,
Klafter Joseph,
Metzler Ralf,
Scher Harvey
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 - 10.1029/2001wr001030
Subject(s) - continuous time random walk , statistical physics , random walk , advection , context (archaeology) , fractional calculus , porous medium , convection–diffusion equation , dispersion (optics) , mathematics , transport phenomena , random field , domain (mathematical analysis) , basis (linear algebra) , physics , mathematical analysis , mechanics , geology , thermodynamics , porosity , geometry , statistics , geotechnical engineering , paleontology , optics
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 Lévy flights is critically examined. In order to determine chemical transport in field‐scale situations, the CTRW approach is generalized to nonstationary 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.