Physical pictures of transport in heterogeneous media: Advection-dispersion, random-walk, and fractional derivative formulations

Brian Berkowitz*, Joseph Klafter, Ralf Metzler, Harvey Scher

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

223 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)9-1-9-12
JournalWater Resources Research
Volume38
Issue number10
DOIs
StatePublished - 1 Oct 2002

Keywords

  • Anomalous transport
  • Fractional derivatives
  • Random walks

Fingerprint

Dive into the research topics of 'Physical pictures of transport in heterogeneous media: Advection-dispersion, random-walk, and fractional derivative formulations'. Together they form a unique fingerprint.

Cite this