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

By Brian Berkowitz, Joseph Klafter, Ralf Metzler, Harvey Scher. 2002

252 citations recorded by the source corpus.

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.

Subfields: Environmental Engineering, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/e4af4aab61e05ee5e52fac90510bb203ff027b1c

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Citations in the corpus

Comparison of alternative models for simulating anomalous solute transport in a large heterogeneous soil column (2009, background): “In this situation, w(s, t) can be expressed as w(s, t) p(s)w(t), where p(s) denotes the transition length probability density function and w(t) denotes the transition-time distribution function (Berkowitz et al., 2002).”
Time behavior of solute transport in heterogeneous media: transition from anomalous to normal transport (2003, methodology): “Recently [ 6 ] a conditional transport model has been suggested which models the transport on the local scale by the CTRW approach while large scale fluctuations of the heterogeneous medium, which lead to a spatially dependent Darcy flow field, are modeled explicitly.”
Upscaling discrete fracture network simulations: An alternative to continuum transport models (2005, background): “Although not explicitly addressing fractured rock, recent work on CTRW [e.g., Berkowitz et al., 2002; Cortis et al., 2004] add to its potential utility as a hybrid model for transport in fractured rock.”
Modeling non‐Fickian transport in geological formations as a continuous time random walk (2006, methodology): “master equation (1) with a Taylor expansion of C(s, t), (23), and w(s, s0) [Berkowitz et al., 2002].”
The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics (2004, background): “…been shown that long-tailed waiting time distributions with a comparably small number of fit parameters can well account for the observed behaviour (Berkowitz et al 2002, Berkowitz and Scher 1995, 1997, Scher et al 2002a); however, also space-fractional models were used to account for the…”
Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion (2013, unknown): “It has been successfully applied to many practical problems, including diﬀusion in media with fractal geometry [19], the dynamics of viscoelastic materials [6], and contaminant transport within underground water ﬂow [3].”
Fractional advection‐dispersion equations for modeling transport at the Earth surface (2009, methodology): “For example, it is possible to generate stochastic processes with nonstationary increments that have heavy tails and satisfy classical ADEs with time-dependent dispersion tensors [Berkowitz et al., 2002].”
Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications (2009, methodology): “The MRMT, CTRW and fADE models do not assume Fickian diffusion at a reference support scale [17], and in almost all cases are closed and have solutions.”
Fractional Calculus as a Mathematical Tool to Improve the Modeling of Mass Transfer Phenomena in Food Processing (2012, background): “For diffusion in porous media, it has been recognized that [10] it is possible to use a time fractional equation to represent the evolution of mass transport as follows:”
Fractional vector calculus for fractional advection–dispersion (2006, background): “The resulting sub-diffusion equation qr=qtb 1⁄4 vqr=qxþ cq(2)r=qx2 describes a plume that spreads away from its center of mass at the rate tb=2, slower than classical diffusion [14,17,40].”