Modeling non‐Fickian transport in geological formations as a continuous time random walk

Abstract

[1] Non-Fickian (or anomalous) transport of contaminants has been observed at field and laboratory scales in a wide variety of porous and fractured geological formations. Over many years a basic challenge to the hydrology community has been to develop a theoretical framework that quantitatively accounts for this widespread phenomenon. Recently, continuous time random walk (CTRW) formulations have been demonstrated to provide general and effective means to quantify non-Fickian transport. We introduce and develop the CTRW framework from its conceptual picture of transport through its mathematical development to applications relevant to laboratory-and field-scale systems. The CTRW approach contrasts with ones used extensively on the basis of the advectiondispersion equation and use of upscaling, volume averaging, and homogenization. We examine the underlying assumptions, scope, and differences of these approaches, as well as stochastic formulations, relative to CTRW. We argue why these methods have not been successful in fitting actual measurements. The CTRW has now been developed within the framework of partial differential equations and has been generalized to apply to nonstationary domains and interactions with immobile states (matrix effects). We survey models based on multirate mass transfer (mobile-immobile) and fractional derivatives and show their connection as subsets within the CTRW framework.

Toward efficiency in heterogeneous multispecies reactive transport modeling: A particle‐tracking solution for first‐order network reactions

“…In this context, Particle Tracking Methods (PTMs) offer a convenient numerical solution particularly efficient in dealing with heterogeneities [e.g., Wen and GomezHernandez, 1996;LaBolle et al, 1996;Salamon et al, 2007;Riva et al, 2008] and a large variety of complex transport processes such as non-Fickian transport [Delay and Bodin, 2001;Cvetkovic and Haggerty, 2002;Berkowitz et al, 2006;Zhang and Benson, 2008;Dentz and Castro, 2009] and multiple porosity systems [Salamon et al, 2006b;Benson and Meerschaert, 2009;Tsang and Tsang, 2001;Huang et al, 2003;Willmann et al, 2013]. Moreover, this methodology, which is always mass conservative, avoids some of the inherent numerical difficulties associated with Eulerian approaches, i.e., numerical dispersion and oscillations due to truncation errors [Salamon et al, 2007;Boso et al, 2013].…”

Reply to comment by V. P. Shkilev on “Non‐Fickian transport and multiple‐rate mass transfer in porous media”

“…In principle we do not have to make this separation; we could continue to develop the transport equations in terms of a memory function (discussed below) including all sites [e.g., Berkowitz et al, 2006]. The separation refers to classes of sites that are amenable to different types of modification in the experimental conditions.…”

“…A recent complete detailing of this ensemble average can be found in work by Scher et al [2010], where an explicit derivation of the relation between the w rates and y(s, t) (the probability density of a tracer displacement s and time t at each transition; the Laplace transform of the 0th spatial moment is (u), which we use below) is displayed and the limitations of the ensemble average are discussed. The literature outlining the ensemble average from Berkowitz et al [2008] includes Appendix B of Berkowitz et al [2006], yielding a generalized master equation [Klafter and Silbey, 1980], as well as section 2.2 of Berkowitz et al [2006]. In this form, we work with c f (s, t) and c % (s, t), which are the (normalized) ensemble-averaged, bulk solute concentrations (tracer per domain volume, analogous to the single-realization definitions given above).…”

“…[12] Within the continuous time random walk (CTRW) framework, we can develop the generalized master equation, with the additional terms of (averaged) capture and release, into a continuous-in-space, partial differential equation formulation analogous to previous developments [Berkowitz et al, 2006, …”

Modifying continuous-time random walks to model finite-size particle diffusion in granular porous media

“…Using CTRW to model diffusion in confined geometries, such as porous media formed by either sintered or unconsolidated granular materials, is very attractive [8][9][10][11] because it alleviates the need to simulate directly the dynamics of particles within the pore space, reducing significantly the computational burden. In addition, it alleviates finite-size errors due to finite samples.…”