Transport behavior of a passive solute in continuous time random walks and multirate mass transfer

Abstract

[1] We consider the transport behavior of a passive solute in a heterogeneous medium, modeled by continuous time random walks (CTRW) and linear multirate mass transfer (MRMT). Within the CTRW framework, we formulate a transport model which is formally equivalent to MRMT. In both approaches the total concentration is divided into mobile and immobile parts. The immobile concentration is given by the convolution in time of the mobile concentration and a memory function. The memory function is a functional of the distribution of transition and trapping times for the CTRW and the MRMT approach, respectively, and determines the transport behavior of the solute. Based on different expressions for the memory function in the two frameworks, we derive conditions for which both approaches describe the same transport behavior. We focus on anomalous transport behavior that can arise if the transition and trapping time distributions behave algebraically in a given time regime. Using an expansion of the Laplace transform of the memory function, we develop explicit expressions for the time behavior of the flux concentration as well as for the center of mass velocity and the (macro) dispersion coefficients of the solute distribution. We observe (anomalous) power law as well as (normal) Fickian transport behavior, depending on the exponents that dominate the trapping and transition time distributions, respectively. The results show that the character of the anomalous transport does not depend on the details of the transport model but only on the exponents dominating the transition or trapping time distributions. The unified transport framework presented here comprising CTRW and MRMT shows new aspects and opens new perspectives for the modeling of transport in heterogeneous media.INDEX TERMS: 1832 Hydrology: Groundwater transport; 1869 Hydrology: Stochastic processes; 3210 Mathematical Geophysics: Modeling; 1829 Hydrology: Groundwater hydrology; KEYWORDS: Groundwater hydrology, groundwater transport, stochastic processes, modeling Citation: Dentz, M., and B. Berkowitz, Transport behavior of a passive solute in continuous time random walks and multirate mass transfer, Water Resour.

Non‐Fickian transport in homogeneous unsaturated repacked sand

“…The principal characteristics of the migration of a tracer are dominated by the late time behavior of the joint probability function ψ(s, t), the probability rate for a displacement s with difference of arrival times t . The CTRW model is able to encompass Fickian, single-rate and multiple-rate MIM transport models [Dentz and Berkowitz, 2003].…”

“…[15] It has recently been shown that multiple-rate, or power law waiting time, MIM models are equivalent to CTRW [Dentz and Berkowitz, 2003;Schumer et al, 2003] when the transition length and time distribution density decouple [Dentz and Berkowitz, 2003]. Both the multiplerate MIM and CTRW models are flexible in the density of waiting times that arise.…”

A general real-time formulation for multi-rate mass transfer problems

“…In solute transport, non-equilibrium has been attributed to diffusion-limited storage into immobile regions, kinetic sorption or heterogeneity (Brusseau et al, 1989;Sudicky, 1989Sudicky, , 1990Valocchi, 1990;Sardin et al, 1991;Cvetkovic et al, 1992;Toride et al, 1993;Tsang, 1995;Haggerty and Gorelick, 1995;Ray et al, 1997;Carrera et al, 1998;Dentz and Berkowitz, 2003;El-Zein et al, 2005;Salamon et al, 2006;Vogel et al, 2006;Zhang et al, 2006Zhang et al, , 2007Alcolea et al, 2008;Willmann et al, 2008;Kumar, 2008;Gouze et al, 2008). Non-equilibrium is intrinsically associated to multicomponent reactive transport at the field scale (Willmann et al, 2009).…”

A random walk solution for modeling solute transport with network reactions and multi-rate mass transfer in heterogeneous systems: Impact of biofilms

“…In this context, Particle Tracking Methods (PTMs) constitute an efficient numerical alternative to simulate reactive transport (Kitanidis, 1994;Henri and Fernàndez-Garcia, 2014). Even though a large variety of methods exist to simulate rate-limited mass transfer processes with particle tracking (Benson and Meerschaert, 2009;Delay and Bodin, 2001;Dentz and Berkowitz , 2003;Tsang and Tsang, 2001), this method is still limited in the type of chemical reactions available, which include sorption (Tompson, 1993;Valocchi and Quinodoz , 1989;Michalak and Kitanidis, 2000), radioactive decay (Wen and Gómez-Hernández , 1996;Painter et al, 2007), first-order network reactions (Burnell et al, 2014;Henri and Fernàndez-Garcia, 2014), and simple bimolecular reactions (Benson and Meerschaert, 2008;Ding et al, 2013;Edery et al, 2009Edery et al, , 2010Paster et al, 2014) among others. None of the methods available nowadays supports multi-porosity systems with network reactions in three-dimensional randomly heterogeneous porous media.…”