Perspective on theories of non-Fickian transport in heterogeneous media

Abstract

Subsurface fluid flow and solute transport take place in a multiscale heterogeneous environment. Neither these phenomena nor their host environment can be observed or described with certainty at all scales and locations of relevance. The resulting ambiguity has led to alternative conceptualizations of flow and transport and multiple ways of addressing their scale and space–time dependencies. We focus our attention on four approaches that give rise to nonlocal representations of advective and dispersive transport of nonreactive tracers in randomly heterogeneous porous or fractured continua. We compare these approaches theoretically on the basis of their underlying premises and the mathematical forms of the corresponding nonlocal advective–dispersive terms. One of the four approaches describes transport at some reference support scale by a classical (Fickian) advection–dispersion equation (ADE) in which velocity is a spatially (and possibly temporally) correlated random field. The randomness of the velocity, which is given by Darcy’s law, stems from random fluctuations in hydraulic conductivity (and advective porosity though this is often disregarded). Averaging the stochastic ADE over an ensemble of velocity fields results in a space–time-nonlocal representation of mean advective–dispersive flux, an approach we designate as stnADE. A closely related space–time-nonlocal representation of ensemble mean transport is obtained upon averaging the motion of solute particles through a random velocity field within a Lagrangian framework, an approach we designate stnL. The concept of continuous time random walk (CTRW) yields a representation of advective–dispersive flux that is nonlocal in time but local in space. Closely related to the latter are forms of ADE entailing fractional derivatives (fADE) which leads to representations of advective–dispersive flux that are nonlocal in space but local in time; nonlocality in time arises in the context of multirate mass transfer models, which we exclude from consideration in this paper. We describe briefly each of these four nonlocal approaches and offer a perspective on their differences, commonalities, and relative merits as analytical and predictive tools.

Introduction

Transport of nonreactive (passive) tracers through porous media has been traditionally described by a deterministic advection–dispersion equation (ADE) based on analogy to Ficks laws of diffusion [5]. Nonreactive tracer transport that is not adequately described by an ADE is therefore said to be non-Fickian. In the special case of Fickian transport at a uniform velocity far from sources or boundaries, a slug of tracer evolves into a Gaussian plume with a variance that grows linearly with time. Correspondingly, in the special case where mean-uniform flow takes place through a heterogeneous porous or fractured continuum, non-Fickian behavior manifests itself through deviations of the mean concentration profile from a Gaussian shape and/or a nonlinear growth rate of a plume’s mean squared displacement. Such behavior (especially but not only when the growth rate is a power-law) is commonly referred to as anomalous transport [53]. The purpose of our paper is to provide a perspective on some modern theories of non-Fickian transport in heterogeneous porous media in velocity fields that are either uniform or nonuniform in the mean. We start by recalling briefly the fundamental tenets of traditional Fickian transport.

The macroscopic description of tracer fate and migration in a uniform porous medium by means of an ADE rests on an assumed analogy to Fick’s laws of diffusive transport. Analogy to Fick’s first law posits that macroscopic tracer mass flux $\mathbf{J}(\mathbf{x}\text{,}t)$ through a fluid–saturated pore space, at any “point” $\mathbf{x}$ within a fictitious continuum representing the fluid–solid mixture at time t, can be expressed as $\mathbf{J}={\mathbf{J}}_{\text{adv}}+{\mathbf{J}}_{\text{dif}}+{\mathbf{J}}_{\text{dis}}$, where ${\mathbf{J}}_{\text{adv}}(\mathbf{x}\text{,}t)$ is advective mass flux attributed to the macroscopic fluid velocity $\mathbf{v}(\mathbf{x}\text{,}t)$ through the pore space (Darcy flux divided by a scalar advective, sometimes called effective or kinematic, porosity), ${\mathbf{J}}_{\text{dif}}(\mathbf{x}\text{,}t)$ is mass flux due to molecular diffusion across the saturated pore space, and ${\mathbf{J}}_{\text{dis}}(\mathbf{x}\text{,}t)$ is dispersive mass flux attributed to random deviations of fluid velocities within this pore space from their macroscopic value $\mathbf{v}$. The analogy further posits that

$${\mathbf{J}}_{\text{adv}}=\mathbf{v}c\text{,}{\mathbf{J}}_{\text{dif}}=-D_{\text{m}}\nabla c\text{,}{\mathbf{J}}_{\text{dis}}=-{\mathbf{D}}_{\text{d}}\nabla c\text{,}$$

where $c(\mathbf{x}\text{,}t)$ is macroscopic concentration (solute mass per unit saturated pore space), $D_{\text{m}}$ is an effective molecular diffusion coefficient (smaller than that in pure fluid due to the obstruction of molecular paths by solids), and ${\mathbf{D}}_{\text{d}}(\mathbf{x}\text{,}t)$ is a dispersion tensor (typically expressed as ${\mathbf{D}}_{\text{d}}=\mathit{\lambda }v$, where v is the magnitude of $\mathbf{v}$ and $\mathit{\lambda }$ is a dispersivity tensor whose principal components are considered to be properties of the medium, parallel and normal to $\mathbf{v}$). Analogy to Fick’s second law posits that solute mass in the saturated pore space is conserved at the macro-scale. In the absence of sinks or sources, this yields a mass balance equation

$$\frac{\mathrm{\partial }c}{\mathrm{\partial }t}=-\nabla ·({\mathbf{J}}_{\text{adv}}+{\mathbf{J}}_{\text{dif}}+{\mathbf{J}}_{\text{dis}})\text{.}$$

Substituting (1) into (2) yields the advection–dispersion equation (ADE)

$$\frac{\mathrm{\partial }c}{\mathrm{\partial }t}=-\nabla ·(\mathbf{v}c)+\nabla ·[(D_{\text{m}}\mathbf{I}+{\mathbf{D}}_{\text{d}})\nabla c]\text{,}$$

where $\mathbf{I}$ is the identity tensor.

The classical ADE (3) often fails to predict observed behavior of solute in the subsurface. Non-Fickian transport behavior has been observed both in the laboratory [78], [79], [47], [13], [58] and in the field [68], [80], [48], [32], [77], [92]. The common feature of these and other experiments is heterogeneity of porous media. Similar non-Fickian behavior has been attributed to heterogeneity of ambient environments in areas as diverse as virus migration in cells [75], protein dynamics [89], transport of lipid granules in the cytoplasm of living yeast cells [85], fluctuations of stocks on financial markets [69], and animal movement in heterogeneous landscapes [39]. Mean non-Fickian behavior has likewise been observed and analyzed in turbulent dispersion (e.g. [43], [44], [72], [31], [45]).

To understand the failure of the ADE to model transport in heterogeneous media, it is important to recall that traditionally (1), (2), (3) have been considered to be deterministic. This required one to assume that all quantities entering into the ADE are defined on a representative elementary volume (REV) large enough to render their space–time variations sufficiently slow to be described deterministically [5]. In dealing with heterogeneous media it is common to consider smaller support volumes (designated here by $\omega$) which, though large enough to be ascribed (directly or indirectly) measurable macroscopic properties, do not represent REVs, as explained in greater detail in [64], [63]. Instead, quantities defined on the scale of $\omega$ vary rapidly enough to justify treating them as random functions of space and/or time over a fictitious macroscopic continuum. When this is the case, the ADE becomes stochastic. The most common (numerical Monte Carlo) method of solving a stochastic ADE is to generate numerous random realizations of the underlying velocity field, solve the ADE numerically for each field and average the results over all realizations. The results are summarized in terms of multivariate (due to the space–time dependence of $\mathbf{v}$ and c) sample statistics such as frequency histograms and sample moments (most commonly mean, variance, auto- and cross-covariance). As the number of realizations increases the sample statistics converge (if all is well) to their theoretical stochastic or ensemble statistics.

It has been demonstrated theoretically [20], [15], [42], [65], [94], [61], [91], [22] and computationally [37], [57], [56] that even if transport in each random realization is governed by the ADE, the ensemble mean deterministic transport through randomly heterogeneous media is generally non-Fickian.

Though the above stochastic theories of transport in porous media are closely related to those underlying turbulent diffusion, there are two fundamental differences between them: (a) the first takes place at small and the second at large Reynolds numbers, and (b) porous flow velocities depend in a known way on medium hydraulic properties coupled with externally imposed head conditions while turbulent velocities fluctuate randomly in space–time. It follows that whereas uncertainty about turbulent velocities is aleatory (controlled by chance), uncertainty about velocities in a porous medium is epistemic (due to incomplete knowledge of medium properties and imposed heads) and hence reducible through conditioning on hydrogeologic data (e.g. [81]). Whereas the aleatory nature of turbulent velocities may justify treating them as being space–time stationary and perhaps even Gaussian, conditioning coupled with the imposition of (at least partially) known forcings (initial conditions, boundary conditions and sources) render porous flow velocities nonstationary and generally non-Gaussian (e.g. [57], [56]). We therefore consider it essential that stochastic theories of flow and transport in porous media be capable of accounting for nonstationary and non-Gaussian behaviors in bounded domains subject to realistic forcings and descriptions of site geology (for the latter see [88], [71], [46] and references therein), though not all presently do so.

In a stationary mean-uniform velocity field, non-Fickian transport manifests itself through an early (pre-asymptotic) linear increase in longitudinal and transverse dispersivities with mean travel distance (or, equivalently, time). Tracer experiments of Peaudecerf and Sauty [68] provided the first documented field evidence that an individual plume may spread at variable rates over long periods of time. That the same can happen over a much longer period of at least 600 days in a mildly heterogeneous geologic medium is vividly demonstrated by the celebrated tracer experiment at Borden, Ontario, Canada [48], [80]. Whereas longitudinal dispersivity (parallel to the mean velocity) eventually stabilizes at a constant “Fickian” asymptote, the transverse dispersivity first reaches a peak and then either decreases to zero or stabilizes at a constant value below the peak, depending on the model. Dentz et al. [26], [27] have demonstrated numerically and Attinger et al. [3] have proven analytically that whereas in two dimensions the transverse dispersivity tends asymptotically to small laboratory-scale values, in three dimensions it stabilizes at much larger field-scale values. Whereas the two-dimensional behavior is consistent with second-order perturbation theories (e.g. [20], [15], [21]), the three-dimensional behavior agrees more closely with a quasi-linear theory based on Corrsin’s conjecture [65], [94].

Non-Fickian pre-asymptotic behavior is explained by the fact that as a plume grows, it gradually encounters (samples) heterogeneities on larger and larger scales. In a statistically homogeneous medium, asymptotic Fickian behavior may eventually be reached due to the limited range of these scales. Yet when one juxtaposes apparent longitudinal dispersivities, determined in the laboratory and in the field for a variety of porous and fractured media on the basis of Fickian models that consider medium properties to be spatially uniform (and thus provide no resolution of the way in which these properties vary in space), their values increase consistently with the scale of observation (plume mean travel distance or travel time) at a rate that is faster than linear [60]. This persistent, supralinear dispersivity scale effect has been interpreted [60], [59], [30] to imply that the juxtaposed media represent a hierarchy of log permeability fields which behave jointly as a random fractal. When one juxtaposes apparent longitudinal dispersivities determined on the basis of Fickian models that do resolve some larger-scale spatial variations in medium properties, one finds [62], [64] that the rate of dispersivity increase with observation scale diminishes.

In the presence of boundaries and/or under conditioning on measurements, the velocity field becomes nonstationary. Though this requires a nonlocal representation of mean transport (see below), localization is sometimes possible as an approximation. Corresponding analyses [57], [56] demonstrate that boundaries tend to increase the rate at which longitudinal and transverse dispersivities vary with mean travel distance (or, equivalently, time) and conditioning tends to decrease it. An asymptotic Fickian regime fails to develop, as the dispersivities vary continuously with mean travel distance and the plume remains non-Gaussian.

Despite some examples to the contrary (e.g. [62], [25]), non-Fickian transport in heterogeneous porous media generally manifests itself through nonlocal mean behavior represented by integro-differential equations or (where such exist) their fractional derivative equivalents. Since subsurface fluid flow and solute transport take place in a multiscale heterogeneous environment, neither these phenomena nor their host environment can be observed or described with certainty at all scales and locations of relevance. The resulting ambiguity allows for alternative conceptualizations of flow and transport and multiple ways of addressing their scale and space–time dependencies.

In this paper, we focus on four conceptualizations and representations of nonlocal mean transport of nonreactive tracers that have gained some prominence in the hydrologic literature over the last two decades: a space–time-nonlocal representation based on the assumption of advective–dispersive behavior at a reference support scale [42], [61], [91], [57], [56] which we designate stnADE; a space–time-nonlocal representation based on a Lagrangian representation of particle motions in a stationary random velocity field [18], [16] which we denote by stnL; a time-nonlocal representation of mean particle transport on a discrete lattice having arbitrarily small cell sizes based on a continuous time random walk (CTRW) concept (e.g. [55], [73], [8]); and space-fractional representations of advection and dispersion (fADE) [50], [7], [6], [93] that are nonlocal in space but local (except in the case of multirate mass transfer) in time.

We describe briefly each of these four nonlocal approaches and offer a perspective on their differences, commonalities, and relative merits as analytical and predictive tools. We compare these approaches theoretically on the basis of their underlying premises with emphasis on the mathematical forms of the corresponding nonlocal advection and dispersion terms. We omit from our discussion other models of non-Fickian transport (see [17]). Some of these, such as models based on spatial averaging (e.g. [70]) and homogenization (e.g. [4]), generally aim at obtaining localized forms of advective–dispersive flux. Others, such as multirate mass transfer [35] and delayed diffusion [28] models, can be related to the four approaches analyzed in this paper [24], [28].