Marco Dentz

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

[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 laboratoryand 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.

Mixing, spreading and reaction in heterogeneous media: A brief review

Geological media exhibit heterogeneities in their hydraulic and chemical properties, which can lead to enhanced spreading and mixing of the transported species and induce an effective reaction behavior that is different from the one for a homogeneous medium. Chemical heterogeneities such as spatially varying adsorption properties and specific reactive surface areas can act directly on the chemical reaction dynamics and lead to different effective reaction laws. Physical heterogeneities affect mixing-limited chemical reactions in an indirect way by their impact on spreading and mixing of dissolved species. To understand and model large-scale reactive transport the interactions of these coupled processes need to be understood and quantified. This paper provides a brief review on approaches of non-reactive and reactive transport modeling in geological media.

Temporal behavior of a solute cloud in a heterogeneous porous medium: 1. Point‐like injection

We investigate the temporal behavior of transport coefficients in a model for transport of a solute through a spatially heterogeneous saturated aquifer. In the framework of a stochastic approach we derive explicit expressions for the temporal behavior of the center-of-mass velocity and the dispersion of the concentration distribution after a point-like injection of solute at time t=0, using a second-order perturbation expansion. The model takes into account local variations in the hydraulic conductivity (which, in turn, induce local fluctuations in the groundwater flow velocities) and in the chemical adsorption properties of the medium (which lead to a spatially varying local retardation factor). In the given perturbation theory approach the various heterogeneity-induced contributions can be systematically traced back to fluctuations in these quantities and to cross correlations between them. We analyze two conceptually different definitions for the resulting dispersion coefficient: the “effective”dispersion coefficient which is derived from the average over the centered second moments of the spatial concentration distributions in every realization and the “ensemble” dispersion coefficient which follows from the second moment of the ensemble-averaged concentration distribution. The first quantity characterizes the dispersion in a typical realization of the medium, whereas the second one describes the (formal) dispersion properties of the ensemble as a whole. We give explicit analytic expressions for both quantities as functions of time and show that for finite times their temporal behavior is remarkably different. The ensemble dispersion coefficient which is usually evaluated in the literature considerably overestimates the dispersion typically found in one given realization of the medium. From our explicit results we identify two relevant timescales separating regimes of qualitatively and quantitatively different temporal behavior: The shorter of the two scales is set by the advective transport of the solute cloud over one disorder correlation length, whereas the second, much larger one, is related to the dispersive spreading over the same distance. Only for times much larger than this second scale, do the effective and the ensemble dispersion coefficient become equivalent because of mixing caused by the local transversal dispersion. The formulae are applied to the Borden experiment data. It is concluded that the observed dispersion coefficient matches the effective dispersion coefficient at finite times proposed in this paper very well.

Multicomponent reactive transport in multicontinuum media

Multicomponent reactive transport in aquifers is a highly complex process, owing to a combination of variability in the processes involved and the inherent heterogeneity of nature. To date, the most common approach is to model reactive transport by incorporating reaction terms into advection-dispersion equations (ADEs). Over the last several years, a large body of literature has emerged criticizing the validity of the ADE for transport in real media, and alternative models have been presented. One such approach is that of multirate mass transfer (MRMT). In this work, we propose a model that introduces reactive terms into the MRMT governing equations for conservative species. This model conceptualizes the medium as a multiple continuum of one mobile region and multiple immobile regions, which are related by kinetic mass transfer processes. Reactants in both the mobile and immobile regions are assumed to always be in chemical equilibrium. However, the combination of local dispersion in the mobile region and the various mass transfer rates induce a global chemical nonequilibrium. Assuming this model properly accounts for transport of reactive species, we derive explicit expressions for the reaction rates in the mobile and immobile regions, and we study the impact of mass transfer on reactive transport. Within this framework, we observe that the resulting reaction rates can be very different from those that arise in a system governed by an ADE-type equation.

Coupling of mass transfer and reactive transport for nonlinear reactions in heterogeneous media

Fast chemical reactions are driven by mixing‐induced chemical disequilibrium. Mixing is poorly represented by the advection‐dispersion equation. Instead, effective dynamics models, such as multirate mass transfer (MRMT), have been successful in reproducing observed field‐scale transport, notably, breakthrough curves (BTCs) of conservative solutes. The objective of this work is to test whether such effective models, derived from conservative transport observations, can be used to describe effective multicomponent reactive transport in heterogeneous media. We use a localized formulation of the MRMT model that allows us to solve general reactive transport problems. We test this formulation on a simple three‐species mineral precipitation problem at equilibrium. We first simulate the spatial and temporal distribution of mineral precipitation rates in synthetic hydraulically heterogeneous aquifers. We then compare these reaction rates to those corresponding to an equivalent (i.e., same conservative BTC) homogenized medium with transport characterized by a nonlocal in time equation involving a memory function. We find an excellent agreement between the two models in terms of cumulative precipitated mass for a broad range of generally stationary heterogeneity structures. These results indicate that mass transfer models can be considered to represent quite accurately the large‐scale effective dynamics of mixing controlled reactive transport at least for the cases tested here, where individual transport paths sample the full range of heterogeneities represented by the BTC.

Non-Fickian dispersion in porous media explained by heterogeneous microscale matrix diffusion

Mobile-immobile mass transfer is widely used to model non-Fickian dispersion in porous media. Nevertheless, the memory function, implemented in the sink/source term of the transport equation to characterize diffusion in the matrix (i.e., the immobile domain), is rarely measured directly. Therefore, the question can be posed as to whether the memory function is just a practical way of increasing the degrees of freedom for fitting tracer test breakthrough curves or whether it actually models the physics of tracer transport. In this paper we first present a technique to measure the memory function of aquifer samples and then compare the results with the memory function fitted from a set of field-scale tracer tests performed in the same aquifer. The memory function is computed by solving the matrix diffusion equation using a random walk approach. The properties that control diffusion (i.e., mobile-immobile interface and immobile domain cluster shapes, porosity, and tortuosity) are investigated by X-ray microtomography. Once the geometry of the matrix clusters is measured, the shape of the memory function is controlled by the value of the porosity at the percolation threshold and of the tortuosity of the diffusion path. These parameters can be evaluated from microtomographic images. The computed memory function compares well with the memory function deduced from the field-scale tracer tests. We conclude that for the reservoir rock studied here, the atypical non-Fickian dispersion measured from the tracer test is well explained by microscale diffusion processes in the immobile domain. A diffusion-controlled mobileimmobile mass transfer model therefore appears to be valid for this specific case.

Temporal behaviour of a solute cloud in a chemically heterogeneous porous medium

In this paper we investigate the temporal behaviour of a solute cloud in a heterogeneous porous medium using a stochastic modelling approach. The behaviour of the plume evolving from a point-like instantaneous injection is characterized by the velocity of its centre-of-mass and by its dispersion as a function of time. In a stochastic approach, these quantities are expressed as appropriate averages over the ensemble of all possible realizations of the medium. We develop a general perturbation approach which allows one to calculate the various quantities in a systematic and unified way. We demonstrate this approach on a simplified aquifer model where only the retardation factor R ( x ) due to linear instantaneous chemical adsorption varies stochastically in space. We analyse the resulting centre-of-mass velocity and two conceptually different definitions for the dispersion coefficient: the ‘effective’ dispersion coefficient which is derived from the average over the centred second moments of the spatial concentration distributions in every realization, and the ‘ensemble’ dispersion coefficient which follows from the second moment of the averaged concentration distribution. The first quantity characterizes the dispersion in a typical realization of the medium as a function of time, whereas the second one describes the (formal) dispersion properties of the ensemble as a whole. We show that for finite times the two quantities are not equivalent whereas they become identical for t →∞ and spatial dimensions d [ges ]2. The ensemble dispersion coefficient which is usually evaluated in the literature considerably overestimates the dispersion typically found in one given realization of the medium. We derive for the first time explicit analytical expressions for both quantities as functions of time. From these, we identify two relevant time scales separating regimes of qualitatively and quantitatively different temporal behaviour: the shorter of the two scales is set by the advective transport of the solute cloud over one disorder correlation length, whereas the second, much larger one, is related to the dispersive spreading over the same distance. Only for times much larger than this second scale, and spatial dimensions d [ges ]2, do the effective and the ensemble dispersion coefficients become equivalent due to mixing caused by the local transversal dispersion. Finally, the formalism is generalized to an extended source. With growing source size the convergence of the effective dispersion coefficient to the ensemble dispersion coefficient happens faster as the extended source already represents an ensemble of point sources. In the limit of a very large source size, convergence occurs on the time scale of advective transport over one disorder length. We derive explicit results for the temporal behaviour in the different time regimes for both point and extended sources.

Temporal behavior of a solute cloud in a heterogeneous porous medium: 2. Spatially extended injection

We investigate the temporal behavior of transport coefficients in a stochastic model for transport of a solute through a spatially heterogeneous saturated aquifer. While the first of these two companion papers [Dentz et al., this issue] investigated a situation characterized by a point-like solute injection, we now focus on the case of spatially extended solute sources. The analysis of the finite time behavior of the transport coefficients makes it necessary to distinguish between two fundamentally different quantities characterizing the solute dispersion. We define an “effective” dispersion coefficient which is derived from the average over the centered second moments of the spatial concentration distributions in every realization and an “ensemble” dispersion coefficient which follows from the second moment of the ensemble-averaged concentration distribution. While the two quantities are equivalent in the asymptotic limit of infinite times or infinitely extended sources, they are qualitatively and quantitatively different for the more realistic situation of finite times and finite source extent. We demonstrate that in this case the ensemble quantity, used more or less implicitly in most of the previous studies, overestimates the true dispersion of the plume. Using a second-order perturbation theory approach, we derive explicit solutions for the temporal behavior of the dispersion coefficients for various types of isotropic and anisotropic initial conditions. We identify the relevant timescales which separate regimes of different temporal behavior and apply our formulae to the Borden experiment data. We find a good agreement between theory and experiment if we compare the observed dispersion with the appropriate effective dispersion coefficient (including the leading effects of the local dispersion), whereas the ensemble dispersion coefficient commonly used in the literature to analyze these data overestimates the experimental results considerably.

Concentration statistics for mixing-controlled reactive transport in random heterogeneous media

Uncertainty in the distribution of hydraulic parameters leads to uncertainty in flow and reactive transport. Traditional stochastic analysis of solute transport in heterogeneous media has focused on the ensemble mean of conservative-tracer concentration. Studies in the past years have shown that the mean concentration often is associated with a high variance. Because the range of possible concentration values is bounded, a high variance implies high probability weights on the extreme values. In certain cases of mixing-controlled reactive transport, concentrations of conservative tracers, denoted mixing ratios, can be mapped to those of constituents that react with each other upon mixing. This facilitates mapping entire statistical distributions from mixing ratios to reactive-constituent concentrations. In perturbative approximations, only the mean and variance of the mixing-ratio distribution are used. We demonstrate that the second-order perturbative approximation leads to erroneous or even physically impossible estimates of mean reactive-constituent concentrations when the variance of the mixing ratio is high and the relationship between the mixing ratio and the reactive-constituent concentrations strongly deviates from a quadratic function. The latter might be the case in biokinetic reactions or in equilibrium reactions with small equilibrium constant in comparison to the range of reactive-constituent concentrations. When only the mean and variance of the mixing ratio is known, we recommend assuming a distribution that meets the known bounds of the mixing ratio, such as the beta distribution, and mapping the assumed distribution of the mixing ratio to the distributions of the reactive constituents.

Mixing and spreading in stratified flow

G. I. Taylor [Proc. R. Soc. London, Ser. A 219, 186 (1953)] quantified enhanced solute mixing in the flow through a tube at asymptotically long times by the constant Taylor dispersion coefficient, which provides a good representation of both the asymptotic dispersion dynamics and evolution of the solute concentration. At preasymptotic times, however, the use of the constant Taylor dispersion coefficient does not facilitate a faithful representation of either the actual mixing or spreading, which are controlling factors for chemical reaction rates. Transport in spatially varying flow fields often displays non-Fickian or anomalous behavior, which is reflected by the fact that effective dispersion evolves in time. Here we study and quantify the mechanisms leading to enhanced solute mixing in spatially nonhomogeneous flow fields using “local” spatial moments, i.e., moments of the transport Green function. On the basis of such a local moment formulation, we define effective dispersion coefficients to character...