Abstract

We investigate effective solute transport in a chemically heterogeneous medium subject to temporal fluctuations of the flow conditions. Focusing on spatial variations in the equilibrium adsorption properties, the corresponding fluctuating retardation factor is modeled as a stationary random space function. The temporal variability of the flow is represented by a stationary temporal random process. Solute spreading is quantified by effective dispersion coefficients, which are derived from the ensemble average of the second centered moments of the normalized solute distribution in a single disorder realization. Using first-order expansions in the variances of the respective random fields, we derive explicit compact expressions for the time behavior of the disorder induced contributions to the effective dispersion coefficients. Focusing on the contributions due to chemical heterogeneity and temporal fluctuations, we find enhanced transverse spreading characterized by a transverse effective dispersion coefficient that, in contrast to transport in steady flow fields, evolves to a disorder-induced macroscopic value (i.e., independent of local dispersion). At the same time, the asymptotic longitudinal dispersion coefficient can decrease. Under certain conditions the contribution to the longitudinal effective dispersion coefficient shows superdiffusive behavior, similar to that observed for transport in s stratified porous medium, before it decreases to its asymptotic value. The presented compact and easy to use expressions for the longitudinal and transverse effective dispersion coefficients can be used for the quantification of effective spreading and mixing in the context of the groundwater remediation based on hydraulic manipulation and for the effective modeling of reactive transport in heterogeneous media in general.

Abstract

This paper presents an approach conducive to an evaluation of the probability density function (pdf) of spatio-temporal distributions of concentrations of reactive solutes (and associated reaction rates) evolving in a randomly heterogeneous aquifer. Most existing approaches to solute transport in heterogeneous media focus on providing expressions for space–time moments of concentrations. In general, only low order moments (unconditional or conditional mean and covariance) are computed. In some cases, this allows for obtaining a confidence interval associated with predictions of local concentrations. Common applications, such as risk assessment and vulnerability practices, require the assessment of extreme (low or high) concentration values. We start from the well-known approach of deconstructing the reactive transport problem into the analysis of a conservative transport process followed by speciation to (a) provide a partial differential equation (PDE) for the (conditional) pdf of conservative aqueous species, and (b) derive expressions for the pdf of reactive species and the associated reaction rate. When transport at the local scale is described by an Advection Dispersion Equation (ADE), the equation satisfied by the pdf of conservative species is non-local in space and time. It is similar to an ADE and includes an additional source term. The latter involves the contribution of dilution effects that counteract dispersive fluxes. In general, the PDE we provide must be solved numerically, in a Monte Carlo framework. In some cases, an approximation can be obtained through suitable localization of the governing equation. We illustrate the methodology to depict key features of transport in randomly stratified media in the absence of transverse dispersion effects. In this case, all the pdfs can be explicitly obtained, and their evolution with space and time is discussed.