A stochastic model of reactive solute transport with time‐varying velocity in a heterogeneous aquifer

Abstract

The cumulant expansion method, used previously by Sposito and Barry (1987) to derive an ensemble average transport equation for a tracer moving in a heterogeneous aquifer, is generalized to the case of a reactive solute that can adsorb linearly and undergo first‐order decay. In the process we also generalize the Van Kampen (1987) result for the cumulant expansion of a multiplicative stochastic differential equation containing a time‐dependent sure matrix. The resulting partial differential equation exhibits terms with field‐scale coefficients that are analogous to those in the corresponding nonstochastic local‐scale transport equation. There are also new terms in the third‐ and fourth‐order spatial derivatives of the ensemble average concentration. It is demonstrated that the effective solute velocity for the aqueous concentration, not that for the total concentration (aqueous plus sorbed), is relevant for a field‐scale description of solute transport. The field‐scale effective solute velocity, dispersion coefficient, retardation factor, and first‐order decay parameters, unlike their local‐scale counterparts, are time‐dependent because of autocorrelations and cross correlations among the random local solute velocity, retardation factor, and first‐order decay constant. It is shown also that negative cross correlations between the random tracer solute velocity and the inverse of the local retardation factor may produce both enhanced dispersion and a temporal growth in the field‐scale retardation factor. These effects are possible in any heterogeneous aquifer for which a stochastic description of aquifer spatial variability is appropriate.

Estimation and upscaling of dual-permeability model parameters for the transport of E. coli D21g in soils with preferential flow

“…It should be emphasized that virtual fields are a simplified representation of the natural environment. Nevertheless, the stream tube model has often been used to approximate flow and transport at the field scale (e.g., Dagan, 1984;Freyberg, 1986;Gelhar and Axness, 1983;Kabala and Sposito, 1991;Russo, 1991;Sposito and Barry, 1987) and available experimental data are generally consistent with this approach (Dean and Foran, 1992;Evans and Owens, 1972;Guzman et al, 2009). Furthermore, the estimated mean and variance in microbial concentrations are needed for risk assessment.…”

“…A variety of stochastic methods have been used to describe solute transport in heterogeneous flow fields (e.g., Dagan, 1984;Freyberg, 1986;Gelhar and Axness, 1983;Kabala and Sposito, 1991;Russo, 1991;Sposito and Barry, 1987). In a simplified stochastic modeling approach the field may be viewed as a series of independent stream tubes with their hydraulic conductivity and/or other selected parameters described using probability density functions (PDFs) (Dagan, 1993;Jury and Roth, 1990).…”

Experimental investigation of acoustically enhanced solute transport in porous media

“…It should be noted that the concept of effective parameters has been applied in numerous groundwater flow and solute transport studies [Valocchi, 1989;Chrysikopoulos et al, 1990Chrysikopoulos et al, , 1992Kabala and Sposito, 1991;Chrysikopoulos, 1995].…”

Stochastic analysis of adsorbing solute transport in three‐dimensional, heterogeneous, unsaturated soils

“…The study shows that solute spreading increases with decreasing water content and with increasing variability of unsaturated soil properties. Analytical solutions for transport of adsorbing solutes in saturated media indicate that macrodispersion is related to the retardation factor, which has a large impact on longitudinal dispersion but not on the transverse dispersion [Kabala and Sposito, 1991 In this paper we study combined effects of physic, al and chemical heterogeneities and variable hydraulic gradient on solute transport in three-dimensional, heterogeneous, unsaturated soils. We analyze the solute transport and unsaturated water flow equations with randomly distributed unsaturated hydraulic conductivity, water content, and reactive solute parameters.…”