Stochastic analysis of dispersion in unsteady flow in heterogeneous aquifers

Abstract

The dispersion of a solute plume resulting from unsteady flow in three-dimensional, heterogeneous porous media was analyzed using stochastic continuum theory. Asymptotic stochastic solutions of the perturbed unsteady flow and solute transport equations were used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three-dimensional, statistically anisotropic input covariance describing the hydraulic conductivity and an input covariance describing the temporal variability of the mean hydraulic gradient. The flow equation was approximated by neglecting the influence of internal storage due to medium and fluid compressibility (specific storage, S s • 0). The predictive expression for the macrodispersivity tensor is the sum of two components: a spatial variability component identical to previous steady state theory and an unsteady component. Two special cases of unsteady flow were examined: variation only in the magnitude and variation only in the direction of the hydraulic gradient. Gradient magnitude variation produces a slightly larger longitudinal macrodispersivity than does the steady flow case, whereas gradient direction variation produces a significantly larger transverse macrodispersivity. Longitudinal and horizontal transverse macrodispersivities predicted with the unsteady stochastic theory were shown to be of a magnitude similar to observed values from the Borden, Cape Cod, and Columbus tracer experiments. 1964; Gelhar and Axness, 1983; Molz et al., 1986; Dagan, 1984, 1989; Neuman et al., 1987]. Near the source of contamination, solute shoots out along pathways of large hydraulic conductivity, but lags behind in regions of small K. Accurate modeling of this advection-dominated nearsource behavior requires a detailed understanding of the local heterogeneity. Much farther from the source, the solute disperses more rapidly than expected from laboratory measurements of dispersion [Anderson, 1979; Gelhat et al., 1985, 1992]. This enhanced field-scale mixing is caused, in large part, by the variability of K. Detailed measurement of the heterogeneous K field for use in advection-dominated models with laboratory-scale dispersivities is an unrealistic alternative for large-scale problems. Deterministic modeling of solute transport in a heterogeneous aquifer with only laboratory-scale dispersivity would require at least four cells per hydraulic conductivity correlation length [Ababou et al., 1989]. Thus, over 2 billion cells would be required to describe the Cape Cod sewage plume [LeBlanc, 1984], which is 3500 m long, 1000 m wide, and 25 m thick in an aquifer with horizontal and vertical correlation lengths of 3.5 and 0.19 m, respectively [Hess et al., 1992]. Of course, one should always obtain as much information about the flow system as resources will allow and incorporate that information into any modeling effort. Nonetheless, some form of advection-dispersion model with field-scale macro-1Now at Illinois State Water Survey, Office of Ground Water Quality,...

Effective two‐phase flow in heterogeneous media under temporal pressure fluctuations

“…[63] Disregarding terms that are quadratic in the fluctuations and assuming flow far away from the injection boundary, the latter gives the well known first-order approximation for q 0 (x, t) [e.g., Gelhar and Axness, 1983;Rehfeldt and Gelhar, 1992;Dentz and Carrera, 2005] …”

“…[10] Kinzelbach and Ackerer [1986] and Rehfeldt and Gelhar [1992] pointed out that temporal pressure fluctuations can lead to enhanced solute dispersion in heterogeneous media. Dentz and Carrera [2003] and Cirpka and Attinger [2003] quantify the interaction of temporal pressure fluctuations and spatial heterogeneity using a perturbation analysis in the fluctuations of the spatial and temporal random fields.…”

Ensemble‐averaged equations for reactive transport in porous media under unsteady flow conditions

“…The time-ordered exponential operator is described by Kavvas and Karakas [1996] and in the sections that follow. Stochastic approaches for nonreactive transport under unsteady flow fields have been studied by other workers [Naff et al, 1988[Naff et al, , 1989Rehfeldt and Gelhar, 1992;, and these works are reviewed by . In each of those works it is assumed that V. q --0, so the transience arises only from boundary conditions.…”

Stochastic analysis of field-scale heat advection in heterogeneous aquifers

“…Gelhar and Axness, 1983;Rehfeldt and Gelhar, 1992) based on Fourier-Stieltjes representation for the perturbed quantities in wave number domain. By using this approach, the random perturbations in Eq.…”

“…Theoretical studies on the field-scale nonreactive solute transport process have been carried out within the Eulerian framework (e.g. Gelhar and Axness, 1983;Neuman et al, 1987;Graham and McLaughlin, 1989;Vomvoris and Gelhar, 1990;Rehfeldt and Gelhar, 1992;Neuman, 1993;Kabala and Sposito, 1994;Kappor and Gelhar, 1994;Rajaram and Gelhar, 1995;Kapoor and Kitanidis, 1997;Guadagnini and Neuman, 1999a;Neuweiler et al, 2001;Cirpka andAttinger, 2003, Attinger et al, 2004;Morales-Casique et al, 2006;Chang and Yeh, 2007;Schwede et al, 2008).…”