Eulerian spatial moments for solute transport in three-dimensional heterogeneous, dual-permeability media

Abstract

A Eulerian analytical method is developed for nonreactive solute transport in heterogeneous, dualpermeability media where the hydraulic conductivities in fracture and matrix domains are both assumed to be stochastic processes. The analytical solution for the mean concentration is given explicitly in Fourier and Laplace transforms. Instead of using the fast fourier transform method to numerically invert the solution to real space (Hu et al., 2002), we apply the general relationship between spatial moments and concentration (Naff, 1990;Hu et al., 1997) to obtain the analytical solutions for the spatial moments up to the second for a pulse input of the solute. Owing to its accuracy and efficiency, the analytical method can be used to check the semi-analytical and Monte Carlo numerical methods before they are applied to more complicated studies. The analytical method can be also used during screening studies to identify the most significant transport parameters for further analysis. In this study, the analytical results have been compared with those obtained from the semi-analytical method (Hu et al., 2002) and the comparison shows that the semi-analytical method is robust. It is clearly shown from the analytical solution that the three factors, local dispersion, conductivity variation in each domain and velocity convection flow difference in the two domains, play different roles on the solute plume spreading in longitudinal and transverse directions. The calculation results also indicate that when the log-conductivity variance in matrix is 10 times less than its counterpart in fractures, it will hardly influence the solute transport, whether the conductivity field is matrix is treated as a homogeneous or random field.

Stochastic analysis of contaminant transport through nonstationary fractured porous media: A dual‐permeability approach

“…This method is also compared with the stationary transport theory developed by Xu and Hu [2004a] for predicting spatial moments of mean concentration. The comparison indicated that the two methods matched very well in predicting first and second spatial moments of mean concentration.…”

“…To compare solute transport in stationary flow fields with the previously used stationary transport theory, we compare the spatial moments of mean concentration calculated by NEMM with results obtained from the analytical method [ Xu and Hu , 2004a]. After conducting this comparison, we analyze the influence of several parameters on concentration moments under stationary cases.…”

“… Xu and Hu [2004a] developed a Eulerian analytical method for nonreactive solute transport in heterogeneous, dual‐permeability media where the hydraulic conductivities in fracture and matrix domains are assumed to be stochastic processes. Owing to its accuracy and efficiency, the analytical method can be used to check NEMM before it is applied to more complicated cases.…”

“…In their study, solute transport was analyzed in three‐dimensional space, but numerical calculation was limited to the two‐dimensional case. Xu and Hu [2004a] developed an analytical solution for three‐dimensional solute transport in heterogeneous, dual‐permeability media. This analytical method is first used to investigate the accuracy of the semianalytical results of Hu et al [2002], and then to study the dimensional effect on solute transport by comparing two‐ and three‐dimensional results of the spatial moments.…”

Correspondence Between One- and Two-Equation Models for Solute Transport in Two-Region Heterogeneous Porous Media

“…Moments are mathematical tools that can be used to analyze the shape of spatial/temporal signals. They have been used for a variety of purposes, including studies of solute breakthrough curves in soils, e.g., in Stagnitti et al (2000); the behavior of fractional advection-diffusion equations (see Zhang 2010); mixing in heterogeneous porous media (see Chiogna et al 2011); model parameter estimations from column experiments, e.g., in Young and Ball (2000); and analytical solutions in dual-permeability media in Xu and Hu (2004).…”