Stochastic flux‐related analysis of transverse mixing in two‐dimensional heterogeneous porous media

Abstract

[1] Transverse mixing of solutes in steady state transport is of utmost importance for assessing mixing-controlled reactions of compounds that are continuously introduced into the subsurface. Classical spatial moments analysis fails to describe mixing because the tortuous streamlines in heterogeneous formations cause plume meandering, squeezing, and stretching, which affect transverse spatial moments even if there is no mass transfer perpendicular to the direction of flow. For transverse solute mixing, however, the decisive process is the exchange of solute mass between adjacent stream tubes. We therefore reformulate the advection-dispersion equation in streamline coordinates (i.e., in terms of the potential and the stream function values) and analyze how flux-related second central moments of plumes increase with dropping hydraulic potential. We compare the ensemble behavior of these second central moments in random two-dimensional heterogeneous flow fields with the moments in an equivalent homogeneous system, thus defining an equivalent effective transverse dispersion coefficient. Unlike transverse macrodispersion coefficients derived by traditional moment analysis, our mixing-relevant, flux-related coefficient does not increase with travel distance. We present closed-form solutions for the mean enhancement of transverse mixing by heterogeneity in two-dimensional isotropic media for linear laws of local-scale transverse dispersion. The mixing enhancement factor increases with the log conductivity variance but remains fairly low. We also evaluate the variance of our cumulative measure of transverse mixing, showing that heterogeneity causes substantial uncertainty of mixing. The analytical expressions are compared to numerical Monte Carlo simulations for various values of log conductivity variance, indicating good agreement with the analytical results at low variability. In the numerical simulations, we also consider nonlinear models of local-scale transverse dispersion.

A modelling investigation of solute transport in permeable porous media containing a discrete preferential flow feature

“…Dense flow lines enhance transverse dispersion (e.g. Werth et al, 2006;Cirpka et al, 2011 ), and the travel distance within the PFF also increases for these cases. Solutes that disperse to the left of the main plume trajectory have a shorter path to exit the PFF and thus, are transversely dispersed less than those that disperse to the right.…”

Stochastic evaluation of mixing-controlled steady-state plume lengths in two-dimensional heterogeneous domains

“…Also, we will keep several important terms expressed in the original coordinate x 1 rather than in φ. This corresponds to the approximation Δx 1 ≈−JΔφ already used in Cirpka et al (2011b).…”

“…For conservative transport in two-dimensional heterogeneous media, Cirpka et al (2011b) reformulated the advectiondispersion equation in streamline coordinates and expressed effective transverse mixing in terms of transverse spatial moments in these coordinates, thus removing the effects of streamline meandering. They performed a rigorous first-order analysis of effective transverse mixing for two-dimensional heterogeneous media with second-order stationary, statistically isotropic, multi-Gaussian log-hydraulic conductivity fields, assuming uniform-in-the-mean flow.…”

“…Our results are limited to steady-state flow and transport conditions in two dimensions. To achieve our goals, we follow the approach by Cirpka et al (2011b) to transform from Cartesian coordinates to streamline coordinates. This removes the effects of streamline meandering.…”

Impact of the spatial structure of the hydraulic conductivity field on vorticity in three-dimensional flows

“…For example, it has been shown that the occurrence of flow focusing (e.g. compression of fluid material elements) in porous formations enhances mass transfer [12][13][14][15].…”