Dispersion variance for transport in heterogeneous porous media

Abstract

[1] We study dispersion in heterogeneous porous media for solutes evolving from point-like and extended source distributions in d ¼ 2 and d ¼ 3 spatial dimensions. The impact of heterogeneity on the dispersion behavior is captured by a stochastic modeling approach that represents the spatially fluctuating flow velocity as a spatial random field. We focus here on the sample-to-sample fluctuations of the dispersion coefficients about their ensemble mean. For finite source sizes, the definition of dispersion coefficients in single realizations is not unique. We consider dispersion measures that describe the extension of the solute distribution, as well as dispersion coefficients that quantify the solute spreading relative to injection points of the partial plumes that constitute the solute distribution. While the ensemble averages of these dispersion quantities may be identical, their fluctuation behavior is found to be different. Using a perturbation approach in the fluctuations of the random flow field, we derive explicit expressions for the temporal evolution of the variances of the dispersion coefficients between realizations. Their evolution is governed by the typical dispersion time over the characteristic heterogeneity scale and the dimensions of the source distribution. We find that the dispersion variance decreases toward zero with time in d ¼ 3 spatial dimensions, while in d ¼ 2 it converges toward a finite long time value that is independent of the source dimensions.Citation: Dentz, M., and F. P. J. de Barros (2013), Dispersion variance for transport in heterogeneous porous media, Water Resour.

Pictures of blockscale transport: Effective versus ensemble dispersion and its uncertainty

“…The fluctuation of the solute plume spreading about its mean value is a measure of heterogeneity-induced uncertainty [26,54,55]. However, as more and more of the flow heterogeneity is sampled by the solute plume, e.g., as the transport histories experienced by the scalar in different realizations, converge, the ensemble average is expected to become representative of dispersion in a single realization [35,36]. If the variance of the dispersion coefficient tends to zero with time, then the dispersion observable is self-averaging [56].…”

“…If the variance of the dispersion coefficient tends to zero with time, then the dispersion observable is self-averaging [56]. More details on self-averaging behavior of solute spreading in spatially heterogeneous porous media can be found in the literature [26,35,36,37,38,39,57,58].…”

“…The authors [35] developed expressions for the blockscale dispersion variance for a finite size solute body released in a stratified geological formation under pure advective transport and, through the minimum relative entropy principle, inferred a model for the probability density function of the blockscale dispersion tensor. Within the context of a fully homogenized aquifer system (e.g., infinite blockscale), Dentz and de Barros [36] derived expressions for the dispersion variance tensor in two-and three-dimensional statistically anisotropic random flow fields in the presence of local-scale dispersion. The evolution of the dispersion variance provides information on the mixing efficiency of the transport processes.…”

“…As the solute plume becomes better mixed, the dispersion variance decreases since the variability between realization reduces. The authors [36] showed that the self-averaging behavior of the solute plume strongly depends on the flow dimensionality and the geometrical properties of the injection source. Selfaveraging implies that the dispersion variance tends to zero for large times (e.g.…”

Ergodic properties of heterogeneous diffusion processes in a potential well

“…such as the Richardson diffusion in turbulence 20 , mesoscopic approaches to transport in heterogenous porous media 21,22 and on random fractals 23,24 . The heterogeneous dynamical behavior is particularly remarkable in biological systems.…”