Time evolution of mixing in heterogeneous porous media

Mixing in heterogeneous media results from the competition between velocity fluctuations and local scale diffusion. Velocity fluctuations create a potential for mixing by generating disorder and large interfacial areas between resident and invading waters. Local scale diffusion smoothes out the disorder while transforming this potential into effective mixing. The effective mixing state is quantified by the integral of concentration squared over the spatial domain. Because it emerges from dispersion, the potential mixing is defined as the mixing state of a Gaussian plume that has the same longitudinal dispersion as the real plume. The difference gamma between effective and potential mixing normalized by the latter traduces the lag of diffusion to homogenize the concentration structure generated by the dispersion processes. This new decomposition of effective mixing into potential mixing and departure rate gamma makes a full use of dispersion for quantifying mixing and restricts the analysis of mixing to gamma. For cases where the mean concentration can be assumed Gaussian, we use the concentration variance equation of Kapoor and Gelhar (1994) to show that gamma depends solely on the macrodispersion coefficient (spreading rate) and the recently developed mixing scale defined as the smallest scale over which concentration can be considered uniform, and which quantifies the internal plume disorder. We use numerical simulations to show that gamma turns out to follow a simple scaling form that depends on neither the heterogeneity level or the Peclet number. A very similar scaling form is recovered for Taylor dispersion. Both derivations of gamma reinforce its relevance to characterize mixing. This generic characterization of mixing can offer new ways to set up transport equations that honor not only advection and spreading but also mixing.