Definition of a mixing scale for transport in heterogeneous media

International audienceMixing in heterogeneous flows depends on the interplay between the generation of gradients by velocity fluctuations and their dissipation by diffusion. One of the key questions concerns the range of scales for which a scalar can be considered well mixed. The spreading scale σ(t) quantifies the global size of the transported plume. Since the heterogeneity induces incomplete mixing inside the plume, this scale is not relevant for quantifying mixing. Yet, no alternative scale have been defined for this purpose. Therefore, we define the mixing scale ε(t) as the upper limit for which the concentration distribution is well mixed. We discuss different methods to estimate ε from a concentration field and study numerically its temporal evolution in heterogeneous porous media. For homogeneous flow fields, ε^2(t) evolves linearly in time and ε(t) is equal to the characteristic plume size σ(t). For highly heterogeneous flows, this behaviour breaks down and ε(t) is much smaller than σ(t). We find that the mixing scale ε(t) evolves subdiffusively over a large range of times. We attribute this behaviour to the wide distribution of velocity gradients, which implies that the competition between diffusion and the velocity field shear action is non-local. Since the characteristic plume size σ(t) evolves superdiffusively, the mixing and spreading scales diverge, implying the persistence of incomplete mixing. This result is of particular importance for upscaling reactive transport, for which the existence of such a mixing scale generally needs to be assumed