Non-Fickian mixing: Temporal evolution of the scalar dissipation rate in heterogeneous porous media

Abstract

We investigate the temporal scaling properties of mixing in heterogeneous permeability fields with variances ranging from very small ( σ ln K 2 = 0.01 ) to very large ( σ ln K 2 = 9 ) . We quantify mixing by the scalar dissipation rate, which we estimate over a large range of temporal scales. For an initial pulse line injection, we find that moderate and strong heterogeneity induce anomalous temporal scaling of the scalar dissipation rate, which we call non-Fickian mixing. This effect is particularly relevant for upscaling reactive transport as it implies a non-Fickian scaling of reactive transport. Although spreading and mixing are intimately coupled, we find that their scaling properties are not directly related in general. In the non-Fickian mixing regime, the temporal scaling of the scalar dissipation rate depends on the complex spatial distribution of the concentration field that generates transverse mixing. For times larger than the characteristic diffusion time associated with one permeability field correlation length, the heterogeneity of concentration in the plume is attenuated and progressively erased by diffusion. Thus, at large times, the temporal scaling of mixing and spreading can be related through a simple analytical expression.