Evolution of solute blobs in heterogeneous porous media

Abstract

We study the mixing dynamics of solute blobs in the flow through saturated heterogeneous porous media. As the solute plume is advected through a heterogeneous porous medium it suffers a series of deformations that determine its mixing with the ambient fluid through diffusion. Key questions are the relation between the spatial disorder and the mixing dynamics and the effect of the initial solute distribution. To address these questions, we formulate the advection–diffusion problem in a coordinate system that moves and rotates along streamlines of the steady flow field. The impact of the medium heterogeneity is quantified systematically within a stochastic modelling approach. For a simple shear flow, the maximum concentration of a blob decays asymptotically as$t^{-2}$. For heterogeneous porous media, the mixing of the solute blob is determined by the random sampling of flow and deformation heterogeneity along trajectories, a mechanism different from persistent shear. We derive explicit perturbation theory expressions for stretching-enhanced solute mixing that relate the medium structure and mixing behaviour. The solution is valid for moderate heterogeneity. The random sampling of shear along trajectories leads to a$t^{-3/2}$decay of the maximum concentration as opposed to an equivalent homogeneous medium, for which it decays as$t^{-1}$.