Analysis of the influence of averaged positive second invariant Q_(av) of deformation tensor ▽u on the maximum dilution index E_(max) in steady Darcy flows through isotropic heterogeneous porous media

摘要

Characterization of flow topology is essential to understand the effects of the heterogeneity and dimensionality of geological formations on the mixing of inert solute clouds in these same geological formations. In this work, we numerically study two indicators of flow topology, the averaged vorticity magnitude vertical bar omega vertical bar(av) and the averaged positive second invariant Q(av) of the deformation tensor del u, in steady Darcy flows through exponentially correlated lognormal hydraulic conductivity fields K. Our numerical results allow us to establish the relationships between the two indicators considered here and the hydraulic conductivity variance sigma(2) in 2D and 3D, highlighting the role played by the spatial structure of these porous media on flow topology, and indirectly on mixing. This work leads us to assess the maximum dilution index E-max, indicator of mixing, theoretically known to increase monotonically in steady Darcy flows through isotropic heterogeneous porous media. Our numerical results allow us to test this hypothesis by establishing the relationship between the slope a of maximum dilution index E-max and the averaged positive second invariant Q(av) of deformation tensor del u. The parameters of this relationship depend on molecular diffusion and dimensionality of problem considered.