Lagrangian scheme to model subgrid-scale mixing and spreading in heterogeneous porous media

Small-scale heterogeneity of permeability controls spreading, dilution, and mixing of solute plumes at large scale. However, conventional numerical simulations of solute transport are unable to resolve scales of heterogeneity below the grid scale. We propose a Lagrangian numerical approach to implement closure models to account for subgrid-scale spreading and mixing in Darcy-scale numerical simulations of solute transport in mildly heterogeneous porous media. The novelty of the proposed approach is that it considers two different dispersion coefficients to account for advective spreading mechanisms and local-scale dispersion. Using results of benchmark numerical simulations, we demonstrate that the proposed approach is able to model subgrid-scale spreading and mixing provided there is a correct choice of block-scale dispersion coefficient. We also demonstrate that for short travel times it is only possible to account for spreading or mixing using a single block-scale dispersion coefficient. Moreover, we show that it is necessary to use time-dependent dispersion coefficients to obtain correct mixing rates. On the contrary, for travel times that are large in comparison to the typical dispersive time scale, it is possible to use a single expression to compute the block-dispersion coefficient, which is equal to the asymptotic limit of the block-scale macrodispersion coefficient proposed by Rubin et al. (1999). Our approach provides a flexible and efficient way to model subgrid-scale mixing in numerical models of large-scale solute transport in heterogeneous aquifers. We expect that these findings will help to better understand the applicability of the advection-dispersion-equation (ADE) to simulate solute transport at the Darcy scale in heterogeneous porous media.