A Novel Lagrangian Approach for Simulating Solute Mixing in Heterogeneous Porous Media

Solute mixing in porous media is the result of transport processes such as local-scale dispersion, which acts at the scale of pore throats; to variations in Darcy or macro-scale velocity, which produce solute spreading at the macro-scale. Variations in macro-scale velocity may enhance the action of local-scale dispersion resulting in higher effective mixing rates. The modeling of these processes that occur at disparate scales represents a formidable challenge for traditional mathematical models based on the advection-dispersion equation with a single dispersion coefficient. We present a new multiscale Lagrangian approach to model mixing in numerical simulations that overcomes some of the issues with traditional models. The proposed approach is based on the use of a Lagrangian particle method. Particles carry solute mass and their locations evolve according to a velocity field, which includes a deterministic component given by the grid-scale velocity and a stochastic component that corresponds to a block-effective macro-dispersion coefficient. Mass transfer between particles due to local-scale dispersion is approximated by a meshless SPH method. Thus, mixing mechanisms that occur at different scales are represented by two different dispersion coefficients. This makes possible to study separately the effect of local-scale dispersion and solute spreading on solute mixing and reaction rates. We demonstrate some of the advantages of the proposed scheme in a set of benchmark simulations that consider the transport of two chemical species that undergo a bimolecular chemical reaction. Concentration of procuct C of bimoleacular reaction A+B->C, controlled by mixing. Top: Origiinal velocity field (left) and filtered coarser velocity field (right). Botom: Irregular solute plume due to velocity fluctuations for original velocity field (left), filtered velocity (right).