Non-reactive transport of solutes in porous media can be modeled (at least for low to moderate degrees of sub-scale heterogeneities) by the classical advection-dispersion-equation (ADE), with an upscaled dispersion coefficient that accounts for the local variability of sub-scale velocities.
However, when chemical reactions are incorporated, reaction rates tend to be over-predicted. This is mainly due to the inability of the upscaled dispersion term in the ADE to treat spreading and mixing separately. That is, the implicit assumption of full-mixing at all scales below the scale of interest is often erroneous. A suitable model should be able to account for the local anti-correlated reactant concentration fluctuations that are generated during the macroscopic transport.
What we propose is a particle-based method to simulate transport of solutes with mixing limitation at the local scale. Besides solute mass, particles also carry a local disequilibrium with respect to the macroscopic (averaged) concentration. The disequilibrium is naturally generated by the random displacements of particles, and relaxed by local mixing. Thus the local concentrations, which ultimately control the reactions, are not defined in the Eulerian space, but on particles instead. On the other hand, the averaged concentrations are well defined in space and controlled by the ADE. We show that the temporal evolution of the local concentration covariance in the proposed model is similar to Kapoor and Gelhar’s (1994) "concentration variance conservation equation". Then, we study the mixing state evolution for two simple cases of initial and boundary conditions, showing that the model reproduces the typical features and temporal scaling of physical systems. The proposed Lagrangian model is implemented to reproduce the laboratory experiment of Gramling et al. (2002). Our numerical results show close agreement with the experimental data for physically meaningful values of the parameters. Finally, we present the Computational Fluid Dynamics simulations that will be performed in order to validate the proposed model and to study the dependence of the model parameters on (i) the local Péclet number and (ii) the pore geometry.