An upscaled Lattice Boltzmann Method (LBM) for flow simulations in heterogeneous porous media at the Darcy scale is proposed in this paper. In the Darcy-scale simulations, the Shan-Chen force model is used to simplify the algorithm. The proposed upscaled LBM uses coarser grids to represent the average effects of the fine-grid simulations. In the upscaled LBM, each coarse grid represents a subdomain of the fine-grid discretization and the effective permeability with the reduced-order models is proposed as we coarsen the grid. The effective permeability is computed using solutions of local problems (e.g., by performing local LBM simulations on the fine grids using the original permeability distribution) and used on the coarse grids in the upscaled simulations. The upscaled LBM that can reduce the computational cost of existing LBM and transfer the information between different scales is implemented. The results of coarse-grid, reduced-order, simulations agree very well with averaged results obtained using a fine grid.
The formation of porous weathering rinds (layers of chemical alteration) on the exterior of rocks is a consequence of dissolution and precipitation of minerals occurring at the mineral–fluid interface within the pores. The speed at which the developed rind advances is controlled by both kinetic reaction rates and the transport of reaction products away from the pore spaces into the outside fluid. We show, using both reaction‐diffusion theory and numerics, that under diffusion limitations, the weathering rate depends on the size and curvature of the sample. This leads to a relationship between rind thickness, δ, and age, t. As the rind thickens, the result in three dimensions differs substantially from the one‐dimensional result of δ˜t. We describe the conditions under which the one‐dimensional and diffusion‐limited approximations apply and how they evolve as the rock weathers. Under chemical kinetic limitations, the rind advances at a constant rate, dδ/dt = v. We defend the application of a spherical approximation to irregular non‐spherical rocks and apply our results to field observations reported in the literature to show consistency with established methods. Finally, we argue that the variability in size, as well as in mineralogy, over ensembles of grains contributes to heterogeneous weathering rates. We demonstrate that this heterogeneity can contribute to the aging, or gradual decrease with time, of weathering rates previously observed in laboratory and field measurements. The formation of porous weathering rinds on the exterior of rocks is a consequence of mineral dissolution and precipitation occurring at the solid–fluid interface within the pores. We describe conditions under which reaction and diffusion‐limited scenarios apply and how, under those conditions, the rind thickness varies with time and rock shape. Furthermore, we argue that variability in weathering over ensembles of grains contributes to the gradual decrease over time of total weathering rates, as previously reported.