Multiscale modelling of solute transport through porous media using homogenization and splitting methods

The aim of this paper is to treat multiscale modelling approaches for solute transport through porous media. This involves coupled systems of convection–diffusion–reaction equations that can be homogenized and solved by splitting methods. The topic is very challenging in the case of multiscale model systems, for example, those arising when the evolution of several chemical species involves time-dependent or non-linear mechanisms. Our proposed modelling approach is based on the idea of homogenization of the diffusion, convection and reaction of chemical species in a porous medium to derive macroscopic equations. Based on the existence of multiple timescales, we introduce multiscale methods to model the evolution and obtain solutions. A more detailed analysis shows that such multiscale methods can be treated via the so-called iterative splitting approach. To solve the multiscale model, we propose exact solutions of some submodels, which can be then be taken into account and play an important role in accelerating the numerical computations of the large coupled model. In the first part, we introduce the model and its application. In the second part, we discuss the analytical solutions of the submodels related to fast and analytically solvable convection–reaction equations. The iterative splitting approaches are then discussed for solving the multiple timescale part. Finally, the last part presents some numerical experiments involving real-life test problems in transport–reaction processes.