A numerical scheme for two-scale phase-field models in porous media

A porous medium is a highly complex domain, in which various processes can take place at different scales. Examples in this sense are the multi-phase flow and reactive transport. Here, due to processes like dissolution or precipitation, or chemical deposition, which are encountered at the scale of pores (the micro-scale), the local structure and geometry of the pores may change, impacting the fluid flow. Since these micro-scale processes depend on the model unknowns (e.g., the solute concentration), free boundaries are encountered, separating the space available for flow from the solid, impermeable part in the medium. Here we consider a phase-field approach to model the evolution of the evolving interfaces at the micro-scale. For mineral precipitation and dissolution, we have evolving fluid-solid interfaces. If considering multi-phase flow, evolving fluid-fluid interfaces are also present. After applying a formal homogenization procedure, a two-scale phase-field model is derived, describing the averaged behavior of the system at the Darcy scale (the macro-scale). In this two-scale model, the micro and the macro scale are coupled through the calculation of the effective parameters. Although the resulting two-scale model is less complex than the original, the numerical strategies based on the homogenization theory remain computationally expensive as they require the computation of several problems over different scales, and in each mesh element. Here, we propose an adaptive two-scale scheme involving different techniques to reduce the computational effort without affecting the accuracy of the simulations. These strategies include iterations between scales, an adaptive selection of the elements wherein effective parameters are computed, adaptive mesh refinement, and efficient non-linear solvers.