Efficiency and Accuracy of Micro-Macro Models for Mineral Dissolution

Micro-macro models for dissolution processes are derived from detailed pore-scale models applying upscaling techniques. They consist of flow and transport equations at the scale of the porous medium (macroscale). Both include averaged time- and space-dependent coefficient functions (permeability, porosity, reactive surface, and effective diffusion). These are in turn explicitly computed from the time- and space-dependent geometry of unit cells and by means of auxiliary cell problems defined therein (microscale). The explicit geometric structure is characterized by a level set. For its evolution, information from the transport equations solutions is taken into account (micro-macro scales). A numerical scheme is introduced, which is capable of evaluating such complex settings. For the level-set equation a second-order scheme is applied, which enables us to accurately determine the dynamic reactive surface. Local mesh refinement methods are applied to evaluate Stokes type cell problems using P-2/P-1 elements and a Uzawa type linear solver. Applications of our permeability solver to scenarios involving static and evolving geometries are presented. Furthermore, macroscopic flow and transport equations are solved applying mixed finite elements. Finally, adaptive strategies to overcome the computational burden are discussed. We apply our approach to the dissolution of an array of dolomite grains in the micro-macro context and validate our numerical scheme.