Upscaling of a tri-phase phase-field model for precipitation in porous media

Abstract

We consider a porous medium with a pore space that is completely filled by three different phases: two immiscible fluids (say water and oil) and a solid phase. One fluid phase contains dissolved ions, which can precipitate at the pore boundary to form the solid phase. The reverse process of dissolution, is also possible. Consequently, the solid phase changes in time; its variation is not known a priori. The second fluid contains no solute and has no interaction with the solid phase. Starting from a standard sharp interface model for the pore-scale dynamics we develop a diffuse interface approach that accounts for the time-dependent spatial distribution of the three species and the overall concentration of the solute. Basic analytical results for this model are presented, including the well posedness of the phase field component of the model. Next we apply matched asymptotic techniques to show that the diffuse interface model converges to the sharp interface one. Further, homogenization is applied to derive a new two-scale model that is valid at the Darcy scale. This leads to a parabolic reaction–diffusion system in a medium with variable, concentration dependent porosity. The diffuse interface approach allows describing the variation in the porosity as phase field type equations at the pore-scale. The last part of the article presents an efficient numerical scheme to approximate the solution of the two-scale model. This scheme has as starting point the algorithm in (Redeker, M. & Eck, C. (2013) A fast and accurate adaptive solution strategy for two-scale models with continuous inter-scale dependencies. J. Comput. Phys., 240, 268–283.). After some test cases validating the method, we finally present computations for several realistic scenarios. The results demonstrate the interdependence of the change of the pore structure due to precipitation/dissolution and the evolution of the averaged, Darcy scale concentration of the dissolved ions in the one fluid.