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We consider a phase-field model for the incompressible flow of two immisciblefluids. This model extends widespread models for two fluid phases by includinga third, solid phase, which can evolve due to e.g. precipitation anddissolution. We consider a simple, two-dimensional geometry of a thin strip,which can still be seen as the representation of a single pore throat in aporous medium. Under moderate assumptions on the Peclet number and thecapillary number, we investigate the limit case when the ratio between thewidth and the length of the strip is going to zero. In this way and employingtransversal averaging, we derive an upscaled model. The result is a multi-scalemodel consisting of the upscaled equations for the total flux and the iontransport, while the phase-field equation has to be solved in cell-problems atthe pore scale to determine the position of interfaces. We also investigate thesharp-interface limit of the multi-scale model, in which the phase-fieldparameter approaches 0. The resulting sharp-interface model consists only ofDarcy-scale equations, as the cell-problems can be solved explicitly. Notablywe find asymptotic consistency, that is the upscaling process and thesharp-interface limit commute. We use numerical results to investigate thevalidity of the upscaling when discontinuities are formed in the upscaledmodel.

Upscaling of a Cahn-Hilliard Navier-Stokes Model with Precipitation and Dissolution in a Thin Strip