  We consider a phase-field model for the incompressible flow of two immiscible
fluids. This model extends widespread models for two fluid phases by including
a third, solid phase, which can evolve due to e.g. precipitation and
dissolution. 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 a
porous medium. Under moderate assumptions on the Peclet number and the
capillary number, we investigate the limit case when the ratio between the
width and the length of the strip is going to zero. In this way and employing
transversal averaging, we derive an upscaled model. The result is a multi-scale
model consisting of the upscaled equations for the total flux and the ion
transport, while the phase-field equation has to be solved in cell-problems at
the pore scale to determine the position of interfaces. We also investigate the
sharp-interface limit of the multi-scale model, in which the phase-field
parameter approaches 0. The resulting sharp-interface model consists only of
Darcy-scale equations, as the cell-problems can be solved explicitly. Notably
we find asymptotic consistency, that is the upscaling process and the
sharp-interface limit commute. We use numerical results to investigate the
validity of the upscaling when discontinuities are formed in the upscaled
model.
