Homogenization of a mineral dissolution and precipitation model involving free boundaries at the micro scale

Abstract

In this work we present the homogenization of a reaction-diffusion model that includes an evolving microstructure. Such type of problems model, for example, mineral dissolution and precipitation in a porous medium. In the initial state, the microscopic geometry is a periodically perforated domain, each perforation being a spherical solid grains. A small parameter ϵ is characterizing both the distance between two neighboring grains, and the radii of the grains. For each grain, the radius depends on the unknown (the solute concentration) at its surface. Therefore, the radii of the grains change in time and are model unknowns, so the model involves free boundaries at the micro scale. In a first step, we transform the evolving micro domain to a fixed, periodically domain. Using the Rothe-method, we prove the existence of a weak solution and obtain a priori estimates that are uniform with respect to ϵ. Finally, letting ϵ to 0, we derive a macroscopic model, the solution of which approximates the micro-scale solution. For this, we use the method of two-scale convergence, and obtain strong compactness results enabling to pass to the limit in the nonlinear terms.