Mathematical analysis and homogenization of a free boundary problem for in situ leaching of rare earths with a special periodic structure

We consider an initial–boundary value problem modeling in situ leaching of rare earths with a special periodic structure by an acid solution, improving some previous studies by the second author and collaborators. At the microscopic scale, fluid motion in the pore space is described by the Stokes equations for a slightly compressible fluid coupled with the deformation of the elastic skeleton, governed by the Lamé system, and the diffusion equation for the acid solution. Due to rock dissolution, the interface between liquid and solid phases is unknown (it is a free boundary) and must be determined as part of the solution. To overcome this difficulty, we introduce a family of approximate microscopic models with prescribed pore geometry and establish their well-posedness in a weak formulation. Using a priori estimates and Galerkin’s method, we obtain existence results and apply the method of two-scale convergence for periodic structures to derive the corresponding homogenized macroscopic model. Finally, a fixed-point argument yields existence and uniqueness for the resulting macroscopic system.