Some free boundary problem arising in rock mechanics

Abstract

An initial boundary value problem for the in situ leaching is considered. We describe physical processes at the microscopic level with a pore size ε ≪ 1 by model A ε , where dynamics of the incompressible solid skeleton is described by the Lamé equations and the physical process in the pore space by the Stokes equations for the incompressible fluid with diffusion equations for the concentration of acid and product of chemical reactions. Since the solid skeleton changes its geometry upon dissolution, the “pore space – solid skeleton” boundary is a free boundary. The goal of the present manuscript is a model H, which is the homogenization of the model A ε . That is, the limit as ε tend to zero, of the model A ε . As usual, free boundary problems are only solvable locally in time. On the other hand, in situ leaching has a very long process duration and there is still no correct microscopic model that describes this process for an arbitrary time interval. To avoid this contradiction, we propose correct approximate microscopic models B ε (r) for this process with a given solid skeleton structure depending on some function r from the set M(0,T ) . Problem B ε (r) is the model A ε without an additional boundary condition at the free boundary that defines this boundary, but with some additional terms in the Stokes and Lame equations that depend linearly on the velocities and disappear upon homogenization. To derive a macroscopic mathematical model H(r) and separately the additional boundary condition at free boundary we use Nguenseng’s two-scale convergence method as ε tends to zero. As a result, we obtain a homogenized model H(r) and an additional equation, possesses construct an operator, which fixed point uniquely defines function r ∗ from the set M(0,T ) and prove the existence and uniqueness theorem for the macroscopic mathematical model H.