Correctness of Biot's model of in situ leaching for incompressible liquid and compressible solid components

TL;DR

This paper analyzes a mathematical model of in situ leaching for rare metals, establishing the existence and uniqueness of solutions by applying homogenization, fixed point theorem, and Lipschitz continuity of an operator.

Contribution

It introduces a new homogenized macroscopic model for in situ leaching with a fixed point approach to handle free boundary problems.

Findings

Proves the operator $ extbf{F}$ is Lipschitz continuous.

Establishes the existence of a unique fixed point for the model.

Derives a macroscopic model with a linear boundary velocity condition.

Abstract

We study a mathematical model of in situ leaching of rare metals, in which the joint filtration of two liquids is governed by the microscopic model $\mathbb{A}^{\varepsilon}$. A key difficulty is the unknown (free) boundary $\Gamma(r)$ between solid and liquid components, determined by an additional condition on $\Gamma(r)$; no standard methods exist for this nonlinear problem. To resolve it, we apply the fixed point theorem. For a given function $r(\boldsymbol{x},t)$ from a set $\mathfrak{M}_{(0,T)}$ of sufficiently smooth functions describing the skeleton structure, we consider the auxiliary problem $\mathbb{B}^{\varepsilon}(r)$: an elliptic system for displacements of the liquid and solid components coupled with parabolic equations for the acid concentration. Selecting the weak solution of minimal smoothness, we apply the homogenization method to pass from the microscopic to the macroscopic description. The resulting macroscopic model $\mathbb{H}(r)$ contains a homogenized boundary condition that expresses the normal boundary velocity $V_{N}=\partial r/\partial t$ as a linear function of the acid concentration $c$. Since $c$ depends on $r$ via an operator $\mathbb{F}\colon\mathfrak{M}_{(0,T)}\to\mathfrak{M}_{(0,T)}$, we prove that $\mathbb{F}$ is Lipschitz continuous and, by Banach's theorem, possesses a unique fixed point $r^{*}$, which yields the unique solution $\mathbb{H}=\mathbb{H}(r^{*})$.