Existence of a Weak Solutionto the Two-Dimensional Filtration Problem in a Thin Poroelastic Layer

Abstract

The paper considers a mathematical model of the joint motion of two immiscible incompressible fluids in a poroelastic medium. This model is a generalization of the classical Musket-Leverett model, in which porosity is considered to be a given function of the spatial coordinate. The model under study is based on the mass conservation equations for liquids and the porous skeleton, Darcy's law for liquids, which takes into account the movement of the porous skeleton, the Laplace formula for capillary pressure, the Maxwell-type rheological equation for porosity, and the "system as a whole" equilibrium condition. In the thin layer approximation, the original problem is reduced to the successive determination of the porosity of the solid skeleton and its velocity. Then an elliptic-parabolic system is derived for the “reduced pressure” and saturation of the wetting phase. Its solution is understood in a generalized sense due to the degeneration on the solution of the equations of the system. The proof of the existence theorem is carried out in four stages: regularization of the problem, proof of the physical maximum principle for saturation, construction of Galerkin approximations, passage to the limit in regularization parameters based on the method of compensated compactness.