Filtration of Two Immiscible Incompressible Fluids in a Thin Poroelastic Layer

The paper considers a mathematical model of the filtration of two immiscibleincompressible fluids in deformable porous media. This model is a generalization of theMusket–Leverett model, in which porosity is a function of the space coordinates. The model understudy is based on the equations of conservation of mass of liquids and porous skeleton, Darcy’s lawfor liquids, accounting for the motion of the porous skeleton, Laplace’s formula for capillarypressure, and a Maxwell-type rheological equation for porosity and the equilibrium condition ofthe “system as a whole.” In the thin layer approximation, the original problem is reduced to thesuccessive determination of the porosity of the solid skeleton and its speed, and then theelliptic-parabolic system for the “reduced” pressure and saturation of the fluid phase is derived. Inview of the degeneracy of equations on the solution, the solution is understood in a weak sense.The proofs of the results are carried out in four stages: regularization of the problem, proof of themaximum principle, construction of Galerkin approximations, and passage to the limit in terms ofthe regularization parameters based on the compensated compactness principle.