Filtration of two immiscible incompressible fluids

The paper considers a mathematical model of the filtration of two immiscible incompressible fluids in deformable porous media. This model is a generalization of the Musket—Leverett model, in which porosity is a function of the space coordinates. The model under study is based on the equations of conservation of mass of liquids and porous skeleton, Darcy's law for liquids, accounting for the motion of the porous skeleton, Laplace's formula for capillary pressure, and a Maxwell-type rheological equation for porosity and the equilibrium condition of the “system as a whole”. In the thin layer approximation, the original problem is reduced to the successive determination of the porosity of the solid skeleton and its speed, and then the elliptic-parabolic system for the “reduced” pressure and saturation of the fluid phase is derived. In view 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 the maximum principle, construction of Galerkin approximations, and passage to the limit in terms of the regularization parameters based on the compensated compactness principle.