On the Solvability of the Problem of Fluid Filtration in a Deformable Porous Medium in the Gravitational Field
The paper deals with the model of filtration of a viscous compressible fluid in a deformable medium, which has predominantly viscous properties with respect to elastic media. In contrast to the earlier works devoted to the substantiation of this model, the present paper gives a justification for a model that takes into account the influence of gravity. A theorem on local solvability of the problem in the field of gravity is proved. In § 1 we give a brief statement of the problem and formulate the main result of the paper. The initial system of equations describing the process consists of the equations of mass conservation for the solid and liquid phases, the law of conservation of momentum for the liquid, which is taken in the form of Darcy’s law and takes into account the motion of the skeleton, the conservation of the momentum of the system as a whole, and the equation linking the effective pressure and porosity, which determines the rheology. After the transition to the Lagrange variables, this system reduces to two equations for finding the porosity functions and the density of the liquid phase. In § 2 we give a proof of the theorem for the system obtained, and we also establish the physical maximum principle for the porosity and density functions of the liquid phase. The proof of the theorem is based on the Tikhonov-Schauder theorem on a fixed point. In paragraph 3 we give a generalization to the case of a complete equation of the balance of forces.
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