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An existence result for nonisothermal immiscible incompressible 2‐phase flow in heterogeneous porous media
Author(s) -
Amaziane Brahim,
Jurak Mladen,
Pankratov Leonid,
Piatnitski Andrey
Publication year - 2017
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4544
Subject(s) - mathematics , porous medium , nonlinear system , capillary pressure , degenerate energy levels , discretization , conservation of mass , compressibility , mathematical analysis , two phase flow , conservation law , flow (mathematics) , mechanics , porosity , physics , materials science , geometry , quantum mechanics , composite material
In the present article, we study the temperature effects on two‐phase immiscible incompressible flow through a porous medium. The mathematical model is given by a coupled system of 2‐phase flow equations and an energy balance equation. The model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy‐Muskat and the capillary pressure laws. The problem is written in terms of the phase formulation; ie, the saturation of one phase, the pressure of the second phase, and the temperature are primary unknowns. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we show the existence of weak solutions with the help of an appropriate regularization and a time discretization. We use suitable test functions to obtain a priori estimates. We prove a new compactness result to pass to the limit in nonlinear terms.