Electron. J. Diff. Equ.,
Vol. 2016 (2016), No. 52, pp. 1-28.
Homogenization of immiscible compressible two-phase
flow in double porosity media
Latifa Ait Mahiout, Brahim Amaziane, Abdelhafid Mokrane, Leonid Pankratov
Abstract:
A double porosity model of multidimensional immiscible compressible
two-phase flow in fractured reservoirs is derived by the mathematical
theory of homogenization. Special attention is paid to developing a general
approach to incorporating compressibility of both phases. The model is written
in terms of the phase formulation, i.e. the saturation of one phase and the
pressure of the second phase are primary unknowns.
This formulation leads to a coupled system consisting of a doubly nonlinear
degenerate parabolic equation for the pressure and a doubly nonlinear degenerate
parabolic diffusion-convection equation for the saturation, subject to appropriate
boundary and initial conditions. The major difficulties related to this model are
in the doubly nonlinear degenerate structure of the equations, as well as in the
coupling in the system. Furthermore, a new nonlinearity appears in the temporal
term of the saturation equation.
The aim of this paper is to extend the results of [9] to this more
general case. With the help of a new compactness result and uniform a priori
bounds for the modulus of continuity with respect to the space and time variables,
we provide a rigorous mathematical derivation of the upscaled model by means of
the two-scale convergence and the dilatation technique.
Submitted February 2, 2016. Published February 18, 2016.
Math Subject Classifications: 35B27, 35K65, 76S05, 76T10.
Key Words: Compressible immiscible; double porous media; two-phase flow;
fractured media homogenization; two-scale convergence.
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Latifa Ait Mahiout
LEDP, École Normale Supérieure de Kouba
BP 92, 16050, Vieux Kouba
Alger, Algeria
email: latifaaitmahiout@gmail.com
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Brahim Amaziane
Laboratoire de Mathématiques et de leurs Applications
CNRS-UMR 5142, Université de Pau
Av. de l'Université, 64000 Pau, France
email: brahim.amaziane@univ-pau.fr
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Abdelhafid Mokrane
LEDP, École Normale Supérieure de Kouba
BP 92, 16050, Vieux Kouba
Alger, Algeria
email: abdelhafid.mokrane@gmail.com
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Leonid Pankratov
Laboratoire de Mathématiques et de leurs Applications
CNRS-UMR 5142, Université de Pau
Av. de l'Université, 64000 Pau, France
email: leonid.pankratov@univ-pau.fr
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