An improved homogenization result for immiscible compressible two-phase flow in porous media

Brahim Amaziane; Leonid Pankratov; Andrey Piatnitski; Brahim Amaziane; Leonid Pankratov; Andrey Piatnitski

doi:10.3934/nhm.2017006

Networks and Heterogeneous Media

2017, Volume 12, Issue 1: 147-171. doi: 10.3934/nhm.2017006

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An improved homogenization result for immiscible compressible two-phase flow in porous media

1.
CNRS / UNIV PAU & PAYS ADOUR, Laboratoire de Mathématiques et de leurs Applications-IPRA, UMR 5142, Av. de l'Université, 64000 Pau, France
2.
Laboratory of Fluid Dynamics and Seismics, 9 Institutskiy per., Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, 141700, Russia
3.
CNRS / UNIV PAU & PAYS ADOUR, Laboratoire de Mathématiques et de leurs Applications-IPRA, UMR 5142, Av. de l'Université, 64000 Pau, France
4.
University of Tromsø, Campus in Narvik, Postbox 385, Narvik, 8505, Norway
5.
Institute for Information Transmission Problems of RAS, Bolshoy Karetny per., 19, Moscow, 127051, Russia

Received: 01 November 2015 Revised: 01 June 2016
Primary: 35B27, 35K65, 74Q15, 76M50; Secondary: 76S05, 76T10

The paper deals with a degenerate model of immiscible compressible two-phase flow in heterogeneous porous media. We consider liquid and gas phases (water and hydrogen) flow in a porous reservoir, modeling the hydrogen migration through engineered and geological barriers for a deep repository for radioactive waste. The gas phase is supposed compressible and obeying the ideal gas law. The flow is then described by the conservation of the mass for each phase. The model is written in terms of the phase formulation, i.e. the liquid saturation phase and the gas pressure phase are primary unknowns. This formulation leads to a coupled system consisting of a nonlinear degenerate parabolic equation for the gas pressure and a nonlinear degenerate parabolic diffusion-convection equation for the liquid saturation, subject to appropriate boundary and initial conditions. 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. The aim of this paper is to extend our previous results to the case of an ideal gas. In this case a new degeneracy appears in the pressure equation. With the help of an appropriate regularization we show the existence of a weak solution to the studied system. We also consider the corresponding nonlinear homogenization problem and provide a rigorous mathematical derivation of the upscaled model by means of the two-scale convergence.
- Heterogeneous porous media,
- homogenization,
- ideal gas,
- immiscible compressible,
- nuclear waste,
- two--phase flow,
- water-hydrogen
Citation: Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. An improved homogenization result for immiscible compressible two-phase flow in porous media[J]. Networks and Heterogeneous Media, 2017, 12(1): 147-171. doi: 10.3934/nhm.2017006

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Abstract

The paper deals with a degenerate model of immiscible compressible two-phase flow in heterogeneous porous media. We consider liquid and gas phases (water and hydrogen) flow in a porous reservoir, modeling the hydrogen migration through engineered and geological barriers for a deep repository for radioactive waste. The gas phase is supposed compressible and obeying the ideal gas law. The flow is then described by the conservation of the mass for each phase. The model is written in terms of the phase formulation, i.e. the liquid saturation phase and the gas pressure phase are primary unknowns. This formulation leads to a coupled system consisting of a nonlinear degenerate parabolic equation for the gas pressure and a nonlinear degenerate parabolic diffusion-convection equation for the liquid saturation, subject to appropriate boundary and initial conditions. 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. The aim of this paper is to extend our previous results to the case of an ideal gas. In this case a new degeneracy appears in the pressure equation. With the help of an appropriate regularization we show the existence of a weak solution to the studied system. We also consider the corresponding nonlinear homogenization problem and provide a rigorous mathematical derivation of the upscaled model by means of the two-scale convergence.

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