Homogenization results for a coupled system modelling immiscible compressible two-phase flow in porous media by the concept of global pressure

Abstract

A model for immiscible compressible two-phase flow in heterogeneous porous media is considered. Such models appear in gas migration through engineered and geological barriers for a deep repository for radioactive waste. The main feature of this model is the introduction of a new global pressure and it is fully equivalent to the original equations. The resulting equations are written in a fractional flow formulation and lead to a coupled degenerate system which consists of a nonlinear parabolic (the global pressure) equation and a nonlinear diffusion–convection one (the saturation) equation with rapidly oscillating porosity function and absolute permeability tensor. 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 obtain a nonlinear homogenized problem with effective coefficients which are computed via a cell problem and give a rigorous mathematical derivation of the upscaled model by means of two-scale convergence.

Keywords:

• compressible immiscible
• global pressure
• heterogeneous porous media
• homogenization
• nuclear waste
• two-phase flow
• water-hydrogen

AMS Subject Classifications::

• 35B27
• 35K65
• 76S05
• 76T05
• 76T10

Acknowledgements

The authors acknowledge the AUF ‘Agence Universitaire de la Francophonie’, Projet de Coopération Scientifique Inter-Universitaire, PHC COGITO 2011–12 PROJET N0 24775XE, and the GnR MoMaS (PACEN/CNRS, ANDRA, BRGM, CEA, EDF, IRSN) France, for partially supporting this research.