Homogenization of coupled immiscible compressible two-phase flow with kinetics in porous media

Abstract

In this paper, we consider a liquid–gas system with two components: water and hydrogen flow model in heterogeneous porous media with periodic microstructure taking into account kinetics in the mass transfer between the two phases. The particular feature in this model is that chemistry effects are taken into account. The microscopic model consists of the usual equations derived from the mass conservation laws of both fluids, along with the Darcy–Muskat and capillary pressure laws and the mass exchange is modeled as a source term in the equations. The problem is written in the terms of the phase formulation; i.e. the saturation of one phase, the pressure of the second phase, and the concentration of dissolved hydrogen in the liquid phase. The mathematical model consists in a system of partial differential equations: two degenerate nonlinear parabolic equations and one diffusion–convection equation. The major difficulties related to this new 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 coupled system of three coupled partial differential equations with effective coefficients which are computed via solving cell problems. We give a rigorous mathematical derivation of the effective model by means of the two-scale convergence.

COMMUNICATED BY:

• Grigory Panasenko

Keywords:

• Homogenization
• flows in porous media
• two-phase and multiphase flows
• reaction effects in flows
• degenerate parabolic equations

AMS Subject Classifications:

• 35B27
• 35K55
• 35K65
• 35Q35
• 74Q15
• 76S05
• 76T10
• 76V05
• 80A30

Acknowledgments

Most of the work on this paper was done when L. Pankratov was visiting the Applied Mathematics Laboratory of the University of Pau & CNRS. L. P. is grateful for the invitations and the hospitality.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partially supported by the Carnot Institute, ISIFoR project (Institute for the Sustainable Engineering of Fossil Resources). The work of L. Pankratov is supported by Russian Foundation for Basic Research [grant number 19-01-00592].