Some remarks on the homogenization of immiscible incompressible two-phase flow in double porosity media

Brahim Amaziane; Mladen Jurak; Leonid Pankratov; Anja Vrbaški

doi:10.3934/dcdsb.2018037

Article Contents

2018, Volume 23, Issue 2: 629-665. Doi: 10.3934/dcdsb.2018037

This issue Previous Article Stability of travelling waves in a Wolbachia invasion Next Article Global Hopf bifurcations of neutral functional differential equations with state-dependent delay

Some remarks on the homogenization of immiscible incompressible two-phase flow in double porosity media

1.
Laboratoire de Mathématiques et de leurs Applications-IPRA, CNRS/UNIV PAU & PAYS ADOUR, UMR 5142, Av. de l'Université, 64000 Pau, France
2.
Faculty of Science, University of Zagreb, Bijenička 30,10000 Zagreb, Croatia
3.
Laboratory of Fluid Dynamics and Seismic (RAEP 5 Top 100), Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141700, Russian Federation
4.
Faculty of Mining, Geology and Petroleum Engineering, University of Zagreb, Pierottijeva 6,10000 Zagreb, Croatia

Received: June 30, 2016

Revised: September 30, 2017

Early access: November 2017

Published: June 2018

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Abstract

This paper presents a study of immiscible incompressible two-phase flow through fractured porous media. The results obtained earlier in the pioneer work by A. Bourgeat, S. Luckhaus, A. Mikelić (1996) and L. M. Yeh (2006) are revisited. The main goal is to incorporate some of the most recent improvements in the convergence of the solutions in the homogenization of such models. The microscopic model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy-Muskat law. The problem 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. We will consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves. The fractured medium consists of periodically repeating homogeneous blocks and fractures, the permeability being highly discontinuous. Over the matrix domain, the permeability is scaled by ${\varepsilon }^θ$, where $\varepsilon$ is the size of a typical porous block and $θ>0$ is a parameter. The model involves highly oscillatory characteristics and internal nonlinear interface conditions. Under some realistic assumptions on the data, the convergence of the solutions, and the macroscopic models corresponding to various range of contrast are constructed using the two-scale convergence method combined with the dilation technique. The results improve upon previously derived effective models to highly heterogeneous porous media with discontinuous capillary pressures.

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Mathematics Subject Classification: Primary: 35B27, 35K55, 35K65, 74Q15; Secondary: 35Q35, 76S05.

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Figure 1. (a) The domain $\Omega$. (b) The reference cell $Y$

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