Homogenized model of immiscible incompressible two-phase flow in double porosity media : A new proof

Abstract: In this paper we give a new proof of the homogenization result for an immiscible incompressible two-phase flow in double porosity media obtained earlier in the pioneer work by A. Bourgeat, S. Luckhaus, A. Mikeli\'c (1996) and in the paper of L. M. Yeh (2006) under some restrictive assumptions. The microscopic model consists of the usual equations derived from the mass conservation laws for both fluids along with the standard Darcy-Muskat law relating the velocities to the pressure gradients and gravitational effects. The problem is written in terms of the phase formulation, i.e. where the phase pressures and the phase saturations are primary unknowns. The fractured medium consists of periodically repeating homogeneous blocks and fractures, where the absolute permeability of the medium is discontinuous. The important difference with respect to the results of the cited papers is that the global pressure function as well as the saturation are also discontinuous. This makes the initial mesoscopic problem more general and, physically, more realistic. 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.