  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.
