Homogenization of immiscible compressible two-phase flow in double porosity media

A double porosity model of multidimensional immiscible com-pressible two-phase flow in fractured reservoirs is derived by the mathematicaltheory of homogenization. Special attention is paid to developing a generalapproach to incorporating compressibility of both phases. The model is writ-ten in terms of the phase formulation, i.e. the saturation of one phase andthe pressure of the second phase are primary unknowns. This formulationleads to a coupled system consisting of a doubly nonlinear degenerate para-bolic equation for the pressure and a doubly nonlinear degenerate parabolicdiffusion-convection equation for the saturation, subject to appropriate bound-ary and initial conditions. The major difficulties related to this model are inthe doubly nonlinear degenerate structure of the equations, as well as in thecoupling in the system. Furthermore, a new nonlinearity appears in the tem-poral term of the saturation equation. The aim of this paper is to extend theresults of [9] to this more general case. With the help of a new compactness re-sult and uniform a priori bounds for the modulus of continuity with respect tothe space and time variables, we provide a rigorous mathematical derivation ofthe upscaled model by means of the two-scale convergence and the dilatationtechnique.