Homogenization of nonisothermal immiscible incompressible two-phase flow in double porosity media

Abstract

In this paper, we establish a homogenization result for a nonlinear degenerate system arising from two-phase flow through fractured porous media with periodic microstructure taking into account the temperature effects. The mathematical model is given by a coupled system of two-phase flow equations, and an energy balance equation. The microscopic model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy–Muskat and the capillary pressure laws. The problem is written in terms of the phase formulation, i.e. the saturation of one phase, the pressure of the second phase and the temperature are primary unknowns. The fractured medium consists of periodically repeating homogeneous blocks and fractures, the permeability being rapidly oscillating discontinuous function. Over the matrix domain, the permeability is scaled by ε2, where ε is the size of a typical porous block. Furthermore, we will consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves. The model involves highly oscillatory characteristics and internal nonlinear interface conditions accounting for discontinuous capillary pressures. We then show by a rigorous mathematical argument that the solution of this microscopic problem converges as ε tends to zero to the solution of a double-porosity model of the global macroscopic flow. Our techniques make use of the two-scale convergence method combined to extension and dilation operators in the homogenization context. The memory effects of usual double porosity media are reproduced by this model. We show how the effective coefficients of the porous medium are determined in a precise way by certain physical and geometric features of the microscopic fracture domain, the microscopic matrix blocks, and the interface between them.