A two-scale model of two-phase ow in porous media ranging from porespace to the macro scale

Abstract

We will derive two-scale models for two-phase flow in porous media, with the microscale given by the porescale. The resulting system will account for balance of mass, momentum and energy. To this aim, we will combine a generalization of Rajagopal’s and Srinivasa’s assumption of maximum rate of entropy production [39, 20, 21] with formal asymptotic expansion. The microscopic model will be based on phase fields, in particular to the full Cahn-Hilliard-Navier- Stokes-Fourier model derived in [23] with the boundary conditions from [20]. Using a generalized notion of characteristic functions, we will show that the solutions to the two-scale model macroscopically behave like classical solutions to a system of porous media flow equations. Relative permeabilities and capillary pressure relations are outcomes of the theory and exist only for special cases. Therefore, the two-scale model can be considered as a true generalization of classical models providing more information on the microscale thereby making the introduction of hysteresis superfluous.