Description of multiphase flows in porous media using an effective convective Cahn-Hilliard equation

Immiscible two-phase flows in porous media find a variety of applications such as microfluidics, oil extraction from reservoirs and chromatography, to name but a few. In this study, we investigate the dynamics of interfaces in porous media using an effective convective Cahn-Hilliard equation which was derived in [1] from a Stokes-Cahn-Hilliard equation for microscopic heterogeneous domains by means of a homogenization methodology. We consider different types of microstructures, including periodic and non-periodic, observing that the macroscopic model is able to retain the microscopic features, hence indicating that our formulation provides an efficient and systematic computational framework to track interfaces in porous media. \\[4pt] [1] M. Schmuck, M. Pradas, G.A. Pavliotis and S. Kalliadasis, 2013 ``Derivation of effective macroscopic Stokes--Cahn--Hilliard equations for periodic immiscible flows in porous media,'' Nonlinearity {\bf 26} 3259-3277.