Derivation of effective macroscopic Stokes–Cahn–Hilliard equations for periodic immiscible flows in porous media

Using thermodynamic and variational principles we examine a basic phase field model for a mixture of two incompressible fluids in strongly perforated domains. With the help of the multiple scale method with drift and our recently introduced splitting strategy for Ginzburg–Landau/Cahn–Hilliard-type equations (Schmuck et al 2012 Proc. R. Soc. A 468 3705–24), we rigorously derive an effective macroscopic phase field formulation under the assumption of periodic flow and a sufficiently large Péclet number. As for classical convection–diffusion problems, we obtain systematically diffusion–dispersion relations (including Taylor–Aris-dispersion). Our results also provide a convenient computational framework to macroscopically track interfaces in porous media. In view of the well-known versatility of phase field models, our study proposes a promising model for many engineering and scientific applications such as multiphase flows in porous media, microfluidics, and fuel cells.