Upscaling a Navier-Stokes-Cahn-Hilliard model for two-phase porous-media flow with solute-dependent surface tension effects

We consider a model for the flow of two immiscible and incompressible fluid phases in a porous medium. A surfactant is dissolved in one of the fluid phases, and its concentration at the interface separating the two fluids can change the surface tension. At the scale of pores, we assume that the flow is governed by the Navier-Stokes equations, while for the phase separation, a Cahn-Hilliard phase-field model is adopted. Using formal homogenization, we derive a two-scale model describing the averaged behaviour of the system at the larger Darcy scale, where effective quantities are found through local (cell) problems at the smaller pore scale. For this two-scale model, we formulate a numerical scheme and present numerical results highlighting the influence of the solute-dependent surface tension.