A phase-field model of two-phase Hele-Shaw flow

Abstract We propose a continuum model of two-phase flow in a Hele-Shaw cell. The model describes the multiphase three-dimensional flow in the cell gap using gap-averaged quantities such as fluid saturation and Darcy flux. Viscous and capillary coupling between the fluids in the gap leads to a nonlinear fractional flow function. Capillarity and wetting phenomena are modelled within a phase-field framework, designing a heuristic free energy functional that induces phase segregation at equilibrium. We test the model through the simulation of bubbles and viscously unstable displacements (viscous fingering). We analyse the model’s rich behaviour as a function of capillary number, viscosity contrast and cell geometry. Including the effect of wetting films on the two-phase flow dynamics opens the door to exploring, with a simple two-dimensional model, the impact of wetting and flow rate on the performance of microfluidic devices and geological flows through fractures.