A front-tracking approach to a two-phase fluid-flow model with capillary forces

We consider a prototype two-phase fluid-flow model with capillary forces. The pressure equation is solved using standard finite-elements and multigrid techniques. The parabolic saturation equation is addressed via a novel corrected operator splitting approach. In typical applications, the importance of advection versus diffusion (capillary forces) may change rapidly during a simulation. The corrected splitting is designed so that any combination of advection and diffusion is resolved accurately. It gives a hyperbolic conservation law for modelling advection and a parabolic equation for modelling diffusion. The conservation law is solved by front tracking, which naturally leads to a dynamically defined residual flux term that can be included in the diffusion equation. The residual term ensures that self-sharpening fronts are given the correct structure. A Petrov-Galerkin finite-element method is used to solve the diffusion equation. We present several examples that demonstrate potential shortcomings of standard viscous operator splitting and highlight the corrected splitting strategy. This is the first time a front-tracking simulator is applied to a flow model including capillary forces.