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Multiphase flow in porous media occurs in several disciplines includingpetroleum reservoir engineering, petroleum systems' analysis, and CO$_2$sequestration. While simulations often use a fully implicit discretization toincrease the time step size, restrictions on the time step often exist due tonon-convergence of the nonlinear solver (e.g. Newton's method). Here thisproblem is addressed for the Buckley-Leverett equations, which modelincompressible, immiscible, two-phase flow with no capillary potential. Theequations are recast as a gradient flow using the phase-field method, and aconvex energy splitting scheme is applied to enable large timesteps, even forhigh degrees of heterogeneity in permeability and viscosity. By using thephase-field formulation as a homotopy map, the underlying hyperbolic flowequations can be solved with large timesteps. For a heterogeneous test problem,the new homotopy method allows the timestep to be increased by more than sixorders of magnitude relative to the unmodified equations while maintainingconvergence.

Simulation of incompressible two-phase flow in porous media with large timesteps