Sequential explicit-implicit algorithm for computing discontinuous flows in porous media

A novel numerical algorithm for computing incompressible, discontinuous, two-phase flows in two-dimensional, inhomogeneous, and isotropic porous media is presented. The algorithm uses Colella et al's. hybrid sequential explicit-implicit approach for both accuracy and speed of the calculations. The explicit part uses a high-order Godunov scheme with a modified Van Leer geometrical slope limiter, similar to those used in shock dynamics. The implicit part is a two-step solver: the first step is a Crank-Nicolson saturation solver and the second one is a Poisson solver for the phase pressure. Both use fast multilevel multigrid solvers with the number of operations of the order of θ[Nlog(N)], where N is the number of grid points. For an implicit simulator, the number of operations is θ(N₃) per time step. Two numerically stiff reservoir engineering problems are presented to demonstrate the low numerical dispersion and second-order accuracy of our method.