Sequential Semi-Implicit Algorithm for Computing Discontinuous Flows in Porous Media

This paper presents a novel numerical algorithm for computing incompressible, discontinuous, two-phase flows in two-dimensional (2D), inhomogeneous, and isotropic porous media. The algorithm uses Bell et al.'s hybrid sequential semi-implicit approach 1 for both accuracy and efficiency of the calculations. The explicit part uses a high-order Godunov 2 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 is a Poisson solver for the phase pressure. Both use fast, multilevel, multigrid solvers with the number of operations of the order of O[N log(N)], where N = number of gridpoints. Two numerically stiff reservoir engineering problems are presented to demonstrate the low numerical dispersion and second-order accuracy of our method.