Convergence of Implicit Monotone Schemes with Applications in Multiphase Flow in Porous Media

Abstract

Phase-based upstreaming, which is a commonly used spatial discretization for multiphase flow in reservoir simulation, naturally gives rise to implicit monotone schemes when implicit time-stepping is used. The nonlinear Gauss-Seidel and Jacobi algorithms are shown to converge to a unique bounded solution when applied to the resulting system of equations. Thus, for one-dimensional problems, we obtain an alternate, constructive proof that such schemes are well-defined and converge to the entropy solution of the conservation law for arbitrary CFL numbers. The accuracy of phase-based upstream solutions is studied for various spatial and temporal grids, revealing the importance of unconditional stability when nonuniform grids and/or variable porosity is involved.