We discuss several fully implicit techniques for solving the nonlinear algebraic system arising in an expanded mixed finite element or cell-centered finite difference discretization of two- and three-phase porous media flow. Every outer nonlinear Newton iteration requires solution of a nonsymmetric Jacobian linear system. Two major types of preconditioners, supercoarsening multigrid (SCMG) and two-stage, are developed for the GMRES iteration applied to the solution of the Jacobian system. The SCMG reduces the three-dimensional system to two dimensions using a vertical aggregation followed by a two-dimensional multigrid. The two-stage preconditioners are based on decoupling the system into a pressure and concentration equations. Several pressure preconditioners of different types are described. Extensive numerical results are presented using the integrated parallel reservoir simulator (IPARS) and indicate that these methods have low arithmetical complexity per iteration and good convergence rates.

We propose a cell-centered finite volume (FV) scheme with the minimal stencil formed by the closest neighbouring cells. The discrete solution satisfies the discrete maximum principle and approximates the exact solution with second-order accuracy. The coefficients in the FV stencil depend on the solution; therefore, the FV scheme is nonlinear. The scheme is applied to a steady state advection-diffusion equation discretized on a general polygonal mesh.