Collocation Simulation of Multiphase Porous-Medium Flow

Abstract

Abstract Hermite orthogonal collocation yields oscillatory and sometimes nonconvergent numerical solutions to convection-dominated oil reservoir flow problems. A new method of choosing collocation points emphasizes upstream values in the convective terms, yielding oscillation-free solutions to the linear convection-dispersion transport equation and convergent solutions to the Buckley-Leverett problem. Introduction Although collocation promises to be an attractive numerical procedure for the simulation of multiphase flows in porous media, efforts to use this method have had limited success. The main problem in such applications is finding a numerical method in such applications is finding a numerical method that respects the mathematical peculiarities of convection-dominated flows. Mercer and Faust,1 for example, examined solutions to the coupled pressure and saturation equations describing immiscible displacements in oil reservoirs. They compared several approaches to solving the nonlinear saturation equation, including the Galerkin method and orthogonal collocation on linear, C0 cubic, and C1 (Hermite) cubic bases. They also compared the benefits of using functional evaluation of coefficients, lumping the time-derivative matrix, weighting upstream permeability, and adding artificial capillary pressure in overcoming the convergence problems associated with the saturation discontinuity. Although these techniques generally smear steep gradients, Mercer and Faust concluded that sacrificing sharp front-tracking capability is necessary in some instances to obtain convergent solutions on uniform grids. Sincovec2 applied orthogonal collocation on Hermite basis functions to compute solutions to a nonlinear gas flow problem and to the linear convection-dispersion transport equation, demonstrating improved efficiency over finite difference methods. In applying his method to the Buckley-Leverett saturation equation, however, he used a formulation of the nonlinear term that failed correctly to propagate the saturation front. Sincovec concluded that his collocation procedure is inappropriate for the solution of the saturation equation for immiscible flow.3