A New Finite Element Technique for the Solution of Two-Phase Flow through Porous Media

A new upstream weighting finite element technique is developed for improved solution of the two-phase immiscible flow equations. Unlike the upstream weighting technique used by previous investigators, the new technique does not employ finite difference concepts to achieve the required upstream weighting of relative permeabilities or mobilities. Instead, upstream weighting is achieved by (1) representing the relative permeabilities or mobilities as continuous functions expressed in terms of the shape functions and nodal values (2) using asymmetric weighting functions to weight the spatial terms in the flow equations. These weighting functions are constructed such that they are dependent on the flow direction along each side of an element. In conjunction with the proposed technique, two solution schemes for treating the resulting set of non-linear algebraic equations are presented. These are the fully-implicit chord slope incremental solution scheme and the Newton-Rapkson solution scheme. Both schemes allow the use of large time steps without being unstable. The proposed numerical technique is applied to two problems (1) the one-dimensional Buckley-Leverett problem (2) the two-dimensional five-spot well flow problem. Results indicate that this technique is superior to not only earlier finite element schemes but also five-point upstream finite difference formulae.