A high-resolution total-variation-diminishing (TVD) finite-difference scheme has been developed and used in our multidimensional multicomponent, multiphase, finite-difference, IMPES-type compositional simulators for years. A variety of simulation results of enhanced oil recovery (EOR) processes have shown that this scheme gives convergent, high-order, accurate solutions. However, a restriction on the timestep size is always needed to ensure stability because of the IMPES formulation. This restriction sometimes can be very severe which means in some cases simulations are of high cost or even infeasible. It is well known that the fully implicit formulation is the most stable method. The standard approach, however, usually adopts lower-order finite-difference schemes for both the temporal and spatial discretizations because of computational requirements and difficulties in the program coding and the implementation of the physical property models. The advantage of the methods thus are overshadowed by the increased amount of numerical dispersion associated with large truncation error, which is especially detrimental to accurate field simulation and process design. A new fully implicit, high-resolution algorithm and a simulator based on the algorithm were developed and are described in this paper. The algorithm is second-order correct in time and uses a third-order finite-difference method to discretize the first-order spatial derivatives and a new total-variation-diminishing (TVD) third-order flux limiter to constrain the gradients of the fluxes to obtain accurate, oscillation-free solutions. All variables and TVD limiter functions are evaluated using the values of the new timestep. The schemes are applied for computing both the interface concentrations and mobilities with flow directions either in or against the coordinate directions and for nonuniform grids. The Jacobian matrix and the residual equations are updated at the end of each iteration. Unlike many numerical schemes in the literature, there are no problems with generalizations of this scheme.

The new algorithm and the simulator are verified by the good agreement between numerical results and analytical solutions. Verification cases with analytical solutions were also used to compare different simulation approaches. The new algorithm has higher resolution than standard methods, is more stable than the IMPES method, and stability is preserved with nonuniform grids.

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