A comparison of the finite-difference and the finite-element methods for simulating unstable displacements

Abstract

The finite-difference and the finite-element methods are the two most commonly used numerical methods in reservoir simulation. A comparative study of the two methods in compositional and noncompositional simulations is presented. The comparisons are based on grid orientation effect and computational efficiency. In the grid orientation study, the finite-element model with Gaussian quadrature showed less grid orientation than either the finite-difference model or the finite-element model with Lobatto quadrature. In heterogeneous media, the finite-element model with Gaussian quadrature suffered from severe numerical oscillations inspite of upstream weighting. The finite-element model with Lobatto quadrature was stable in heterogeneous media; however, the results were essentially identical to those obtained by the finite-difference method. In simulating two-phase noncompositional displacements, the finite-element models took twice as much computing time as the finite-difference model due to the lengthy process of formulating and assembling the transmissibility matrices. In compositional systems, however, the finite-element method took only 50% more time than the finite-difference method because the bulk of the computational effort was concentrated in the phase behavior calculations rather than in the formulation of the system matrices.