Improvement in the Time Accuracy of Numerical Methods in Petroleum Engineering Problems—A New Combination

Abstract

Convective dispersion is one of the most important phenomena that takes place during miscible and chemical displacements in porous media. This has application both in enhanced oil recovery and ground water modeling. The presence of the parabolic and hyperbolic terms in the convective dispersion equation makes it a difficult equation to solve. Solving the convective dispersion equation by finite difference schemes is always the first choice due to the simplicity of the technique. However, current techniques suffer from the lack of numerical accuracy, especially for a reasonable time step, Δt; an accuracy of θ(Δt) is used in most cases. In this article, a central difference model for the time term having the inherent accuracy of the order of Δt 2 is used for the first time along with the Barakat-Clark scheme to target the overall accuracy in time of the order of Δt 4. The effect of this new combination on the solution is analyzed with the help of a case study, in where a 2-D convective-dispersion equation is solved by using the Barakat-Clark scheme with given boundary conditions. The time term in the equation is approximated by a central difference scheme. A comparison is also presented among the DuFort-Frankel scheme, Barakat-Clark scheme with the forward time difference term (FTD) and the proposed scheme of Barakat-Clark with the central time difference term (CTD). The Barakat-Clark scheme is proved to be stable even for this new combination. These findings will be helpful in applying different numerical schemes in the areas of petroleum engineering and ground water modeling, for which larger time steps (while maintaining the same accuracy) can be very useful. The utilization of the central difference scheme for the time term in parallel with the Barakat-Clark scheme does not make any significant difference on the solution of the equation although the accuracy in time becomes approximately of the order Δt 4. It is also found that the DuFort-Frankel scheme gives the highest and most consistent accuracy in its stability range. This scheme is not found to be unconditionally stable as compared to what has been professed in literature especially for the equations having hyperbolic and parabolic terms. The use of central weighing scheme for the time term (proposed) in Barakat-Clark's finite difference approximation gives almost the identical results as compared to the original Barakat-Clark scheme. This new combination is proved to be unconditionally stable even for this accuracy in time. Finally it is recommended to extend the application of the proposed Barakat-Clark scheme in 3-D problems and higher order equations relating to other phenomena with an attempt to increase the spatial accuracy as well.