Improved Pressure Response Representation And Reduction Of Numerical Dispersion Effects In Reservoir Simulation
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AbstractA study has been made of the effects of numerical dispersion in reservoir simulation. A one-dimensional two-phase (water-oil) numerical model has been used in this investigation.Instead of the regular single-point upstream transmissibilities, a new technique is proposed which consists of keeping track of the position of the displacement front, and using an interblock pseudo transmissibility. It is shown that a very significant reduction of the numerical dispersion of flood fronts and much improved representation of pressure history are obtained when using these modifications.The applicability and effectiveness of this technique is illustrated by comparisons of numerical examples with theoretical and data published in the literature. The method proposed in this study can be implemented in any finite-difference reservoir simulator without much reprogramming effort, provided the displacement process to be simulated can be described by the Buckley-Leverett theory.IntroductionThe flow of fluids in porous media is described by non-linear partial differential equations which have analytical solutions only for a limited number cases. Since each reservoir is a unique system, general reservoir simulators have been developed which use finite difference methods in solving these equations.In these numerical simulators, the reservoir is divided into a number of discrete blocks (cells) and a system of mathematical equations is used to calculate pressures and saturations at the grid block faces. pressures and saturations at the grid block faces. Several different schemes have been proposed in order to evaluate the fluid mobilities; these are: single-point upstream, downstream, mid-point weighting, and two-point upstream. Of these, the usual procedure adopted is to evaluate these terms by using the procedure adopted is to evaluate these terms by using the single-point scheme, either because of the oscillation and instabilities that some of them have, or because of the programming efforts involved.The use of the single-point upstream mobility has been proved to cause excessive numerical dispersion of flood fronts for some problems. Also, when simulating unfavorable mobility ratio displacements, the pressure profile will present a physically unrealistic pressure profile will present a physically unrealistic stepwise behavior when the displacing phase enters a new cell. Thus, the calculated pressure drops and the saturation profiles will be in error if upstream mobilities are used.Todd, et al., presented an alternative to the single-point scheme by introducing a two-point upstream weighting scheme. They demonstrated that an improvement in accuracy is obtained when using this method om implicit pressure-explicit saturation type models.A modified version of the two-point upstream weighting scheme has been introduced by Wheatley which allows its implementation in fully implicit or IMPES simulators. The results were compared with an analytical solution and it was shown that the differences between the analytical and the numerical results is reduced when using this technique. However, no results have been reported at adverse mobility ratios when using the two-point upstream weithting scheme.Other methods have been proposed to alleviate the numerical dispersion problem. These nine-point finite difference schemes are in disadvantage with respect to the regular five point finite difference approximation because of the extra computer time that is required to solve the resultant system of equations.Leventhal describes the use of the fourth order operator compact implicit (OCI) method in the simulation of an immiscible water flood. The results were compared with the analytical solution of Buckley and Leverett, conventional finite difference methods using the single-point upstream method and also with variational methods.
