Mixed Five-Point/Nine-Point Finite-Difference Formulation of Multiphase Flow in Petroleum Reservoirs

Abstract

Summary Nine-point finite-difference formulation of multiphase flow in the x-y plane is well-known to reduce grid-orientation effects. This paper proposes a new implicit pressure, explicit saturation (IMPES) formulation that uses implicit pressures for the customary five points of the x-y grid system, explicit pressures for the four points on the diagonals, and explicit saturations for all nine points. This proposed split-operator scheme produces virtually identical results as the standard nine-point formulation of Yanosik and McCracken, but can be easily implemented in existing five-point simulators without requiring any changes in the commonly used solution algorithms, such as the D-4 Gauss. The use of this new technique should enhance the accuracy of existing large-scale IMPES simulators with little increase in storage requirement or computing time. The method can be used with any set of nine-point coefficients and can be extended to fully implicit simulators. Introduction Grid-orientation effects1 result from the low-order accuracy of the finite-difference approximation of the partial-differential equations describing multiphase flow in porous media. Fig. 1 shows that the numerical gridblocks may be aligned either parallel to or at a 45° angle (diagonal) to the line joining an injection well and a production well in an areal five-spot flood pattern. The standard five-point formulation predicts different recovery performance for these two grid orientations because fluid tends to flow more easily along the grid lines of the grid network used. It seems intuitively obvious that this grid-orientation effect should decrease as the grid cell size decreases. In practical use of reservoir simulation models, however, it is rarely economically justifiable, and in many cases impossible from a computer hardware standpoint, to increase the number of gridblocks enough to eliminate the grid-orientation effect. Many methods to improve the accuracy of the finite-difference approximation have been investigated over the past 20 years. Yanosik and McCracken2 introduced one of these methods, a nine-point finite-difference formulation that used the IMPES solution method. They showed that the nine-point scheme was significantly less sensitive to grid orientation than the five-point scheme. Fig. 2 shows a comparison of recovery performance for different grid network systems when the five-point and the nine-point schemes are used. The figure illustrates how much less sensitive the nine-point scheme is to grid orientation than the five-point scheme. In this paper, a split-operator scheme is proposed and evaluated with respect to grid-orientation effects and compared with the five-point and Yanosik and McCracken nine-point schemes. It will become obvious that the split-operator scheme can be implemented easily in existing five-point simulators. Proposed Split-Operator Scheme The proposed technique, a modification of the Yanosik and McCracken2 nine-point method, is called the split-operator scheme. The idea behind the scheme is to connect the (i, j) grid cell and its eight surrounding blocks (Fig. 3) in both the pressure and the saturation equations, as Yanosik and McCracken did, without the stability problems observed by Ko and Au.3 The difference between the Yanosik and McCracken scheme and the split-operator scheme is the formulation of the pressure equation. In the split-operator scheme, the pressure in the (i, j) gridblock is tied to four north-south, east-west gridblock pressures at present time level and to four diagonal gridblock pressures at the old time level. The nine-point scheme requires all nine pressure points to be at the present time level. For the split-operator scheme to be implemented into an existing IMPES five-point finite-difference code, it is necessary to modify only the right side of the pressure equation and to extend the explicit saturation equation to nine points. The modified five-point IMPES model, in addition to the calculation of north-south and east-west transmissibilities, takes on the calculation of diagonal transmissibilities, and the storage requirement increases by four arrays of dimension (imax,jmax). Note that the implicit part of the pressure equation remains essentially as given by the original five-point formulation; thus, the split-operator scheme uses the same pressure-solution routine as the five-point scheme. Testing the Split-Operator Scheme The split-operator scheme was tested for the same reservoir and fluid data as used by Yanosik and McCracken2(Table 1). Also, in accordance with Yanosik and McCracken, it is assumed that rock compressibility is zero and that capillary forces are negligible. Table 2 gives relative permeability and fractional-flow curves for a waterflood example. This study focuses on the performance of the split-operator scheme relative to the standard IMPES five-point and the. Yanosik and McCracken nine-point schemes. Variables in the simulations are mobility ratio, M, gridblock size, and reservoir permeability. Results Unfavorable Mobility Ratio. The simulations of the waterflood example for unfavorable mobility ratios cover from M=1 to 50. Fig. 4 shows the recovery performance of the split-operator scheme as the mobility ratios vary between 1 and 50. Note that no grid-orientation effects are detectable. Fig. 5 shows the recovery curves for M=50 for the five-point, nine-point, and split-operator schemes. The five-point scheme shows significant grid-orientation effects, while the nine-point and split-operator schemes are identical and show no sign of such effects. Fig. 6 shows the 40% water saturation contours (fronts) for M=50. All three finite-difference schemes exhibit grid-orientation effects in this figure. The five-point scheme yields a large difference between the diagonal and the parallel grid saturation contours. The split-operator scheme produces identical results to the nine-point scheme. The diagonal and the parallel grid fronts are closer together for the split-operator scheme than for the five-point scheme. Nonuniform Grid Network. Two nonuniform grid systems were used to test the grid-orientation behavior of the split-operator scheme relative to the five-point and the nine-point schemes. The first grid system has grid-size ratios of 1:2:4, such that the first two and the last two grid cells in each direction have the ratio of 1:2, while the remaining cells have a size four times the first grid cell's size. This grid system was studied by Bertiget and Padmanabhan.4 p. 439-444 Unfavorable Mobility Ratio. The simulations of the waterflood example for unfavorable mobility ratios cover from M=1 to 50. Fig. 4 shows the recovery performance of the split-operator scheme as the mobility ratios vary between 1 and 50. Note that no grid-orientation effects are detectable. Fig. 5 shows the recovery curves for M=50 for the five-point, nine-point, and split-operator schemes. The five-point scheme shows significant grid-orientation effects, while the nine-point and split-operator schemes are identical and show no sign of such effects. Fig. 6 shows the 40% water saturation contours (fronts) for M=50. All three finite-difference schemes exhibit grid-orientation effects in this figure. The five-point scheme yields a large difference between the diagonal and the parallel grid saturation contours. The split-operator scheme produces identical results to the nine-point scheme. The diagonal and the parallel grid fronts are closer together for the split-operator scheme than for the five-point scheme. Nonuniform Grid Network. Two nonuniform grid systems were used to test the grid-orientation behavior of the split-operator scheme relative to the five-point and the nine-point schemes. The first grid system has grid-size ratios of 1:2:4, such that the first two and the last two grid cells in each direction have the ratio of 1:2, while the remaining cells have a size four times the first grid cell's size. This grid system was studied by Bertiget and Padmanabhan.4 p. 439-444