A Unique Grid-Block System for Improved Grid Orientation

Abstract

Abstract The grid orientation effect is a long-standing problem plaguing reservoir simulators that employ finite difference schemes. A rotation of the computational grids yields a substantially different solution under certain circumstances. For example, in a five-spot pattern, the predicted recovery, water cut performance and the locations of the fronts depend on the type of grid system used. A Cartesian grid with one axis parallel to the line joining an injector and producer gives a solution significantly different from a grid that has the axes oriented at 45° to this line. The objective of this study is to develop a grid system that can improve the representation of the configuration, thus minimizing the grid orientation effect. This paper presents a method for reducing the effect of grid orientation on computed numerical results in finite difference reservoir simulation. This method involves using a unique grid-block assignment where rectangular grid blocks are interspersed with octagonal grid blocks. The boundaries are then populated with triangular grid blocks. Thus, the entire domain will consist of a "structured" grid block system referred to as the Hybrid Grid Block (HGB). To test the viability of HGB, a finite difference IMPES-formulated two-dimensional black oil simulator was developed in this study, while retaining the familiar finite-difference discretization of the flow equations. Simulation cases were conducted to further examine the grid orientation effect in conventional grid block systems. Then, similar cases are run using HGB grid system. The outcome shows that HGB gives very close agreement between the parallel and diagonal grid orientation. This innovative grid block assignment will help to reduce the grid orientation effects compared to conventional grid block system. Introduction It has been demonstrated by several authors1–5 that two-dimensional simulations of immiscible displacements with unfavorable mobility ratio exhibit grid orientation effect. In fact, despite the fact that the reservoir is isotropic and homogeneous with favorable mobility ratio, there can still be an effect of grid orientation. To examine this effect, we conducted simulations of a quarter five-spot waterflood using parallel and diagonal grid systems, as illustrated in Fig. 1. A parallel grid system is a grid that is oriented parallel to injector-producer pairs. Meanwhile, a diagonal grid system is a grid oriented at 45° between injector and producer pairs. The distance of a producer to an injector and the size of the grid blocks are the same for both grid systems. Waterflood simulations were performed for oil/water mobility ratios (M) of 0.5, 1.0 and 10. The input data and the grid sizes are shown in Tables 1 and 2. Since the distance of injector to producer is the same, we expect to get similar recovery performance from both grid systems. However, when we compare the recovery performance of parallel grid blocks of 8×8 and diagonal of 6×6, the recovery performances from both grid blocks are different as seen in Fig. 2. This is because rotation of the coordinate axes results in differing amounts of truncation error.11 Increasing the resolution of grid blocks may reduce truncation error and lessen dispersion for a favorable mobility ratio (M<1.0) as pointed out by previous authors. Thus, we increased the number of grid blocks in diagonal and parallel grid blocks at M = 0.5. We found that recovery performance is not sensitive to the number of diagonal grid blocks in the model (Fig. 3). However, as the number of the parallel grid blocks is increased, the recovery performance changes gradually until it converges to a single recovery curve (Fig. 4). The recovery performances of finer grid blocks in both models (diagonal 21×21 vs. parallel 29×29), were compared. We found that the grid orientation effect was minimized (Fig. 5), as compared to those results from a parallel 8×8 and diagonal 6×6 grid blocks. The saturation distribution map at VPinj = 1.0 (Fig. 6) shows similar front profile of both parallel and diagonal grids indicating that the grid orientation effect is at its minimal.