A new method for the determination of finely gridded reservoir simulation pressures has been developed. It is estimated to be as much as tens to hundreds of times faster than other methods for very large reservoir simulation grids. The method extends the work of Weber et al. (CIPC Paper 2004–170). Weber demonstrated accuracies normally requiring millions of cells using traditional finite-difference equations, using only hundreds of cells. This was accomplished through the use of finite-difference equations that incorporate the physics of the flow. Although these coarse-grid solutions achieve accuracies normally requiring orders of magnitude more resolution, their coarse resolution does not resolve local pressure variations resulting from fine-grid permeability variations.

The present investigation demonstrates a method for obtaining the full, fine-grid solution with significantly reduced computer times by incorporating the accurate course-grid solution. The method involves two steps:

  1. using Weber's equations to obtain an accurate pressure solution on a coarse grid, and

  2. refining the grid to obtain detailed pressures that honor the course-grid pressures.

The performance of various linear algebraic solvers is considered to maximize the speed of calculations of both the coarse and fine grid steps. The speed of the new method is compared with the traditional, single-step solution using these solvers.


Throughout the history of computers, reservoir simulators have always taxed the very fastest machines. Despite the tremendous increases in computer speeds in the past decades, the need for faster reservoir simulation remains as critical today as it has ever been. This need results from emerging technologies such as:

  • Geo-statistics, which results in many very detailed models of the same field,

  • Automatic history matching, which requires many simulations of the same field to determine a data set that matches the production history of the field,

  • Optimization, which uses repeated simulations to automatically determine the best well location, geometry, completion intervals, and rates.

  • Smart wells, which use real-time simulations to control the production rate from various well segments.1

This work proposes a new linear algebra technique for thesolving of finely gridded reservoir pressures. The new method is based on the work of Weber et al.2 who proposed that finite-difference equations, used to represent the pressure equation, be based on mathematical expressions that incorporate the physics of the process instead of on traditional polynomial expressions. In modeling the reservoir pressures, equations incorporating the physically realistic ln(r) dependence on pressure for reservoirs with straight line wells, and a 1/r dependence for reservoirs with more complex well geometries were used (r is the distance to the well). The results of Weber's 1/r based finite-difference equations are shown in Figure 1. The figure compares the accuracy of the pressures calculated by various methods for an 11×11×22 grid of a rectangular reservoir of the same geometry used in this study. The new finite-difference equations showed a four-order-of-magnitude improvement in accuracy compared with traditional equations.

This work uses these previous results of Weber et al. tocreate a nested-grid method that uses both a coarse and fine grid to obtain a full, fine-grid solution with significantly reduced computer times.

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