# Application of Singularity Programming in Reservoir Simulation

## Abstract/Contents

- Abstract
- Finite difference approximations of differential equations are usually inaccurate when the equations being solved exhibit mathematical singularities. In reservoir simulators these singularities can be observed in pressure and saturation equations. In many cases involving singularities, it is a common practice to increase the accuracy of the simulation by grid refinement, which also increases the computational effort. Sometimes, to avoid costly simulations, other mathematical techniques are used instead of grid refinement. Singularity programming is a superposition procedure that has been used in many engineering applications to improve the accuracy of the solution at, or near, singularities, without increasing the cost of calculation. In this report, the application of this method to reservoir simulation is evaluated. The procedure for applying singularity programming involves decoupling the finite difference (or finite element) equation into a singular part and a regular part. The singular part, which includes the effects of the singularity, is solved analytically. The regular part is solved by the usual numerical methods, ignoring the presence of the singularity. The results of both parts are then added to give the complete solution. In the first part of this study, singularity programming is used to improve the accuracy of the solution of the pressure equation in the presence of wells and the results of a test case are presented. It is observed that the pressure equation is explicitly separable into a singular part and a regular part, and can thus be handled easily by this technique. The results of the model problem show a reduction in simulation effort for the same accuracy. In the second part of this report, singularity programming is applied to the saturation equation in the presence of capillary pressure effects. The saturation equation is not explicitly separable and can be subjected to singularity programming by using numerical methods for solving both the singular part and the regular part. Since an analytical expression is not used to model the singularity, the accuracy of using singularity programming for the saturation equation is the same as that obtained without its use. It is recommended that a method to decouple the saturation equation should be devised before singularity programming can be considered a useful tool in reservoir simulation.

## Description

Type of resource | text |
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Date created | March 1995 |

## Creators/Contributors

Author | Aftab, Taher | |
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Primary advisor | Horne, Roland N. | |

Degree granting institution | Stanford University, Department of Petroleum Engineering |

## Subjects

Subject | School of Earth Energy & Environmental Sciences |
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Genre | Thesis |

## Bibliographic information

## Access conditions

- Use and reproduction
- User agrees that, where applicable, content will not be used to identify or to otherwise infringe the privacy or confidentiality rights of individuals. Content distributed via the Stanford Digital Repository may be subject to additional license and use restrictions applied by the depositor.

## Preferred citation

- Preferred Citation
- Aftab, Taher. (1995). Application of Singularity Programming in Reservoir Simulation. Stanford Digital Repository. Available at: https://purl.stanford.edu/sp396vv5239

## Collection

Master's Theses, Doerr School of Sustainability

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