A New Method of Formulating Finite Difference Equations -Some Reservoir Simulation Examples

Abstract

Finite difference approximations to partial derivatives are generally based on Taylor series, which are polynomial expressions for the unknown variable as a function of the grid locations. In many problems, approximate analytical solutions are known that incorporate the physics of the process. It is proposed that such expressions be used to derive finite difference equations. Increased accuracy is anticipated, particularly when the solutions are highly non-linear, singular, or discontinuous.Reservoir simulation is such a problem. Flow in petroleum reservoirs results from injection and productions from wells, which are relatively small sources and sinks. Near singularities in the pressure around the wells result. The immiscibility of the fluids causes an oil bank to form in front of displacing water, and near discontinuities in the saturations occur. This paper investigates the utility and accuracy of finite difference equations for reservoir pressures based on two new functional forms: ln(r) and 1/r, where r is the distance to the well. The ln(r) form is based on pressures from line sources, and thus is effective at representing straight line wells. The 1/r form is based on pressures from point sources. The sum of many points represent more complex wells. Both are found to greatly increase the accuracy of the simulated reservoir pressures relative to solutions based on the polynomial approach.

A New Method for the Rapid Calculation of Finely Gridded Reservoir Simulation Pressures

“…extends the work of Weber et al 27 Weber demonstrated accuracies for the pressure solution normally requiring millions of cells using traditional finite-difference equations with only hundreds of cells. This was accomplished through the use of finite-difference equations that incorporate the physics of the flow.…”

“…The new pressure-solution method developed in this thesis is based on the work of Weber et al 27 They 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 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 wells).…”

A Streamline Reservoir Simulator With Dynamic Gridding

“…These results are intuitively satisfying, but may be surprising to anyone who has simulated with such a course grid using a traditional reservoir simulator. The precision of the circular saturation contours, as well as the smoothness of the stereamlines, results from the new Weber 13 finite difference equations that include ln(r)-terms in their formulation. Figure 4 shows that the dynamic grid solution for saturation on the stream lines provides a sharp flood front.…”

“…Details of the derivation can be found in Weber et al 13 , who show several orders of magnitude improvement in the error in calculated pressures. A pressure drop proportional to ln(r) occurs around infinite, straight-line sources and sinks.…”

“…Such is the case for the pressure equation where nearsingularities appear at the wells. Weber et al 13 have…”