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.

Introduction

The fact that traditional, Taylor-series based, finite difference equations are inaccurate representing reservoir pressures near the wells in petroleum reservoirs, has long been known. Most simulators do not simulate wellbore pressures directly with finite difference equations, but instead correct simulated well cell pressures to obtain the actual wellbore pressures with a "well equation". Many use an empirical productivity index, PI:

Equation (1) (Available in full paper]

In 1978, Peaceman1 was perhaps the first to suggest a method of calculating the PI, or the difference in the well bore and well cell pressures:

Equation (2) (Available in full paper

Equation (3) (Available in full paper

This expression is based on the pressures in a 2-D, homogeneous, isotropic, reservoir with vertical, fully penetrating wells arranged in a five-spot pattern. The finite difference grid consists of square cells. This expression, "Peaceman's correction", is still widely used despite errors that occur when the geometry and reservoir properties differ substantially those of his study. However, since that pioneering work, Peaceman2,3 and others4–7 have proposed alternative well equations to accommodate non-square well cells, anisotropic permeabilities, and off-center and multiple wells within a grid cell. Ding et al. 8 proposed altered transmissibilities between the well cell and neighboring cells, as a companion to the well equation. However, none of the proposed well equations are adequate for all wells, and the growing complexity of well geometries, including horizontal wells, slant wells, and multilateral completions, makes it difficult to know which if any of the well equation is adequate.

This paper proposes the use of finite difference equations that incorporate the singularities in pressure at the wells. The new finite difference equations accurately represent the actual pressures at the wellbore and elsewhere in the well cells. No well equations are required.

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