Abstract

The Method of Characteristics was proposed as a fast method of reservoir simulation many years ago. When gravitational effects are absent, the method results in a reservoir simulation technique identical to that of currently popular streamline models. However, the characteristic method rigorously includes gravity, whereas the streamline method does not. The characteristic method has never become popular, probably because of the lack of a method for locating the flood fronts in multiple dimensions. This problem is widely discussed in the mathematical literature, and has become a paradox of some renown.

This paper describes the use of a local material balance at points throughout the reservoir as a criterion for determining the location of the fronts. The availability of this criterion makes the characteristic simulation method practical. The front location technique is fast, taking approximately the same amount of time as determining the saturations elsewhere. Hence the method offers the speed of streamline simulators combined with the greater accuracy and versatility resulting from a rigorous treatment of gravitational effects.

Introduction

The Method of Characteristics provides a means of solving partial differential equations by providing an equivalent set of simultaneous ordinary differential equations. It is an exact, analytical method, although approximate, numerical methods are often required to solve the resulting ODE's. The method has been used for a long time. It is discussed in classical mathematical textbooks such as Currant and Hilbert.i However, it has remained unpopular, and relatively unknown, probably because of its limited applicability. It is applicable only to hyperbolic PDE's. Partial differential equations can be divided into three classes, elliptic, parabolic, and hyperbolic. Determination of a particular equation's type can be made through a sometimes complicated analysis of its descriminant. However, its type is more readily determined from the behavior of its solution to initial value problems. If the solution dissipates abrupt changes in the initial values of the unknown as it progresses, the equation is parabolic. If such changes move about the solution space without variations in their magnitude, the equation is elliptic. If the solution produces discontinuities as it progresses, when none existed originally, the equation is hyperbolic. The most widespread use of the method of characteristics is probably in the study of supersonic fluid flows and their associated shock waves. Meteorological atmospheric pressures spontaneously form fronts, as well. Petroleum reservoir saturations also form fronts as one reservoir phase displaces another. Hence, the saturation equation used for reservoir simulation has the characteristics of a hyperbolic equation. Strictly speaking, the saturation equation is hyperbolic only when capillary pressures are neglected. However, capillary forces tend to disperse the front over a few inches to a few feet depending on the velocity of the front. In most instances, particularly for large reservoirs, capillary effects on the flood front can be neglected.

The first use of the method of characteristics for reservoir simulation was probably by Buckley and Leverett. 2 Their classical solution is now familiar to almost every reservoir engineering student, as it is covered in detail in popular reservoir engineering texts.3,4

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