A Characteristic-Based Reservoir Simulator with Gravitational Effects

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 criterion for locating the flood fronts in multiple dimensions. This problem is widely discussed in the mathematical literature, and seems to have become a classical paradox.

This paper describes the use of a local material balance at oints throughout the reservoir to establish the location of the front. 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 locating the constant saturation lines or characteristics. Hence, the method offers the speed of streamline simulators combined with the greater accuracy and versatility resulting from a rigorous treatment of gravitational effects.

Examples of simulations from the new characteristic simulator are provided.

Introduction

The method of characteristics provides a means of solving partial differential equations (PDE) in which a set of simultaneous ordinary differential equations is derived which is equivalent tothe original PDE. This is an exact, analytical method, although approximate, numerical methods are often required to solve the resulting ordinary differential equations (ODE). The method has been used for a long time. It is discussed in classical mathematical textbooks such as Currant and Hilbert ⁽¹⁾. 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 behaviour 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 elliptic. If such changes move about the solution space without variations in their magnitude, the equation is parabolic. 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⁾.