ABSTRACT

A brief review is made of the state of the art in computing the motion of sharp flood fronts in immiscible flow. The mathematical and computational difficulties are underlined and methods which are feasible for two-dimensional simulators are selected for further study. The strengths and weaknesses of these methods are discussed and a detailed example given.

INTRODUCTION

Numerical simulation can be an extremely useful tool for the reservoir engineer. However, naive numerical techniques are either inaccurate or prohibitively expensive in cases where there are phase interfaces or fronts which are sharp on a length scale small compared to the scale of the reservoir. It is clearly desirable to find suitable methods for these cases since they can be of real practical importance, as in waterflooding of reservoirs in conditions of nearly constant capillary pressure. Although a number of methods have been proposed, very often they are only suitable for simple one-dimensional problems, and either cannot be extended at all or only at great computing cost.

We present here a discussion of some of the difficulties inherent in computing the motion of sharp fronts in oil reservoirs and of some methods designed to overcome them. Only methods which are suitable for at least two-dimensional problems are considered. It must be emphasized that although we have chosen to illustrate the methods with a simple two-dimensional problem in this paper, they are in no way limited to this and may be generalised quite easily to more complicated systems, such as three-dimensional or compressible flows.

THE EQUATIONS

The essential problems of modelling sharp fronts can already be brought out in the relatively simple case of two-dimensional incompressible flow of oil and water in a horizontal isotropic and homogeneous reservoir.

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