Analog Computer Study of Horizontal Evolution Of Interfaces in a Three-Phase Reservoir
Authors: Goguel D. , Delor M. , Mizzi F. , Ploton Y. 
Published: December 2007

American Institute of Mining, Metallurgical and Petroleum Engineers Inc.
Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers Office. Such discussions may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines.

Abstract
This paper presents a study of the initial single-phase production period in a three-phase, two-dimensional horizontal reservoir. All phases are assumed to be viscous and phases are assumed to be viscous and incompressible. By means of an analog computer, approximate complex potentials are elaborated in every phase, and used to represent the stream lines of particles located on interfaces, Their successive positions are observed by periodically interrupting the strokes of the periodically interrupting the strokes of the pen. pen. Also presented, and using the same mathematical expressions, are the influences of some parameters—well spacing, well location between parameters—well spacing, well location between initial interfaces—upon time of first breakthrough and corresponding areal sweep efficiency.
The results obtained are well suited to an attempt of an economical optimization of oilfield development.

Introduction
The main purpose of this work is to study the influence of relative mobility of a driving fluid with respect to oil over some development parameters of a hydrocarbon reservoir. An parameters of a hydrocarbon reservoir. An attempt was made to achieve this by means of a formal resolution, made possible by several simplifications of physical mechanisms. This method was chosen in order to obtain explicit expressions, at least for some classes of reservoirs. Thus a better knowledge was acquired of the specific influence of spatial distribution parameters of a well line in an oil ring over-sweep areal efficiency at the end of the single-phase production period, and over the length of this period. The method applies to long-shaped reservoirs or to thin oil rings.

STATEMENT OF THE PROBLEM
To be studied is the horizontal evolution [Fig. 1] of a three-phase, two-dimensional reservoir with one oil phase located between two phases [gas and water]. It is assumed that the fluids are incompressible and have a constant viscosity. The saturations are 0 or 1.
An infinite line of identical,, steadily spaced wells is set in the oil phase. Oil is drawn off under a pressure p. Periodicity allows the interest to be concentrated on a single mesh [Fig. 2]. The well line is parallel with the initial rectilinear interface S between oil and gas, and S' between oil and water. Our purpose is to set out the successive positions occupied by the interfaces S and S' positions occupied by the interfaces S and S' between oil and both driving fluids, and to calculate some characteristics of reservoir development: single-phase draw-off time, quantity of oil supplied by each well during this time, instantaneous output, and areal sweep efficiency. The final instant is the first breakthrough.