Solution of the Equations of Unsteady State Two-Phase Flow in Oil Reservoirs

Abstract

Published in Petroleum Transactions, AIME, Volume 201, 1954, pages 217–229. Abstract One of the most important problems of reservoir mechanics is the prediction of the behavior of a gas and oil producing reservoir for a variety of production programs. In the absence of a knowledge of the behavior of even idealized systems, it has been difficult, if not impossible, to determine which production program would yield maximum economic recovery. The problem was formulated mathematically many years ago, but a solution of the equations had never been obtained because of their extreme complexity. The solution of the problem became possible, however, with the advent of modern computing machines. This paper describes a digital method of solving the problem and discusses some of the results which have been obtained to date. The IBM Type 701 Electronic Data Processing Machines located at IBM's New York Scientific Computing Service were used successfully to perform the computations. The flow of gas and oil through a reservoir can be described mathematically by two second-order, nonlinear partial differential equations which must be solved simultaneously. All derivatives were replaced by finite differences to bring the equations into a form suitable for digital calculations. A number of finite difference formulations were investigated and one form was found which resulted in stable, convergent solutions. This form was used to obtain solutions for both radial and linear systems producing by solution gas drive. The results presented in this paper are:saturation profile within the hypothetical reservoir vs. time,pressure profile within the hypothetical reservoir vs. time,gas-oil ratio vs. cumulative recovery, andultimate recovery. The saturation profile clearly shows the effect of critical gas saturation. The saturation near the wellbore drops quickly to the critical gas saturation and remains essentially at that value until the remainder of the reservoir also has dropped to the critical gas saturation. The pressure profile shows a smooth variation of pressure across the reservoir. The gas-oil ratio drops slightly during the early stage of production until the critical gas saturation is reached, rises rapidly from then on and eventually passes through a sharp maximum just before the economic limit of production is reached. For the radial example presented in this paper, the average oil saturation of the reservoir at a time corresponding to the limit of economic production was 56.5 per cent and the ultimate recovery was 6.86 per cent. The difference between change in saturation and ultimate recovery is caused by shrinkage of the oil phase.