Reservoir Engineering–General - A Numerical Solution to the Unsteady-State Partial-Water-Drive Reservoir Performance Problem

Solutions to the unsteady-state partial water-drive reservoir performance problem can be obtained through the use of analogue computers or high-speed electronic digital computers. The solutions that have previously been resolved for use on digital computers, however, demand a knowledge of the aquifer parameters. Generally, the analogue computers now in use do not require this knowledge. A solution is presented herein where the aquifer performance, expressed in terms of difference equations, is related to the reservoir performance as expressed by the modified Scbiltbuis material-balance equation. A numerical procedure for a medium-sized digital computer also is presented in which a solution to the set of equations defining the aquifer and reservoir performance is obtained and the aquifer parameters (permeability and sand thickness) are automatically optimized while simultaneously matching the known pressure behavior. Predicted pressure behavior can be calculated using rates from any assumed future production practice. The procedure provides an output format which presents the cumulative, incremental and average rate of natural water influx, the per cent gas-cap expansion, the calculated reservoir pressure, the measured reservoir pressure, and the difference between the measured and calculated pressures. Results of a test problem are presented in comparison with results obtained by the Bruce Analyzer, Ohio Oil Co. 's Pace General Purpose Analogue Computer, and Sun Oil Co.'s Single-Pool Electronic Reservoir Analyzer. These results indicate that unsteady-state reservoir perlormance for a single-pool system can be adequately simulated by a numerical method employing a digital computer and that the special-purpose analogue computer can be supplanted by this method. INTRODUCTION The numerical approach to the unsteady-state partial water-drive reservoir performance problem heretofore has been limited in that solutions have been resolved only for certain fixed parameters. Also, prior to the advent of high-speed electronic digital computers in the last few years, calculations were so tedious and time-consuming that they ordinarily were neglected, or some other means of obtaining the solutions were used. The analogue approach had been used for obtaining solutions to the transmission-of-heat problem and, consequently, means were developed for applying the electrical analogue to the unsteady-state reservoir performance problem. Bruce submitted a paper for AIME publication in Sept. 1942, in which he described an electronic device (now called the Bruce Analyzer) for analyzing the water-drive performance of any reservoir, based on either steady or unsteady-state flow characteristics. 1 Sun Oil Co. adopted the analogue approach in 1949 and now has two special-purpose high-speed electronic reservoir analyzers in use.2,3 Two main features of Sun's analyzers are (1) the high repetitive rate of reproducing the entire history of reservoir performance cyclically a number of times per second, and (2) the elimination of the necessity of knowing the properties of the associated aquifer. Since the advent of larger and faster digital computers, the possibility of incorporating this second feature into a numerical approach appeared feasible. Previous work has been done and material published covering the application of numerical methods to the behavior of the pressure with respect to time at any point in an associated aquifer? However, the aquifer parameters must be known from some source. The material presented in Ref. 4 has been used to a great degree in developing the numerical method described herein. DIFFERENTIAL EQUATION AND BOUNDARY CONDITIONS The problem of simulating the performance of a reservoir and the associated aquifer is to determine the characteristics of the aquifer such that the