Optimization OfThe Material Balance Equation

Abstract

Abstract The material balance equation is usually optimized on the basic of the original oil or gas in place. In this paper, an alternate basis for optimization is presented. This involves using reservoir withdrawals as the basis for optimization. Two hypothetical reservoir examples are examined:an oil reservoir with a gas cap; anda gas reservoirs under water drive. It is shown that for reservoir with a large gas cap (no water drive), available method are inadequate and optimization on the basis of cumulative reservoir withdrawals rather than oil-in-place is a more satisfactory approach to determining system parameters. For the gas reservoir examined in this study, it is shown that the approach suggested in this paper compares favorably with arrangements suggested in the literature. The effect of system parameters and degree of reservoir depletion, as well as uncertainties in reservoir pressure, are presented. INTRODUCTION THE ZERO-DIMENSIONAL material balance equation has been the historical basis for performance-matching estimation of the initial oil-in-place, the ratio of the initial gas-cap volume to the initial oil-column volume and the quantitative nature of water influx. Generally, the material balance equation proposed by Schilthuis (1) is arranged as a straight-line equation. This was first recognized by van Everdingen, Timmerman and McMahon(3). The straight-line method involves plotting one group of variables (the dependent variable, expansion) versus another group (the independent variable, withdrawals). The specific terms in each group, of course, are dependent on the problem under consideration. Havlena and Odeh(2) provide an excellent documentation of the straight-line approach and examine its application to various hypothetical oil and gas reservoirs. The straight-line technique implicitly assumes that all of the dependent variable. Thus, as pointed out by McEwen(4), the material balance equation must be arranged so as to insure that this condition is satisfied. (Agarwal, AlHussainy and Ramer(3) have also pointed out that some forms of the material balance equation are more reliable in the computational sense than others.) In most practical cases, however, both the dependent andNow with Petrobras, Brazil.Now at the University at Tulsa. independent variables are subject to error. In order to account for both the dependent and independent variables, Wall and Craven- Walker(C) proposed an alternate straight-line method whereby both the variables in the material balance equation can be subject to error. The straight-line approach suggested in Ref. 2 uses the oil (or gas) in place as one of the parameters for optimization. The principal disadvantage of this approach is that no observed values of this parameter, which incidentally happens to be the principal parameter of interest, are available. To eliminate this limitation, Tehrani(2) proposed that cumulative reservoir withdrawals rather than the original oil-in-place be used as the basis for optimization. This proposal has the advantage of using observed values as the basis for optimization. However, he has not presented detailed calculations using either real or hypothetical reservoir system to demonstrate that the use of reservoir cumulative withdrawals is more advantageous than the conventional approach.