Reservoir Engineering-General - The Material Balance as an Equation of a Straight Line

The material balance equation used by reservoir engineers is arranged algebraically, resulting in an equation of a straight line. The straight line method of analysis imposes an additional necessary condition that a successful solution of the material balance equatiott should meet. In addition, this algebraic arrangement attaches a dynamic ineuning to the otherwise static material balance equation. The straight line method requires the plotting of one variable group vs mother variable group. The sequence of the plotted points as well as the general shape of the resulting plot is of utmost importance. Therefore, one cannot progrm the method entirely on a digital computer ar is usually done in the routine solution of the material balance equation. If this method is applied, then plotting and anaIysis are asential. Only the appropriate equations and the method of analysis and interpremtion with comments and discussion are presented in this paper. Illustrative field examples for the various cases treated are deferred to a subsequent writing. INTRODUCTION One of the fundamental principles utilized in engineering work is the law of conservation of matter. The application of this principle to hydrocarbon reservoirs for the purpose of quantitative deductions and prediction is termed "the material balance method of reservoir analysis". While the construction of the material balance equation (MBE) and the computations that go with its application are not difficult tasks, the criteria that a successful solution of the MBE should fulfill have always been a problem facing the reservoir engineer. True and complete criteria should embody necessary and dcient conditions. The criteria which the reservoir engineer uses possess a few necessary but no sufficient conditions. Because of this, the answers obtained from the MBB are always open to question. However, the degree of their acceptability should increase with the increase in the number of the necessary conditions that they should satisfy. Generally, the necessary conditions commonly used are (1) an unspecified consistency of the results and (2) the agreement between the MBE results and those determined volumetrically. This second criterion is usually overemphasized. Actually, the volumetrically determined results are based on geological and petrophysical data of unknown accuracy. In addition, the oil-in-place obtained by the MBE is that oil which contributes to the pressure-production history,' while the volumetrically calculated oil-in-place refers to the total oil, part of which may not contribute to said history. Because of this difference, the disagreement between the two answers might be of paramount importance, and the concordance between them should not be overemphasized as the measure of correctness of either one. In this paper, a third necessary condition of mathematical as well as physical significance is discussed. It is not subject to any geological or petrophysical interpretation, and as such, it is probably the most important necessary condition. It consists essentially of rearranging the MBE to result in an equation of a straight line. This straight line method of the MBE solution has invalidated a few long time accepted concepts. For instance, it has always been advocated that if a water drive exists, but one neglects to take it into account in the MBE, the calculated oil-in-place should increase with time. The straight line method shows that in some cases, depending on the size of the neglected aquifer, the calculated oil-in-place might decrease with time. The straight line method requires the plotting of a variable group vs another variable group, with the variable group selection depending on the mechanism of production under which the reservoir is producing. The most important aspect of this method of solution is that it attaches a significance to the sequence of the plotted points, the direction in which they plot, and to the shape of the resulting plot. Thus, a dynamic meaning has been introduced into the picture in arriving at the final answer. Since the emphasis of this method is placed on the interpretation of the sequence of the points and the shape of the plot, one cannot completely automate the whole sequence to obtain "the best value" as normally done in the routine application of the MBE. If one uses the straight line method, then plotting and analysis are musts. The straight line method was first recognized by van Everdingen, et al,2 but for some reason it was never fully exploited. The advantages and the elegance of this method can be more appreciated after a few cases are carefully treated and worked out by it. SOLUTION OF THE MATERIAL BALANCE EQUATION SATURATED RESERVOIRS The MBE for saturated reservoirs written in AIME symbols is