Statistical Basis for Choice among Hyperbolic Decline Curves and Computer Application in Calculating Confidence Limits of Reserve Predictions
Authors: McCray A.W. , Comer A.G. 
Published: December 2007

American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc.
This paper was prepared for the 42nd Annual Fall Meeting of the Society of Petroleum Engineers of AIME, to be held in Houston, Tex., Oct. 1–4, 1967. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal provided agreement to give proper credit is made.
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Abstract
The extrapolation of oil well production decline data into the future has long been accomplished through the use of semi-log and log-log plots of the data. These commonly used methods encompass a family of hyperbolic decline curves. The object of such extrapolations is to determine the future production capacity of the oil producing unit.
Concepts from statistics were applied to the analysis of oil well production decline data. Means were developed to manipulate the data so that it presented an array of representative possible positions in the family of hyperbolic decline curves. This permitted a comparison of the degree of fit between the data and the several hyperbolic curves and the determination of the particular hyperbolic curve which would produce the best possible fit. These concepts were incorporated into a computer program. The method is equivalent to finding the position on a semi-log or log-log plot where the data points form a straight line.
The statistical analysis of the production decline data permitted the determination of upper and lower confidence limits of the extrapolated estimates of future production. Thus it was possible to estimate not only the expected amount of future production but to add to this knowledge the upper and lower confidence limits, such as 90 per cent confidence limits, of the maximum and minimum amounts of the reserves indicated by the analysis of the data.
In order to illustrate the techniques presented by the authors, five actual field cases are included. At least two of these cases illustrate the pit falls that are encountered when using only the apparent best fit.

Introduction
One of the basic methods of estimating the most probable future life and production of an oil well is the use of decline curves. There is a long history of the use of this method, wherein generally the production rate is plotted vs. time, and the plotting of the past productions per unit time permits extrapolating the trend of the past into the future. The method is particularly useful when the well is produced at its natural capacity, which generally decreases with time.