New York Paper - Mathematical Determination of Production Decline Curves

Numerous papers have been published on the use of graphic methods for determining the best curve for estimating the production decline of oil wells but, as far as the writer has been able to ascertain, nothing has been published showing how curves may be worked out mathematically that will conform most closely to the production data under consideration in any given instance; hence the writer has endeavored to show just how such application can be made. Among the valuation engineers, two types of curves are in current use for the representation of the decline in production of oil wells. One may be represented by the equation, y = krx; this is sometimes known as a percentage curve, for the production for any period is multiplied by a constant rate or percentage to obtain the production for the succeeding period. It appears as a straight line on semilogarithmic cross-section paper. The other curve may be represented by the equation, y = kxn, and is shown as a straight line on logarithmic cross-section paper. In each equation, y represents the number of barrels produced in a given period of time, and x the successive periods of time. The other letters are constants to be determined for each well, or group of wells considered in arriving at the most probable curve to represent the rate of production for a given property. From this the underground reserves may be estimated by extrapolation, or by extending the curve beyond the last known period of production to show the most probable production values for the future. Application to the Burbank Pool Fig. 1 relates to a property in the Burbank Pool, Osage County, Okla. The production is shown as an average per well per month, is plotted to the vertical scale of barrels on the left-hand margin and to the horizontal scale of months at the bottom of the chart. In column y, the number of barrels is listed beginning with May, 1922, as the first month on which to base a decline curve. The type of curve used is of the form, y = krx, or, expressed in logarithms, log y = log k + x log r, in which log k and log r are constants to be determined so that the curve will most closely approximate the production data given.