A New Approach to the Hyperbolic Curve (includes associated papers 18728 and 18942 )
Authors: Long D.R. , Davis M.J. 
Published: December 2007

Summary.
An efficient and timesaving approach to hyperbolic-decline-curveanalysis has been developed that has made it possible to determine thehyperbolic b exponents characterized by thousands of wells. Afterextensive use of the technique, we concluded that the range of the bexponent previously prescribed by Arps is too narrow.

Introduction
The extrapolation of decline curves is a valuable yet often misused tool for predicting the future production rate and ultimaterecovery of a producing well. Despite all the evidence of the commonapplicability of hyperbolic curves to the extrapolation of historicalproduction performance, many engineers continue to use only theconstant-percentage-type decline analysis, mainly because of thdifficultyhistorically associated with use of the hyperbolic curve. While some engineers may use a French curve or a series ofconstant-percentage declines with sequentially decreasing declinrates tocompensate for the conservative results of the constant-percentage-decline assumption, these procedures have neither theexperience nor the theoretical background documentationavailable for hyperbolic declines.
This paper emphasizes several circumstances, among manyobserved throughout the U.S., where the production curve exhibitsa decline exponent b is greater than 1.0 and demonstrates the ease withwhich engineers may use hyperbolic decline curves by using overlays inestimating future production rate and ultimate recovery. Relatively high b exponent values make the correct use of the hyperboliccurve more essential than ever before. When this overlay techniqueis used to match the history of a producing well. the engineer acquires a visual perception of the uniqueness of the solution that is not readily apparent in a computer's numerical solution.
When Arps presented his analysis of decline curves in 1944, he concluded that all hyperbolic decline exponents ranged from0- b- 1.0. About 25 years passed before authors began to question thisconclusion, even then restricting themselves to special casesin specific areas. Gentry and McCray demonstrated thetheoretical consistency of b exponents &gt;1.0 with a reservoir model. Brown et al. attributed high b exponents to transient flow behavior in the Hay Reservoir of Wyoming. McNuity and Knapp also sawthis in a selected well in Oklahoma. Bailey noted a wide range(0. 2 - b - 3.4) of the b exponent in his analysis of producing gaswells in the Wattenberg field in Colorado. From experience withthe technique described here, widespread evidence exists that theb exponent exceeds 1.0 in many areas.
We hope that the observations presented in this paper and thehyperbolic-curve-analysis technique described will lead to a betterunderstanding of the hyperbolic decline equations and will encouragepracticing engineers, where applicable, to make greater use of thehyperbolic decline curve.

Decline-Curve Analysis
The extrapolation of production decline performance with rate/time plots of oil and gas volumes historically produced is one of theprimary tools used by the petroleum engineer to evaluateproducing properties. The most common decline-curve technique involvesplotting monthly volumes on a logarithmic scale vs. time on a linearscale and extrapolating the observed trend into the future as a straightline. This extrapolation. or constant-percentage-decline projection, portrays a decline exponent b = 0 and accurately characterizes theproduction performance of many producing wells. The b exponentis the rate of change of the slope function, a, of the curve withrespect to time. Hyperbolic declines frequently and with betterreliability describe the anticipated trend of future production for seltwells. These declines exhibit a concave upward appearance ongraphic production plots of semilog rate vs. time.
The curvature or bending feature of the hyperbolic decline curve has not been conducive to widespread use in extrapolatingproduction rates because the procedures have been complicated, not wellunderstood, and/or time consuming. The prevailing technique usedin the past to fit historical data to a hyperbolic curve by graphicmethods involved repetitive plotting of data points usingtrial-and- error techniques to obtain a straight line. One overlaytechnique commonly requires many different overlays. Computer softwarehas generated a variety of curve-fitting solutions that are difficult to carry around in a briefcase.
In addition to the excessive amount of time or awkwardnessgenerally inherent in hyperbolic-decline-curve-analysis techniques, there are other restraining factors. Some of the petroleum economic cashflow programs do not support b exponents &gt;1.0. Another restraintstems from the continued flattening of the hyperbolic curve withtime, which may lead to unrealistically high projected lifetimes andreserve estimates. To compensate for these prevailing criticisms, many engineers use a series of sequentially decreasing exponentialdecline rates in the construction of type curves; however. this doesnot depict the true character of a well's performance or provide for determination of the b exponent that may be characteristic andvaluable in the analysis of other nearby wells.
An overlay technique was developed to permit the engineer toextrapolate the hyperbolic decline curve of a production plot ofsemilog rate vs. time with almost as much ease as a straight lineconstant-percentage-decline projection. This technique uses a suite ontype curves plotted on a clear plastic overlay that has been customdesigned for the scale of graph paper being used. The type curvesare marked to exhibit instantaneous equivalent annual decline rates. In addition to its simplicity, this technique provides the engineerwith a straightforward visual understanding of historical performance and a clear grasp of potentially expected future production trends.

Type-Curve Overlay
The type-curve-overlay technique is similar in theory to the methods presented by Sliders and Fetkovich. Each technique recognizesthe importance of a unique b exponent in describing the historicalproduction trends of a well. The new overlay, however, offersseveral advantages. Slider's approach requires a different overlay foreach b exponent, which causes the engineer to sift through anynumber of overlays in his attempt to match historical data to theindigenous b exponent.
JPT
P. 909^