A New Approach to the Hyperbolic Curve (includes associated papers 18728 and 18942 )
- D.R. Long (Williamson Petroleum Consultants Inc.) | M.J. Davis (Williamson Petroleum Consultants Inc.)
- Document ID
- Society of Petroleum Engineers
- Journal of Petroleum Technology
- Publication Date
- July 1988
- Document Type
- Journal Paper
- 909 - 912
- 1988. Society of Petroleum Engineers
- 5.7 Reserves Evaluation, 4.3.4 Scale, 2.4.3 Sand/Solids Control
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- 494 since 2007
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Summary. An efficient and timesaving approach to hyperbolic-decline-curve analysis has been developed that has made it possible to determine the hyperbolic b exponents characterized by thousands of wells. After extensive use of the technique, we concluded that the range of the b exponent previously prescribed by Arps is too narrow.
The extrapolation of decline curves is a valuable yet often misused tool for predicting the future production rate and ultimate recovery of a producing well. Despite all the evidence of the common applicability of hyperbolic curves to the extrapolation of historical production performance, many engineers continue to use only the constant-percentage-type decline analysis, mainly because of thdifficulty historically associated with use of the hyperbolic curve. While some engineers may use a French curve or a series of constant-percentage declines with sequentially decreasing declinrates to compensate for the conservative results of the constant-percentage-decline assumption, these procedures have neither the experience nor the theoretical background documentation available for hyperbolic declines.
This paper emphasizes several circumstances, among many observed throughout the U.S., where the production curve exhibits a decline exponent b is greater than 1.0 and demonstrates the ease with which engineers may use hyperbolic decline curves by using overlays in estimating future production rate and ultimate recovery. Relatively high b exponent values make the correct use of the hyperbolic curve more essential than ever before. When this overlay technique is 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 from 0- b- 1.0. About 25 years passed before authors began to question this conclusion, even then restricting themselves to special cases in specific areas. Gentry and McCray demonstrated the theoretical consistency of b exponents >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 saw this 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 gas wells in the Wattenberg field in Colorado. From experience with the technique described here, widespread evidence exists that the b exponent exceeds 1.0 in many areas.
We hope that the observations presented in this paper and the hyperbolic-curve-analysis technique described will lead to a better understanding of the hyperbolic decline equations and will encourage practicing engineers, where applicable, to make greater use of the hyperbolic decline curve.
The extrapolation of production decline performance with rate/time plots of oil and gas volumes historically produced is one of the primary tools used by the petroleum engineer to evaluate producing properties. The most common decline-curve technique involves plotting monthly volumes on a logarithmic scale vs. time on a linear scale and extrapolating the observed trend into the future as a straight line. This extrapolation. or constant-percentage-decline projection, portrays a decline exponent b = 0 and accurately characterizes the production performance of many producing wells. The b exponent is the rate of change of the slope function, a,, of the curve with respect to time. Hyperbolic declines frequently and with better reliability describe the anticipated trend of future production for selt wells. These declines exhibit a concave upward appearance on graphic 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 extrapolating production rates because the procedures have been complicated, not well understood, and/or time consuming. The prevailing technique used in the past to fit historical data to a hyperbolic curve by graphic methods involved repetitive plotting of data points using trial-and- error techniques to obtain a straight line. One overlay technique commonly requires many different overlays. Computer software has 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 awkwardness generally inherent in hyperbolic-decline-curve-analysis techniques, there are other restraining factors. Some of the petroleum economic cash flow programs do not support b exponents >1.0. Another restraint stems from the continued flattening of the hyperbolic curve with time, which may lead to unrealistically high projected lifetimes and reserve estimates. To compensate for these prevailing criticisms, many engineers use a series of sequentially decreasing exponential decline rates in the construction of type curves; however. this does not depict the true character of a well's performance or provide for determination of the b exponent that may be characteristic and valuable in the analysis of other nearby wells.
An overlay technique was developed to permit the engineer to extrapolate the hyperbolic decline curve of a production plot of semilog rate vs. time with almost as much ease as a straight line constant-percentage-decline projection. This technique uses a suite on type curves plotted on a clear plastic overlay that has been custom designed for the scale of graph paper being used. The type curves are marked to exhibit instantaneous equivalent annual decline rates. In addition to its simplicity, this technique provides the engineer with a straightforward visual understanding of historical performance and a clear grasp of potentially expected future production trends.
The type-curve-overlay technique is similar in theory to the methods presented by Sliders and Fetkovich. Each technique recognizes the importance of a unique b exponent in describing the historical production trends of a well. The new overlay, however, offers several advantages. Slider's approach requires a different overlay for each b exponent, which causes the engineer to sift through any number of overlays in his attempt to match historical data to the indigenous b exponent.
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