A Simplified Method of Hyperbolic Decline Curve Analysis
Authors: Slider H.C. 
Published: December 2007

Present decline curve analysis methods encompass both mathematical and graphical approaches. Most engineers now use only the constant percent decline (exponential) analysis.
The method presented herein proposes the use of several simple hyperbolic curves plotted on transparent paper. Once these basic curves have been prepared, a curve-matching technique can be used to obtain hyperbolic extrapolations with about the same effort required for a constant percentage decline extrapolation.
In hyperbolic decline the decline rate varies with the production rate.(1)
where a is the decline rate when the production rate is q, and ai and qi are the decline rate and production rate when time t=0. The constant n is between 0 and 1.0 and is characteristic of a particular hyperbolic decline. This n can be shown to be equal to the change in (1/a) with time,
(2)
It also can be shown mathematically that for hyperbolic decline
(3)
Also, the total production Delta Np during the hyperbolic decline from a rate qi to a rate q is
(4)
The proposed curve-matching method offers a simple, efficient and effective solution to the curve-fitting problem. A plot of production rate vs time is prepared on semilog paper. Using a set of general, preconstructed decline curves, each representing a different combination of n and ai, the best fit to the production curve can be found.
Fig. 1 illustrates a group of such curves representing an n of 0.5 and a variety of initial decline rates. Note that data for such curves can be calculated from Eq. 3 by using negative times and defining ai as the decline rate when q/q, is 1.0.
Once the constants it and a, and the producing rate corresponding to a q/q, of 1.0 has been obtained by curve fitting, the future production rates, reserves and remaining life can be calculated from Eqs. 3 and 4 or obtained graphically from a series of curves illustrated by Fig. 2.
The procedure for determining the best fit of the rate-time curve is relatively simple. Make transparent overlays of a series of figures (such as Fig. 1) representing various values of n between 0 and 1.0. Place these overlays over actual production rate-time curves plotted on graph paper of the same size and then shift them, keeping the axis of the overlay and actual data parallel. Find the curve that best fits and note n and a, and the actual rate q. corresponding to a q/q, of 1.0.
The proposed curve-fitting technique is illustrated in Fig. 3. The plotted points are actual production vs time. The curve shown is the fit obtained using an overlay of the n = 0.5, a, = 0.001/day curve from Fig. 1. The position of the axis of the subject plot necessary to give this curve fit indicates that the decline rate of 1.5 months after Jan., 1941, is 0.001/day. This corresponds to the (q/qi) = 1.0 or 0 time of the overlay curve. The actual production rate corresponding to this point is about 1,800 bbl of oil a month (BOPM). This curve fit was chosen so that Fig. 3 would be readable. In actual practice it is advisable to fit the data with a curve whose zero point is as close as possible to the last data point to obtain the greatest extrapolation accuracy. The zero time may actually correspond to some time beyond the last data point.
With the following data fixed by the curve match of Fig. 3 (n = 0.5, a, = 0.001/day and qi = 1,800 BOPM), a production rate can be calculated for any future time.
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