Universal Approach to the Decline Curve Analysis
Authors: Shirman E.I. 
Published: March 2010

Abstract
The paper presents a new approach to decline curve analysis. In contrast to conventional methods, no trial-and-error or curve matching techniques are required to find the decline curve coefficients even for the hyperbolic decline case. To develop the method, a differential equation expressing the decline rate was manipulated to produce a unique linear relation between production rate and its first derivative with respect to time. A linear regression or an eyeball curve fitting may be performed on these variables to obtain decline curve parameters. This method allows the graphical presentation of dimensionless production rate versus dimensionless time in a manner that makes prediction of future production possible by extrapolation of a linear trend. This makes the presentation of hyperbolic decline practical and easier to grasp by non-technical personnel. The accuracy of the proposed method is equal or superior to conventional trial-and-error methods. A field example is presented in the paper.

Introduction
Decline curve analysis is one of the most extensively used techniques of oil and gas reserve evaluation. The principle of the method is to find an equation, which fits the observed rate-time performance graph, and then to use this formula to extrapolate the graph into future. Three different rate-time equations are used to match production history: exponential, hyperbolic, and harmonic.
In most cases production will decline at a decreasing rate. The rate of decline is expressed by:
Equation 1 (available in full paper)
The decline rate is also approximated by a power function:
Equation 2 (available in full paper)
Substitution Equation (2) into Equation (1) yields a differential equation, which can be solved for production rate, q.
Equation 3 (available in full paper)
When n = 0, production rate decreases exponentially; when n=1, the decline becomes harmonic. For the hyperbolic decline 0 &lt; n &lt; 1. Thus, both exponential and harmonic decline curves are the special cases of the generic hyperbolic decline.
Cutler(1) stated that most decline curves, normally encountered, are hyperbolic with values of n between 0 and 0.7. Arps(2) reduced the maximum of this interval to 0.4. Lefkovits and Matthews(3) found that for certain conditions of gravity drainage, n = 0.5. Fetckovich(4) established that hyperbolic decline curve analysis has theoretical basis. He also uses hyperbolic decline as a diagnostic technique showing that decline exponent, n, ranges from 0 to 1/2 for gas reservoirs and from 1/3 to 2/3 for gas driven reservoirs.
The conventional equation, of hyperbolic decline is given by Arps(2):
Equation 4 (available in full paper)
Even though, the hyperbolic decline is the most common and typical decline, other types of decline analyses are more frequently used in practice due to their simplicity. Namely these are exponential and harmonic declines. Equations (5) and (6) present the production-time relation for these types of decline, respectively.
Equation 5,6 (available in full paper)
To apply one of the above-mentioned equations to the production data, an analyst should first assume applicability of the corresponding types of decline.