Analysis Of Production-Performance Graphs
Authors: Purvis R.A. 
Published: March 2010

Abstract
This paper presents graphical techniques for interpreting decline curves which utilize all available production, performance and operational data. The historical data is subdivided into correlation segments (time intervals) during which a constant choke size, pump speed, line pressure or plant capacity constrains production. The slope (decline or incline) of the production and performance graphs in a correlation segment are interdependent. Forecasts are similarly interdependent and are subject to appropriate constraints. The techniques are applicable to both single-well and aggregated data for a pool, project or lease, and are demonstrated by the Alberta examples which emphasize identification of the factors contributing to the observed production decline.

Introduction
The primary application of decline curve analysis is to forecast future production which in turn is used to estimate reserves and property values. Such forecasts are usually linear extrapolations of historical trends. However, any extrapolation is strongly affected by the variable transformations used to obtain a linear relationship. It is also implicitly assumed that the factors which caused any discernible trend to develop in the past will continue into the future. Production declines are commonly classified as exponential, hyperbolic or harmonic, depending on the value of the decline exponent(1, 2).
For exponential decline, n = 0 and the rate of decline is a constant fraction of the production rate. For this special type of decline
q = qi e−D i t
Therefore, exponential decline graphs as a straight line when the logarithm of production rate is plotted vs time. A graph of rate vs cumulative production is also a straight line on linear coordinates with a slope of D i,
With hyperbolic decline, 0 &lt; n &lt; 1, and the rate of change of production rate is proportional to a fractional power of the instantaneous production rate. In practice, n is seldom known apriori and a trial and error approach is required to determine the value of n. Therefore hyperbolic decline is not well suited for graphical (visual) analysis.
For harmonic decline, n = 1 and the rate of change in production rate is directly proportional to the instantaneous production rate. The rate-time relationship is
q = q i/ (1+D i t)
The apparent reciprocal relationship between q and t requires a trial and error method to estimate D i and therefore is seldom used in routine graphical methods.
Schematic rate-time and rate-cumulative graphs for the three types of decline are illustrated on linear, semi-log and log-log coordinates in Figure 1. The production rate was assumed to decline to one-tenth ts initial value. The initial decline factor of 0.23 was calculated to result in a ten-year life for the exponential decline nd was applied to all three types of decline. Figure 1(A) shows that for the same initial decline, initial and final rates, the producing lives are 10, 19 and 39 years for the exponential, hyperbolic and harmonic declines, respectively. It should be noted that during the first two years the decline types are indistinguishable. The corresponding rate-cumulative graphs are shown in Figure 1(B).