Decline-Curve Analysis

Abstract

This paper explains a simple and effective method for graphically solving all three types of production decline. The three types of declines are: exponential hyperbolic, and harmonic. The mathematical development of these curves was by Arps. Decline curves are one of the most extensively used forms of data analysis employed in the evaluation of oil properties. Often future production is extrapolated as a straight line on semilog paper (exponential or constant-percentage decline) because this type of decline is the easiest to handle mathematically and graphical. This is done irrespective of the fact that several investigators have reported that this type of decline is rare and that actual oil production usually follows a hyperbolic decline. production usually follows a hyperbolic decline. However, the hyperbolic decline is difficult to analyze mathematically or graphically. The most recent method utilizes transparencies, as proposed by Slider. The method outlined below greatly simplifies the solution and extrapolation of decline curves. The first four columns of Table 1 list the rate:time and cumulative-production:rate relationships as developed by Arps. The equations are all solutions of the differential equation D = Kq = - (dq/dt)/q. In each instance two unknowns must be calculated from the two relationships. They are the decline exponent n and the initial decline rate Di. The third unknown, qi, can be obtained from the production history of the well. First. the rate:time relationship is manipulated to solve for the value of Dit in terms of the ratio (qi/qt). These relationships are shown in Column 5 of Table 1. Next, the rate:time relationship is solved for Di, and this value of Di is substituted into the cumulative-production:rate relationship. This relationship is then solved for the value of Qt/(qit) in terms of (qi/qt), These relationships are shown in Column 6 of Table 1. Two graphs can then be constructed by selecting a value for n and then substituting values of (qi/qt into the relationships. A curve on each graph for the selected value of n will be produced. This can be done for any desired number of n values from 0 n 1. (See Figs. 1 and 2.) These curves can then be used to analyze and extrapolate decline curves from actual production history. P. 38