A Graphical Presentation of Well Productivity And an Application to Gas Lift
Authors: Haskett C.E. 
Published: December 2007

Abstract
A method of graphical presentation of the relation between reservoir pressure, productivity index, flowing bottom-hole pressure, flow rate and tubing pressure which permits the illustration of multiple calculations on a single sheet of graph paper and simplifies the representation of well productivities under changing conditions of reservoir pressure, productivity index, etc., is demonstrated. A variation of the graph is used to illustrate its application to gas lift.

Introduction
Combination of the productivity index equation J = q/p(ws)-p(wf) with a vertical flow calculation such as that of Poettmann and Carpenter allows the engineer to determine flow rate as a function of wellhead pressure. Often, this combination involves a trial-and-error solution to insure compatibility of all the factors involved; more frequently today, computer programs resolve these factors. However, the concise presentation of these data presents a problem, particularly if it is desired to illustrate a number of wells of different productivity indices at the same time, and with varying reservoir conditions. Such a presentation is very useful when, for example, the application of gas lift to part or all of a large field must be demonstrated in view of future as well as present conditions. This paper endeavors to resolve this problem of presenting the results of such calculations in a direct and practical fashion.

Graphical Solution of the Productivity-Index Equation
The productivity index is generally defined as the barrels per day of stock-tank oil production per pound of pressure differential between the wellbore opposite the producing horizon and the static reservoir pressure, which is presumably the pressure at the well's radius of drainage. Usually it is represented by the equation:
qJ =p (ws) - p (wf)
where
J = productivity index, B/D/psi
q = flow rate, B/D
p(ws) = reservoir pressure, psi
p(wf) = flowing bottom-hole pressure, psi.
If one rearranges this as p(ws) p(wf) = q/J, it can be seen that the ratio of production rate to productivity index is equal to the difference between reservoir and flowing bottom-hole pressures. It is possible to plot these four variables around a sheet of coordinate paper (Fig. 1). The ratio of productivity index to flow rate on the ordinates is equal to the difference in pressures on the abscissas.
To illustrate the use of the graph, assume a well productivity index equal to 1.5 B/D/psi and a reservoir pressure of 3,000 psi and plot Point A, as shown. When the flowing bottom-hole pressure is equal to the reservoir pressure, the flow rate must be zero, which yields Point B. A straight line extending through these two points defines this well for all combinations of flow rate and flowing bottom-hole pressures. Thus, a flowing bottom-hole pressure of 2,600 psi would yield 600 B/D, using Line AB. It should be noted that the representation of productivity index as a straight line for varying pressures and flow rates is only true for the flow of a monophasic fluid. As production proceeds below the bubble point, or in a water drive reservoir as water encroachment increases, the productivity index would develop a curve in a downward direction at higher flow rates and lower bottom-hole pressures. Where this becomes critical, it would be necessary to plot the appropriate curve in place of the straight line described above; but, in general, the straight line representation will probably suffice.