Two-rate IPR Testinga Practical Production Tool
Authors: Richardson J.M. , Shaw Albert H. 
Published: March 2010

Abstract
The Inflow Performance Relationship (IPR) of W.E. Gilbert and J. V. Vogel is expanded to include both saturated and undersaturated reservoirs. The equations derived have the flexibility to vary the IPR curvature to account for differing fluid properties of crude oils. The effects of water production are included.
A method is presented to define the IPR curve using data from two or three production tests. This permits the calculation of effective formation pressure for the well concerned. It eliminates the need for extended shut-in production periods to obtain reservoir pressure. The use of this technique in stratified reservoir is discussed, showing how it can detect high- or low-pressure water stringers and explain water/oil ratios which increase or decrease with varying production rates.

Introduction
In calculating the productivity of an oil well, we employ the concept that inflow into the well depends on the difference in pressure between the reservoir and the well bore.
With ps (static reservoir pressure) fixed and Pwf(bottom-hole flowing pressure) variable, then ps – Pwf (drawdown) will determine q (the flow rate). Flow rate will increase with drawdown, with qmax (maximum flow rate) occurring at 100% drawdown or zero bottom-hole flowing pressure. There will be a locus connecting the points labelled ps and qmax on the vertical and horizontal axes, providing the relationship (Fig. 1) of flow rate versus pressure.
We have developed a functional relationship relating flow rate to bottom-hole flowing pressure. The relationship has been made simple, involving only a straight line and I or a quadratic curve. However, it is more general than some earlier models, and does seem to provide a satisfactory fit for the situations we have encountered. Should our model not be sufficiently flexible for certain purposes, more complex formulas can be employed.
Evinger and Muskat(1) (1942) introduced the concept of Productivity Factor, closely related to Productivity Index (PI), the rate at which flow increases with drawdown, and pointed out that it would not be constant for two-phase flow (fluid and gas). Gilbert(2) (1954) introduced the term Inflow Performance
(Figure in full paper)
Relationship (IPR) for the relationship between flow rate and bottom-hole flowing pressure. Vogel(3) (1968) provided an easily understood method for the application of IPR techniques for solution-gas drive reservoirs. Patton(4) (1980) employed a straight line for the locus when bottom-hole pressure exceeds bubble-point pressure, and a Vogel (quadratic) locus for bottom-hole pressure below bubble-point pressure.
References are given at the end of this paper to others who contributed to the evolution of the present concepts.

Downhole Flowing Pressure Exceeds Bubble Point
If both Ps and Pwf are greater than Pb (the bubble-point pressure), we are dealing with flow in the undersaturated region where production is due mainly to fluid and rock compressibility. There, the rate of flow varies directly with drawdown. Accordingly, the PI is constant. The IPR will take the form shown in Figure 2.
The equation can be determined from any two points on the line. One point can be provided by a shut-in pressure test, but that is not essential.