Generalized Inflow Performance Relationship For Stimulated Wells
Authors: Lekia Solomon D.L. , Evans Ronald D. 
Published: March 2010

Abstract
This theoretical study presents a new way of constructing the Inflow Performance Relationship (IPR) for stimulated wells. It is shown that the new method not only accurately reproduces and extends Standing's IPR curves for flow efficiencies greater than one but also eliminates the non-physical behaviour previously associated with the extrapolation of the curves.
This paper also examines and categorizes present-time IPR equations according to their domain of applicability. The equations, including the new method, are generalized for the construction of IRP curves from one flow-test point regardless of whether the flow test is taken below or above the bubble point pressure.
The new method consists of an easy-to-use IPR equation that is a function of bath the flow efficiency and bottomhole pressure ratio. A set of reference IPR curves for a stimulated well is developed using the new equation and an example problem is used to demonstrate the utility of the method.

Introduction
This paper focusses on the Inflow Performance Relationship (IPR) for stimulated wells. First IPR equations already existing in the literature are generalized using techniques analogous to those of Patton and GoIan(1). The equations are then examined and classified according to whether the flow efficiency (FE) for reservoir fluid flow across the sandface is unity, less than, or greater than, unity. The IPR equations of Vogel(2) and Fetkovich(3) apply to flow efficiencies equal to one, which corresponds to a skin factor of zero.
For FEs not equal to one, corresponding to negative or positive values of the skin factor, Standing(4) published curves that have enjoyed widespread use. The representation of these curves by a simple easy-to-use equation have been sought by many workers. Dias-Couto and Golan(5) succeeded in reproducing the original FE curves by an equation which exactly reproduces Standing's curves for FE &lt; I but suffers from a rate-prediction reversal for qFE&gt;1/qmaxFE = 1&gt; 1.125 or for Pwf/P &lt; (1-1.125/FE). This Dias-Couto et al.'s equation cannot be used for stimulated wells for which FE&gt; 1.
Harrison in an unpublished work(6) attempted to fit Standing's FE&gt; I curves with an empirical equation but the equation has been Observed(6) to unrlerpredict flow rates. Furthermore, his generalized IPR curves, which are based on obtaining slopes and intercepts from a log-log plot of (P2 - Pwf2) vs q require interpolation for nondiscrete values of FE, thus making it cumbersome to use this approach.
This paper presents a reasonably accurate reproduction of, and an extension to, Standing's FE&gt; 1 curves by a simple easy-to-use equation that is quadratic in Pwf/P. Also, it is shown how the equation can be applied to construct an IPR curve from one flowtest point taken either above or below the bubble point pressure of the reservoir oil. The equation is finally used to construct a set of reference curves for stimulated wells (FE&gt; 1.0 cases).
In the following sections, present-time IPR equations already in the literature are examined and their region of applicability are determined to enable categorization based on whether the flow efficiency is one or less than, or greater than, one.