A Simple Procedure to Develop Analytical Curves of IPR from Reservoir Simulators with Application in Production Optimization
Authors: Cadena T. J.R. , Schiozer D.J. , Triggia A.A. 
Published: December 2007

Abstract
This paper describes a simple procedure, based in the work of Vogel and Wiggins, to develop IPR curves at any stage of depletion or at any time, for solution gas-drive reservoirs, for two or three phase flow, from results of a Black-Oil reservoir simulator.
In many applications, the use of numerical simulators to represent reservoir behavior would require an excessive computational time. In such a case, one alternative is to use analytical Inflow Performance Relationships (IPR).
One application of IPR is a production optimization method implemented in this work to choose the best value of production parameters such as tubing diameter and choke size. In this procedure, only discrete values are considered and a economic factor is used to optimize an objective function which is the present value of cumulative oil production.
This procedure is compared with the results obtained from simultaneous simulation of reservoir and production facilities which requires much greater computational time.
This work shows:a simple automatic procedure to generate IPR curves from reservoir simulators results,an optimization procedure where only available values of each production parameter are considered; andthat dynamic IPR curves that vary with depletion can be used to represent reservoirs in optimization procedures.
Problems related to the development of analytical IPR curves are also discussed.

Introduction
The estimation of individual performance of oil wells can be used to determine, for instance, an optimum method of production, adequate design of artificial lift, success design stimulation, and treatments and forecast of production performance. In many of these cases, the utilization of numerical simulators results in a very high time consumption while IPR curves can be utilized to represent reservoir performance with low computation effort.
Gilbert utilized curves which related flow rate and pressure. He was the first who called them IPR curves.
Weller developed one method to calculate depletion performance in solution-gas drive reservoirs applicable for all saturation conditions considering steady state flow and variable gas-oil ratio.
Vogel presented an empirical method to estimate the pressure-production behavior of oil wells producing from solution-gas drive reservoirs based on reservoir simulator results. He used Weller method to calculate IPR curves with a variety of PVT properties and relative permeability data.
These methods considered only two-phase flow (oil and gas). Brown presented a solution developed in Petrobras to calculate three-phase flow IPR curves. This correlation uses a combination of Vogel type equation for the oil IPR and a constant productivity index (PI) for the water IPR. Oil and water fractions were held constant for all flowing bottom hole pressures.
Wiggins, Russell and Jennings developed analytical IPR curves based on the physical nature of the multiphase flow system. They extended these ideas from two to three-phase flow systems.
In this paper, it is described a simple procedure, based in the works of Vogel and Wiggins, to developed IPR curves at any stage of depletion for solution gas-drive reservoirs using results obtained from a Black-Oil simulator.
IPR curves are used here to optimize production parameters such as tubing diameter and choke sizes. Most of the published work about production optimization, for instance, Caroll, developed non-linear techniques to optimize production performance using some continuous variables. In this work, only discrete values are considered to optimize the objective function which is the present value of cumulative oil production. Therefore, this procedure can be used to optimize several parameters simultaneously with low computation effort.

MATHEMATICAL FORMULATION
The development of the mathematical model is based on the following assumptions:homogeneous limited reservoir;reservoir are initially above bubble point pressure;radial flow,Darcy's law for multiphase flow applies;gravity and capillary effects are neglected;isothermal conditions;no gas solubility in water; andfully penetrating wellbore.
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