Inflow Performance Relationships for Solution-Gas Drive Wells
Authors: Vogel J.V. 
Published: December 2007

Abstract
In calculating oil well production, it has commonly been assumed that producing rates are proportional to drawdowns. Using this assumption, a well's behavior can be described by its productivity index (PI). This PI relationship was developed from Darcy's law for the steady-state radial flow of a single, incompressible fluid. Although Muskat pointed out that the relationship is not valid when both oil and gas flow in a reservoir, its use has continued for lack of better approximations. Gilbert proposed methods of well analysis utilizing a curve of producing rates plotted against bottom-hole well pressures; he termed this complete graph the inflow performance relationship (IPR) of a well. The calculations necessary to compute IPR's from two-phase flow theory were extremely tedious before advent of the computer. Using machine computations, IPR curves were calculated for wells producing from several fictitious solution-gas drive reservoirs that covered a wide range of oil PVT properties and reservoir relative permeability characteristics. Wells with hydraulic fractures were also included. From these curves, a reference IPR curve was developed that is simple to apply and, it is believed, can be used for most solution-gas drive reservoirs to provide more accurate calculations for oil well productivity than can be secured with PI methods. Field verification is needed.

Introduction
In calculating the productivity of oil wells, it is commonly assumed that inflow into a well is directly proportional to the pressure differential between the reservoir and the wellbore - that production is directly proportional to drawdown. The constant of proportionally is the PI, derived from Darcy's law for the steady-state radial flow of a single, incompressible fluid. For cases in which this relationship holds, a plot of the producing rates vs the corresponding bottom-hole pressures results in a straight line (Fig. 1). The PI of the well is the inverse of the slope of the straight line. However, Muskat pointed out that when two-phase liquid and gas flow exists in a reservoir, this relationship should not be expected to hold; he presented theoretical calculations to show that graphs of producing rates vs bottom-hole pressures for two-phase flow resulted in curved rather than straight lines. When curvature exists, a well cannot be said to have a single PI because the value of the slope varies continuously with the variation in drawdown. For this reason, Gilbert proposed methods of well analysis that could utilize the whole curve of producing rates plotted against intake pressures. He termed this complete graph the inflow performance relationship (IPR) of a well. Although the straight-line approximation is known to have limitations when applied to two-phase flow in the reservoir, it still is used primarily because no simple substitutes have been available. The calculations necessary to compute IPR's from two-phase flow theory have been extremely tedious. However, recently the approximations of Weller for a solution-gas drive reservoir were programmed for computers. The solution involved the following simplifying assumptions:the reservoir is circular and completely bounded with a completely penetrating well at its center;the porous medium is uniform and isotropic with a constant water saturation at all points;gravity effects can be neglected;compressibility of rock and water can be neglected;the composition and equilibrium are constant for oil and gas;the same pressure exists in both the oil and gas phases; andthe semisteady-state assumption that the tank-oil desaturation rate is the same at all points at a given instant.
Weller's solution did not require the constant-GOR assumption. The resulting computer program proved convenient to use and gave results closely approaching those furnished by the more complicated method of West, Garvin and Sheldon.