Predicting Two-Phase Pressure Drops in Vertical Pipe
Authors: Orkiszewski J. 
Published: December 2007

Abstract
A method is presented which can accurately predict, with a precision of about 10 percent, two-phase pressure drops in flowing and gas-lift production wells over a wide range of well conditions. The method is an extension of the work done by Griffith and Wallis and was found to be superior to five other published methods. The precision of the method was verified when its predicted values were compared against 148 measured pressure drops. The unique features of this method over most others are that liquid holdup is derived from observed physical phenomena, the pressure gradient is related tot eh geometrical distribution of the liquid and gas phase (flow regimes), and the method provides a good analogy of what happens inside the pipe. It takes less than a second to obtain a prediction on the IBM 7044 computer.

Introduction
The problem of accurately predicting pressure drops in flowing or gas-lift wells has given rise to many specialized solutions for limited conditions, but not to any generally accepted one for broad conditions. The reason for these many solutions is that the two-phase flow is complex and difficult to analyze even for the limited conditions studied. Under some conditions. the gas moves at a much higher velocity than the liquid. As a result, the down-hole flowing density of the gas-liquid mixture is greater than the corresponding density, corrected for down-hole temperature and pressure, that would be calculated from the produced gas-liquid ratio. Also, the liquid's velocity along the pipe wall can vary appreciably over a short distance and result in a variable friction loss. Under other conditions, the liquid is almost completely entrained in the gas and has very little effect on the wall friction loss. The difference in velocity and the geometry of the two phases strongly influence pressure drop. These factors provide the basis for categorizing two-phase flow. The generally accepted categories (flow regimes) of two-phase flow are bubble, slug, (slug-annular) transition and annular-mist. They are ideally depicted in Fig. 1 and briefly described as follows.

BUBBLE FLOW (FIG. 1A)
The pipe is almost completely, filled with the liquid and the free-gas phase is small. The gas is present as small bubbles, randomly distributed, whose diameters also very randomly. The bubbles move at different velocities depending upon their respective diameters. The liquid moves up the pipe at a fairly uniform velocity and, except for its density, the gas phase has little effect on the pressure gradient.

SLUG FLOW (FIG. 1B)
In this regime, the gas phase is more pronounced. Although the liquid phase is still continuous, the gas bubbles coalesce and form stable bubbles of approximately the same size and shape which are nearly the diameter of the pipe. They are separated by slugs of liquid. The bubble velocity is greater than that of the liquid and can be predicted in relation to the velocity of the liquid slug. There is a film of liquid around the gas bubble. The liquid velocity is not constant whereas the liquid slug always moves upward (in the direction of bulk flow); the liquid in the film may move upward but possibly at a lower velocity, or it may move downward. These varying liquid velocities will result not only in varying wall friction losses, but also in a "liquid holdup" which will influence flowing density. At higher flow velocities, liquid can even be entrained in the gas bubbles. Both the gas and liquid phases have significant effects on the pressure gradient.

TRANSITION FLOW (FIG. 1C)
The change from a continuous liquid phase to a continuous gas phase occurs in this region. The liquid slug between the bubbles virtually disappears, and a significant amount of liquid becomes entrained in the gas phase.
JPT
P. 829ˆ