Experimental Study of Pressure Gradients Occurring During Continuous Two-Phase Flow in Small-Diameter Vertical Conduits
Authors: Hagedorn Alton R. , Brown Kermit E. 
Published: December 2007

Abstract
A 1,500-ft experimental well was used to study the pressure gradients occurring during continuous, vertical, two-phase flow through 1-in., 1 1/4-in. and 1 1/2-in. nominal size tubing. The test well was equipped with two gas-lift valves and four Maihak electronic pressure transmitters as well as with instruments to measure the liquid production rate, air injection rate, temperatures and surface pressures. Tests were conducted for widely varying liquid flow rates, gas-liquid ratios and liquid viscosities. From these data, an accurate pressure-depth traverse was constructed for each test in each of the three tubing sizes. From the results of these tests, correlations have been developed which allow the accurate prediction of flowing pressure gradients for a wide variety of tubing sizes, flow conditions, and liquid properties. Also, the correlations and equations which are developed satisfy the necessary condition that they reduce to the relationships appropriate to single-phase flow when the flow rate of either the gag or the liquid phase becomes zero. All the correlations involve only dimensionless groups, which is a condition usually sought for in similarity analysis but not always achieved. The correlations developed in this study have been used to calculate pressure gradients for pipes of larger diameter than those upon which the correlations are based. Comparisons of these calculated gradients with experimentally determined gradients for the same flow conditions obtained from the literature indicate that extrapolation to these larger pipe sizes is possible with a degree of accuracy sufficient for engineering calculations. The extent of this extrapolation can only be determined with additional data from larger pipe diameters.

Introduction
The accurate prediction of the pressure drop expected to occur during the multiphase flow of fluids in the flow string of a well is a widely recognized problem in the petroleum industry. The problem has been brought even more into prominence with the advent of tubingless or slim-hole completions which use small-diameter tubing. Many of the correlations which give reasonably accurate results in the larger tubing sizes are greatly in error when applied to small-diameter conduits. Small-diameter conduits are defined as 1 1/2-in. nominal size tubing or smaller. The study of the pressure gradients which occur during multiphase flow of fluids in pipes is exceedingly complex because of the large number of variables involved. Further difficulties relate to the possibility of numerous flow regimes of widely varying geometry and mechanism and the instabilities of the fluid interfaces involved. Consequently, a solution to the problem by the approach normally used in classical fluid dynamics based on the formulation and solution of the Navier-Stokes equation has not been forthcoming. This is primarily the result of the nonlinearities involved and the difficulty of adequately describing the boundary conditions. As a result of the foregoing, most investigators have chosen semi-empirical or purely empirical approaches in an effort to obtain a practical solution to the problem. Much of the previous work in this area was done in short-tube models in the laboratory. A number of problems arise, however, when attempts are made to extrapolate these laboratory results to oilfield conditions where a much longer tube is encountered. In those few studies where data taken in long tubes were utilized, the data covered only a limited range of the variables, and as a result inaccuracies are introduced when the correlations are extended outside the range of the original data. Also, as a consequence of the limited amount of data available for these studies, the effects of several important variables were overlooked. The problem of predicting the pressure drop which occurs in multiphase flow differs from that of single-phase flow in that another source of pressure loss is introduced, namely, those pressure losses arising from slippage between the phases. This slippage is a result of the difference between the integrated average linear velocities of the two phases, which in turn is due to the physical properties of the fluids involved. In contrast to single-phase flow, the pressure losses in multiphase flow do not always increase with a decrease in the size of the conduit or an increase in production rate. This is attributed to the presence of the gas phase which tends to slip by the liquid phase without actually contributing to its lift. Many investigators have attempted to correlate both the slippage losses and the friction losses by means of a single energy-loss factor analogous to the one used in the single-phase flow problem.
JPT
P. 475ˆ