Hall  carried out experimental study on gas-oil-water three-phase flow in horizontal pipes. He modeled the three-phase stratified flow by using the obtained hold-up to calculate the transition from stratified flow to slug flow. The model was compared with experimental data which showed that the transition occurred at higher gas velocities than those predicted by the model. The oil layer was believed to be the reason, because it travels at a higher mean velocity since its lower interface was in contact with a moving water layer and not a fixed wall.
Hold-ups of stratified three-phase flow pattern of gas-oil-water was calculated by Taitel et al. . Three steady state solutions for the upward inclined case were obtained. The only stable configuration was the one with the thinnest liquid layer. The essential step for the calculation of the hold-up, pressure drop, and transition criteria of the flow pattern was found to be the information regarding the liquid and oil levels in the pipe.
Zhang and Sarica  developed a model called unified model to predict the flow pattern and pressure gradient of three-phase gas-oil-water which was an improvement on the earlier unified model of Zhang et al. . The model was compared with experimental measurements of three-phase gas/oil/water pipe flows. The three-phase unified model gave better predictions than the unified model of gas/liquid two-phase pipe flow when compared with the experimental measurements of Khorr  for stratified gas/oil/water flow in horizontal and 1.5° downward pipes. Similar performance was seen when the two models were also compared with the experimental measurements of Hall  on pressure gradients for three-phase slug flow in a horizontal pipe.
Liu, Z., Liao, R., Luo, W., & Ribeiro, J. (2019). A new model for predicting liquid holdup in two-phase flow under high gas and liquid velocities. Scientia Iranica, 26(3), 1529-1539. doi: 10.24200/sci.2018.50040.1480
Z. Liu; R. Liao; W. Luo; J.X.F. Ribeiro. "A new model for predicting liquid holdup in two-phase flow under high gas and liquid velocities". Scientia Iranica, 26, 3, 2019, 1529-1539. doi: 10.24200/sci.2018.50040.1480
Liu, Z., Liao, R., Luo, W., Ribeiro, J. (2019). 'A new model for predicting liquid holdup in two-phase flow under high gas and liquid velocities', Scientia Iranica, 26(3), pp. 1529-1539. doi: 10.24200/sci.2018.50040.1480
Liu, Z., Liao, R., Luo, W., Ribeiro, J. A new model for predicting liquid holdup in two-phase flow under high gas and liquid velocities. Scientia Iranica, 2019; 26(3): 1529-1539. doi: 10.24200/sci.2018.50040.1480
Multiphase flow occurs in oil/gas, chemical, civil, and nuclear industries. The dominant occurrence of gas-oil-water three-phase flow in the petroleum industry requires sound knowledge of the behavior of multiphase flow. The most important characteristic of multiphase flow is its flow pattern (physical distribution of the phases within the enclosure they flow through) and the pressure gradient along the horizontal pipeline. In this regard, it is imperative to fully understand and study the flow rates, flow regimes/patterns, liquid-hold-up/water cut (WC), pressure gradients, and volume fractions of gas, oil, and water going into the pipelines during transportation of petroleum products. The water cut (WC) is the water quantity at the pipe inlet as volume percentage of the total inlet volumetric flow rate. The water cut is always the basis for pipelines and equipment design. During the transportation of the multiphase flow, water in the system starts separation and thereby accumulates at the pipe bottom and that amount of water is being referred to as local water contents, local water, or water hold-up. Also, it is important to better understand/predict/investigate the flow characteristics during petroleum production at different flow conditions such as the geometrical configuration of the pipeline, the physical properties of the fluids, and flow rates. There is a need to accurately investigate and predict the flow configurations and the pressure drop [1, 2].
The experiments were carried out in an acrylic pipe to visualize the flow patterns. The test fluids used were Safrasol D80 oil, tap water and air (properties of these fluids are mentioned earlier in Introduction). The three different fluids were passed into the horizontal pipeline and the flow patterns were observed while the pressure gradients were measured/recorded (using pressure transducers and U-tube manometers). A total of 377 data points were acquired and studied. The matrix range for three-phase flow of air-oil-water experiments is shown in Table 1. The effects of water cut, liquid velocity, gas velocity on flow patterns, and pressure drop have been studied.
Preface 1 Introduction 1.1 Multi-phase flow assurance 1.1.1 General 1.1.2 Nuclear reactor multi-phase models 1.1.3 Multi-phase flow in the petroleum industry 1.2 Two-phase flow 1.2.1 Flow regimes in horizontal pipes 1.2.2 Slugging 1.2.3 Flow regimes in vertical pipes 1.2.4 Flow regime maps 1.2.5 Flow in concentric and eccentric annulus 1.3 Three and four-phase flow 1.3.1 Types of three-phase and quasi four-phase flow 1.3.2 Three-phase flow regimes 1.4 Typical flow assurance tasks 1.5 Some definitions 1.5.1 General 1.5.2 Volume fraction, holdup and water cut 1.5.3 Superficial velocity 1.5.4 Mixture velocity and density 1.5.5 Various sorts of pipes
6 Solving the two-phase three-fluid equations 6.1 Steady-state incompressible isothermal flow 6.2 Comparing with measurements 6.3 Steady-state compressible flow 6.4 Transient three-fluid two-phase annular flow model
8 Including boiling and condensation 8.1 Extending the three-fluid two-phase model 8.2 Mass conservation 8.3 Momentum conservation 8.3.1 Main equations 8.3.2 Some comments on interface velocity 8.4 Energy equation 8.5 Pressure equation 8.6 Mass transfer from liquid (film and droplets) to gas 8.7 Slip between gas and droplets in annular flow 8.8 Droplet deposition in annular flow 8.8.1 The Wallis-correlation 8.8.2 The Oliemans, Pots, and Trope-correlation 8.8.3 The Ishii and Mishima-correlation 8.8.4 The Sawant, Ishii, and Mori-correlation 8.9 Dispersed bubble flow 8.10 Slug flow
10 Multi-phase flow heat exchange 10.1 Introduction 10.2 Classical, simplified mixture correlations 10.3 Improved correlations for all flow regimes in horizontal two-phase flow 10.4 Flow regime-dependent approximation for horizontal flow 10.5 Flow-regime dependent two-phase correlations for inclined pipes 10.6 Dispersed bubble flow 10.7 Stratified flow 10.8 Slug flow
11 Flow regime determination 11.1 The Beggs & Brill flow regime map 11.2 The Taitel & Duckler horizontal flow model 11.3 Flow regimes in vertical flow 11.3.1 Bubble to slug transition 11.3.2 Transition to dispersed-bubble flow 11.3.3 Slug to churn transition 11.3.4 Transition to annular flow 11.4 Flow regimes in inclined pipes 11.4.1 Bubble to slug transition 11.4.2 Transition to dispersed-bubble flow 11.4.3 Intermittent to annular transition 11.4.4 Slug to churn transition 11.4.5 Downward inclination 11.5 The minimum-slip flow regime criterion
13 Two-phase liquid-liquid flow 13.1 General 13.2 Emulsion viscosity 13.3 Phase inversion criteria 13.4 Stratified flow friction modeling 14 Two-phase liquid-solid flow 14.1 General about liquid-solid flow 14.2 The building up of solids in the pipeline 14.3 Minimum transport velocity
15 Three-phase gas-liquid-liquid flow 15.1 Introduction 15.2 Main equations 15.3 Three-layer stratified flow 15.4 Incompressible steady-state slug flow model 15.5 Combining the different flow regimes into a unified model
19 Various subjects 19.1 Multi-phase flowmeters and flow estimators 19.2 Gas lift 19.2.1 General 19.2.2 Oil & water-producing well with gas lift: Simulation example 19.3 Slug catchers 2b1af7f3a8