# Gas–Liquid Two-Phase Upward Flow through a Vertical Pipe: Influence of Pressure Drop on the Measurement of Fluid Flow Rate

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Arrangements and Measurement Procedure

#### 2.1. Required Input Data

#### 2.2. Computational Algorithm

_{1}to c

_{6}are shown in Table 2.

_{1}, C

_{2}and C

_{3}are given by

Coefficient | °API ≤ 30 | °API ≥ 30 |

C_{1} | 27.64 | 56.060 |

C_{2} | 1.0937 | 1.187 |

C_{3} | 11.172 | 10.393 |

_{L}and H

_{g})

## 3. Results and Discussion

^{2}) explained exactly how the data points were fitted close to the regression line ($y=x$). Figure 8, Figure 9 and Figure 10 displayed the regression model for oil, water, and gas flow rate measurements. It can be seen that the plots show that most data points lie on or close to the unit slope line (e.g., best fit line), indicating that the predicted and actual values were in excellent agreement and illustrated an accurate flow rate prediction for oil, water, and gas with good correlating coefficients of 0.994, 0.993, and 0.966, respectively. This means that 99.4%, 99.3%, and 96.6% of the variance in the oil, water, and gas data, respectively, was explained by the line and 0.6%, 0.7%, and 3.4% of the variance was due to unexplained effects. The figures show that the predicted wells flow rates fell within the accepted uncertainty when compared with the measured flow rates.

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A | cross-sectional area, (sq ft) |

API | American Petroleum Institute |

B_{o} | oil formation volume factor, (bbl/stb) |

B_{ob} | oil formation volume (at bubble point pressure), (bbl/STB) |

B_{g} | gas formation volume factor, (cf/scf) |

Cnt | count |

dp/dz | pressure gradient, (psi/ft) |

d | inside diameter, (ft) |

ESP | electrical submersible pump |

f | friction factor, (unitless) |

g | Gravity, (ft/s^{2}) |

H_{L} | liquid hold-up |

H_{G} | gas hold-up |

H1 | bubble point pressure (at location depth before shut-in the well head valve), (ft) |

H2 | bubble point pressure (at location depth after shut-in the well head valve), (ft) |

mt | mass flow rate, (lb/day) |

N_{gv} | gas velocity number, (unitless) |

N_{Lv} | liquid velocity number, (unitless) |

N_{d} | pipe diameter number, (unitless) |

N_{CL} | coefficient number of viscosity correction, (unitless) |

N_{L} | liquid viscosity number, (unitless) |

q_{o} | oil rate, (stb/day) |

q_{g} | gas rate, (stb/day) |

q_{w} | water rate, (stb/day) |

q_{L} | liquid rate, (stb/day) |

q_{m} | measured flow rate, (stb/day) |

Q_{C} | quality check |

P | pressure, (psia) |

P_{r} | pseudo-critical pressure (for gas mixture), (psia) |

P_{b} | bubble point pressure, (psia) |

P_{sc} | pressure at standard conditions (P = 14.7 atm, T = 60 °F), (psia) |

PSD | pump setting depth |

SGG | gas specific gravity |

STB | stock tank barrel (for liquid) |

r_{w} | wellbore radius, (ft) |

R_{s} | gas-oil ratio, (scf/stb ) |

R_{sb} | gas oil ratio at bubble point pressure, (cf/scf) |

R_{e} | Reynolds number, (unitless) |

T | temperature, (°F) |

t | total shut-in time, (min) |

T_{r} | pseudo-critical temperature (for gas mixture), (psia) |

T_{sc} | temperature at standard condition, (°R) |

T_{r} | reservoir fluid temperature, (°F) |

VR | gas volume at reservoir conditions, (ft^{3}) |

V_{sc} | gas volume at standard condition, (ft^{3}) |

V_{SL} | superficial velocity for liquid, (ft/sec) |

V_{Sg} | superficial velocity for gas, (ft/sec) |

V_{m} | total mixture velocity, (ft/sec) |

WHP_{a} | well head pressure (after shut-in the well), (psia) |

WHP_{b} | well head pressure (before shut-in the well), (psia) |

WC | water cut (unitless) |

WHT | well head temperature, (°F) |

W | water vapor density, (unitless) |

Z | gas compressibility factor (unitless) |

Greek Symbols | |

ΔP | pressure drop, (psia) |

H_{L}/ψ | hold-up correlation factor |

γ_{o} | oil gravity |

γ_{w} | water gravity |

γ_{g} | gas gravity |

σ | surface tension, (dyne/m) |

ΔH | differences between bubble point pressure location depths (before and after shut-in the well head valve), (ft) |

ρ_{o} | oil density, (lbm/ft^{3}) |

ρ_{g} | gas density, (lbm/ft^{3}) |

ρ_{w} | water density, (lbm/ft^{3}) |

ρ_{L} | liquid density, (lb/ft^{3}) |

ρ_{m} | mixture density, (lbm/ft^{3}) |

µ_{o} | oil viscosity, cP |

µ_{g} | gas viscosity, cP |

µ_{L} | liquid viscosity, cP |

Subscripts | |

g_{sc} | gas (at standard condition) |

h | hydrostatic |

L | liquid |

m | mixture (liquid and gas) |

o | oil |

sc | standard condition |

w | water |

## References

- Saeb, M.B.; Philip, D.M.; David, C.C.; Ali, J. Modeling friction factor in pipeline flow using a GMDH-type neural network. Cogent Eng.
**2015**, 2. [Google Scholar] [CrossRef] - Shannak, B.A. Frictional pressure drop of gas liquid two-phase flow in pipes. Nuclear Eng. Des.
**2008**, 238, 3277–3284. [Google Scholar] [CrossRef] - Jiang, J.Z.; Zhang, W.M.; Wang, Z.M. Research progress in pressure-drop theories of gas-liquid two-phase pipe flow. In Proceedings of the China International Oil & Gas Pi (CIPC 2011), Langfang, China, 5 September 2011. [Google Scholar]
- Xu, Y.; Fang, X.D.; Su, X.H.; Zhou, Z.R.; Chen, W.W. Evaluation of frictional pressure drop correlations for two-phase flow in pipes. Nuclear Eng. Des.
**2012**, 253, 86–97. [Google Scholar] [CrossRef] - Mittal, G.S.; Zhang, J. Friction factor prediction for Newtonian and non-Newtonian fluids in pipe flows using neural networks. Int. J. Food Eng.
**2007**, 3, 1–18. [Google Scholar] [CrossRef] - Fadare, D.A.; Ofidhe, U.I. Artificial neural network model for prediction of friction factor in pipe flow. J. Appl. Sci.
**2009**, 5, 662–670. [Google Scholar] - Bilgil, A.; Altun, H. Investigation of flow resistance in smooth open channels using artificial neural networks. Flow Meas. Instrum.
**2008**, 19, 404–408. [Google Scholar] [CrossRef] - Özger, M.; Yildirim, G. Determining turbulent flow friction coefficient using adaptive neuro-fuzzy computing techniques. Adv. Eng. Softw.
**2009**, 40, 281–287. [Google Scholar] [CrossRef] - Shayya, W.H.; Sablani, S.S.; Campo, A. Explicit calculation of the friction factor for non-Newtonian fluids using artificial neural networks. Dev. Chem. Eng. Miner. Process.
**2005**, 13, 5–20. [Google Scholar] [CrossRef] - Yuhong, Z.; Wenxin, H. Application of artificial neural network to predict the friction factor of open channel flow. Commun. Nonlinear Sci. Numer. Simul.
**2009**, 14, 2373–2378. [Google Scholar] [CrossRef] - Yazdi, M.; Bardi, A. Estimation of friction factor in pipe flow using artificial neural networks. Can. J. Autom. Control Intell. Syst.
**2011**, 2, 52–56. [Google Scholar] - Sablani, S.S.; Shayya, W.H. Neural network based non-iterative calculation of the friction factor for power law fluids. J. Food Eng.
**2003**, 57, 327–335. [Google Scholar] [CrossRef] - Griffith, P. Two-Phase Flow in Pipes. In Special Summer Program; Massachusetts Institute of Technology: Cambridge, MA, USA, 1962. [Google Scholar]
- Raxendell, P.B. The Calculation of Pressure Gradients in High-Rate Flowing Wells. J. Pet. Technol.
**1961**, 13, 1023. [Google Scholar] [CrossRef] - Fancher, G.H.; Brown, K.E. Prediction of Pressure Gradients for Multiphase Flow in Tubing. Soc. Pet. Eng. J.
**1963**, 3, 59. [Google Scholar] [CrossRef] - Hagedorn, A.R.; Brown, K.E. The Effect of Liquid Viscosity in Vertical Two-Phase Flow. J. Pet. Technol.
**1964**, 16, 203. [Google Scholar] [CrossRef] - Poettmann, F.H.; Carpenter, P.G. The Multiphase Flow of Gas, Oil and Water through Vertical Flow Strings with Application to the Design of Gas-Lift Installations. In Proceedings of the Drilling and Production Practice, New York, NY, USA, 1 January 1952; Volume 52, p. 257. [Google Scholar]
- Tek, M.R. Multiphase Flow of Water, Oil and Natural Gas Through Vertical Flow Strings. J. Pet. Technol.
**1961**, 13, 1029. [Google Scholar] [CrossRef] - Shaban, H.; Tavoularis, S. Identification of flow regime in vertical upward air–water pipe flow using differential pressure signals and elastic maps. Int. J. Multiph. Flow
**2014**, 61, 62–72. [Google Scholar] [CrossRef] - Daev, Z.A.; Kairakbaev, A.K. Measurement of the Flow Rate of Liquids and Gases by Means of Variable Pressure Drop Flow Meters with Flow Straighteners. Meas. Tech.
**2017**, 59, 1170–1174. [Google Scholar] [CrossRef] - Cai, B.; Guo, D.X.; Jing, F.W. Study on gas-liquid two-phase flow patterns and pressure drop in a helical channel with complex section. In Proceedings of the 23rd International Compressor Engineering Conference, West Lafayette, IN, USA, 11–14 July 2016. [Google Scholar]
- Oliveira, J.L.G.; Passos, J.C.; Verschaeren, R.; Van der Geld, C. Mass flow rate measurements in gas-liquid flows by means of a venturi or orifice plate coupled to a void fraction sensor. Exp. Therm. Fluid Sci.
**2009**, 33, 253–260. [Google Scholar] [CrossRef] - Brown, G.O. The history of the Darcy-Weisbach equation for pipe flow resistance. Environment. Available online: https://ascelibrary.org/doi/abs/10.1061/40650 (accessed on 17 September 2018).
- Blasius, H. Das Ähnlichkeitsgesetz bei Reibungsvorgängen in Flüssigkeiten, Mitteilung 131 über Forschungsarbeiten auf dem Gebiete des Ingenieurwesens; Springer: Berlin, Germany, 1913. [Google Scholar]
- Colebrook, C.F. Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws. J. Inst. Civ. Eng.
**1939**, 11, 133–156. [Google Scholar] [CrossRef] - Lee, A.L.; Gonzalez, M.H.; Eakin, B.E. The Viscosity of Natural Gases. J. Pet. Technol.
**1966**, 18, 997–1000. [Google Scholar] [CrossRef] - Beggs, D.H.; Brill, J.P. A study of two-phase flow in inclined pipes. J. Pet. Technol.
**1973**, 25, 607–617. [Google Scholar] [CrossRef] - Standing, M.B.; Katz, D.L. Density of natural gases. Trans. AIME
**1942**, 146, 140–149. [Google Scholar] [CrossRef] - Sloan, E.; Khoury, F.; Kobayashi, R. Measurement and interpretation of the water content of a methane-propane mixture in the gaseous state in equilibrium with hydrate. Ind. Eng. Chem. Fundam.
**1982**, 21, 391–395. [Google Scholar] - Vazquez, M.; Beggs, H.D. Correlations for Fluid Physical Property Prediction. J. Pet. Technol.
**1980**, 32, 968–970. [Google Scholar] [CrossRef] - Hagedorn, A.R.; Brown, K.E. Experimental Study of Pressure Gradients Occurring during Continuous Two-Phase Flow in Small Diameter Vertical Conduit. J. Pet. Technol.
**1965**, 17, 475–484. [Google Scholar] [CrossRef] - Duns, H., Jr.; Ros, N. Vertical Flow of Gas and Liquid Mixtures in Wells. In Proceedings of the 6th World Petroleum Congress, Frankfurt am Main, Germany, 19–26 June 1963; p. 451. [Google Scholar]
- Orkiszewski, J. Predicting Two-Phase Pressure Drops in Vertical Pipe. J. Pet. Technol.
**1967**, 19, 829–838. [Google Scholar] [CrossRef] - Aziz, K.; Govier, G.; Fogarasi, M. Pressure Drop in Wells Producing Oil and Gas. J. Can. Pet. Technol.
**1972**, 11, 38. [Google Scholar] [CrossRef]

**Figure 1.**Schematic of the flow measurement stages in a vertical pipe before and after the well head wing valve shut-in.

Well Name | WHPb | WHPa | WHT | GOR | WC | Total Shut-in Time |
---|---|---|---|---|---|---|

(PSIA) | (PSIA) | (F) | (SCE/STB) | (%) | (min) | |

A33 | 140 | 200 | 98 | 360.92 | 93 | 1.55 |

A125 | 100 | 200 | 95 | 360.92 | 91.62 | 0.83 |

A64 | 180 | 250 | 107 | 360.92 | 81.52 | 1.06 |

A29 | 250 | 270 | 107 | 360.92 | 84.88 | 0.80 |

A23 | 210 | 260 | 127.7 | 360.92 | 82.11 | 0.24 |

A135 | 210 | 250 | 100 | 360.92 | 59.88 | 0.56 |

A126 | 250 | 300 | 98 | 360.92 | 66.91 | 0.20 |

A12 | 175 | 270 | 107 | 360.92 | 82.9 | 0.84 |

A108 | 260 | 300 | 97 | 360.92 | 81.31 | 0.28 |

5J5 | 150 | 300 | 95 | 360.92 | 4 | 3.36 |

5J2 | 100 | 170 | 101 | 360.92 | 52 | 1.39 |

5J4 | 250 | 300 | 101.6 | 360.92 | 58.95 | 0.27 |

5J7 | 250 | 300 | 98 | 360.92 | 30 | 0.47 |

E89 | 150 | 190 | 140 | 300 | 79 | 0.27 |

E210 | 80 | 120 | 110 | 300 | 83 | 0.33 |

E211 | 80 | 120 | 129 | 300 | 74 | 0.96 |

E286 | 70 | 100 | 146 | 300 | 90 | 0.25 |

E192 | 80 | 110 | 146 | 300 | 83 | 0.24 |

E327 | 80 | 110 | 115.5 | 300 | 77 | 0.35 |

E325 | 70 | 110 | 124.6 | 300 | 83 | 0.44 |

E197 | 90 | 110 | 146.1 | 300 | 82 | 0.11 |

E208 | 95 | 110 | 146.8 | 300 | 81 | 0.07 |

E226 | 80 | 110 | 138.4 | 300 | 91 | 0.25 |

E284 | 80 | 120 | 124.5 | 300 | 76 | 0.33 |

E258 | 65 | 90 | 142 | 300 | 86 | 0.23 |

E326 | 60 | 100 | 113 | 300 | 82 | 0.48 |

E227 | 100 | 150 | 142 | 300 | 84 | 0.36 |

4E_3 | 130 | 300 | 146 | 300 | 87 | 1.18 |

B56 | 120 | 170 | 120 | 384 | 42 | 2.3 |

B70 | 160 | 230 | 120 | 384 | 29.9 | 2.9 |

B121 | 100 | 160 | 120 | 364 | 67.29 | 0.5 |

B119 | 100 | 160 | 120 | 364 | 76.18 | 1.1 |

B50 | 180 | 250 | 110 | 364 | 68.65 | 0.55 |

B88 | 100 | 160 | 110 | 364 | 63.59 | 2.1 |

B14 | 250 | 270 | 110 | 364 | 76.28 | 0.15 |

B151 | 180 | 230 | 110 | 364 | 55.78 | 0.66 |

B164 | 100 | 170 | 120 | 364 | 26.3 | 2.1 |

B51 | 240 | 310 | 120 | 364 | 59.05 | 0.44 |

Q89 | 100 | 150 | 120 | 364 | 0 | 3.7 |

Q21 | 80 | 150 | 120 | 364 | 79.22 | 2.1 |

Q53 | 80 | 150 | 120 | 364 | 71.41 | 2.3 |

Q14 | 75 | 130 | 120 | 364 | 74.83 | 1.4 |

Q100 | 80 | 130 | 110 | 364 | 78.27 | 0.55 |

Q12 | 80 | 150 | 110 | 364 | 80.33 | 0.58 |

Q85 | 100 | 150 | 110 | 364 | 18.18 | 2.5 |

Q82 | 100 | 150 | 110 | 364 | 75.3 | 0.5 |

Q78 | 80 | 150 | 120 | 364 | 37.27 | 2.5 |

Q76 | 80 | 150 | 120 | 364 | 80.5 | 1.3 |

Constants | Value |
---|---|

c_{1} | 28.911 |

c_{2} | −9668.146 |

c_{3} | −1.663 |

c_{4} | −130,823.5 |

c_{5} | 205.323 |

c_{6} | 0.0385 |

Prediction Method | Average Error | Standard Deviation |
---|---|---|

(%) | (%) | |

Duns and Ros | −1.06 | 13.06 |

Hagedorn and Brown | −0.86 | 11.57 |

Orkiszewski | 1.8 | 16.52 |

Aziz et al. | 2.9 | 16.66 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ganat, T.A.; Hrairi, M.
Gas–Liquid Two-Phase Upward Flow through a Vertical Pipe: Influence of Pressure Drop on the Measurement of Fluid Flow Rate. *Energies* **2018**, *11*, 2937.
https://doi.org/10.3390/en11112937

**AMA Style**

Ganat TA, Hrairi M.
Gas–Liquid Two-Phase Upward Flow through a Vertical Pipe: Influence of Pressure Drop on the Measurement of Fluid Flow Rate. *Energies*. 2018; 11(11):2937.
https://doi.org/10.3390/en11112937

**Chicago/Turabian Style**

Ganat, Tarek A., and Meftah Hrairi.
2018. "Gas–Liquid Two-Phase Upward Flow through a Vertical Pipe: Influence of Pressure Drop on the Measurement of Fluid Flow Rate" *Energies* 11, no. 11: 2937.
https://doi.org/10.3390/en11112937