Heat and Mass Transfer

, Volume 55, Issue 8, pp 2137–2151 | Cite as

Analytical model for predicting frictional pressure drop in upward vertical two-phase flowing wells

  • Tarek A. GanatEmail author
  • Meftah Hrairi
  • Belladonna Maulianda
  • Eghbal Motaei


In multiphase flow engineering operations, the pipelines that convey viscous fluids are subjected to interior friction where the pipe wall meets the fluid. The friction on the inner surface of the pipe causes energy losses. The losses are exhibited as a progressive pressure drop over the length of the pipe that varies with the fluid flow rate. This study develops a computational method to estimate the pressure change at any flow condition of multiphase flow (oil, gas, and water) inside a vertical pipe by developing fluid mechanics equations and using empirical correlations. Darcy and Colebrook friction factor correlations were used to ratify the predicted frictional pressure drop by computational method outcomes. OLGA dynamic simulation software was used to validate the accuracy of the computational method results. A sensitivity analysis was performed to evaluate the performance of the developed computational method, by using different well flow rate, pipe size diameter, and fluid properties. The frictional pressure drop estimation by computational method has acceptable accuracy and it is located within the accepted average relative error band (±20%). The overall performance of the method is satisfactory when compared with other observations.



Oil specific gravity


Pipe cross-sectional area, sq ft


Gas formation vol. Factor, res. CF/SCF


Oil formation vol. Factor, res. BBL/STB


Oil formation volume at bubble point pressure, BBL/STB




Inside pipe diameter, ft


Total pressure gradient (friction pressure loss is considered).


Friction losses factor


Formation volume factor




Gas holdup


Liquid holdup


Bubble point pressure location depth before closing the wellhead valve, ft


Bubble point pressure location depth after closing the wellhead valve, ft


Mass flow rate, lb/day


Liquid velocity number


Gas velocity number


Liquid viscosity number


Pipe diameter number


Correction for viscosity number coefficient


Oil flow rate STB/day


Water flow rate STB/day


Gas flow rate STB/day


Liquid flow rate STB/day


Measured flow rate STB/day


Quality check


Average pressure, psia


Bubble point pressure, psia


Pseudo-critical pressure of gas mixture, psia


Pressure at standard conditions, psia


Pump setting depth


Specific gravity of gas


Stock tank barrel


Wellbore radius, ft


Solution gas-oil ratio, SCF/STB


Solution gas at bubble point pressure, (CF/SCF)


Reynolds number


Average temperature, °F


Shut-in time, min


Pseudo-critical temperature of gas mixture, psia


Temperature at standard condition, °R


Reservoir temperature, °F


Gas volume at down-hole conditions, ft3


Gas volume at standard condition, ft3


Superficial liquid velocity, ft/sec


Superficial gas velocity, ft/sec


Mixture velocity, ft/sec


Wellhead pressure after closing the well, psia


Wellhead pressure before closing the well, psia


Water cut (non-dimensional)


Wellhead temperature, °F


Water vapour density


Gas compressibility factor

Greek symbols


Drawdown pressure, psia


Holdup factor correlation


Oil gravity


Water gravity


Gas gravity


Surface Tension


The differences between bubble point pressure location depth before and after closing the wellhead valve, ft


Oil density lbm/ cu ft


Gas density lbm/ cu ft


Water density lbm/ cu ft


Liquid density, Ib/cu ft


Mixture density, Ibm/ cu ft


Oil viscosity, cp


Gas viscosity, cp


Liquid viscosity, cp



Gas at standard condition






Mixture of liquid and gas




Standard condition



SI Metric conversion factors


141.5/(131.5+°API) E –00

bbl × 1.589873 E –0


cp. × 1.0 E – 03


ft × 3.048 E – 01


°F (°F-32)/1.8 E – 00


psia × 6.894757 E + 00




The authors would like to thank the production technology and reservoir engineering staff of Waha oil Company in Libya, for their generous assistance and for providing technical support, collaboration and words of encouragement on the success of this paper.

Compliance with ethical standards

I certify that no funding has been received for the conduct of this study and/or preparation of this manuscript.

Conflict of interest statement

On behalf of all authors, the corresponding author states that there is no conflict of interest.


  1. 1.
    Abdul-Majeed GH, Al-Mashat AM (2000) A Mechanistic Model for Vertical and Inclined Two-Phase Slug Flow. J Pet Sci Eng 59–67:27CrossRefGoogle Scholar
  2. 2.
    Awwad A, Xin RC, Dong ZF, Ebadian M, Soliman H (1995) Measurement and correlation of the pressure drop in air-water two-phase flow in horizontal helicoidal pipes. Int J Multiphase Flow 21:607–619Google Scholar
  3. 3.
    Aziz K, Govier GW, Fogarasi M (1972) Pressure drop in wells producing oil and gas. J Can Pet Technol 11:38–48Google Scholar
  4. 4.
    Beggs DH, Brill JP (1973) A study of two-phase flow in inclined pipes. J Pet Technol 25(5):607–617CrossRefGoogle Scholar
  5. 5.
    Boyce BE, Collier JG, Levy J (1969) Hold-up and pressure drop measurement in the two-phase flow of air-water mixing tubes in helical coils. In: Rhodes E, Scott DS (eds) Proceedings, International Symposium on Research in Concurrent Gas and Liquid Flow--1969. Plenum Press, New York, pp 203–231Google Scholar
  6. 6.
    Banerjee S, Rhodes E, Scott DS (1969) Studies on concurrent gas-liquid flow in helically coiled tubes. I--Flow patterns, pressure drop, and holdup. Can J Chem Eng 47:445–453CrossRefGoogle Scholar
  7. 7.
    Blasius H (1913) Das Ähnlichkeitsgesetz bei Reibungsvorgängen in Flüssigkeiten, Forschungs-Arbeit des Ingenieur-Wesens 131Google Scholar
  8. 8.
    Berger SA, Talbot L (1983) Flow in curved pipes. Annu Rev Fluid Mech 15:461–512CrossRefzbMATHGoogle Scholar
  9. 9.
    Chawla JM (1967) Wärmeübertragung und Druckabfall in waagerechten Rohren bei der Strömung von erdampfenden Kältemitteln VDI Forschungsheft, p. 523Google Scholar
  10. 10.
    Chen NH (1979) An explicit equation for friction factor in pipes. Ind Eng Fund 18:296–297CrossRefGoogle Scholar
  11. 11.
    Colebrook CF (1939) Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws. Journal of the ICE 11:133–156. CrossRefGoogle Scholar
  12. 12.
    Cazarez-Candia D, Montoya G (2009) VitalMathematical model for bubbly water–heavy oil–gas flow in vertical pipes. J Pet Sci Technol 27:1715–1726CrossRefGoogle Scholar
  13. 13.
    Ding GL, Hu HT, Huang XC, Deng B, Gao YF (2009) Experimental Investigation and Correlation of Two-Phase Frictional Pressure Drop of R410A—Oil Mixture Flow Boiling in a 5 mm Microfin Tube. Int J Refrig 32:150–161CrossRefGoogle Scholar
  14. 14.
    Dogen, H.A. 2006. A Comprehensive bit hydraulics model for gasified drilling fluids.
  15. 15.
    Duns H Jr, Ros NCJ (1963) Vertical flow of gas and liquid mixtures in wells. Proc. 6th World Petroleum. Congr., Frankfurt, II, pp. 451–465Google Scholar
  16. 16.
    Fan J, Hu CB, He GQ, Yang YX (2009) Study of Two-Phase Loss in Gas-Solid Flow. 45th AIAA/ASME/ SAE/ASEE Joint Propulsion Conference & Exhibit.
  17. 17.
    Glenn OB (2002) The history of the Darcy-Weisbach equation for pipe flow resistance. In A. Fredrich and J. Rogers, editors, Proceedings of the 150th Anniversary Conference of ASCE, Washington D.C., pages 34–43. American Society of Civil EngineersGoogle Scholar
  18. 18.
    Hagedorn AR, Brown KE (1965) Experimental Study of Pressure Gradients Occurring During Continuous Two-Phase Flow in Small Diameter Vertical Conduit. J Pet Technol 17(4):475–484CrossRefGoogle Scholar
  19. 19.
    Hasan AR, Kabir CS (1988) A study of multiphase flow behaviour in vertical wells. SPE Production Engineering. AIME 285:263–272Google Scholar
  20. 20.
    Hasan A.R. and Kabir C.S., 1992. Two-phase flow in vertical and inclined annuli. Volume 18, Issue 2, March 1992, Pages 279–293.
  21. 21.
    Hasan AR (1995) Void fraction in bubbly and slug flow in downward two-phase flow in vertical and inclined wellbores. SPE Prod & Facil 10(3):172–176Google Scholar
  22. 22.
    Lee AL, Gonzalez MH, Eakin BE (1966) The Viscosity of Natural Gases. J Pet Technol 18(8):997–1000. SPE-1340-PA. CrossRefGoogle Scholar
  23. 23.
    Lockhart RW, Martinelli RC (1949) Proposed correlation of data for isothermal two-phase component flow in pipes. Chem Eng Prog 45:39–45Google Scholar
  24. 24.
    Kabir CS, Hasan HR (1990) Performance of a Two-Phase Gas/Liquid FlowModel in Vertical Wells. J Pet Sci Eng 4(3):273–289. CrossRefGoogle Scholar
  25. 25.
    Kabir CS, Hasan HR (2010) Modeling two-phase fluid and heat flows in geothermal wells. J Pet Sci Eng 71:77–86CrossRefGoogle Scholar
  26. 26.
    Kabir CS, Hasan HR (1994) Aspects of wellbore heat transfer during two-phase flow. SPE Production & Facilities 9(3):211–216Google Scholar
  27. 27.
    Kabir CS, Hasan HR, Sayarpour M (2007) A basic approach to wellbore two-phase flow modelling. Proceedings of the SPE Annual Technical Conference and Exhibition, Anaheim California, pp 1–9Google Scholar
  28. 28.
    Kabir CS, Hasan HR (2009) Modeling two-phase fluid and heat flows in geothermal wells. Proceedings of the SPE Western Regional Meeting, San Jose, California, pp. 1–13. (SPE121351)Google Scholar
  29. 29.
    Kabir CS, Hasan HR, Sayarpour M (2010) Simplified two-phase flow modeling in wellbores. J Pet Sci Eng 72:42–49CrossRefGoogle Scholar
  30. 30.
    Kabir CS, Hasan HR (2012) Wellbore heat-transfer modeling and applications. J Pet Sci Eng 86–87:127–136Google Scholar
  31. 31.
    Kaya AS, Sarica C, Brill, JP (1999) Comprehensive Mechanistic Modeling of TwoPhase Flow in Deviated Wells. Paper SPE 56522 presented at the 1999 Annual and Technical Conferences and Exhibition, HoustonGoogle Scholar
  32. 32.
    Mortensen PH, Andersson HI, Gillissen JJJ, Boersma BJ (2008) On the Orientation of Ellipsoidal Particles in a Turbulent Shear Flow. Int J Multiphase Flow 34:678–683. CrossRefGoogle Scholar
  33. 33.
    Moreno-Quiben J (2005) Experimental and analytical study of two-phase pressure drops during evaporation in horizontal tubes. Ph.D. Thèsen No. 3337, Ècole Polytechique Fèdèrale de Lusanne, pp. 1–159Google Scholar
  34. 34.
    Müller-Steinhagen H, Heck K (1986) A simple friction pressure drop correlation for two-phase flow in pipes. Chem Eng Process 20:297–308CrossRefGoogle Scholar
  35. 35.
    Nathan JE, Satish GK (2005) An experimental investigation into the effect of surfactants on air–water two-phase flow in mini channels Proceedings of 3rd International Conference on Microchannels and Minichannels (ICMM), Canada, pp. 1–10Google Scholar
  36. 36.
    Ould Didi MB, Kattan N, Thome JR (2002) Correlation of two-phase friction for refrigerants in small-diameter tubes Exp. Thermal Fluid Sci 25:131–139Google Scholar
  37. 37.
    Orkiszewski J (1967) Predicting two-phase pressure drop in vertical pipe. J. Pet. Technol. (Jun.): 829–838. Trans., AIME, 240Google Scholar
  38. 38.
    Peter G, Chun WL, Pou CH, John FP (1973) Two phase pressure drop in inclined and vertical pipes. Technical report number 80063–81.
  39. 39.
    Rippel GR, Eidt CR, Jornan HB (1966) Two-phase flow in a coiled tube. Ind Eng Chem 5:32–39CrossRefGoogle Scholar
  40. 40.
    Shah RK, Joshi SD (1987) Convective heat transfer in curved ducts. In Handbook of Single-Phase Convective Heat Transfer-- 1987, New York, Chapter 3Google Scholar
  41. 41.
    Schmidt J, Giesbrecht H, van der Geld CWM (2008) Phase and velocity distributions in vertically upward high-viscosity two-phase flow. Int J Multiphase Flow 34:363–374CrossRefGoogle Scholar
  42. 42.
    Standing MB, Katz DL (1942) Density of natural gases. Trans AIME 146:140–149CrossRefGoogle Scholar
  43. 43.
    Sergio PF, Marcela BG (2006) A numerical model for multiphase flow on oil production wells, Center for Industrial Research, Tenaris– Campana, ArgentinaGoogle Scholar
  44. 44.
    Vazquez M, Beggs HD (1980) Correlations for Fluid Physical Property Prediction. J Pet Technol 32 (6): 968–970. SPE-6719-PA.
  45. 45.
    Thome JR (2006) Engineering Data Book IIIWolverine Tube, Inc. (Chapter 13)Google Scholar
  46. 46.
    Tran MCC, Wambsganss MW, France DM (2002) Two-phase pressure drop of refrigerants during flow boiling in small channels: an experimental investigation and correlation development. Int J Multiphase Flow 20:1739–1754zbMATHGoogle Scholar
  47. 47.
    Wallis GB (1969) One-Dimensional Two-Phase Flow. pp. 408.
  48. 48.
    Xin RC, Awwad A, Dong ZF, Ebadian MA, Soliman HM (1996) An investigation and comparative study of the pressure drop in air-water two-phase flow in vertical helicoidal pipes. Int J Heat Mass Transf 39:735–743 488 Int. J. Heat and Fluid Flow, Vol. 18, No. 5CrossRefGoogle Scholar
  49. 49.
    Zhang M, Webb RL (2001) Correlation of two-phase friction for refrigerants in small-diameter tubes Exp. Thermal Sci 25:131–139Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Petroleum EngineeringUniversiti Teknologi PetronasKuala LumpurMalaysia
  2. 2.Department of Mechanical EngineeringInternational Islamic UniversityKuala LumpurMalaysia
  3. 3.Department of Reservoir EngineeringPetronas Carigali SDN BHDKuala LumpurMalaysia

Personalised recommendations