Fluid Dynamics

, Volume 51, Issue 1, pp 56–69 | Cite as

On the hyperbolicity of one-dimensional models for transient two-phase flow in a pipeline

  • V. D. ZhibaedovEmail author
  • N. A. Lebedeva
  • A. A. Osiptsov
  • K. F. Sin’kov


Characteristic properties of one-dimensionalmodels of transient gas-liquid two-phase flows in long pipelines are investigated. The methods for studying the hyperbolicity of the systems of equations of multi-fluid and drift-fluxmodels are developed. On the basis of analytical and numerical studies, the limits of the hyperbolicity domains in the space of governing dimensionless parameters are found, and the impact of the closure relations on the characteristic properties of the models is analyzed. The methods of ensuring the global unconditional hyperbolicity are proposed. Explicit formulas for the eigenvelocities of the system of the drift-flux model equations are obtained and the conclusions about their sign-definiteness are drawn.


hyperbolicity multiphase flows multi-fluid approach drift flux model pipe flow 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    O. Bratland, Pipe Flow 2: Multi-Phase Flow Assurance. 2009. Scholar
  2. 2.
    J.D. Ramshaw and J.A. Trapp, “Characteristics, Stability, and Short-Wavelength Two-Phase Flows in Systems of Equations,” Nuclear Sc. Eng. 66 (1), 93–102 (1978).Google Scholar
  3. 3.
    K.H. Bendiksen, D. Maines, R. Moe, and S. Nuland, “The Dynamic Two-Fluid Model OLGA; Theory and Applications,” SPE Technology 6 (2), (1991).Google Scholar
  4. 4.
    D. Lhuillier, C.H. Chang, and T.G. Theofanous, “On the Quest for a Hyperbolic Effective-FieldModel of Disperse Flows,” J. Fluid Mech. 731, 184–194 (2013).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kh.A. Rakhmatulin, “Fundamentals of Gas Dynamics of Interpenetrating Continua,” Prikl. Matem. Mekh. 20 (2), 184–195 (1956).Google Scholar
  6. 6.
    R.I. Nigmatulin, Dynamics of Multiphase Media. V.1. (Hemisphere, New York, 1989).Google Scholar
  7. 7.
    H. Stuhmiller, “The Influence of Interfacial Pressure Forces on the Character of Two-Phase Flow Model Equations,” Int. J. Multiphase Flow 3 (6), 551–560 (1977).CrossRefzbMATHGoogle Scholar
  8. 8.
    M. Ndjinga, “Influence of Interfacial Pressure on the Hyperbolicity of the Two-Fluid Model,” Comptes Rendus Mathematique 344 (6), 407–412 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    A. Kumbaro and M. Ndjinga, “Influence of Interfacial Pressure Term on the Hyperbolicity of a General Multifluid Model,” J. Comput. Multiphase Flows 3 (3), 177–196 (2011).MathSciNetCrossRefGoogle Scholar
  10. 10.
    B. Théron, Ecoulements Diphasiques Instationnaires en Conduite Horizontale, Thése INP Toulouse, France, 1989.Google Scholar
  11. 11.
    S. Benzoni-Gavage, Analyse Numérique des Modéles Hydrodynamiques d’écoulements Diphasiques Instationnaires dans les Réseaux de Production Pétroliére, Thése ENS Lyon, France, 1991.Google Scholar
  12. 12.
    S.L. Gavrilyuk and J. Fabre, “Lagrangian Coordinates for a Drift-Flux Model of a Gas-Liquid Mixture,” Int. J. Multiphase Flow 22 (3) 453–460 (1996).CrossRefzbMATHGoogle Scholar
  13. 13.
    S. Evje and T. Flåtten, “On the Wave Structure of Two-Phase Models,” Siam J. Appl. Math. 67 (2), 487–511 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    B.L. Rozhdestvenskii and N.N. Yanenko, Systems of Quasi-Linear Equations and Their Applications to Gas Dynamics [in Russian] (Nauka, Moscow, 1978).zbMATHGoogle Scholar
  15. 15.
    A.G. Kulikovskii, N.V. Pogorelov, and A.Yu. Semenov, Mathematical Problems of Numerical Solution of Hyperbolic Systems of Equations [in Russian] (Fizmatlit, Moscow, 2001).zbMATHGoogle Scholar
  16. 16.
    D. Barnea and Y. Taitel, “Kelvin–Helmholtz Stability Criteria for Stratified Flow: Viscous Versus Non-Viscous (Inviscid) Approaches,” Int. J. Multiphase Flow 19, 639–649 (1993).CrossRefzbMATHGoogle Scholar
  17. 17.
    D. Barnea and Y. Taitel, “Interfacial and Structural Stability of Separated Flow,” Int. J. Multiphase Flow 20 387–414 (1994).CrossRefzbMATHGoogle Scholar
  18. 18.
    M.R. Baer and J.W. Nunziato, “Two-Phase Mixture Theory for the Deflagration to Detonation Transition (DDT) in Reactive Granular Materials,” Int. J. Multiphase Flow 12 (6), 861–889 (1986).CrossRefzbMATHGoogle Scholar
  19. 19.
    E. Romenski and E.F. Toro, “Compressible Two-Phase Flows: Two-Pressure Models and Numerical Methods,” Comput. Fluid Dyn. J. 13, 403–416 (2004).zbMATHGoogle Scholar
  20. 20.
    T. Galloüet, J.M. Hérard, and N. Seguin, “Numerical Simulation of Two-Phase Flows Using Two-Fluid Two-Pressure Approach,” Mathem. Model. Meth. Appl. Sci. 14 (5), 663–700 (2004).CrossRefzbMATHGoogle Scholar
  21. 21.
    M. Bonizzi, P. Andreussi, and S. Banerjee, “Flow Regime Independent, High Resolution Multi-Field Modelling of Near-Horizontal Gas-Liquid Flows in Pipelines,” Int. J. Multiphase Flow 35 (1), 34–46 (2009).CrossRefGoogle Scholar
  22. 22.
    U. Kadri, R.F. Mudde, R.V.A. Oliemans, et al., “Prediction of the Transition from Stratified to Slug Flow or Roll-Waves in Gas-Liquid Horizontal Pipes,” Int. J. Multiphase Flow 35 (11) 1001–1010 (2009).CrossRefGoogle Scholar
  23. 23.
    A.A. Osiptsov, “A Self-Similar Solution to the Problem of Lava Dome Growth on an Arbitrary Conical Surface,” Fluid Dynamics 39 (1), 53–68 (2004).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    M.A. Jenkins and J.F. Traub, “A Three-Stage Algorithm for Real Polynomials Using Quadratic Iteration,” SIAM J. Numer. Analysis 7 (4), 545–566 (1970).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    N. Zuber and J.A. Findlay, “Average Volumetric Concentration in Two-Phase Flow Systems,” Trans. ASME. Ser. C. J. Heat Transfer 87 (4), 453–468 (1965).CrossRefGoogle Scholar
  26. 26.
    A.R. Hasan and C.S. Kabir, Fluid Flow and Heat Transfer in Wellbores (Soc. Petrol. Eng., Richardson, Texas, 2002).Google Scholar
  27. 27.
    J.M. Masella, Q.H. Tran, D. Ferre, and C. Pauchon, “Transient Simulation of Two-Phase Flows in Pipes,” Int. J. Multiphase Flow 24 (5), 739–755 (1998).CrossRefzbMATHGoogle Scholar
  28. 28.
    A.A. Osiptsov, K.F. Sin’kov, and P.E. Spesivtsev, “Justification of the Drift-Flux Model for Two-Phase Flow in a Circular Pipe,” Fluid Dynamics 49 (5), 614–626 (2014).CrossRefzbMATHGoogle Scholar
  29. 29.
    P.A. Varadarajan and P.S. Hammond, “Numerical Scheme for Accurately Capturing Gas Migration Described by 1D Multiphase Drift Flux Model,” Int. J. Multiphase Flow 73 57–70 (2015).MathSciNetCrossRefGoogle Scholar
  30. 30.
    P. Spesivtsev, K. Sinkov, and A.A. Osiptsov, “The Hyperbolic Nature of a System of Equations Describing Three-Phase Flows in Wellbores,” in: 14th European Conference on the Mathematics of Oil Recovery, Catania, Italy, 2014.Google Scholar
  31. 31.
    H. Shi, J.A. Holmes, L.J. Durlofsky, et al. “Drift-Flux Modeling of Two-Phase Flow in Wellbores,” SPE Journal 10 (1), 24–33 (2005).CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  • V. D. Zhibaedov
    • 1
    • 2
    Email author
  • N. A. Lebedeva
    • 1
  • A. A. Osiptsov
    • 1
  • K. F. Sin’kov
    • 1
    • 3
  1. 1.Moscow Research Center of the Schlumberger CompanyMoscowRussia
  2. 2.Lomonosov Moscow State UniversityMoscowRussia
  3. 3.Moscow Institute of Physics and Technology (State University)Dolgoprudnyi, Moscow oblastRussia

Personalised recommendations