Transport in Porous Media

, Volume 84, Issue 1, pp 177–200 | Cite as

Numerical Simulation of Two-Phase Inertial Flow in Heterogeneous Porous Media

  • Azita Ahmadi
  • Ali Akbar Abbasian Arani
  • Didier LasseuxEmail author


In this study, non-Darcy inertial two-phase incompressible and non-stationary flow in heterogeneous porous media is analyzed using numerical simulations. For the purpose, a 3D numerical tool was fully developed using a finite volume formulation, although for clarity, results are presented in 1D and 2D configurations only. Since a formalized theoretical model confirmed by experimental data is still lacking, our study is based on the widely used generalized Darcy–Forchheimer model. First, a validation is performed by comparing numerical results of the saturation front kinetics with a semi-analytical solution inspired from the Buckley–Leverett model extended to take into account inertia. Second, we highlight the importance of inertial terms on the evolution of saturation fronts as a function of a suitable Reynolds number. Saturation fields are shown to have a structure markedly different from the classical case without inertia, especially for heterogeneous media, thereby, emphasizing the necessity of a more complete model than the classical generalized Darcy’s one when inertial effects are not negligible.


Inertial two-phase flow Heterogeneous porous media Numerical simulations Generalized Darcy–Forchheimer model 

List of Symbols


Section of the medium, m2


κ-Region capillary number


Grain size, m


Fractional flow for the α-phase


Gravitational acceleration, m s−2


Unit tensor


Intrinsic permeability tensor (= k I for an isotropic case), m2


α-Phase effective permeability tensor, m2


Intrinsic permeability in the κ-region, m2


Relative permeability tensor for the α-phase (= k r α I for an isotropic case)


Characteristic scale of the problem, m


Length of the medium, m


Total mobility tensor (= M o  + M w ), m3 kg−1 s


α-Phase mobility tensor (= M α I for an isotropic case), m3 kg−1 s


Number of grid blocks


Unit vector normal to the outlet face


Unit vector normal to the inlet face


Unit vector normal to the lateral surfaces


Unit vector normal to the ωη interface pointing from the ω-region toward the η-region


Fluid pressure for one-phase flow, Pa


Atmospheric pressure, Pa


α-Phase pressure, Pa


Initial oil-phase pressure, Pa


Capillary pressure, Pa


Maximum capillary pressure at S w  = S wi, Pa


Capillary pressure in the κ-region, Pa


Flow rate of water injected at the inlet of the medium, m3 s−1


Position vector, m


Position vector relative to the outlet face, m


Reynolds number


Reynolds number associated to the α-phase, \({\left(=\frac{\rho_{\alpha}\left\Vert {\bf u}_{\alpha}\right\Vert l}{\mu_{\alpha}}\right)}\)


Classical Reynolds number associated to the α-phase, \({(=\underset{\alpha}{\max}(\rho_{\alpha}/\mu_{\alpha})\left\Vert {\bf u}_{t}\right\Vert d)}\)


α-Phase saturation


Irreducible water saturation


Residual oil saturation


Initial water-phase saturation


Reduced saturation \({\left(=\frac{S_{w}-S_{\rm wi}}{1-S_{\rm wi}-S_{\rm or}}\right)}\)


Time, s


Seepage velocity for one-phase flow, m s−1


α-Phase seepage velocity, m s−1

\({{\bf u}_{\alpha}^{\kappa}}\)

α-Phase seepage velocity in the κ-region, m s−1


Total velocity (=u o  + u w ), m s−1

utx, uty, utz

Components of the total velocity, m s−1


Front velocity, m s−1


Position variable, m

Greek Letters


α-Phase effective inertial resistance tensor (= β α I for the isotropic case), m−1


Intrinsic inertial resistance factor, m−1


α-Phase relative inertial resistance tensor (= β r α I for the isotropic case)


Intrinsic inertial resistance factor for the κ-region, m−1


Interface between the ω-region and the η-region, m2


Time step, s

Δx, Δy, Δz

Grid sizes in the x, y and z directions, m




α-Phase dynamic viscosity, Pas


α-Phase density, kg/m3


Interfacial tension, N m−1

ξ, γ, θ

Constant exponents




Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abdobal, S.: Ecoulements diphasiques en milieux poreux: Etude expérimentale des écoulements liquide-gaz et liquide-liquide à forts débits. PhD thesis, Institut National Polytechnique de Lorraine (2002)Google Scholar
  2. Ahmadi A., Quintard M.: Calculation of large-scale properties for multiphase flow in random porous media. Iran. J. Sci. Technol. 9(1 Transaction A), 11–37 (1995)Google Scholar
  3. Aziz K., Settari A.: Petroleum Reservoir Simulation. Elsevier Applied Science, London (1979)Google Scholar
  4. Bear J., Braester C., Menier P.C.: Effective and relative permeabilities of anisotropic porous media. Transp. Porous Media 2(3), 301–316 (1987)CrossRefGoogle Scholar
  5. Brooks R.H., Corey A.T.: Properties of porous media affecting fluid flow. J. Irrig. Drain. Div. Proc. ASCE 92(IR 2), 61–88 (1966)Google Scholar
  6. Buckley S.E., Leverett M.C.: Mechanism of fluid displacement in sands. Trans. Am. Inst. Min. Petrol. Eng. 146, 107–116 (1942)Google Scholar
  7. Chang, J., Yortsos, Y.: Effect of heterogeneity on Buckley–Leverett displacement. SPE paper 18798 (1989)Google Scholar
  8. Chavent, G.: A new formulation of diphasic incompressible flows in porous media. Lecture Note in Math 503, pp. 258–270. Springer-Verlag, Berlin (1976)Google Scholar
  9. Chen Z., Huan G., Li B.: An improved IMPES method for two-phase flow in porous media. Transp. Porous Media 54, 361–376 (2004)CrossRefGoogle Scholar
  10. Corey A.T.: The interrelation between gas and oil relative permeabilities. Prod. Mon. 19(1), 38–41 (1954)Google Scholar
  11. Cornell D., Katz D.L.: Flow of gases through consolidated porous media. Ind. Eng. Chem. 45, 2145–2152 (1953)CrossRefGoogle Scholar
  12. Dale M., Ekrann S., Mykkeltveit J., Virnovsky G.: Effective relative permeabilities and capillary pressure for one-dimensional heterogeneous media. Transp. Porous Media 26, 229–260 (1997)CrossRefGoogle Scholar
  13. Ergun S.: Fluid flow through packed columns. Chem. Eng. Prog. 48, 89–94 (1952)Google Scholar
  14. Evans E.V., Evans R.D.: Influence of an immobile or mobile saturation on non-Darcy compressible flow of real gases in propped fractures. J. Petrol. Technol. 40(10), 1343–1351 (1988)Google Scholar
  15. Evans R.D., Hudson C.S., Greenlee J.E.: The effect of an immobile liquid saturation on the non-Darcy flow coefficient in porous media. J. SPE Prod. Eng. Trans. AIME 283, 331–338 (1987)Google Scholar
  16. Forchheimer P.: Wasserbewegung durch boden. Z. Ver. Deutsch. Ing. 45, 1781–1788 (1901)Google Scholar
  17. Fourar M., Lenormand R., Karimi-Fard M., Horne R.: Inertia effects in high-rate flow through heterogeneous porous media. Transp. Porous Media 60, 353–370 (2005)CrossRefGoogle Scholar
  18. Geertsma J.: Estimating the coefficient of inertial resistant in fluid flow through porous media. SPE J. 14, 445–450 (1974)Google Scholar
  19. Honarpour M., Koederitz L., Harvey A.H.: Relative Permeability of Petroleum Reservoirs. CRC Press, Boca Raton (1986)Google Scholar
  20. Hubbert M.K.: Darcy’s law and the field equation of the flow of underground fluids. Trans. SPE AIME 207, 222–239 (1956) (JPT)Google Scholar
  21. Jamiolahmady M., Danesh A., Sohrabi M., Duncan D.: Measurement and modelling of gas condensate around rock perforation. Transp. Porous Media 63, 323–347 (2006)CrossRefGoogle Scholar
  22. Kalaydjian, F., Bourbiaux, J.M., Lombard, J.M.: Predicting gas-condensate reservoir performance: how flow parameters are altered when approaching producing wells. SPE paper 36715 (1996)Google Scholar
  23. Katz D.L., Lee R.L.: Natural Gas Engineering, Production and Storage, Chemical Engineering Series. McGraw-Hill, New York (1990)Google Scholar
  24. Khashan S., Al-Amiri A., Pop I.: Numerical simulation of natural convection heat transfer in a porous cavity heated from below using a non-darcian and thermal non-equilibrium model. Int. J. Heat Mass Transf. 49(5–6), 1039–1049 (2006)CrossRefGoogle Scholar
  25. Kim M., Park E.: Fully discrete mixed finite element approximations for non-Darcy flows in porous media. Comput. Math. Appl. 38, 113–129 (1999)CrossRefGoogle Scholar
  26. Koederitz L.F., Harvey A.H., Honarpour M.: Introduction to Petroleum Reservoir Analysis. Gulf, Houston (1989)Google Scholar
  27. Lasseux D., Whitaker S., Quintard M.: Determination of permeability tensors for two phase flow in homogeneous porous media: theroy. Transp. Porous Media 24(1), 103–137 (1996)Google Scholar
  28. Lasseux D., Ahmadi A., Abbasian Arani A.: Two-phase inertial flow in homogeneous porous media: a theoretical derivation of a macroscopic model. Transp. Porous Media 75, 371–400 (2008)CrossRefGoogle Scholar
  29. Lee, H.S., Catton, I.: Two-phase flow in stratified porous media. In: 6th Information Exchange Meeting on Debris Coolability, Los Angeles (1984)Google Scholar
  30. Li, D., Engler, T.W.: Literature review on correlations of the non-Darcy coefficient. SPE paper 70015 (2001)Google Scholar
  31. Lipinski, R.J.: A model for boiling and dryout in particle beds. Report SAND 82-0756 (NUREG/CR-2646) Sandia Labs (1982)Google Scholar
  32. Liu X., Civan F., Evans R.D.: Correlations of the non-Darcy flow coefficient. J. Can. Petrol. Technol. 34(10), 50–54 (1995)Google Scholar
  33. Marle C.M.: Cours de Production, Tome IV, Les Écoulements Polyphasiques En Milieu Poreux. Editions Technip, Paris (1972)Google Scholar
  34. Mazaheri A., Zerai B., Ahmadi G., Kadambi J., Saylor B., Oliver M., Bromhal G., Smith D.: Computer simulation of flow through a lattice flow-cell model. Adv. Water Res. 28(12), 1267–1279 (2005)CrossRefGoogle Scholar
  35. Mei C.C., Auriault J.L.: The effect of weak inertia on flow through a porous medium. J. Fluid Mech. 222, 647–663 (1991)CrossRefGoogle Scholar
  36. Muskat M.: The Flow of Homogeneous Fluids Through Porous Media. McGraw-Hill, New York (1937)Google Scholar
  37. Saez A.E., Carbonnell R.G.: Hydrodynamic parameters for gas-liquid co-current flow in packed beds. AIChE J. 31, 52–62 (1985)CrossRefGoogle Scholar
  38. Sanchez M., Luna E., Medina A., Méndez F.: Simultaneous imbibition-heat convection process in a non-Darcian porous medium. J. Colloid Interface Sci. 288, 562–569 (2005)CrossRefGoogle Scholar
  39. Scheidegger A.E.: The Physics of Flow Through Porous Media. University of Toronto Press, Toronto (1972)Google Scholar
  40. Schulenberg T., Muller V.: An improved model for two-phase flow through beds of coarse particles. Int. J. Multiph. Flow 13, 87–97 (1987)CrossRefGoogle Scholar
  41. Skjetne E., Auriault J.L.: High-velocity laminar and turbulent flow in porous media. Transp. Porous Media 36(2), 131–147 (1999)CrossRefGoogle Scholar
  42. Tek M.R., Coats K.H., Katz D.L.: The effect of turbulence on flow of natural gas through porous reservoirs. J. Petrol. Technol. Trans. AIME 222, 799–806 (1962)Google Scholar
  43. Virnovsky G., Friis H., Lohne A.: A steady-state upscaling approach for immiscible two-phase flow. Transp. Porous Media 54, 167–192 (2004)CrossRefGoogle Scholar
  44. Wahyudi I., Montillet A., Khalifa A.O.A.: Darcy and post-Darcy flows within different sands. J. Hydraul. Res. 40(4), 519–525 (2000)CrossRefGoogle Scholar
  45. Wang X., Thauvin F., Mohanty K.K.: Non-Darcy flow through anisotropic porous media. Chem. Eng. Sci. 54, 1859–1869 (1999)CrossRefGoogle Scholar
  46. Welge H.J.: A simplified method for computing oil recovery by gas or water drive. Trans. Am. Inst. Min. Metall. Petrol. Eng. 195, 91–98 (1952)Google Scholar
  47. Whitaker S.: The Forchheimer equation: a theoretical development. Transp. Porous Media 25(1), 27–61 (1996)CrossRefGoogle Scholar
  48. Whitney, D.D.: Characterization of the non-Darcy flow coefficient in propped hydraulic fractures. Master’s thesis, University of Oklahoma (1988)Google Scholar
  49. Wu Y.S.: Non-Darcy displacement of immiscible fluids in porous media. Water Resour. Res. 37(12), 2943–2950 (2001)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Azita Ahmadi
    • 1
  • Ali Akbar Abbasian Arani
    • 1
  • Didier Lasseux
    • 1
    Email author
  1. 1.TREFLE, UMR CNRS 8508, Arts et Métiers ParisTechTalence CedexFrance

Personalised recommendations