Transport in Porous Media

, Volume 98, Issue 2, pp 401–426 | Cite as

Modeling Non-Darcian Single- and Two-Phase Flow in Transparent Replicas of Rough-Walled Rock Fractures

  • Giovanni RadillaEmail author
  • Ali Nowamooz
  • Mostafa Fourar


While it is generally assumed that in the viscous flow regime, the two-phase flow relative permeabilities in fractured and porous media depend uniquely on the phase saturations, several studies have shown that for non-Darcian flows (i.e., where the inertial forces are not negligible compared with the viscous forces), the relative permeabilities not only depend on phase saturations but also on the flow regime. Experimental results on inertial single- and two-phase flows in two transparent replicas of real rough fractures are presented and modeled combining a generalization of the single-phase flow Darcy’s law with the apparent permeability concept. The experimental setup was designed to measure injected fluid flow rates, pressure drop within the fracture, and fluid saturation by image processing. For both fractures, single-phase flow experiments were modeled by means of the full cubic inertial law which allowed the determination of the intrinsic hydrodynamic parameters. Using these parameters, the apparent permeability of each fracture was calculated as a function of the Reynolds number, leading to an elegant means to compare the two fractures in terms of hydraulic behavior versus flow regime. Also, a method for determining the experimental transition flow rate between the weak inertia and the strong inertia flow regimes is proposed. Two-phase flow experiments consisted in measuring the pressure drop and the fluid saturation within the fractures, for various constant values of the liquid flow rate and for increasing values of the gas flow rate. Regardless of the explored flow regime, two-phase flow relative permeabilities were calculated as the ratio of the single phase flow pressure drop per unit length divided by the two-phase flow pressure drop per unit length, and were plotted versus the measured fluid saturation. Results confirm the dependence of the relative permeabilities on the flow regime. Also the proposed generalization of Darcy’s law shows that the relative permeabilities versus fluid saturation follow physical meaningful trends for different liquid and gas flow rates. The presented model fits correctly the liquid and gas experimental relative permeabilities as well as the fluid saturation.


Rough fracture Two-phase flow Non-Darcian flow  Inertial effect Relative permeability 

List of Symbols



Cross-sectional area of the fracture or porous media (\(\text{ m }^{2}\))


Corey saturation power law exponent


Corey saturation power law exponent


Concentration (g/l)


Capillary number




Local aperture (m)


Hydraulic aperture of the fracture (m)




Intrinsic permeability (\(\text{ m }^{2}\))


Apparent permeability (\(\text{ m }^{2}\))


Relative permeability


Volumetric flow rate (\(\text{ m }^{3}/\text{ s }\))


Reynolds number (\(Re = K\beta \rho Q/\mu A\))


Weak inertia cubic law Reynolds number \(Re_{\mathrm{c}} =\sqrt{K\gamma }{\rho Q}/{\mu A}\)


Fluid saturation


Width of the fracture (m)

\(\Delta P/L\)

Pressure drop per unit length (Pa/m)

\(\beta \)

Inertial coefficient (\(\text{ m }^{-1}\))

\(\beta _{\mathrm{r}}\)

Relative inertial coefficient

\(\gamma \)

Inertial coefficient

\(\varepsilon \)

Solute absorptivity (\(\text{ m }^{2}/\text{ kg }\))

\(\mu \)

Dynamic viscosity (Pa s)

\(\rho \)

Density (\(\text{ kg/m }^{3}\))







Single-phase flow




  1. Andrade, J.S., Costa, U.M.S., Almeida, M.P., Makse, H.A., Stanley, H.E.: Inertial effects on fluid flow through disordered porous media. Phys. Rev. Lett. 82, 5249–5252 (1999). doi: 10.1103/PhysRevLett.82.5249 CrossRefGoogle Scholar
  2. Auriault, J.-L.: Nonsaturated deformable porous media: quasistatics. Transp. Porous Media 2, 45–64 (1987)CrossRefGoogle Scholar
  3. Barrère, J.: Modélisation des écoulements de Stokes et Navier–Stokes en milieu poreux. Ph.D. thesis, University of Bordeaux I, Bordeaux (1990)Google Scholar
  4. Bauget, F., Fourar, M.: Non-Fickian dispersion in a single fracture. J. Contam. Hydrol. 100(3–4), 137–148 (2008). doi: 10.1016/j.jconhyd.2008.06.005 CrossRefGoogle Scholar
  5. Bear, J.: Dynamics of Fluids in Porous Media. Dover, Mineola (1972)Google Scholar
  6. Brown, S.R.: Fluid flow through rock joints: the effect of surface roughness. J. Geophys. Res. 92(B2), 1337–1347 (1987). doi: 10.1029/JB092iB02p01337 CrossRefGoogle Scholar
  7. Brush, D.J., Thomson, N.R.: Fluid flow in synthetic roughwalled fractures: Navier–Stokes, Stokes, and local cubic law simulations. Water Resour. Res. 39(4), 1085 (2003). doi: 10.1029/2002WR001346 CrossRefGoogle Scholar
  8. Buchlin, J.M., Stubos, A.: Phase change phenomena at liquid saturated self heated particulate beds. In: Bear, J., Buchlin, J.M. (eds.) Modeling and Applications of Transport Phenomena in Porous Media. Kluwer Acad. Pub, Dordrecht (1987)Google Scholar
  9. Buès, M., Panfilov, M., Oltean, C.: Macroscale model and inertia-viscous effects for Navier–Stokes flow in a radial fracture with corrugated walls. J. Fluid Mech. 504, 41–60 (2004). doi: 10.1017/S002211200400816X CrossRefGoogle Scholar
  10. Chauveteau, G., Thirriot, C.: Régimes d’écoulement en milieu poreux et limite de la loi de Darcy. La Houille Blanche 2, 141–148 (1967). doi: 10.1051/lhb/1967009 CrossRefGoogle Scholar
  11. Chen, C., Horne, R.N., Fourar, M.: Experimental study of liquid–gas flow structure effect on relative permeabilities in a fracture. Water Resour. Res. 40, W08301 (2004). doi: 10.1029/2004WR003026 Google Scholar
  12. Chen, C., Horne, R.N.: Two-phase flow in rough-walled fractures: experiments and a flow structure model. Water Resour. Res. 42, W03430 (2006). doi: 10.1029/2004WR003837 Google Scholar
  13. Chen, Z., Lyons, S.L., Qin, G.: Derivation of the Forchheimer law via homogenization. Transp. Porous Media 44(2), 325–335 (2001). doi: 10.1023/A:1010749114251 CrossRefGoogle Scholar
  14. Corey, A.T.: The interrelationship between gas and oil relative permeabilities. Prod. Mon. 19(1), 38–41 (1954)Google Scholar
  15. Cornell, D., Katz, D.L.: Flow of gases through consolidated porous media. Ind. Eng. Chem. 45(10), 2145–2153 (1953). doi: 10.1021/ie50526a021 CrossRefGoogle Scholar
  16. Cvetkovic, V.D.: A continuum approach to high velocity flow in a porous medium. Transp. Porous Media 1(1), 63–97 (1986). doi: 10.1007/BF01036526 CrossRefGoogle Scholar
  17. Darcy, H.: Les fontaines publiques de la ville de Dijon. Dalmont, Paris (1854)Google Scholar
  18. de Gennes, P.G.: Theory of slow biphasic flows in porous media. Physico-Chem. Hydrodyn. 4, 175–185 (1983)Google Scholar
  19. Detwiler, R.L., Pringle, S.E., Glass, R.J.: Measurement of fracture aperture fields using transmitted light: an evaluation of measurement errors and their influence on simulations of flow and transport through a single fracture. Water Resour. Res. 35(9), 2605–2617 (1999). doi: 10.1029/1999WR900164 CrossRefGoogle Scholar
  20. Diomampo, G.P.: Relative permeability through fractures. MS thesis, Stanford University, Stanford (2001)Google Scholar
  21. Dullien, A.L., Azzam, M.I.S.: Flow rate-pressure gradient measurement in periodically nonuniform capillary tube. AIChE J. 19, 222–229 (1973). doi: 10.1002/aic.690190204 CrossRefGoogle Scholar
  22. Firdaouss, M., Guermond, J.-L., Le-quéré, P.: Nonlinear correction to Darcy’s law at low Reynolds numbers. J. Fluid Mech. 343, 331–350 (1997). doi: 10.1017/S0022112097005843 CrossRefGoogle Scholar
  23. Forchheimer, P.: Wasserberwegng durch Boden. Forschtlft ver. D. Ing. 45(50), 1782–1788 (1901)Google Scholar
  24. Fourar, M., Bories, S.: Experimental study of air–water two-phase flow through a fracture (narrow channel). Int. J. Multiphase Flow 21(4), 621–637 (1995). doi: 10.1016/0301-9322(95)00005-I CrossRefGoogle Scholar
  25. Fourar, M., Bories, S., Lenormand, R., Persoff, P.: Two-phase flow in smooth and rough fractures: measurement and correlation by porous-media and pipe-flow models. Water Resour. Res. 29(11), 3699–3708 (1993). doi: 10.1029/93WR01529 CrossRefGoogle Scholar
  26. Fourar, M., Lenormand, R.: A viscous coupling model for relative permeabilities in fractures. Paper SPE 49006 presented at the 1977 SPE annual technical conference and exhibition, New Orleans, 27–30 Sept 1998. doi: 10.2118/49006-MS
  27. Fourar, M., Lenormand, R.: Inertial effects in two-phase flow through fractures. Oil Gas Sci. Tech. Rev. IFP 55(3), 259–268 (2000). doi: 10.2516/ogst:2000018 CrossRefGoogle Scholar
  28. Fourar, M., Lenormand, R.: A new model for two-phase flows at high velocities through porous media and fractures. J. Pet. Sci. Eng. 30(2), 121–127 (2001). doi: 10.1016/S0920-4105(01)00109-7 CrossRefGoogle Scholar
  29. Fourar, M., Radilla, G., Lenormand, R., Moyne, C.: On the non-linear behavior of a laminar single-phase flow through two and three-dimensional porous media. Adv. Water Resour. 27(6), 669–677 (2004). doi: 10.1016/j.advwatres.2004.02.021 CrossRefGoogle Scholar
  30. Geertsma, M.: Estimating the coefficient of inertial resistance fluid flow through porous media. SPE J. 14(5), 445–450 (1974). doi: 10.2118/4706-PA Google Scholar
  31. Giorgi, T.: Derivation of the Forchheimer law via matched asymptotic expansions. Transp. Porous Media 29(2), 191–206 (1997). doi: 10.1023/A:1006533931383 CrossRefGoogle Scholar
  32. Hubbert, M.K.: Darcy law and the field equations of the flow of underground fluids. Trans. Am. Inst. Min. Mandal. Eng. 207, 222–239 (1956)Google Scholar
  33. Isakov, E., Ogilvie, S.R., Taylor, C.W., Glover, P.W.J.: Fluid flow through rough fractures in rocks I: high resolution aperture determinations. Earth Planet. Sci. Lett. 191(3–4), 267–282 (2001). doi: 10.1016/S0012-821X(01)00424-1 CrossRefGoogle Scholar
  34. Kalaydjian, F., Legait, B.: Perméabilités relatives couplées dans les écoulements en capillaries et en milieux poreux. C. R. Acad. Sci. Paris 304(série II), 1035–1038 (1987)Google Scholar
  35. Lee H.S., Catton I. (1984) Two-phase flow in stratified porous media. 6th Information exchange meanding on debris coolability, Los AngelesGoogle Scholar
  36. Lipinski, R.J.: A one-dimensional particle bed dryout. Model. Trans. Am. Nucl. Soc. 38, 386–387 (1981)Google Scholar
  37. Lipinski, R.J.: A model for boiling and dryout in particle beds. Report SAND 82–0756 (NUREG/CR-2646), Sandia Labs., Albuquerque (1982)Google Scholar
  38. Lockhart, R.W., Martinelli, R.C.: Proposed correlation of data for isothermal two-phase, two-component flow in pipes. Chem. Eng. Prog. 45, 39–48 (1949)Google Scholar
  39. MacDonald, I.F., El-Sayed, M.S., Mow, K., Dullien, F.A.L.: Flow through porous media, the Ergun equation revisited. Ind. Eng. Chem. Fundam. 18(3), 199–208 (1979). doi: 10.1021/i160071a001 CrossRefGoogle Scholar
  40. Mahoney, D., Doggett, K.: Multiphase flow in fractures. In: Proceedings from the international symposium of the society of core analysts in Calgary, Calgary (1997)Google Scholar
  41. Mei, C.C., Auriault, J.-L.: The effect of weak inertia on flow through a porous medium. J. Fluid Mech. 222, 647–663 (1991). doi: 10.1017/S0022112091001258 CrossRefGoogle Scholar
  42. Miskimins, J.L., Lopez-Hernandez, H.D., Barree, R.D.: Non-Darcy flow in hydraulic fractures: does it really matter? SPE 96389 annual technical conference and exhibition, Dallas, 9–12 Oct 2005Google Scholar
  43. Murphy, J.R., Thomson, N.R.: Two-phase flow in a variable aperture fracture. Water Resour. Res. 29(10), 3453–3476 (1993). doi: 10.1029/93WR01285 CrossRefGoogle Scholar
  44. Muskat, M.: The flow of Homogeneous Fluids Through Porous Media. International Human Resources Development Corporation (reprint from McGraw-Hill), Boston (1937)Google Scholar
  45. Neasham, J.W.: The morphology of dispersed clay in sandstone reservoirs and its effects on sandstone shaliness, pore space and fluid flow properties. Paper SPE 6858 presented at the 1977 SPE annual technical conference and exhibition, Denver, 9–12 Oct 1977 doi: 10.2118/6858-MS
  46. Nicholl, M.J., Glass, R.J.: Wetting phase permeability in a partially saturated horizontal fracture. Proceedings of the 5th annual international high-level radioactive waste management conference 2007–2019, Las Vegas, 22–26 May 1994Google Scholar
  47. Noman, R., Archer, M.S.: The effect of pore structure on Non-Darcy gas flow in some low permeability reservoir rocks. Paper SPE 16400 presented at the SPE/DOE low permeability reservoirs symposium, Denver, 18–19 May 1987. doi: 10.2118/16400-MS
  48. Nowamooz, A., Radilla, G., Fourar, M.: Non-Darcian flow in transparent replica of rough-walled rock fractures. Water Resour. Res. 45, W07406 (2009). doi: 10.1029/2008WR007315 CrossRefGoogle Scholar
  49. Pyrak-Nolte, L.J., Helgeston, D., Haley, G.M., Morris, J.W.: Immiscible fluid flow. In: Tillersson and Wawersik, Balkema A.A. (eds) Fracture, Proceeding of the 33rd U.S. Rock mechanics symposium, pp. 571–578, Rotterdam (1992)Google Scholar
  50. Raats, D.A.C., Klute, A.: Transport in soils: the balance of momentum. Soil Sci. Soc. Am. J. 32(4), 161–166 (1968). doi: 10.2136/sssaj1968.03615995003200040013x CrossRefGoogle Scholar
  51. Rasoloarijaona, M., Auriault, J.L.: Nonlinear seepage flow through a rigid porous medium. Eur. J. Mech. B/Fluids 13(2), 177–195 (1994)Google Scholar
  52. Rocha, R.P.A., Cruz, M.E.: Calculation of the permeability and apparent permeability of three-dimensional porous media. Transp. Porous Media 83(2), 349–373 (2010). doi: 10.1007/s11242-009-9445-7 CrossRefGoogle Scholar
  53. Romm, E.S.: Fluid flow in fractured rocks. Translated from the Russian, English translation: Blake, W.R., Bartlesville, O.K., 1972, Nedra Publishing House, Moscow (1966)Google Scholar
  54. Rose, W.: Petroleum reservoir engineering at the crossroads (ways of thinking and doing). Iran Petroleum Inst. Bull. 46, 23–27 (1972)Google Scholar
  55. Rossen, W.R., Kumar, A.T.A.: Single and two-phase flow in natural fractures. Paper SPE 24195 presented at the 67th SPE annual technical conference and exhibition, Washington, DC, 4–7 Oct 1992. doi: 10.2118/24915-MS
  56. Saez, A.E., Carbonell, R.G.: Hydrodynamic parameters for gas–liquid co-current flow in packed beds. AIChE J. 31(1), 52–62 (1985). doi: 10.1002/aic.690310105 CrossRefGoogle Scholar
  57. Sanchez-Palencia, E.: Non homogeneous media and vibration theory. Lecture Notes in Physics, Springer, New York (1980). doi: 10.1007/3-540-10000-8
  58. Schneebeli, G.: Expériences sur la limite de validité de la loi de Darcy et l’apparition de la turbulence dans un écoulement de filtration. La Houille Blanche 2, 141–149 (1955). doi: 10.1051/lhb/1955030 CrossRefGoogle Scholar
  59. Scheidegger, A.E.: The physics of flow through porous media. Macmillan, New York (1960)Google Scholar
  60. Schulenberg, T., Muller, U.: A refined model for the coolability of core debris with flow entry from bottom. 6th Information Exchange Meanding on Debris Coolability, EPRI NP-4455, 108–113 Los Angeles (1984)Google Scholar
  61. Turland, B.D., Moore, K.A.: One-dimensional models of boiling and dryout. Post accident debris cooling. Paper presented at 5th Post Accident Heat Removal Information Exchange Mtg., Karlsruhe (1983)Google Scholar
  62. Whitaker, S.: Flow in porous media I: a theoretical derivation of Darcy’s law. Transp. Porous Media 1(1), 3–25 (1986). doi: 10.1007/BF01036523 CrossRefGoogle Scholar
  63. Witherspoon, P.A., Wang, J.S.Y., Iwai, K., Gale, J.E.: Validity of cubic law for fluid flow in a deformable rock fracture. Water Resour. Res. 16(6), 1016–1024 (1980). doi: 10.1029/WR016i006p01016 CrossRefGoogle Scholar
  64. Zarcone, C., Lenormand, R.: Détermination expérimentale du couplage visqueux dans les écoulements diphasiques en milieu poreux. C. R. Acad. Sci. Paris 318(série II), 1429–1435 (1994)Google Scholar
  65. Zimmerman, R.W., Yeo, I.W.: Fluid flow in rock fractures: From the Navier–Stokes equations to the cubic law. In: Faybishenko, B., Witherspoon, P.A., Benton, S.M. (eds.) Dynamics of Fluids in Fractured Rock, Geophys. Monogr. (122), pp. 213–224. AGU, Washington, DC (2000)Google Scholar
  66. Zimmerman, R.W., Al-Yaarubi, A.H., Pain, C.S., Grattoni, C.A.: Nonlinear regimes of fluid flow in rock fractures. Int. J. Rock Mech. Min. Sci. 41, 163 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Giovanni Radilla
    • 1
    • 2
    Email author
  • Ali Nowamooz
    • 1
    • 2
  • Mostafa Fourar
    • 1
  1. 1.LEMTA, Nancy-UniversityVandœuvre CedexFrance
  2. 2.Arts et Métiers ParisTechChâlons-en-ChampagneFrance

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