Advertisement

Transport in Porous Media

, Volume 88, Issue 1, pp 133–148 | Cite as

Comparison of Two-Phase Darcy’s Law with a Thermodynamically Consistent Approach

  • Jennifer Niessner
  • Steffen BergEmail author
  • S. Majid Hassanizadeh
Open Access
Article

Abstract

The extended Darcy’s law is a commonly used equation for the description of immiscible two-phase flow in porous media. It dates back to the 1940s and is essentially an empirical relationship. According to the extended Darcy’s law, pressure gradient and gravity are the only driving forces for the flow of each fluid. Within the last two decades, more advanced and physically based descriptions for multiphase flow in porous media have been developed. In this work, the extended Darcy’s law is compared to a thermodynamically consistent approach which explicitly takes the important role of phase interfaces into account, both as entities and as parameters. In this theoretically derived approach, forces related to capillarity and interfaces appear as driving/resisting forces, in addition to the traditional terms. It turns out that the extended Darcy’s law and the thermodynamically based approach are compatible if either (i) relative permeabilities are a function of saturation only, but capillary pressure is a function of saturation and specific interfacial area or (ii) relative permeabilities are a function of saturation and saturation gradients. Theoretical considerations suggest that the former alternative is only valid in case of reversible displacement while in the general case (irreversible displacement), the latter alternative is relevant.

Keywords

Porous media Two-phase flow Interfacial area Thermodynamically consistent approach 

List of Symbols

Latin Variables

aαβ

Specific interfacial area of αβ-interface [m−1]

\({\underline{g}}\)

Gravity [m s−2]

krα

Relative permeability of phase α [−]

pα

Pressure of phase α [Pa]

pc

Capillary pressure [Pa]

t

Time [s]

\({\underline{v}_{\alpha}, \underline{v}_{\alpha\beta}}\)

Velocity of phase α or αβ-interface, respectively [m s−1]

A

Area [m2]

Ed

Net efficiency [−]

H

Specific Helmholtz free energy [m2 s−1]

\({\underline{\underline{K}}}\)

Intrinsic permeability tensor [m2]

L

Length [m]

Qα

External source or sink of phase α [s−1]

Q

Volumetric flux [m3 s−1]

\({\underline{\underline{R}}}\)

Resistance tensor [Pa s m−2]

Sα

Saturation of phase α [−]

Tα

Temperature of phase α [K]

Vb

Bulk volume [m3]

W

Work [J]

Greek Symbols

Symbol

Meaning

γαβ

Macroscopic interfacial tension [Pa m]

μα

Dynamic viscosity of phase α [Pa s]

\({\phi}\)

Porosity [−]

ρα

Density of phase α [kg m−3]

σαβ

Pore-scale interfacial tension of αβ-interface [Pa m]

Γαβ

Areal mass density of αβ-interface [kg m−2]

Θ

Contact angle [rad]

Subscripts

c

Capillary

n

Non-wetting

w

Wetting

α

Phase

αβ

Interface

Notes

Acknowledgments

The authors acknowledge A. W. Cense, J. G. Maas, W. Scherpenisse, P. Doe, and J. Jennings from Shell for their helpful discussions. J. Niessner and S.M. Hassanizadeh are members of the International Research Training Group NUPUS, financed by the German Research Foundation (DFG) and The Netherlands Organisation for Scientific Research (NWO), and thank the DFG (GRK 1398) and NWO/ALW for their support.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. Ataie-Ashtiani B., Hassanizadeh S.M., Celia M.A.: Effects of heterogeneities on capillary pressure–saturation–relative permeability relationships. J. Contam. Hydrol. 56, 175–192 (2002)CrossRefGoogle Scholar
  2. Auriault, J.: Transport in Porous Media: Upscaling by Multiscale Asymptotic Expansions, pp. 3–56. Springer, Berlin (2005)Google Scholar
  3. Avraam D., Payatakes A.: Flow regimes and relative permeabilities during steady-state two-phase flow in porous media. J. Fluid Mech. Digit. Arch. 293, 207–236 (1995)Google Scholar
  4. Ayub M., Bentsen R.G.: Interfacial viscous coupling: a myth or reality?. J. Petrol Sci. Eng. 23, 13–29 (1999)CrossRefGoogle Scholar
  5. Ayub M., Bentsen R.G.: Experimental testing of interfacial coupling in two-phase flow in porous media. Pet. Sci. Technol. 23, 863–897 (2005)CrossRefGoogle Scholar
  6. Bacri J.C., Rosen M., Salin D.: Capillary hyperdiffusion as a test of wettability. Europhys. Lett. 2(2), 127–132 (1990)CrossRefGoogle Scholar
  7. Bartley, J.T., Ruth, D.W.: Experimental investigation of unsteady-state relative permeability in sand-packs. Society of Core Analysis Conference Paper SCA, vol. 22, pp.1–14 (2001)Google Scholar
  8. Bear J., Bachmat Y.: Introduction to Modeling of Transport Phenomena in Porous Media. Kluwer Academic Publishers, The Netherlands (1990)Google Scholar
  9. Berg S., Cense A.W., Hofman J.P., Smits R.M.M.: Two-phase flow in porous media with slip boundary condition. Transp. Porous Media 74(3), 275–292 (2008)CrossRefGoogle Scholar
  10. Boom, W., Wit, K., Schulte, A.M., Oedai, S., Zeelenberg, J.P.W., Maas, J.G.: Experimental evidence for improved condensate mobility at near-wellbore flow conditions. Society of Petroleum Engineers (SPE 30766):667–675 (1995)Google Scholar
  11. Boom, W., Wit, K., Zeelenberg, J.P.W., Weeda, H.C., Maas, J.G.: On the use of model experiments for assessing improved gas-condensate mobility under near-wellbore flow conditions. Society of Petroleum Engineers (SPE 36714):343–353 (1996)Google Scholar
  12. Bottero, S., Hassanizadeh, S., Kleingeld, P., Bezuijen A.: Experimental study of dynamic capillary pressure effect in two-phase flow in porous media. In: Proceedings of the XVI International Conference on Computational Methods in Water Resources (CMWR), Copenhagen, Denmark (2006)Google Scholar
  13. Bowen R.: Compressible porous media models by use of the theory of mixtures. Int. J. Eng. Sci. 20(6), 697–735 (1982)CrossRefGoogle Scholar
  14. Brinkman H.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A1, 27–33 (1947)Google Scholar
  15. Buckley S.E., Leverett M.C.: Mechanism of fluid displacements in sands. Trans. AIME 146, 107–116 (1942)Google Scholar
  16. Cable, A., Mott, R., Spearing, M.: X-ray in-situ saturation in gas condensate relative permeability studies. Society of Core Analysis Conference Paper SCA 2000-39, pp. 1–13 (2000)Google Scholar
  17. Chen, A.L., Wood, A.C.: Rate effects on water-oil relative permeability. Society of Core Analysis Conference Paper SCA 2001-19, pp. 1–12 (2001)Google Scholar
  18. Chen, D., Pyrak-Nolte, L.J., Griffin, J., Giordano, N.J.: Measurement of interfacial area per volume for drainage and imbibition. Water Resour. Res. 43, W12504 (2007)Google Scholar
  19. Cueto-Felgueroso L., Juanes R.: A phase field model of unsaturated flow. Water Resour. Res. 43, W10–409 (2009)Google Scholar
  20. Culligan K., Wildenschild D., Christensen B., Gray W., Rivers M., Tompson A.: Interfacial area measurements for unsaturated flow through a porous medium. Water Resour. Res. 40, 1–12 (2004)CrossRefGoogle Scholar
  21. Culligan K.A., Wildenschild D., Christensen B.S.B., Gray W.G., Rivers M.L.: Pore-scale characteristics of multiphase flow in porous media: a comparison of air–water and oil–water experiments. Adv. Water Resour. 29, 227–238 (2006)CrossRefGoogle Scholar
  22. Dalla E., Hilpert M., Miller C.T.: Computation of the interfacial area for two-fluid porous media systems. J. Contam. Hydrol. 56, 25–48 (2002)CrossRefGoogle Scholar
  23. Darcy, H.: Détermination des lois d’écoulement de l’eau à travers le sable. In: Victor Dalmont P (ed.) Les Fontaines Publiques de la Ville de Dijon, pp. 590–594. V. Dalmont, Paris (1856)Google Scholar
  24. Das D.B., Gauldie R., Mirzaei M.: Dynamic effects for two-phase flow in porous media: fluid property effects. AIChE J. 53(10), 2505–2520 (2007)CrossRefGoogle Scholar
  25. Di Carlo D.: Experimental measurements of saturation overshoot on infiltration. Water Resour. Res. 40, W04215 (2004)CrossRefGoogle Scholar
  26. Di Carlo D.: Modeling observed saturation overshoot with continuum additions to standard unsaturated theory. Adv. Water Resour. 28(10), 1021–1027 (2005)CrossRefGoogle Scholar
  27. Dong M., Dullien F.A.L.: A new model for immiscible displacement in porous media. Transp. Porous Media 27, 185–204 (1997)CrossRefGoogle Scholar
  28. Dullien F.A.L.: Porous Media—Fluid Transport and Pore Structure, 2nd edn. Academic Press, New York (1992)Google Scholar
  29. Eastwood J.E., Spanos T.J.T.: Steady-state contercurrent flow in one dimension. Transp. Porous Media 6, 173–182 (1991)CrossRefGoogle Scholar
  30. Gladkikh M., Bryant S.: Prediction of interfacial areas during imbibition in simple porous media. Adv. Water Resour. 26, 609–622 (2003)CrossRefGoogle Scholar
  31. Gray W., Hassanizadeh S.: Averaging theorems and averaged equations for transport of interface properties in multiphase systems. Int. J. Multi-Phase Flow 15, 81–95 (1989)CrossRefGoogle Scholar
  32. Gray W., Miller C.: Thermodynamically constrained averaging theory approach for modeling of flow in porous media: 1. Motivation and overview. Adv. Water Resour. 28(2), 161–180 (2005)CrossRefGoogle Scholar
  33. Hassanizadeh S.M., Gray W.G.: General conservation equations for multi-phase systems: 3. Constitutive theory for porous media flow. Adv. Water Resour. 3, 25–40 (1980)CrossRefGoogle Scholar
  34. Hassanizadeh S.M., Gray W.G.: High velocity flow in porous media. Transp. Porous Media 2, 521–531 (1987)CrossRefGoogle Scholar
  35. Hassanizadeh S.M., Gray W.G.: Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Adv. Water Resour. 13(4), 169–186 (1990)CrossRefGoogle Scholar
  36. Hassanizadeh S.M., Gray W.G.: Thermodynamic basis of capillary pressure in porous media. Water Resour. Res. 29(10), 3389–3405 (1993a)CrossRefGoogle Scholar
  37. Hassanizadeh S.M., Gray W.G.: Toward an improved description of the physics of two-phase flow. Adv. Water Resour. 16(1), 53–67 (1993b)CrossRefGoogle Scholar
  38. Hassanizadeh S.M., Celia M.A., Dahle H.K.: Dynamic effect in capillary pressure-saturation relationship and its impact on unsaturated flow. Vadose Zone J. 1, 38–57 (2002)Google Scholar
  39. Henderson G.D., Danesh A., Teharani D.H., Al-Shaidi S., Peden J.M.: Measurement and correlation of gas condensate relative permeability by the steady-state method. SPE J. 1(2), 191–201 (1996)Google Scholar
  40. Henderson G.D., Danesh A., Al-kharusi B., Tehrani D.: Generating reliable gas condensate relative permeability data used to develop a correlation with capillary number. J. Pet. Sci. Eng. 25, 79–91 (2000)CrossRefGoogle Scholar
  41. Huang, D.D., Honarpour, M.M.: Capillary end effects in coreflood calculations. Society of Core Analysis Conference Paper SCA 96-34, pp. 1–10 (1996)Google Scholar
  42. Huang H., Lu Xy.: Relative permeabilities and coupling effects in steady-state gas-liquid flow in porous media: a lattice boltzmann study. Phys. Fluids 21, 092–104 (2009)Google Scholar
  43. Huyghe J., Oomens C., Van Campen D., Heethaar R.: Low reynolds number steady state flow through a branching network of rigid vessels: I. A mixture model. Biorheology 26, 73–84 (1989)Google Scholar
  44. Jackson A., Miller C., Gray W.: Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 6. Two-fluid-phase flow. Adv. Water Resour. 32(6), 779–795 (2009)CrossRefGoogle Scholar
  45. Joekar-Niasar V., Hassanizadeh S., Leijnse A.: Insights into the relationship among capillary pressure, saturation, interfacial area and relative permeability using pore-scale network modeling. Transp. Porous Media. 74(2), 201–219 (2008)CrossRefGoogle Scholar
  46. Joekar-Niasar, V., Hassanizadeh, S., Pyrak-Nolte, L., Berentsen, C.: Simulating drainage and imbibition experiments in a high-porosity micro-model using an unstructured pore-network model. Water Resour. Res. 45, W02430 (2009). doi: 10.1029/2007WR006641
  47. Kalam, Z., Obeida, T., Al Masaabi, A.: Acceptable water-oil and gas-oil relative permeability measurements for use in reservoir simulation models. Society of Core Analysis Conference Paper SCA. 11, pp. 1–12 (2007)Google Scholar
  48. Kalaydjian F.: A macroscopic description of multiphase flow in porous media involving spacetime evolution of fluid/fluid interface. Transp. Porous Media 2, 537–552 (1987)CrossRefGoogle Scholar
  49. Killough, J.E.: Reservoir simulation with history dependent saturation functions. Society of Petroleum Engineers SPE 5106 (1976)Google Scholar
  50. Kyte, J.R., Rapoport, L.A.: Linear waterflood behavior and end effects in water-wet porous media. Society of Petroleum Engineers Conference Paper SPE 929 G, pp. 47–50 (1958)Google Scholar
  51. Lake L.W.: Enhanced Oil Recovery. Prentice Hall, Upper Saddle River, NJ (1989)Google Scholar
  52. Langaas, K., Ekrann, S., Ebeltoft, E.: The impact of using composite cores on core analysis results. Society of Core Analysis Conference Paper SCA 96-02, pp. 1–10 (1996)Google Scholar
  53. Li H., Pan C., Miller C.T.: Pore-scale investigation of viscous coupling effects for two-phase flow in porous media. Phys. Rev. E 72, 026–705 (2005)Google Scholar
  54. Manthey S., Hassanizadeh M.S., Helmig R.: Macro-scale dynamic effects in homogeneous and heterogeneous porous media. Transp. Porous Media 58, 121–145 (2005)CrossRefGoogle Scholar
  55. Marle C.M.: On macroscopic equations governing multiphase flow with diffusion and chemical reactions in porous media. Int. J. Eng. Sci. 20(5), 643–662 (1982)CrossRefGoogle Scholar
  56. Miller C., Gray W.: Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 2. Foundation. Adv. Water Resour. 28(2), 181–202 (2005)CrossRefGoogle Scholar
  57. Mirzaei M., Das D.B.: Dynamic effects in capillary pressure–saturations relationships for two-phase flow in 3d porous media: implications of micro-heterogeneities. Chem. Eng. Sci. 62(7), 1927–1947 (2007)CrossRefGoogle Scholar
  58. Morrow N.: Physics and thermodynamics of capillary action in porous media. Ind. Eng. Chem. 62(6), 32–56 (1970)CrossRefGoogle Scholar
  59. Niessner, J., Hassanizadeh, S.: A model for two-phase flow in porous media including fluid–fluid interfacial area. Water Resour. Res. 44, W08439. doi: 10.1029/2007WR006721 (2008)
  60. Niessner J., Hassanizadeh S.: Modeling kinetic interphase mass transfer for two-phase flow in porous media including fluid–fluid interfacial area. Transp. Porous Media 80(2), 329–344 (2009)CrossRefGoogle Scholar
  61. Niessner J., Hassanizadeh S.: Non-equilibrium interphase heat and mass transfer during two-phase flow in porous media—theoretical considerations and modeling. Adv. Water Resour. 32(12), 1756–1766 (2009)CrossRefGoogle Scholar
  62. Oostrom M., White M.D., Brusseau M.L.: Theoretical estimation of free and entrapped nonwetting-wetting fluid interfacial areas in porous media. Adv. Water Resour. 24, 887–898 (2001)CrossRefGoogle Scholar
  63. Poulsen, S., Skauge, T., Dyrhol, S.O., Skauge, E. Stenby, A.: Including capillary pressure in simulations of steady state relative permeability experiments. Society of Core Analysis Conference Paper SCA 2000-14, pp. 1–12 (2000)Google Scholar
  64. Seth, S., Morrow, N.R.: Efficiency of the conversion of work of drainage to surface energy for sandstone and carbonate. SPE Reserv. Eval. Eng. SPE 102490, 338–347 (2007)Google Scholar
  65. Skauge, A., Thorsen, T., Sylte, A.: Rate selection for waterflooding of intermediate wet cores. Society of Core Analysis Conference Paper SCA 2001-20, pp. 1–14 (2001)Google Scholar
  66. Stauffer, F.: Time dependence of the relations between capillary pressure, water content and conductivity during drainage of porous media. In: On scale effects in porous media, IAHR, Thessaloniki, Greece (1978)Google Scholar
  67. Tiab D., Donaldson E.C.: Petrophysics: Theory and Practice of Measuring Reservoir Rock and Fluid Transport Properties, 2nd edn. Gulf Professional Publishing, Burlington, MA (2004)Google Scholar
  68. Tsakiroglou, C.D., Avraam, D.G., Payatakes, A.C.: Simulation of the immiscible displacement in porous media using capillary pressure and relative permeability curves from transient and steady-state experiments. Society of Core Analysis Conference Paper SCA 2004-12, pp. 1–13 (2004)Google Scholar
  69. Urkedal H., Ebeltoft E., Nordtvedt J.E., Watson A.T.U.: A new design of steady-state type experiments for simultaneous estimation of two-phase flow functions. SPE Reserv. Eng. 3(3), 230–238 (2000)Google Scholar
  70. Vankan W., Huyghe J., Janssen J., Huson A.: Poroelasticity of satured solids with an application to blood perfusion. Int. J. Eng. Sci. 34, 1019–1031 (1996)CrossRefGoogle Scholar
  71. Zhang X.Y., Bentsen R.G., Cunha L.B.: Investigations of interfacial coupling phenomena and its impact on recovery factor. J. Can. Pet. Technol. 47(7), 26–32 (2008)Google Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  • Jennifer Niessner
    • 1
  • Steffen Berg
    • 2
    Email author
  • S. Majid Hassanizadeh
    • 3
  1. 1.Institute of Hydraulic EngineeringUniversity of StuttgartStuttgartGermany
  2. 2.Shell International Exploration and Production B.V.Rijswijk (ZH)The Netherlands
  3. 3.Department of Earth Sciences, Faculty of GeosciencesUtrecht UniversityUtrechtThe Netherlands

Personalised recommendations