Skip to main content
SpringerLink
Log in

Fluid Displacement in a Dual-Permeability Medium with Local Capillary Equilibrium

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

The solution to the Buckley–Leverett problem in fractured porous media is investigated by applying the dual-porosity dual-permeability model. It is shown that the dual-permeability medium is equivalent to an upscaled homogenized medium at long injection times when local capillary equilibrium is established. Simple analytical relationships for the petrophysical properties and saturation functions of the equivalent medium are derived. The upscaled relative permeability and the fractional flow function can exhibit kinks, resulting in more complicated solutions to the Buckley–Leverett problem as compared to the classical case. It is found that immiscible displacement in fractured porous media leads to the appearance of saturation profiles with several displacement fronts and rarefaction waves. Up to four solution types are possible, which are constrained in a solution map constructed in the space of the dimensionless parameters of the problem. It is shown that, for particular parameters, the displacing fluid moves faster through the matrix than through the fractures.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Afanasyev, A.: Reservoir simulation with the MUFITS code: extension for horizontal wells and fractured reservoirs. Energy Procedia 125, 596–603 (2017)

    Article  Google Scholar 

  • Afanasyev, A.A.: Effective asymptotic model of two-phase flow through fractured-porous media. Fluid Dyn. 54(5), 671–680 (2019)

    Article  Google Scholar 

  • Amaziane, B., Panfilov, M., Pankratov, L.: Homogenized model of two-phase flow with local nonequilibrium in double porosity media. Adv. Math. Phys. 3058710, (2016)

  • Barenblatt, G., Zheltov, I., Kochina, I.: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata]. J. Appl. Math. Mech. 24(5), 1286–1303 (1960)

    Article  Google Scholar 

  • Bourgeat, A., Mikelić, A.: Homogenization of two-phase immiscible flows in a one-dimensional porous medium. Asymptot. Anal. 9(4), 359–380 (1994)

    Article  Google Scholar 

  • Brooks, R., Corey, A.: Hydraulic properties of porous media. Hydrol. Pap. Color. State Univ. (1964)

  • Buckley, S., Leverett, M.: Mechanism of fluid displacement in sands. Trans. AIME 146(1), 107–116 (1942)

    Article  Google Scholar 

  • Chang, Y., Mohanty, K., Huang, D., Honarpour, M.: The impact of wettability and core-scale heterogeneities on relative permeability. J. Pet. Sci. Eng. 18(1–2), 1–19 (1997)

    Article  Google Scholar 

  • Corey, A.T., Rathjens, C.H.: Effect of stratification on relative permeability. J. Pet. Technol. 8(12), 69–71 (1956)

    Article  Google Scholar 

  • de Swaan, A.: Theory of waterflooding in fractured reservoirs. Soc. Pet. Eng. J. 18(02), 117–122 (1978)

    Article  Google Scholar 

  • Dietrich, P., Helmig, R., Sauter, M., Teutsch, G., Hötzl, H., Köngeter, J.: Flow and Transport in Fractured Porous Media. Springer, Berlin (2005)

    Book  Google Scholar 

  • Douglas, J., Paes-Leme, P.J., Hensley, J.L.: A limit form of the equations for immiscible displacement in a fractured reservoir. Transp. Porous Media 6(5–6), 549–565 (1991)

    Article  Google Scholar 

  • Ehrlich, R.: Relative permeability characteristics of vugular cores - their measurement and significance. In: Proc. Fall Meet. Soc. Pet. Eng. AIME. Society of Petroleum Engineers 3553 (1971)

  • Fanchi, J.R.: Principles of Applied Reservoir Simulation. Elsevier, New York (2006)

    Google Scholar 

  • Firoozabadi, A., Thomas, L.K.: Sixth SPE comparative solution project: dual-porosity simulators. J. Pet. Technol. 42(06), 710–763 (1990)

    Article  Google Scholar 

  • Homsy, G.M.: Viscous fingering in porous media. Ann. Rev. Fluid Mech. 19(1), 271–311 (1987)

    Article  Google Scholar 

  • Kazemi, H., Merrill, L.S., Porterfield, K.L., Zeman, P.R.: Numerical simulation of water-oil flow in naturally fractured reservoirs. Soc. Pet. Eng. AIME J. 16(06), 317–326 (1976)

    Article  Google Scholar 

  • Khaliavin, S.A., Afanasyev, A.A.: Numerical modelling of transient two-phase flows in a fractured-porous medium. Phys. Kinet. Gas Dyn. 20(3), 1–12 (2019). [in Russian]

    Article  Google Scholar 

  • Lim, K.T., Aziz, K.: Matrix-fracture transfer shape factors for dual-porosity simulators. J. Pet. Sci. Eng. 13, 169–178 (1995)

    Article  Google Scholar 

  • Matthai, S.K., Nick, H.M.: Upscaling two-phase flow in naturally fractured reservoirs. Am. Assoc. Pet. Geol. Bull. 93(11), 1621–1632 (2009)

    Google Scholar 

  • Mohanty, K., Miller, A.: Factors influencing unsteady relative permeability of a mixed-wet reservoir rock. SPE Form. Eval. 6(03), 349–358 (1991)

    Article  Google Scholar 

  • Panfilov, M.: Macroscale Models of Flow Through Highly Heterogeneous Porous Media. Springer, Amsterdam (2000)

    Book  Google Scholar 

  • Panfilov, M.: Homogenized model with memory for two-phase compressible flow in double-porosity media. Phys. Fluids 31(9), 093105 (2019)

    Article  Google Scholar 

  • Peters, E.J., Khataniar, S.: The effect of instability on relative permeability curves obtained by the dynamic-displacement method. SPE Form. Eval. 2(04), 469–474 (1987)

    Article  Google Scholar 

  • Riaz, A., Tchelepi, H.A.: Influence of relative permeability on the stability characteristics of immiscible flow in porous media. Transp. Porous Media 64(3), 315–338 (2006)

    Article  Google Scholar 

  • Saffman, P.G., Taylor, S.G.: The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. Ser. A. Math. Phys. Sci. 245(1242), 312–329 (1958)

    Google Scholar 

  • Sahimi, M.: Flow and transport in porous media and fractured rock: from classical methods to modern approaches, 2nd edn. Wiley, Weinheim (2011)

    Book  Google Scholar 

  • Terez, I.E., Firoozabadi, A.: Water injection in water-wet fractured porous media: experiments and a new model with modified Buckley–Leverett theory. SPE J. 4(02), 134–141 (1999)

    Article  Google Scholar 

  • Voloshin, A., Pankratov, L., Konyukhov, A.: Homogenization of Kondaurov’s non-equilibrium two-phase flow in double porosity media. Appl. Anal. 98(8), 1429–1450 (2019)

    Article  Google Scholar 

  • Warren, J.E., Root, P.J.: The behavior of naturally fractured reservoirs. Soc. Pet. Eng. J. 3(3), 245–255 (1963)

    Article  Google Scholar 

  • Welge, H.J.: A simplified method for computing oil recovery by gas or water drive. J. Pet. Technol. 4(04), 91–98 (1952)

    Article  Google Scholar 

  • Yeh, L.M.: Homogenization of two-phase flow in fractured media. Math. Models Methods Appl. Sci. 16(10), 1627–1651 (2006)

    Article  Google Scholar 

  • Zhongxiang, C., Ciqun, L.: Theory of fluid displacement in a medium with double porosity. Energy Sources 6(3), 193–214 (1982)

    Article  Google Scholar 

Download references

Acknowledgements

Funding for this work was provided by the RF President’s Council on Grants (MD-3567.2018.1).

Author information

Authors and Affiliations

  1. Institute of Mechanics, Moscow State University, 1 Michurinskiy Prospekt, Moscow, Russia, 119192

    Andrey Afanasyev

Authors
  1. Andrey Afanasyev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrey Afanasyev.

Ethics declarations

Conflict of interest

The author declares that he has no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Afanasyev, A. Fluid Displacement in a Dual-Permeability Medium with Local Capillary Equilibrium. Transp Porous Med 135, 513–533 (2020). https://doi.org/10.1007/s11242-020-01486-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-020-01486-x

Keywords

Search

Navigation