Skip to main content
SpringerLink
Log in

On Darcy- and Brinkman-type models for two-phase flow in asymptotically flat domains

  • Original Paper
  • Published:
Computational Geosciences Aims and scope Submit manuscript

Abstract

We study two-phase flow for Darcy and Brinkman regimes. To reduce the computational complexity for flow in vertical equilibrium, various simplified models have been suggested. Examples are dimensional reduction by vertical integration, the multiscale model approach in Guo et al. (Water Resour. Res. 50(8), 6269–6284, 2014) or the asymptotic approach in Yortsos (Transport Porous Med. 18, 107–129, 1995). The latter approach uses a geometrical scaling. In this paper, we provide a comparative study on efficiency of the asymptotic approach and the relation to the other approaches. First, we prove that the asymptotic approach is equivalent to the multiscale model approach. Then, we demonstrate its accuracy and computational efficiency over the other approaches and with respect to the full two-phase flow model. We apply then asymptotic analysis to the two-phase flow model in Brinkman regimes. The limit model is a single nonlocal evolution law with a pseudo-parabolic extension. Its computational efficiency is demonstrated using numerical examples. Finally, we show that the new limit model exhibits overshoot behaviour as it has been observed for dynamical capillarity laws (Hassanizadeh and Gray, Water Resour. Res. 29, 3389–3406, 1993).

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

Similar content being viewed by others

References

  1. Armiti-Juber, A., Rohde, C.: Almost parallel flows in porous media. Finite Volumes for Complex Applications VII-Elliptic Parabolic and Hyperbolic Problems, pp. 873–881. Springer International Publishing (2014)

  2. Auriault, J.L.: On the domain of validity of Brinkman’s equation. Transport Porous Med. 79(2), 215–223 (2009)

    Article  Google Scholar 

  3. Auriault, J.L., Geindreau, C., Boutin, C.: Filtration law in porous media with poor separation of scales. Transport Porous Med. 60(1), 89–108 (2005)

    Article  Google Scholar 

  4. Bear, J: Dynamics of Fluids in Porous Media. Dover (1988)

  5. Becker, B., Guo, B., Bandilla, K., Celia, M.A., Flemisch, B., Helmig, R.: A pseudo vertical equilibrium model for slow gravity drainage dynamics. Water Resour. Res. (2017)

  6. Brinkman, H.C.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. 1(1), 27–34 (1949)

    Article  Google Scholar 

  7. Cao, X., Pop, I.S.: Uniqueness of weak solutions for a pseudo-parabolic equation modeling two phase flow in porous media. Appl. Math Lett. 46, 25–30 (2015)

    Article  Google Scholar 

  8. Coats, K.H., Dempsey, J.R., Henderson, J.H.: The use of vertical equilibrium in two-dimensional dimulation of three-dimensional reservoir performance. Soc. Petrol. Eng. 11, 63–71 (1971)

    Article  Google Scholar 

  9. Coclite, G.M., Mishra, S., Risebro, N.H., Weber, F.: Analysis and numerical approximation of Brinkman regularization of two-phase flows in porous media. Computat. Geosci. 18(5), 637–659 (2014)

    Article  Google Scholar 

  10. Court, B., Bandilla, K.W., Celia, M.A., Janzen, A., Dobossy, M., Nordbotten, J.M.: Applicability of vertical-equilibrium and sharp-interface assumptions in CO2 sequestration modeling. Int. J. Greenh. Gas Con. 10, 134–147 (2012)

    Article  Google Scholar 

  11. Debbabi, Y., Jackson, M.D., Hampson, G.J., Fitch, P.J.R., Salinas, P.: Viscous crossflow in layered porous media. Transport Porous Med. 117(2), 281–309 (2017)

    Article  Google Scholar 

  12. Fan, Y., Pop, I.S.: A class of pseudo-parabolic equations: existence, uniqueness of weak solutions, and error estimates for the Euler-implicit discretization. Math. Meth. Appl. Sci. 34(18), 2329–2339 (2011)

    Google Scholar 

  13. Gasda, S.E., Nordbotten, J.M., Celia, M.A.: Vertical equilibrium with sub-scale analytical methods for geological CO2, sequestration. Computat. Geosci. 13, 469–481 (2009)

    Article  Google Scholar 

  14. Gasda, S.E., Nordbotten, J.M., Celia, M.A.: Vertically averaged approaches for CO2, migration with solubility trapping. Water Resour. Res., 47 (2011)

  15. Guo, B., Bandilla, K.W., Doster, F., Keilegavlen, E., Celia, M.A.: A vertically integrated model with vertical dynamics for CO2, storage. Water Resour. Res. 50(8), 6269–6284 (2014)

    Article  Google Scholar 

  16. Hassanizadeh, S.M., Gray, W.G.: Thermodynamic basis of capillary pressure in porous media. Water Resour. Res. 29, 3389–3406 (1993)

    Article  Google Scholar 

  17. Helmig, R: Multiphase Flow and Transport Processes in the Subsurface. Springer-Verlag (1997)

  18. Hornung, U.: Homogenization and Porous Media. Springer, New York (1997)

    Book  Google Scholar 

  19. Huber, R., Helmig, R.: Multiphase flow in heterogeneous porous media :A classical finite element method versus an implicit pressure–explicit saturation-based mixed finite element–finite volume approach. Int. J. Numer. Meth. Fluids 29(8), 899–920 (1999)

    Article  Google Scholar 

  20. Kissling, F., Rohde, C.: The computation of nonclassical shock waves in porous media with a heterogeneous multiscale method: The multidimensional case. Multiscale Model Simul. 13(4), 1507–1541 (2015)

    Article  Google Scholar 

  21. Krotkiewski, M., Ligaarden, I.S., Lie, K.-A., Schmid, D.W.: On the importance of the Stokes-Brinkman equations for computing effective permeability in Karst reservoirs. Commun. Computat. Phys. 10(5), 1315–1332 (2011)

    Article  Google Scholar 

  22. Lake, L.W.: Enhanced Oil Recovery. Prentice Hall, Englewood Cliffs (1989)

    Google Scholar 

  23. Marusic-Paloka, E., Pazanin, I., Marusic, S.: Comparison between Darcy and Brinkman laws in a fracture. Applied Math. and Computat. 218, 7538–7545 (2012)

    Article  Google Scholar 

  24. Menon, G., Otto, F.: Dynamic scaling in miscible viscous fingering. Commun. Math. Phys. 257(2), 303–317 (2005)

    Article  Google Scholar 

  25. Qin, C.Z., Hassanizadeh, S.M.: Multiphase flow through multilayers of thin porous media General balance equations and constitutive relationships for a solid–gas–liquid three-phase system. Int. J. Heat Mass Transf. 70, 693–708 (2014)

    Article  Google Scholar 

  26. van Duijn, C.J., Fan, Y., Peletier, L.A., Pop, I.S.: Travelling wave solutions for degenerate pseudo-parabolic equations modelling two-phase flow in porous media. Nonlinear Anal. Real World Appl. 14(3), 1361–1383 (2013)

    Article  Google Scholar 

  27. Yang, Z., Yortsos, Y.C.: Asymptotic solutions of miscible displacements in geometries of large aspect ratio. Phys. Fluids 9(2), 286–298 (1997)

    Article  Google Scholar 

  28. Yokoyama, Y., Lake, L.W.: The effects of capillary pressure on immiscible displacements in stratified porous media. Soc Petrol Eng. (1981)

  29. Yortsos, Y.C.: A theoretical analysis of vertical flow equilibrium. Transport Porous Med. 18, 107–129 (1995)

    Article  Google Scholar 

  30. Yortsos, Y.C., Salin, D.: On the selection principle for viscous fingering in porous media. J. Fluid Mech. 557, 225–236 (2006)

    Article  Google Scholar 

  31. Zapata, V.J., Lake, L.W.: A theoretical analysis of viscous crossflow. Soc Petrol Eng. (1981)

  32. Zhang, X., Shapiro, A., Stenby, E.H.: Upscaling of two-phase immiscible flows in communicating stratified reservoirs. Transport Porous Med. 87(3), 739–764 (2011)

    Article  Google Scholar 

  33. Zhang, X., Shapiro, A., Stenby, E.H.: Gravity effect on two-phase immiscible flows in communicating layered reservoirs. Transport Porous Med. 92(3), 767–788 (2012)

    Article  Google Scholar 

Download references

Funding

The first author would like to thank the DAAD for the financial support.

Author information

Authors and Affiliations

  1. Institute for Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Germany

    Alaa Armiti-Juber & Christian Rohde

Authors
  1. Alaa Armiti-Juber
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christian Rohde
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alaa Armiti-Juber.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Armiti-Juber, A., Rohde, C. On Darcy- and Brinkman-type models for two-phase flow in asymptotically flat domains. Comput Geosci 23, 285–303 (2019). https://doi.org/10.1007/s10596-018-9756-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-018-9756-2

Keywords

Search

Navigation