Validation and calibration of coupled porous-medium and free-flow problems using pore-scale resolved models

Abstract

The correct choice of interface conditions and effective parameters for coupled macroscale free-flow and porous-medium models is crucial for a complete mathematical description of the problem under consideration and for accurate numerical simulation of applications. We consider single-fluid-phase systems described by the Stokes–Darcy model. Different sets of coupling conditions for this model are available. However, the choice of these conditions and effective model parameters is often arbitrary. We use large-scale lattice Boltzmann simulations to validate coupling conditions by comparison of the macroscale simulations against pore-scale resolved models. We analyse three settings (lid-driven cavity over a porous bed, infiltration problem and general filtration problem) with different geometrical configurations (channelised and staggered distributions of solid grains) and different sets of interface conditions. Effective parameters for the macroscale models (permeability tensor, boundary layer constants) are computed numerically for each geometrical configuration. Numerical simulation results demonstrate the sensitivity of the coupled Stokes–Darcy problem to the location of the sharp fluid–porous interface, the effective model parameters and the interface conditions.

References

  1. 1.

    Blunt, M.J.: Multiphase flow in permeable media: A pore-scale perspective. Cambridge University Press, Cambridge (2017)

    Google Scholar 

  2. 2.

    Wildenschild, D., Sheppard, A.P.: X-ray imaging and analysis techniques for quantifying pore-scale structure and processes in subsurface porous medium systems. Adv. Water Res. 51, 217–246 (2013)

    Google Scholar 

  3. 3.

    Beavers, G.S., Joseph, D.D.: Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197–207 (1967)

    Google Scholar 

  4. 4.

    Goyeau, B., Lhuillier, D., Gobin, D., Velarde, M.: Momentum transport at a fluid-porous interface. Int. J. Heat Mass Transfer 46, 4071–4081 (2003)

    Google Scholar 

  5. 5.

    Nield, D.A.: The Beavers–Joseph boundary condition and related matters: a historical and critical note. Transp. Porous Media 78, 537–540 (2009)

    Google Scholar 

  6. 6.

    Ochoa-Tapia, A.J., Whitaker, S.: Momentum transfer at the boundary between a porous medium and a homogeneous fluid. I: Theoretical development. Int. J. Heat Mass Transfer 38, 2635–2646 (1995)

    Google Scholar 

  7. 7.

    Saffman, P.G.: On the boundary condition at the surface of a porous medium. Stud. Appl. Math. 50, 93–101 (1971)

    Google Scholar 

  8. 8.

    Discacciati, M., Miglio, E., Quarteroni, A.: Mathematical and numerical models for coupling surface and groundwater flows. Appl. Num. Math. 43, 57–74 (2002)

    Google Scholar 

  9. 9.

    Discacciati, M., Quarteroni, A.: Navier–Stokes/Darcy coupling: modeling, analysis, and numerical approximation. Rev. Mat. Complut. 22, 315–426 (2009)

    Google Scholar 

  10. 10.

    Jackson, A.S., Rybak, I., Helmig, R., Gray, W.G., Miller, C.T.: Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 9. Transition region models. Adv. Water Res. 42, 71–90 (2012)

    Google Scholar 

  11. 11.

    Jäger, W, Mikelić, A: Modeling effective interface laws for transport phenomena between an unconfined fluid and a porous medium using homogenization. Transp. Porous Media 78, 489–508 (2009)

    Google Scholar 

  12. 12.

    Layton, W., Schieweck, F., Yotov, I.: Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40, 2195–2218 (2003)

    Google Scholar 

  13. 13.

    Dawson, C.: A continuous/discontinuous Galerkin framework for modeling coupled subsurface and surface water flow. Comput. Geosci. 12, 451–472 (2008)

    Google Scholar 

  14. 14.

    Sochala, P., Ern, A., Piperno, S.: Mass conservative BDF-discontinuous Galerkin/explicit finite volume schemes for coupling subsurface and overland flows. Comput. Methods Appl. Mech. Engrg. 198, 2122–2136 (2009)

    Google Scholar 

  15. 15.

    Reuter, B., Rupp, A., Aizinger, V., Knabner, P.: Discontinuous Galerkin method for coupling hydrostatic free surface flows to saturated subsurface systems. Comput. Math. Appl. 77, 2291–2309 (2019)

    Google Scholar 

  16. 16.

    Brinkman, H.C.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. 1, 27–34 (1947)

    Google Scholar 

  17. 17.

    Cimolin, F., Discacciati, M.: Navier–Stokes/Forchheimer models for filtration through porous media. Appl. Numer. Math. 72, 205–224 (2013)

    Google Scholar 

  18. 18.

    Mosthaf, K., Baber, K., Flemisch, B., Helmig, R., Leijnse, A., Rybak, I., Wohlmuth, B.: A coupling concept for two-phase compositional porous-medium and single-phase compositional free flow. Water Resour. Res. 47, W10522 (2011)

    Google Scholar 

  19. 19.

    Rybak, I., Magiera, J., Helmig, R., Rohde, C.: Multirate time integration for coupled saturated/unsaturated porous medium and free flow systems. Comput. Geosci. 19, 299–309 (2015)

    Google Scholar 

  20. 20.

    Carraro, T., Goll, C., Marciniak-Czochra, A., Mikelić, A: Effective interface conditions for the forced infiltration of a viscous fluid into a porous medium using homogenization. Comput. Methods Appl. Mech. Engrg. 292, 195–220 (2015)

    Google Scholar 

  21. 21.

    Angot, P., Goyeau, B., Ochoa-Tapia, J.A.: Asymptotic modeling of transport phenomena at the interface between a fluid and a porous layer: Jump conditions. Phys. Rev. E 95, 063302 (2017)

    Google Scholar 

  22. 22.

    Arbogast, T., Brunson, D.S.: A computational method for approximating a Darcy–Stokes system governing a vuggy porous medium. Comput. Geosci. 11, 207–218 (2007)

    Google Scholar 

  23. 23.

    Hanspal, N.S., Waghode, A.N., Nassehi, V., Wakeman, R.J.: Numerical analysis of coupled Stokes/Darcy flows in industrial filtrations. Transp. Porous Media 64, 383–411 (2006)

    Google Scholar 

  24. 24.

    Iliev, O., Laptev, V.: On numerical simulation of flow through oil filters. Comput. Vis. Sci. 6, 139–146 (2004)

    Google Scholar 

  25. 25.

    Rivière, B: Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems. J. Sci. Comput. 22, 479–500 (2005)

    Google Scholar 

  26. 26.

    Girault, V., Rivière, B: DG approximation of coupled Navier–Stokes and Darcy equations by Beavers–Joseph–Saffman interface condition. SIAM J. Numer. Anal. 47, 2052–2089 (2009)

    Google Scholar 

  27. 27.

    Jäger, W, Mikelić, A: On the interface boundary conditions by Beavers, Joseph and Saffman. SIAM J. Appl. Math. 60, 1111–1127 (2000)

    Google Scholar 

  28. 28.

    Valdés-Parada, FJ, Alvarez-Ramìrez, J, Goyeau, B., Ochoa-Tapia, J.A.: Computation of jump coefficients for momentum transfer between a porous medium and a fluid using a closed generalized transfer equation. Transp. Porous Media 78, 439–457 (2009)

    Google Scholar 

  29. 29.

    Lācis, U, Bagheri, S.: A framework for computing effective boundary conditions at the interface between free fluid and a porous medium. J. Fluid Mech. 812, 866–889 (2017)

    Google Scholar 

  30. 30.

    Zampogna, G.A., Bottaro, A.: Fluid flow over and through a regular bundle of rigid fibres. J. Fluid Mech. 792, 5–35 (2016)

    Google Scholar 

  31. 31.

    Le Bars, M., Worster, M.: Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification. J. Fluid Mech. 550, 149–173 (2006)

    Google Scholar 

  32. 32.

    Yang, G., Coltman, E., Weishaupt, K., Terzis, A., Helmig, R., Weigand, B.: On the Beavers–Joseph interface condition for non-parallel coupled channel flow over a porous structure at high Reynolds numbers. Transp. Porous Media 128, 431–457 (2019)

    Google Scholar 

  33. 33.

    Eggenweiler, E., Rybak, I.: Unsuitability of the Beavers–Joseph interface condition for filtration problems. J. Fluid Mech. 892, A10 (2020)

    Google Scholar 

  34. 34.

    Discacciati, M., Gerardo-Giorda, L.: Optimized Schwarz methods for the Stokes–Darcy coupling. IMA J. Numer. Anal. 38, 1959–1983 (2018)

    Google Scholar 

  35. 35.

    Jones, I.P.: Low Reynolds number flow past a porous spherical shell. Proc. Camb. Phil. Soc. 73, 231–238 (1973)

    Google Scholar 

  36. 36.

    Jäger, W., Mikelić, A., Neuss, N.: Asymptotic analysis of the laminar viscous flow over a porous bed. SIAM J. Sci. Comput. 22, 2006–2028 (2001)

    Google Scholar 

  37. 37.

    Hornung, U.: Homogenization and porous media. Springer, Berlin (1997)

    Google Scholar 

  38. 38.

    Wolf-Gladrow, D.: Lattice-gas cellular automata and lattice Boltzmann models. Springer, Berlin (2000)

    Google Scholar 

  39. 39.

    He, X., Luo, L-S: A priori derivation of the lattice Boltzmann equation. Phys. Rev. E 55, R6333 (1997)

    Google Scholar 

  40. 40.

    Krüger, T, Kusumaatmaja, H., Kuzmin, A., Shardt, O., Silva, G., Viggen, E.M.: The lattice Boltzmann method: Principles and practice. Springer, Berlin (2017)

    Google Scholar 

  41. 41.

    Rettinger, C., Godenschwager, C., Eibl, S., Preclik, T., Schruff, T., Frings, R., Rüde, U: Fully resolved simulations of dune formation in riverbeds. In: Proceedings of the International Conference on High Performance Computing, pp 3–21. Springer International Publishing (2017)

  42. 42.

    Kuron, M., Rempfer, G., Schornbaum, F., Bauer, M., Godenschwager, C., Holm, C., de Graaf, J.: Moving charged particles in lattice Boltzmann-based electrokinetics. J. Chem. Phys. 145(21), 214102 (2016)

    Google Scholar 

  43. 43.

    Huang, H., Sukop, M.C., Lu, X-Y: Multiphase lattice Boltzmann methods: Theory and application. Wiley, New York (2015)

    Google Scholar 

  44. 44.

    Fattahi, E., Waluga, C., Wohlmuth, B., Rüde, U: Large scale lattice Boltzmann simulation for the coupling of free and porous media flow, pp 1–18. Springer (2016)

  45. 45.

    Fattahi, E., Waluga, C., Wohlmuth, B., Rüde, U, Manhart, M., Helmig, R.: Lattice Boltzmann methods in porous media simulations: From laminar to turbulent flow. Comput. Fluids 140, 247–259 (2016)

    Google Scholar 

  46. 46.

    Pan, C., Luo, L., Miller, C.T.: An evaluation of lattice Boltzmann schemes for porous medium flow simulation. Comput. Fluids 35, 898–909 (2006)

    Google Scholar 

  47. 47.

    Ginzburg, I., Verhaeghe, F., d’Humières, D.: Two-relaxation-time lattice Boltzmann scheme: About parametrization, velocity, pressure and mixed boundary conditions. Commun. Comput. Phys. 3, 427–478 (2008)

    Google Scholar 

  48. 48.

    Ginzburg, I., Verhaeghe, F., d’Humières, D.: Study of simple hydrodynamic solutions with the two-relaxation-times lattice Boltzmann scheme. Commun. Comput. Phys. 3, 519–581 (2008)

    Google Scholar 

  49. 49.

    Hecht, F.: New development in FreeFem++. J. Numer. Math. 20, 251–265 (2012)

    Google Scholar 

  50. 50.

    Bauer, M., Eibl, S., Godenschwager, C., Kohl, N., Kuron, M., Rettinger, C., Schornbaum, F., Schwarzmeier, C., Thönnes, D., Köstler, H., Rüde, U.: waLBerla: A block-structured high-performance framework for multiphysics simulations. CAMWA (2020)

  51. 51.

    Godenschwager, C., Schornbaum, F., Bauer, M., Köstler, H., Rüde, U.: A framework for hybrid parallel flow simulations with a trillion cells in complex geometries. In: Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis, pp 1–12. ACM (2013)

  52. 52.

    Schornbaum, F., Rüde, U.: Extreme-scale block-structured adaptive mesh refinement. SIAM J. Sci. Comput. 40(3), C358–C387 (2018)

    Google Scholar 

Download references

Acknowledgements

Open Access funding provided by Projekt DEAL. The authors thank Ivan Yotov, Jim Magiera and Christoph Rettinger for the valuable discussions. The authors are grateful to the Regionales Rechenzentrum Erlangen (www.rrze.de) for providing access to supercomputing facilities.

Funding

I. Rybak and E. Eggenweiler received financial support for this work from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) by funding SFB 1313, Project Number 327154368. C. Schwarzmeier and U. Rüde received funding from the DFG for supporting project RU 422/27 and the Bundesministerium für Bildung und Forschung (BMBF, Federal Ministry of Education and Research) for supporting the projects HPC2SE (01ICH16003D) and SKAMPY (01IH15003A).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Iryna Rybak.

Additional information

Publisher’s note

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

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Rybak, I., Schwarzmeier, C., Eggenweiler, E. et al. Validation and calibration of coupled porous-medium and free-flow problems using pore-scale resolved models. Comput Geosci (2020). https://doi.org/10.1007/s10596-020-09994-x

Download citation

Keywords

  • Stokes equations
  • Darcy’s law
  • Interface conditions
  • Lattice Boltzmann method

Mathematics Subject Classification (2010)

  • 68N99
  • 76D07
  • 76M50
  • 76S05