Skip to main content
SpringerLink
Log in

Numerical investigation of macroscopic permeability of biporous solids with elliptic vugs

  • Original Article
  • Published:
Theoretical and Computational Fluid Dynamics Aims and scope Submit manuscript

Abstract

In this paper, we aim to determine the effective permeability of biporous solids containing fractures in a computational homogenization framework. Precisely, at the local scale, we use the Brinkman law for the porous solids and the Stokes equations within fractures. The resolution of the Brinkman/Stokes coupled equations is performed with the fast Fourier transform iterative scheme on periodic unit cells. The role of the dimensions and orientation of the cracks is investigated. The simulation results are also compared with an analytical formulation based on Mori–Tanaka estimates. The findings indicate that, depending on the considered flow direction, the dimensions and orientation of cracks strongly affect the effective permeability of fractured porous media. Besides, we also determine the macroscopic permeability for a population of fractures with regular and random orientations, dimensions and shapes. The results are then given and discussed.

Graphic Abstract

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
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Arbogast, T., Brunson, D.S.: A computational method for approximating a Darcy–Stokes system governing a vuggy porous medium. Comput. Geosci. 11(3), 207–218 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Auriault, J.L., Boutin, C.: Deformable porous media with double porosity. Quasi-statics. I: Coupling effects. Transp. Porous Med. 7, 63–82 (1992)

    Article  Google Scholar 

  3. Auriault, J.L., Boutin, C.: Deformable porous media with double porosity. Quasi-statics. II: Memory effects. Transp. Porous Med. 10, 153–169 (1993)

    Article  Google Scholar 

  4. Auriault, J.L., Boutin, C.: Deformable porous media with double porosity. III: Acoustics. Transp. Porous Med. 14, 143–162 (1994)

    Article  Google Scholar 

  5. Auriault, J.L., Sanchez-Palencia, E.: Etude du comportement macroscopique d’un milieu poreux saturé déformable. J. Méc. 16, 575–603 (1977)

    MATH  Google Scholar 

  6. Bachu, S.: Co2 storage in geological media: role, means, status and barriers to deployment. Prog. Energy Combust. Sci. 34(2), 254–273 (2008)

    Article  Google Scholar 

  7. Beavers, G.S., Joseph, D.D.: Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30(1), 197–207 (1967)

    Article  Google Scholar 

  8. Beckermann, C.H., Viskanta, R., Ramadhyani, S.: Natural convection in vertical enclosures containing simultaneously fluid and porous layers. J. Fluid Mech. 186, 257–284 (1988)

    Article  MATH  Google Scholar 

  9. Bose, S., Roy, M., Bandyopadhyay, A.: Recent advances in bone tissue engineering scaffolds. Trends Biotechnol. 30(10), 546–554 (2012)

    Article  Google Scholar 

  10. Boutin, C., Royer, P., Auriault, J.L.: Acoustic absorption of porous surfacing with dual porosity. Int. J. Solids Struct. 35, 4709–4737 (1998)

    Article  MATH  Google Scholar 

  11. Boutin, C., Royer, P.: Time analysis of the three characteristic behaviours of dual-porosity media. i: fluid flow and solute transport. Transp. Porous Med. 95, 603–626 (2012)

    Article  MathSciNet  Google Scholar 

  12. Brinkman, H.C.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Flow Turbul. Combust. 1(1), 27 (1949)

    Article  MATH  Google Scholar 

  13. Celle, P., Drapier, S., Bergheau, J.M.: Numerical modelling of liquid infusion into fibrous media undergoing compaction. Eur. J. Mech. A/Solids 27(4), 647–661 (2008)

    Article  MATH  Google Scholar 

  14. de Borst, R.: Fluid flow in fractured and fracturing porous media: a unified view. Mech. Res. Commun. 80, 47–57 (2017)

    Article  Google Scholar 

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

    Book  Google Scholar 

  16. Gerke, H.H., Van Genuchten, M.T.: A dual-porosity model for simulating the preferential movement of water and solutes in structured porous media. Water Resour. Res. 29(2), 305–319 (1993)

    Article  Google Scholar 

  17. Golfier, F., Lasseux, D., Quintard, M.: Investigation of the effective permeability of vuggy or fractured porous media from a Darcy–Brinkman approach. Comput. Geosci. 19, 63–78 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  19. Herzig, J.P., Leclerc, D.M., Le Goff, P.: Flow of suspensions through porous media-application to deep filtration. Ind. Eng. Chem. 62(5), 8–35 (1970)

    Article  Google Scholar 

  20. Hornung, U.: Homogenization and Porous Media. Springer Science, Berlin (1997)

    Book  MATH  Google Scholar 

  21. Jones, I.P.: Low Reynolds number flow past a porous spherical shell. In: Mathematical proceedings of the Cambridge philosophical society, volume 73(1), pp. 231–238. Cambridge University Press (1973)

  22. Joseph, D.D., Tao, L.N.: Lubrication of a porous bearing-Stokes’ solution. J. Appl. Mech. 33(4), 753–760 (1966)

    Article  Google Scholar 

  23. Le Droumaguet, B., Lacombe, R., Ly, H.B., Guerrouache, M., Carbonnier, B., Grande, D.: Engineering functional doubly porous phema-based materials. Polymer 55(1), 373–379 (2014)

    Article  Google Scholar 

  24. Le, L.M., Ly, H.B., Pham, B.T., Le, V.M., Pham, T.A., Nguyen, D.H., Tran, X.T., Le, T.T.: Hybrid artificial intelligence approaches for predicting buckling damage of steel columns under axial compression. Materials 12(10), 1670 (2019)

    Article  Google Scholar 

  25. Lévy, T.: Fluid flow through an array of fixed particles. Int. J. Eng. Sci. 21, 11–23 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lewallen, K.T., Wang, H.F.: Consolidation of a double-porosity medium. Int. J. Solids Struct. 35, 4845–4867 (1998)

    Article  MATH  Google Scholar 

  27. Lim, K.T., Aziz, K.: Matrix-fracture transfer shape factors for dual-porosity simulators. J. Pet. Sci. Eng. 13(3–4), 169–178 (1995)

    Article  Google Scholar 

  28. Lundgren, T.S.: Slow flow through stationary random beds and suspensions of spheres. J. Fluid Mech. 51(2), 273–299 (1972)

    Article  MATH  Google Scholar 

  29. Ly, H.B., Le Droumaguet, B., Monchiet, V., Grande, D.: Designing and modeling doubly porous polymeric materials. Eur. Phys. J. Spec. Top. 224(9), 1689–1706 (2015)

    Article  Google Scholar 

  30. Ly, H.B., Le Droumaguet, B., Monchiet, V., Grande, D.: Facile fabrication of doubly porous polymeric materials with controlled nano-and macro-porosity. Polymer 78, 13–21 (2015)

    Article  Google Scholar 

  31. Ly, H.B., Le Droumaguet, B., Monchiet, V., Grande, D.: Tailoring doubly porous poly (2-hydroxyethyl methacrylate)-based materials via thermally induced phase separation. Polymer 86, 138–146 (2016)

    Article  Google Scholar 

  32. Ly, H.B., Poupart, R., Carbonnier, B., Monchiet, V., Le Droumaguet, B., Grande, D.: Versatile functionalization platform of biporous poly (2-hydroxyethyl methacrylate)-based materials: application in heterogeneous supported catalysis. React. Funct. Polym. 121, 91–100 (2017)

    Article  Google Scholar 

  33. Ly, H.-B., Christophe Desceliers, L., Le, T.-T., Pham, B.T., Nguyen-Ngoc, L., Doan, V.T., Le, M., Le, M., et al.: Quantification of uncertainties on the critical buckling load of columns under axial compression with uncertain random materials. Materials 12(11), 1828 (2019)

    Article  Google Scholar 

  34. Ly, H.B., Monteiro, E., Le, T.T., Le, V.M., Dal, M., Regnier, G., Pham, B.T.: Prediction and sensitivity analysis of bubble dissolution time in 3d selective laser sintering using ensemble decision trees. Materials 12(9), 1544 (2019)

    Article  Google Scholar 

  35. Markov, M., Kazatchenko, E., Mousatov, A., Pervago, E.: Permeability of the fluid-filled inclusions in porous media. Transp. Porous Med. 84, 307–317 (2010)

    Article  Google Scholar 

  36. Masuoka, T.: Convective currents in a horizontal layer divided by a permeable wall. Bull. JSME 17(104), 225–232 (1974)

    Article  Google Scholar 

  37. Mezhoud, S., Monchiet, V., Bornert, M., Grande, D.: Computation of macroscopic permeability of doubly porous media with fft based numerical homogenization method. Eur. J. Mech. B/Fluids 83, 141–155 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  38. Moench, A.F.: Double-porosity models for a fissured groundwater reservoir with fracture skin. Water Resour. Res. 20(7), 831–846 (1984)

    Article  Google Scholar 

  39. Monchiet, V., Bonnet, G., Lauriat, G.: A fft-based method to compute the permeability induced by a stokes slip flow through a porous medium. C. R. Méc. 337(4), 192–197 (2009)

    Article  MATH  Google Scholar 

  40. Monchiet, V., Ly, H.B., Grande, D.: Macroscopic permeability of doubly porous materials with cylindrical and spherical macropores. Meccanica 54(10), 1583–1596 (2019)

    Article  MathSciNet  Google Scholar 

  41. Moulinec, H., Suquet, P.: A fft-based numerical method for computing the mechanical properties of composites from images of their microstructures. In: IUTAM Symposium on Microstructure-Property Interactions in Composite Materials, pp. 235–246. Springer (1995)

  42. Moulinec, H., Suquet, P.: A fast numerical method for computing the linear and nonlinear mechanical properties of composites. C. R. Acad. Sci. 318(11), 1417–1423 (1994)

    MATH  Google Scholar 

  43. Moulinec, H., Suquet, P.: A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comput. Methods Appl. Mech. Eng. 157(1), 69–94 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  44. Neale, G.H., Nader, W.K.: Prediction of transport processes within porous media: creeping flow relative to a fixed swarm of spherical particles. AIChE J. 20(3), 530–538 (1974)

    Article  Google Scholar 

  45. Neale, G., Epstein, N., Nader, W.: Creeping flow relative to permeable spheres. Chem. Eng. Sci. 28(10), 1865–1874 (1973)

    Article  Google Scholar 

  46. Nguyen, T.K., Monchiet, V., Bonnet, G.: A Fourier based numerical method for computing the dynamic permeability of periodic porous media. Eur. J. Mech. B/Fluids 37, 90–98 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  47. Nguyen, H.L., Le, T.H., Pham, C.T., Le, T.T., Ho, L.S., Le, V.M., Pham, B.T., Ly, H.B.: Development of hybrid artificial intelligence approaches and a support vector machine algorithm for predicting the marshall parameters of stone matrix asphalt. Appl. Sci. 9(15), 3172 (2019)

    Article  Google Scholar 

  48. Ochoa-Tapia, J.A., Whitaker, S.: Momentum transfer at the boundary between a porous medium and a homogeneous fluid. i. Theoretical development. J. Heat Mass Transf. 38(14), 2635–2646 (1995)

    Article  MATH  Google Scholar 

  49. Olny, X., Boutin, C.: Acoustic wave propagation in double porosity media. J. Acoust. Soc. Am. 114, 73–89 (2003)

    Article  Google Scholar 

  50. Pham, B.T., Nguyen, M.D., Van Dao, D., Prakash, I., Ly, H.B., Le, T.T., Ho, L.S., Nguyen, K.T., Ngo, T.Q., Hoang, V., et al.: Development of artificial intelligence models for the prediction of compression coefficient of soil: an application of Monte Carlo sensitivity analysis. Sci. Total Environ. 679, 172–184 (2019)

    Article  Google Scholar 

  51. Poulikakos, D., Kazmierczak, M.: Forced convection in a duct partially filled with a porous material. J. Heat Transf. 109(3), 653–662 (1987)

    Article  Google Scholar 

  52. Rahm, D.: Regulating hydraulic fracturing in shale gas plays: the case of Texas. Energy Policy 39(5), 2974–2981 (2011)

    Article  Google Scholar 

  53. Rasoulzadeh, M., Kuchuk, F.J.: Effective permeability of a porous medium with spherical and spheroidal vug and fracture inclusions. Transp. Porous Med. 116, 613–644 (2017)

    Article  MathSciNet  Google Scholar 

  54. Royer, P., Auriault, J.L., Boutin, C.: Macroscopic modeling of double-porosity reservoirs. J. Pet. Sci. Eng. 16, 187–202 (1996)

    Article  Google Scholar 

  55. Sahimi, M.: Flow and Transport in Porous Media and Fractured Rock: From Classical Methods to Modern Approaches. Wiley, New York (2011)

    Book  MATH  Google Scholar 

  56. Salinger, A.G., Aris, R., Derby, J.J.: Modeling the spontaneous ignition of coal stockpiles. AIChE J. 40(6), 991–1004 (1994)

    Article  Google Scholar 

  57. Sanchez-Palencia, E.: Non-homogeneous media and vibration theory. In: Non-homogeneous media and vibration theory, vol. 127 (1980)

  58. Soize, C.: A nonparametric model of random uncertainties for reduced matrix models in structural dynamics. Probab. Eng. Mech. 15(3), 277–294 (2000)

    Article  Google Scholar 

  59. Soize, C.: Random matrix theory for modeling uncertainties in computational mechanics. Comput. Methods Appl. Mech. Eng. 194(12–16), 1333–1366 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  60. Tang, H.T., Fung, Y.C.: Fluid movement in a channel with permeable walls covered by porous media: a model of lung alveolar sheet. J. Appl. Mech. 42(1), 45–50 (1975)

    Article  Google Scholar 

  61. Whitaker, S., Quintard, M.: Ecoulement monophasique en milieu poreux: Effet des hétérogénéités locales. J. Theor. Appl. Mech. 6, 691–726 (1987)

    MATH  Google Scholar 

  62. Xie, X., Xu, J., Xue, G.: Uniformly-stable finite element methods for Darcy–Stokes–Brinkman models. J. Comput. Math. 26(3), 437–455 (2008)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 107.03-2019.23.

Author information

Authors and Affiliations

  1. University of Transport Technology, Hanoi, 100000, Vietnam

    Hai-Bang Ly & Hoang-Long Nguyen

  2. University of Transport and Communications, Hanoi, 100000, Vietnam

    Viet-Hung Phan & Long Nguyen-Ngoc

  3. Laboratoire Modẽlisation et Simulation Multi Echelle, MSME UMR8208 CNRS, Universitẽ Paris-Est, 5 boulevard Descartes, 77454, Marne la Vallẽe Cedex, France

    Viet-Hung Phan & Vincent Monchiet

Authors
  1. Hai-Bang Ly
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Viet-Hung Phan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vincent Monchiet
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hoang-Long Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Long Nguyen-Ngoc
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Viet-Hung Phan.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by Vassilios Theofilis.

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

Ly, HB., Phan, VH., Monchiet, V. et al. Numerical investigation of macroscopic permeability of biporous solids with elliptic vugs. Theor. Comput. Fluid Dyn. 36, 689–704 (2022). https://doi.org/10.1007/s00162-022-00614-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00162-022-00614-1

Keywords

Search

Navigation