Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Computation of Saturation Dependence of Effective Diffusion Coefficient in Unsaturated Argillite Micro-fracture by Lattice Boltzmann Method

  • 280 Accesses

  • 2 Citations


Getting access to the effective diffusion coefficient is a key point to provide realistic predictions of migration of radionuclides from radioactive waste repository in deep argillaceous geological formations. In the present work, the effective diffusion coefficient was computed inside an argillite micro-fracture as a function of its saturation level. The micrometric fracture geometry was extracted from the X-ray \(\mu \)-tomography image (\(0.7\,\upmu \mathrm{m}\) voxel resolution) of an Opalinus clay sample. It was collected in the host rock excavated damaged zone surrounding a borehole in the Mont Terri laboratory. The computations were performed using two two-relaxation-time lattice Boltzmann models. The first one, a phase separation model, was used to extract the connected liquid phase inside the fracture for given saturations. The second, a diffusion model, was used to compute non-reactive tracer diffusion in the connected liquid phase of the fracture and to calculate the effective diffusion coefficient for the associated saturations. The dependence of the effective diffusion coefficient on saturation was found to be quasi-linear and to qualitatively match the Maxwell expression for saturations lower than 0.8.

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

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


  1. Alonso, U., Missana, T., Garcia-Gutierrez, M., Patelli, A., Albarran, N., Rigato, V.: Colloid diffusion coefficients in compacted and consolidated clay barriers: compaction density and colloid size effects. Phys. Chem. Earth 36, 1700–1707 (2011)

  2. Appert, C., Zaleski, S.: Lattice-gas with a liquid–gas transition. Phys. Rev. Lett. 64, 1–4 (1990)

  3. Archie, G.E.: The electrical resistivity log as an aid in determining some reservoir characteristics. Pet. Trans. AIME 146, 54–62 (1942)

  4. Balberg, I.: Excluded-volume explanation of Archie’s law. Phys. Rev. B 33, 3618–3620 (1986)

  5. Bear, J., Bachmat, Y.: Introduction to Modeling of Transport Phenomena in Porous Media. Kluwer, Dordrecht (1991)

  6. Bertei, A., Nucci, B., Nicolella, C.: Effective transport properties in random packings of spheres and agglomerates. Chem. Eng. Trans. 32, 1531–1536 (2013)

  7. Boudreau, B.P.: The diffusive tortuosity of fine-grained unlithified sediments. Geochim. Cosmochim. Acta 60(16), 3139–3142 (1996)

  8. Chau, J.F., Or, D., Sukop, M.C.: Simulation of gaseous diffusion in partially saturated porous media under variable gravity with lattice Boltzmann methods. Water Resour. Res. 41, W08410 (2005). doi:10.1029/2004WR003821

  9. Croisé, J., Mayer, G., Talandier, J., Wendling, J.: Impact of water consumption and saturation-dependent corrosion rate on hydrogen generation and migration from an intermediate-level radioactive waste repository. Transp. Porous Media 90, 59–75 (2011)

  10. Dagnelie, R.V.H., Arnoux, P., Radwan, J., Lebeau, D., Nerfie, P., Beaucaire, C.: Perturbation induced by EDTA on HDO, Br- and Eu\(^{III}\) diffusion in a large-scale clay rock sample. Appl. Clay Sci. 105–106, 142–149 (2015)

  11. Delay, J., Vinsot, A., Krieguer, J.M., Rebours, H., Armand, G.: Making of the underground scientific experimental programme at the Meuse/Haute-Marne underground research laboratory. Phys. Chem. Earth 32, 2–18 (2007)

  12. de Marsily, G.: Quantitative Hydrogeology. Academic Press, San Diego (1986)

  13. d’Humières, D., Ginzburg, I.: Viscosity independent numerical errors for Lattice Boltzmann models: from recurrence equations to “magic” collision numbers. Comput. Math. Appl. 58(5), 823–840 (2009)

  14. Epstein, N.: On tortuosity and the tortuosity factor in flow and diffusion through porous media. Chem. Eng. Sci. 44(3), 777–779 (1989)

  15. Friedman, S.P.: Soil properties influencing apparent electrical conductivity: a review. Comput. Electron. Agric. 46, 45–70 (2005)

  16. Genty, A., Pot, V.: Numerical simulation of 3D liquid–gas distribution in porous media by a two-phase TRT lattice Boltzmann method. Transp. Porous Media 96, 271–294 (2013)

  17. Genty, A., Pot, V.: Numerical calculation of effective diffusion in unsaturated porous media by the TRT lattice Boltzmann method. Transp. Porous Media 105, 391–410 (2014)

  18. Ghanbarian, B., Hunt, A.G., Ewing, R.P., Sahimi, M.: Tortuosity in porous media: a critical review. Soil Sci. Soc. Am. J. 77, 1461–1477 (2013)

  19. Ghanbarian, B., Daigle, H., Hunt, A.G., Ewing, R.P., Sahimi, M.: Gas and solute diffusion in partially saturated porous media: percolation theory and Effective Medium Approximation compared with lattice Boltzmann simulations. J. Geophys. Res. Solid Earth 120, 182–190 (2015)

  20. Ginzburg, I.: Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation. Adv. Water Resour. 28, 1171–1195 (2005)

  21. Ginzburg, I., d’Humières, D.: Multireflection boundary conditions for lattice Boltzmann models. Phys. Rev. E 68(6), 066614 (2003)

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

  23. Ginzburg, I., d’Humières, D., Kuzmin, A.: Optimal stability of advection-diffusion lattice Boltzmann models with two relaxation times for positive/negative equilibrium. J. Stat. Phys. 139(6), 1090–1143 (2010)

  24. Guillon, V., Fleury, M., Bauer, D., Neel, M.C.: Superdispersion in homogeneous unsaturated porous media using NMR propagators. Phys. Rev. E 87(0430007), 1–10 (2013)

  25. Hamamoto, S., Moldrup, P., Kawamoto, K., Komatsu, T.: Excluded-volume expansion of Archie’s law for gas and solute diffusivities and electrical and thermal conductivities in variably saturated porous media. Water Resour. Res. 46, W06514 (2010). doi:10.1029/2009WR008424

  26. Hu, Q., Wang, J.S.Y.: Aqueous-phase diffusion in unsaturated geologic media: a review. Crit. Rev. Environ. Sci. Technol. 33(3), 275–297 (2003)

  27. Jones, S.B., Or, D., Bingham, J.E.: Gas diffusion measurement and modeling in coarse-textured porous media. Vadose Zone J. 2, 602–610 (2003)

  28. Marschall, P., Horseman, S., Gimmi, T.: Characterization of gas transport properties of the Opalinus Clay, a potential host rock formation for radioactive waste disposal. Oil Gas Sci. Technol. Rev. IFP 60, 121–139 (2005)

  29. Martys, N.S.: Diffusion in partially-saturated porous materials. Mater. Struct. 32, 555–562 (1999)

  30. Martys, N.S., Chen, H.: Simulation of multicomponent fluids in complex three-dimensional geometries by the lattice Boltzmann method. Phys. Rev. E 53(1), 743–750 (1996)

  31. Matray, J.-M., Savoye, S., Cabrera, J.: Desaturation and structure relationships around drifts excavated in the well-compacted Tournemire’s argillite (Aveyron, France). Eng. Geol. 90(1–2), 1–16 (2007)

  32. Maxwell, J.C.: A Treatise on Electricity and Magnetism. Chap IX, vol. I. Claredon Press, Oxford (1873)

  33. Mayor, J.-C., Velasco, M., García-Siñeriz, J.-L.: Ventilation experiment in the Mont Terri underground laboratory. Phys. Chem. Earth 32(8–14), 616–628 (2007)

  34. Moldrup, P., Olesen, T., Yoshikawa, S., Komatsu, T., Rolston, D.E.: Three-porosity model for predicting the gas diffusion coefficient in unsaturated soil. Soil Sci. Soc. Am. J. 68, 750–759 (2004)

  35. Patriarche, D., Michelot, J.-L., Ledoux, E., Savoye, S.: Diffusion as the main process for mass transport in very low water content argillites: 1. Chloride as a natural tracer for mass transport—diffusion coefficient and concentration measurements in interstitial water. Water Resour. Res. 40, W01516. doi:10.1029/2003WR002600

  36. Poller, A., Enssle, C.P., Mayer, G., Croisé, J., Wendling, J.: Repository-scale modeling of the long-term hydraulic perturbation induced by gas and heat generation in a geological repository for high-and intermediate-level radioactive waste: methodology and example of application. Transp. Porous Media 90, 77–94 (2011)

  37. Pot, V., Hammou, H., Elyeznasmi, N., Ginzburg, I.: Role of soil heterogeneities onto pesticide fate: a pore-scale study with lattice Boltzmann. In: Proceedings of the 1st International Conference and Exploratory Workshop on Soil Architecture and Physico-Chemical Functions “CESAR”, ISBN 87 91949-59-9, Nov. 30–Dec. 2: Research Centre Foulum. Tjele, Denmark (2010)

  38. Pot, V., Peth, S., Monga, O., Vogel, L.E., Genty, A., Garnier, P., Vieublé-Gonod, L., Ogurreck, M., Beckmann, F., Baveye, P.C.: Three-dimensional distribution of water and air in soil pores: comparison of two-phase two-relaxation-times lattice-Boltzmann and morphological model outputs with synchrotron X-ray computed tomography data. Adv. Water Resour. 84, 87–102 (2015)

  39. Raiskinmäki, P., Koponen, A., Merikoski, J., Timonen, J.: Spreading dynamics of three-dimensional droplets by the lattice-Boltzmann method. Comput. Mater. Sci. 18, 7–12 (2000)

  40. Rübel, A.P., Sonntag, C., Lippmann, J., Pearson, F.J., Gautschi, A.A.: Solute transport in formations of very low permeability: Profiles of stable isotope and dissolved noble gas contents of pore water in the Opalinus Clay, Mont Terri, Switzerland. Geochim. Cosmochim. Acta 66(8), 1311–1321 (2002)

  41. Saripalli, K.P., Serne, R.J., Meyer, P.D., McGrail, B.P.: Prediction of diffusion coefficients in porous media using tortuosity factors based on interfacial areas. Ground Water 40(4), 346–352 (2002)

  42. Savoye, S., Page, J., Puente, C., Imbert, C., Coelho, D.: New experimental approach for studying diffusion through an intact and unsaturated medium: a case study with Callovo-Oxfordian argillite. Environ. Sci. Technol. 44(10), 3698–3704 (2010)

  43. Shan, X., Chen, H.: Lattice Boltzmann model for simulating flows with multiple phases and components. Phys. Rev. E 47(3), 1815–1820 (1993)

  44. Shan, X., Chen, H.: Simulation of nonideal gases and liquid–gas phase transitions by the lattice Boltzmann equation. Phys. Rev. E 49(4), 2941–2948 (1994)

  45. Shen, L., Chen, Z.: Critical review of the impact of tortuosity on diffusion. Chem. Eng. Sci. 62, 3748–3755 (2007)

  46. van Brakel, J., Heertjes, P.M.: Analysis of diffusion in macroporous media in terms of a porosity, a tortuosity and a constrictivity factor. Int. J. Heat Mass Transf. 17, 1093–1103 (1974)

  47. Voutilainen, M., Sardini, P., Siitari-Kauppi, M., Kekaäläinen, P., Aho, V., yllys, M., Timonen, J.: Diffusion of tracer in altered tonalite: experiments and simulations with heterogeneous distribution of porosity. Transp. Porous Media 96, 319–336 (2013)

  48. Xu, T., Senger, R., Finsterle, S.: Corrosion-induced hydrogen generation in a nuclear waste repository: reactive geochemistry and multiphase flow effects. Appl. Geochem. 23, 3423–3433 (2008)

  49. Xuan, Y.M., Zhao, K., Li, Q.: Investigation on mass diffusion process in porous media based on lattice Boltzmann method. Int. J. Heat Mass Transf. 46, 1039–1051 (2010)

  50. Yang, R., Lemarchand, E., Fen-Chong, T.: A micromechanics model for solute diffusion coefficient in unsaturated granular materials. Transp. Porous Media 111, 347–368 (2016)

  51. Zhang, M., Ye, G., van Breugel, K.: Modeling of ionic diffusivity in non-saturated cement-based materials using lattice Boltzmann method. Cem. Concr. Res. 42(11), 1524–1533 (2012)

Download references


The authors acknowledge the financial support of the NEEDS-MIPOR program within “Mission for Interdisciplinarity” of the French National Centre for Scientific Research which partially funded Soukaina Gueddani’s graduate engineer internship at IRSN.

Author information

Correspondence to Alain Genty.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Genty, A., Gueddani, S. & Dymitrowska, M. Computation of Saturation Dependence of Effective Diffusion Coefficient in Unsaturated Argillite Micro-fracture by Lattice Boltzmann Method. Transp Porous Med 117, 149–168 (2017).

Download citation


  • Lattice Boltzmann method
  • Effective diffusion
  • Clay
  • Unsaturated
  • Fracture