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Transport in Porous Media

, Volume 124, Issue 3, pp 803–824 | Cite as

Old and New Approaches Predicting the Diffusion in Porous Media

  • Nadja RayEmail author
  • Andreas Rupp
  • Raphael Schulz
  • Peter Knabner
Article
  • 503 Downloads

Abstract

An accurate quantitative description of the diffusion coefficient in a porous medium is essential for predictive transport modeling. Well-established relations, such as proposed by Buckingham, Penman, and Millington–Quirk, relate the scalar diffusion coefficient to the porous medium’s porosity. To capture the porous medium’s structure in more detail, further models include fitting parameters, geometric, or shape factors. Some models additionally account for the tortuosity, e.g., via Archie’s law. A validation of such models has been carried out mainly via experiments relating the proposed description to a specific class of porous media (by means of parameter fitting). Contrary to these approaches, upscaling methods directly enable calculating the full, potentially anisotropic, effective diffusion tensor without any fitting parameters. As input only the geometric information in terms of a representative elementary volume is needed. To compute the diffusion–porosity relations, supplementary cell problems must be solved numerically and their (flux) solutions must be integrated. We apply this approach to provide easy-to-use quantitative diffusion–porosity relations that are based on representative single grain, platy, blocky, prismatic soil structures, porous networks, or random porous media. As a discretization method, we use the discontinuous Galerkin method on structured grids. To make the relations explicit, interpolation of the obtained data is used. We furthermore compare the obtained diffusion–porosity relations with the well-established relations mentioned above and also with the well-known Voigt–Reiss or Hashin–Shtrikman bounds. We discuss the ranges of validity and further provide the explicit relations between the diffusion and surface area and comment on the role of a tortuosity–porosity relation.

Keywords

Diffusion in porous media Upscaling Tortuosity 

Notes

Acknowledgements

This research was kindly supported by the DFG RU 2179 “MAD Soil - Microaggregates: Formation and turnover of the structural building blocks of soils.”

References

  1. Aizinger, V., Rupp, A., Schütz, J., Knabner, P.: Analysis of a mixed discontinuous galerkin method for instationary darcy flow. Comput. Geosci. 22(1), 179–194 (2018).  https://doi.org/10.1007/s10596-017-9682-8 CrossRefGoogle Scholar
  2. Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518 (1992).  https://doi.org/10.1137/0523084 CrossRefGoogle Scholar
  3. Allaire, G., Brizzi, R., Dufrêche, J.F., Mikelić, A., Piatnitski, A.: Ion transport in porous media: derivation of the macroscopic equations using upscaling and properties of the effective coefficients. Comput. Geosci. 17(3), 479–495 (2013).  https://doi.org/10.1007/s10596-013-9342-6 CrossRefGoogle Scholar
  4. Auriault, J.L., Lewandowski, J.: Effective diffusion coefficient: from homogenization to experiment. Transp. Porous Media 27(2), 205–223 (1997).  https://doi.org/10.1023/A:1006599410942 CrossRefGoogle Scholar
  5. Bensoussan, A., Lions, J.L., Papanicolau, G.: Asymptotic Analysis of Periodic Structures. North-Holland, Amsterdam (1978)Google Scholar
  6. Blunt, M.J., Bijeljic, B., Dong, H., Gharbi, O., Iglauer, S., Mostaghimi, P., Paluszny, A., Pentland, C.: Pore-scale imaging and modelling. Adv. Water Resour. 51, 197–216 (2013).  https://doi.org/10.1016/j.advwatres.2012.03.003 CrossRefGoogle Scholar
  7. Buckingham, E.: Contributions to Our Knowledge of the Aeration of Soils, vol. 25. U.S. Dept. of Agriculture, Bureau of Soils, Washington, D.C. (1904)Google Scholar
  8. Chang, H.C.: Multi-scale analysis of effective transport in periodic heterogeneous media. Chem. Eng. Commun. 15(1–4), 83–91 (1982).  https://doi.org/10.1080/00986448208911060 CrossRefGoogle Scholar
  9. Crawford, J.W., Deacon, L., Grinev, D., Harris, J.A., Ritz, K., Singh, B.K., Young, I.: Microbial diversity affects self-organization of the soil-microbe system with consequences for function. J. R. Soc. Interface 71(9), 1302–1310 (2012).  https://doi.org/10.1098/rsif.2011.0679 CrossRefGoogle Scholar
  10. Currie, J.A.: Gaseous diffusion in porous media. Part 2. Dry granular materials. Br. J. Appl. Phys. 11(1), 318–324 (1960).  https://doi.org/10.1088/0508-3443/11/8/303 CrossRefGoogle Scholar
  11. De Vries, D.A.: Some remarks on gaseous diffusion in soils. Trans. 4th Int. Congr. Soil Sci. 2, 41–43 (1950)Google Scholar
  12. Frank, F., Liu, C., Alpak, F.O., Berg, S., Riviére, B.: Direct numerical simulation of flow on pore-scale images using the phase-field method. SPE J (2018).  https://doi.org/10.2118/182607-PA Google Scholar
  13. Fujii, N., Ichikawa, Y., Kawamura, K., Suzuki, S., Kitayama, K.: Micro-structure of bentonite clay and diffusion coefficient given by multiscale homogenization analysis. J. Soc. Mater. Sci. Jpn. 52(6 Appendix), 117–124 (2003).  https://doi.org/10.2472/jsms.52.6Appendix_117 CrossRefGoogle Scholar
  14. Galindo-Rosales, F.J., Campo-Deaño, L., Pinho, F.T., van Bokhorst, E., Hamersma, P.J., Oliveira, M.S.N., Alves, M.A.: Microfluidic systems for the analysis of viscoelastic fluid flow phenomena in porous media. Microfluid Nanofluid 12(1), 485–498 (2012).  https://doi.org/10.1007/s10404-011-0890-6 CrossRefGoogle Scholar
  15. Gray, W.G., Miller, C.T.: Introduction to the Thermodynamically Constrained Averaging Theory for Porous Medium Systems. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-04010-3
  16. Hallett, P., Karim, K., Bengough, A., Otten, W.: Biophysics of the vadose zone: From reality to model systems and back again. Vadose Zone J. (2013).  https://doi.org/10.2136/vzj2013.05.0090
  17. Hornung, U.: Homogenization and Porous Media. Springer, Cham (1997)CrossRefGoogle Scholar
  18. Hsinyi, C., Laosheng, W., Lingzao, Z., Andrew, C.: Evaluation of solute diffusion tortuosity factor models for variously saturated soils. Water Resour. Res. (2012).  https://doi.org/10.1029/2011WR011653
  19. Huang, X., Yue, W., Liu, D., Yue, J., Li, J., Sun, D., Yang, M., Wang, Z.: Monitoring the intracellular calcium response to a dynamic hypertonic environment. Sci. Rep. 6, 23591 (2016).  https://doi.org/10.1038/srep23591 CrossRefGoogle Scholar
  20. Ijioma, E., Muntean, A., Ogawa, T.: Effect of material anisotropy on the fingering instability in reverse smoldering combustion. Int. J. Heat Mass Transf. 81, 924–938 (2015).  https://doi.org/10.1016/j.ijheatmasstransfer.2014.11.021 CrossRefGoogle Scholar
  21. Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Cham (1994)CrossRefGoogle Scholar
  22. Kim, J.H., Ochoa, J.A., Whitaker, S.: Diffusion in anisotropic porous media. Transp. Porous Media 2(4), 327–356 (1987).  https://doi.org/10.1007/BF00136440 CrossRefGoogle Scholar
  23. Marshall, T.: The diffusion of gases through porous media. J. Soil Sci. 10(1), 79–82 (1959).  https://doi.org/10.1111/j.1365-2389.1959.tb00667.x CrossRefGoogle Scholar
  24. Maxwell, J.C.: A Treatise on Electricity and Magnetism, vol. 1, 2 edn. Clarendon Press, London (1981)Google Scholar
  25. Miller, C.T., Valdés-Parada, F.J., Ostvar, S., Wood, B.D.: A priori parameter estimation for the thermodynamically constrained averaging theory: species transport in a saturated porous medium. Transp. Porous Media 122(3), 611–632 (2018).  https://doi.org/10.1007/s11242-018-1010-9 CrossRefGoogle Scholar
  26. Millington, R.J.: Gas diffusion in porous media. Science 130(3367), 100–102 (1959).  https://doi.org/10.1126/science.130.3367.100-a CrossRefGoogle Scholar
  27. Moldrup, P., Olesen, T., Komatsu, T., Schjonning, P., Rolston, D.E.: Tortuosity, diffusivity, and permeability in the soil liquid and gaseous phases. Soil Sci. Soc. Am. J. 65, 613–623 (2001)CrossRefGoogle Scholar
  28. Moldrup, P., Olesen, T., Yamaguchi, T., Schjønning, P., Rolston, D.: Modeling diffusion and reaction in soils: Ix. The Buckingham–Burdine–Campbell equation for gas diffusivity in undisturbed soil. Soil Sci. 164(8), 542–551 (1999)CrossRefGoogle Scholar
  29. Moldrup, P., Olesen, T., Yoshikawa, S., Komatsu, T., Rolston, D.E.: Three-porosity model for predicting the gas diffusion coefficient in undisturbed soil. Soil Sci. Soc. Am. J. 68(3), 750–759 (2004)CrossRefGoogle Scholar
  30. Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20(3), 608–623 (1989).  https://doi.org/10.1137/0520043 CrossRefGoogle Scholar
  31. Nimmo, J.: Porosity and pore size distribution. Encycl. Soils Environ. 3, 295–303 (2004)Google Scholar
  32. Ochoa-Tapia, J.A., Stroeve, P., Whitaker, S.: Diffusive transport in two-phase media: spatially periodic models and maxwell’s theory for isotropic and anisotropic systems. Chem. Eng. Sci. 49(5), 709–726 (1994).  https://doi.org/10.1016/0009-2509(94)85017-8 CrossRefGoogle Scholar
  33. Penman, H.L.: Gas and vapor movements in soil, 1. The diffusion of vapors through porous solids. A. Agric. Sci. 30, 437–463 (1940).  https://doi.org/10.1017/S0021859600048164 CrossRefGoogle Scholar
  34. Quintard, M.: Diffusion in isotropic and anisotropic porous systems: three-dimensional calculations. Transp. Porous Media 11(2), 187–199 (1993).  https://doi.org/10.1007/BF01059634 CrossRefGoogle Scholar
  35. Randall, C.L., Kalinin, Y.V., Jamal, M., Manohar, T., Gracias, D.H.: Three-dimensional microwell arrays for cell culture. Lab Chip 11(1), 127–131 (2011).  https://doi.org/10.1039/c0lc00368a CrossRefGoogle Scholar
  36. Ray, N., van Noorden, T., Frank, F., Knabner, P.: Multiscale modeling of colloid and fluid dynamics in porous media including an evolving microstructure. Transp. Porous Media 95(3), 669–696 (2012).  https://doi.org/10.1007/s11242-012-0068-z CrossRefGoogle Scholar
  37. Ray, N., Rupp, A., Prechtel, A.: Discrete-continuum multiscale model for transport, biomass development and solid restructuring in porous media. Adv. Water Resour. 107(Supplement C), 393–404 (2017).  https://doi.org/10.1016/j.advwatres.2017.04.001
  38. Rupp, A., Knabner, P.: Convergence order estimates of the local discontinuous galerkin method for instationary darcy flow. Numer. Methods Partial Differ. Equ. 33(4), 1374–1394 (2017).  https://doi.org/10.1002/num.22150 CrossRefGoogle Scholar
  39. Rupp, A., Knabner, P., Dawson, C.: A local discontinuous Galerkin scheme for Darcy flow with internal jumps. Comput. Geosci. (2018).  https://doi.org/10.1007/s10596-018-9743-7 Google Scholar
  40. Ryan, D., Carbonell, R., Whitaker, S.: Effective diffusivities for catalyst pellets under reactive conditions. Chem. Eng. Sci. 35(1), 10–16 (1980).  https://doi.org/10.1016/0009-2509(80)80064-9 CrossRefGoogle Scholar
  41. Schmuck, M., Bazant, M.Z.: Homogenization of the Poisson–Nernst–Planck equations for ion transport in charged porous media. SIAM J. Appl. Math. 75(3), 1369–1401 (2015).  https://doi.org/10.1137/140968082 CrossRefGoogle Scholar
  42. Schulz, R., Ray, N., Frank, F., Mahato, H.S., Knabner, P.: Strong solvability up to clogging of an effective diffusion-precipitation model in an evolving porous medium. Eur. J. Appl. Math. (2016).  https://doi.org/10.1017/S0956792516000164
  43. Shen, L., Chen, Z.: Critical review of the impact of tortuosity on diffusion. Chem. Eng. Sci. 62(14), 3748–3755 (2007).  https://doi.org/10.1016/j.ces.2007.03.041 CrossRefGoogle Scholar
  44. Smith, D., Pivonka, P., Jungnickel, C., Fityus, S.: Theoretical analysis of anion exclusion and diffusive transport through platy-clay soils. Transp. Porous Media 57(3), 251–277 (2004).  https://doi.org/10.1007/s11242-003-4056-1 CrossRefGoogle Scholar
  45. Troeh, F.R., Jabro, J.D., Kirkham, D.: Gaseous diffusion equations for porous materials. Geoderma 27(3), 239–253 (1982).  https://doi.org/10.1016/0016-7061(82)90033-7 CrossRefGoogle Scholar
  46. Tuli, A.: Pore geometry effect on gaseous diffusion and convective fluid flow in soils. Ph.D. thesis, University of California Davis (2001)Google Scholar
  47. Ungureanu, A., Statescu, F.: Models for predicting the gas diffusion coefficient in undisturbed soil. Trans. Hydrotech. 55(69), 168–172 (2010)Google Scholar
  48. van Noorden, T.: Crystal precipitation and dissolution in a porous medium: effective equations and numerical experiments. Multiscale Model. Simul. 7, 1220–1236 (2009).  https://doi.org/10.1137/080722096 CrossRefGoogle Scholar
  49. Valdés-Parada, F., Alvarez-Ramirez, J.: On the effective diffusivity under chemical reaction in porous media. Chem. Eng. Sci. 65(13), 4100–4104 (2010).  https://doi.org/10.1016/j.ces.2010.03.040 CrossRefGoogle Scholar
  50. Valdés-Parada, F.J., Lasseux, D., Whitaker, S.: Diffusion and heterogeneous reaction in porous media: the macroscale model revisited. Int. J. Chem. React. Eng. 15(6), 24 (2017).  https://doi.org/10.1515/ijcre-2017-0151 Google Scholar
  51. Valdés-Parada, F.J., Porter, M.L., Wood, B.D.: The role of tortuosity in upscaling. Transp. Porous Media 88(1), 1–30 (2011).  https://doi.org/10.1007/s11242-010-9613-9 CrossRefGoogle Scholar
  52. Weissberg, H.L.: Effective diffusion coefficient in porous media. J. Appl. Phys. 34(9), 2636 (1963).  https://doi.org/10.1063/1.1729783 CrossRefGoogle Scholar
  53. Whitaker, S.: The Method of Averaging. Springer, Cham (1999)CrossRefGoogle Scholar
  54. Wieners, C.: Distributed point objects. A new concept for parallel finite elements. In: Barth, T., Griebel, M., Keyes, D., Nieminen, R., Roose, D., Schlick, T., Kornhuber, R., Hoppe, R., Périaux, J., Pironneau, O., Widlund, O., Xu, J. (eds.) Domain Decomposition Methods in Science and Engineering, Lecture Notes in Computational Science and Engineering, vol. 40, pp. 175–182. Springer, Berlin (2005)Google Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Nadja Ray
    • 1
    Email author
  • Andreas Rupp
    • 1
  • Raphael Schulz
    • 1
  • Peter Knabner
    • 1
  1. 1.Department of MathematicsFriedrich-Alexander University of Erlangen-NürnbergErlangenGermany

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