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


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.


Diffusion in porous media Upscaling Tortuosity 



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


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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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