Topological Characterization of Porous Media

  • Hans-Jörg Vogel
Part of the Lecture Notes in Physics book series (LNP, volume 600)


It is an attractive approach to predict flow and in based on direct investigations of their structure. The most crucial property is the of the structure because it is difficult to measure. This is true both at the pore scale, which may be represented as a binary structure, and at a larger scale defined by continuous macroscopic state variables as phase density or. At the pore scale a function is introduced which is defined by the as a function of the pore diameter. This function is used to generate of the porous structure that allow to predict bulk hydraulic properties of the material. At the continuum scale the structure is represented on a grey scale representing the porosity of the material with a given resolution. Here, topology is quantified by a connectivity function defined by the Euler characteristic as a function of a porosity threshold. Results are presented for the structure of natural soils measured by. The significance of topology at the continuum scale is demonstrated through numerical simulations. It is found that the effective permeabilities of two heterogeneous having the same auto-covariance but different topology differ considerably.


Porous Medium Capillary Pressure Representative Elementary Volume Euler Characteristic Topological Characteristic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    P. M. Adler, C. G. Jacquin, J. A. Quiblier, Flow in simulated porous media, Int. J. Multiphase Flow 16 (1990) 691–712zbMATHCrossRefGoogle Scholar
  2. 2.
    C. Arns, M. Knackstedt, K. Mecke, Characterising the morphology of disordered materials, Lecture Notes in Physics (2002), this volumeGoogle Scholar
  3. 3.
    C. Beisbart, R. Dahlke, K. Mecke, H. Wagner, Vector-and tensor-valued descriptors of spatial patterns, Lecture Notes in Physics (2002), this volumeGoogle Scholar
  4. 4.
    B. Berkowitz, I. Balberg, Percolation theory and its application to groundwater hydrology, Water Resources Res. 29 (1993) 775–794CrossRefADSGoogle Scholar
  5. 5.
    M. A. Celia, P. C. Reeves, L. A. Ferrand, Recent advances in pore scale models, Rev.Geophys. 33 (1995) 1049–1057CrossRefADSGoogle Scholar
  6. 6.
    R. Chandler, J. Koplik, K. Lerman, J. F. Willemsen, Capillary displacement and percolation in porous media, J. Fluid Mech. 119 (1982) 249–267zbMATHCrossRefADSGoogle Scholar
  7. 7.
    I. Chatzis, F. A. L. Dullien, Modeling pore structure by 2-D and 3-D networks with application to sandstone, J. Can. Pet. Technol. 16 (1977) 97–108Google Scholar
  8. 8.
    R. T. DeHoff, Use of the disector to estimate the Euler characteristic of three dimensional microstructures, Acta Stereol. 6 (1987) 133–140Google Scholar
  9. 9.
    I. Fatt, The network model of porous media. I. Capillary pressure characteristics, Pet. Trans. AIME 207 (1956) 144–159Google Scholar
  10. 10.
    I. Fatt, The network model of porous media. II. Dynamic properties of a single size tube network, Pet. Trans. AIME 207 (1956) 160–163Google Scholar
  11. 11.
    I. Fatt, The network model of porous media. III. Dynamic properties of networks with tube radius distribution, Pet. Trans. AIME 207 (1956) 164–181Google Scholar
  12. 12.
    L. A. Ferrand, M. A. Celia, The effect of heterogeneity on the drainage capillary pressure-saturation relation, Water Resour. Res. 28(3) (1992) 859–870CrossRefADSGoogle Scholar
  13. 13.
    S. Friedman, N. Seaton, On the transport properties of anisotropic networks of capillaries, Water Resour. Res. 32 (1996) 339–347CrossRefADSGoogle Scholar
  14. 14.
    H. Hadwiger, Vorlesung über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, Berlin, 1957Google Scholar
  15. 15.
    R. Hilfer, Geometric and dielectric characterization of porous media, Phys. Rev. B 44 (1991) 60CrossRefADSGoogle Scholar
  16. 16.
    R. Hilfer, Local-porosity theory for flow in porous media, Physical Review B 45 (1992) 7115–7121CrossRefADSGoogle Scholar
  17. 17.
    R. Hilfer, Transport and relaxation phenomena in porous media, Adv. Chem. Phys. XCII (1996) 299–424CrossRefGoogle Scholar
  18. 18.
    R. Hilfer, T. Rage, B. Virgin, Local percolation probabilities for a natural sandstone, Physica A 241 (1997) 105–110CrossRefADSGoogle Scholar
  19. 19.
    J.W. Hopmans, M. Cislerova, T. Vogel, X-ray tomography of soil properties, In: Tomography of Soil-Water-Root Processes. Eds.: Ande S.H. and J.W. Hopmans. ASA, SSSA Madison, Wisconsin, USA (1994) 17–28Google Scholar
  20. 20.
    G. R. Jerauld, S. J. Salter, The effect of pore-structure on hysteresis in relative permeability and capillary pressure: pore-level modeling, Transort in Porous Media 5(2) (1990) 103–151CrossRefGoogle Scholar
  21. 21.
    M. A. Knackstedt, A. P. Sheppard, M. Sahimi, Pore network modelling of two-phase flow in porous rock: the effect of correlated heterogeneity, Adv. Water Res. 24 (2001) 257–277CrossRefGoogle Scholar
  22. 22.
    R. Magerle, Nanotomography, Lecture Notes in Physics (2002), this volumeGoogle Scholar
  23. 23.
    J. Mecke, D. Stoyan, The specific connectivity number of random networks, Adv. Appl. Prob. (SGSA) 33 (2001) 576–583zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    K. Mecke, H. Wagner, Euler characteristic and related measures for random geometric sets, J. Stat. Phys. 64 (1991) 843zbMATHCrossRefADSMathSciNetGoogle Scholar
  25. 25.
    K. Mecke, Additivity, convexity, and beyond: applications of Minkowski Functionals in statistical physics, Lecture Notes in Physics 554 (2000) 111–184Google Scholar
  26. 26.
    K. Mecke, D. Stoyan, Statistical physics and spatial statistics-the art of analyzing and modeling spatial structures and pattern formation, Lecture Notes in Physics 554.Google Scholar
  27. 27.
    J. Ohser, W. Nagel, K. Schladitz, The Euler number of discretized sets, Lecture Notes in Physics (2002), this volumeGoogle Scholar
  28. 28.
    J. Ohser, W. Nagel, The estimation of the Euler-poincaré characteristic from observations on parallel sections, J. Microsc. 184 (1996) 117–126CrossRefGoogle Scholar
  29. 29.
    W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C. The Art of Scientific Computing, second edition Edition, Cambridge University Press, Cambridge, 1992zbMATHGoogle Scholar
  30. 30.
    J. Quiblier, A new three dimensional modeling technique for studying porous media, J. Colloid Interface Sci. 98 (1984) 84–102Google Scholar
  31. 31.
    H. Rajaram, L. A. Ferrand, M. A. Celia, Prediction of relative permeabilities for unconsolidated soils using pore-scale network models, Water Resour. Res. 33 (1997) 43–52CrossRefADSGoogle Scholar
  32. 32.
    A. P. Roberts, S. Torquato, Chord-distribution functions of three-dimensional random media: approximate first-passage times of gaussian processes, Phys. Rev. E 59 (1999) 4953–4963CrossRefADSMathSciNetGoogle Scholar
  33. 33.
    M. J. L. Robin, A. L. Gutjahr, E. A. Sudicky, J. L. Wilson, Cross-correlated random field generation with the direct fourier transform method, Water Resour. Res. 29 (1993) 2385–2397CrossRefADSGoogle Scholar
  34. 34.
    V. Robns, Computational topology for point data: Betti numbers and α-shapes, Lecture Notes in Physics (2002), this volumeGoogle Scholar
  35. 35.
    H. Rogasik, J.W. Crawford, O. Wendroth, I. M. Young, M. Joschko, K. Ritz, Discrimination of soil phases by dual energy X-ray tomography, Soil Sci. Soc. Am. J. 63 (1999) 741–751CrossRefGoogle Scholar
  36. 36.
    K. Roth, Steady state flow in an unsaturated, two-dimensional, macroscopically homogeneous, miller-similar medium, Water Resources Res. 31 (1995) 2127–2140CrossRefADSGoogle Scholar
  37. 37.
    J. Serra, Image Analysis and Mathematical Morphology, Academic Press, London, 1982zbMATHGoogle Scholar
  38. 38.
    D. C. Sterio, The unbiased estimation of number and sizes of arbitrary particles using the disector, J. of Microsc. 134 (1984) 127–136Google Scholar
  39. 39.
    V. C. Tidwell, L. C. Meigs, T. Christian-Frear, C.M. Boney, Effects of spatially heterogeneous porosity on matrix diffusion as investigated by X-ray absorption imaging, J. Contam. Hydrol. 42 (2000) 285–302CrossRefGoogle Scholar
  40. 40.
    H. J. Vogel, Digital unbiased estimation of the Euler-Poincar’e characteristic in different dimensions, Acta Stereol. 16/2 (1997) 97–104Google Scholar
  41. 41.
    H. J. Vogel, Morphological determination of pore connectivity as a function of pore size using serial sections, Europ. J. Soil Sci. 48 (1997) 365–377CrossRefGoogle Scholar
  42. 42.
    H. J. Vogel, K. Roth, A new approach for determining effective soil hydraulic functions, Europ. J. Soil Sci. 49 (1998) 547–556CrossRefGoogle Scholar
  43. 43.
    H. J. Vogel, A numerical experiment on pore size, pore connectivity, water retention, permeability, and solute transport using network models, Europ. J. Soil Sci. 51 (2000) 99–105CrossRefGoogle Scholar
  44. 44.
    H.-J. Vogel, K. Roth, Quantitative morphology and network representation of soil pore structure, Adv. Water Res. 24 (2001) 233–242CrossRefGoogle Scholar
  45. 45.
    A. W. Western, G. Blöschl, R. B. Grayson, Toward capturing hydrologcally significant connectivity in spatial patterns, Water Resour. Res. 37 (2001) 83–97CrossRefADSGoogle Scholar
  46. 46.
    J. Widjajakusuma, C. Manwart, B. Biswal, R. Hilfer, Exact and approximate calculations for the conductivity of sandstones, Physica A 270 (1999) 325–331CrossRefADSGoogle Scholar
  47. 47.
    W. R. Wise, A new insight on pore structure and permeability, Water Resour. Res. 28 (1992) 189–198CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Hans-Jörg Vogel
    • 1
  1. 1.Institute of Environmental Physics, INF 229University of HeidelbergHeidelbergGermany

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