Flow, Diffusion, Dispersion, and Thermal Convection in Percolation Clusters: NMR Experiments and Numerical FEM/FVM Simulations


Based on computer-generated templates, percolation objects were fabricated. Random-site, semi-continuous swiss cheese, and semi-continuous inverse swiss-cheese percolation models above the percolation threshold were considered. The water-filled pore space was investigated by nuclear magnetic resonance (NMR) imaging and in the presence of a pressure gradient, by NMR velocity mapping. The percolation backbones were determined using velocity maps. The fractal dimension of the backbones turned out to be smaller by about 17 % than that of the complete cluster. As a further relation of interest, the volume-averaged velocity was calculated as a function of the probe volume radius. In a certain scaling window, the resulting dependence can be represented by a power law. The experimental results favorably compare to computer simulations with the finite-element method (FEM) or the finite-volume method (FVM). Thermal convection in percolation clusters of different porosities was studied using the NMR velocity mapping technique. The velocity distribution is related to the convection roll size distribution. The maximum velocity as a function of the porosity clearly visualizes a closed-loop percolation transition if the Rayleigh number conditions are appropriate. Percolation theory suggests a relationship between the anomalous diffusion exponent and the fractal dimension of the cluster, i.e. between a dynamic and a structural parameter. Interdiffusion between two compartments initially filled with H2O and D2O, respectively, was examined by proton imaging. The results confirm the theoretical expectation. Finally, advection driven dispersive transport was investigated in the large Péclet number limit. The superdiffusive transport anomaly was demonstrated and discussed in terms of the non-local advection-diffusion and the fractional diffusion theories.

This is a preview of subscription content, access via your institution.


  1. 1.

    D. Stauffer and A. Aharony, Introduction to Percolation Theory (Taylor Francis, 1992).

    Google Scholar 

  2. 2.

    A. Bunde and S. Havlin, (Eds.), Fractals and Disordered Systems (Springer-Verlag, 1996).

    Google Scholar 

  3. 3.

    H. Hermann, Stochastic Models of Heterogeneous Materials (Trans Tech Publ., 1991).

    Google Scholar 

  4. 4.

    S. Feng, B. I. Halperin, and P. N. Sen, Phys. Rev. B 35, 197 (1987).

    CAS  Article  Google Scholar 

  5. 5.

    A. Klemm, H.-P. Müller, R. Kimmich, Phys. Rev. E 55, 4413 (1997).

    CAS  Article  Google Scholar 

  6. 6.

    R. Kimmich, NMR Tomography, Diffusometry, Relaxometry (Springer-Verlag, 1997).

    Google Scholar 

  7. 7.

    D. A. Nield and A. Bejan, Convection in Porous Media (Springer-Verlag, 1992).

    Google Scholar 

  8. 8.

    H.-P. Müller, J. Weis, and R. Kimmich, Phys. Rev. E 52, 5195 (1995).

    Article  Google Scholar 

  9. 9.

    H.-P. Müller, R. Kimmich, and J. Weis, Phys. Rev. E 54, 5278 (1996).

    Article  Google Scholar 

  10. 10.

    J. S. Andrade Jr, M. P. Almeida, J. Mendes Filho, S. Havlin, B. Suki, and H. E. Stanley, Phys. Rev. Letters 79, 3901 (1997).

    CAS  Article  Google Scholar 

  11. 11.

    F. Klammler and R. Kimmich, Phys. Med. Biol. 35, 67 (1990).

    CAS  Article  Google Scholar 

  12. 12.

    S. L. Codd, B. Manz, J. D. Seymour, and P. T. Callaghan, Phys. Rev. E 60, R3491 (1999).

    CAS  Article  Google Scholar 

  13. 13.

    A. Kapitulnik, A. Aharony, G. Deutscher, and D. Stauffer, J. Phys. A: Math. Gen. 16, L269 (1983).

    Article  Google Scholar 

  14. 14.

    M. Porto, A. Bunde, S. Havlin, and H. E. Roman, J. Phys. Rev. E 56, 1667 (1997).

    CAS  Article  Google Scholar 

  15. 15.

    M. D. Shattuck, R. P. Behringer, G. A. Johnson, and J. G. Georgiadis, Phys. Rev. Letters 75, 1934 (1995).

    CAS  Article  Google Scholar 

  16. 16.

    S. Alexander and R. Orbach, J. Physique-Lettres (Paris) 43, L625 (1982).

    Article  Google Scholar 

  17. 17.

    D. C. Hong, S. Havlin, H. J. Herrmann, and H. E. Stanley, Phys. Rev. B 30, 4083 (1984).

    Article  Google Scholar 

  18. 18.

    J. G. Zabolitzky, Phys. Rev. B 30, 4077 (1984).

    Article  Google Scholar 

  19. 19.

    G. I. Taylor, Proc. Roy. Soc. Lond. A 219, 186 (1953).

    CAS  Article  Google Scholar 

  20. 20.

    R. Aris, Proc. Roy. Soc. Lond. A 235, 67 (1956).

    Article  Google Scholar 

  21. 21.

    H. Brenner, J. Stat. Phys. 62, 1095 (1991).

    Article  Google Scholar 

  22. 22.

    C. van den Broeck, Physica A 168, 677 (1990).

    Article  Google Scholar 

  23. 23.

    J. H Cushman, B.X. Hu and T.R. Ginn, J. Stat. Phys. 75, 859 (1994).

    Article  Google Scholar 

  24. 24.

    D.L. Koch and J.F. Brady, J. Fluid Mech. 200, 173 (1987).

    Article  Google Scholar 

  25. 25.

    J. Koplik, S. Redner and D. Wilkinson, Phys. Rev. A 37, 2619 (1988).

    CAS  Article  Google Scholar 

  26. 26.

    M. Sahimi, Rev. Mod. Phys, 65, 1393 (1993).

    Article  Google Scholar 

  27. 27.

    H.A. Makse, J.S. Andrade and H.E. Stanley, Phys. Rev. E 61, 583 (2000).

    CAS  Article  Google Scholar 

  28. 28.

    R. Metzler and J. Klafter, Europhys. Lett. 51, 492 (2000).

    CAS  Article  Google Scholar 

  29. 29.

    W. R. Schneider and W. Wyss, J. Math. Phys. 30, 134 (1989).

    Article  Google Scholar 

  30. 30.

    J. D. Seymour and P. T. Callaghan, Al ChE J. 43, 2096 (1997).

    CAS  Google Scholar 

  31. 31.

    O. J. Poole and D. W. Salt, J. Phys. A; Math. Gen. 29, 7959 (1996).

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Rainer Kimmich.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kimmich, R., Klemm, A., Weber, M. et al. Flow, Diffusion, Dispersion, and Thermal Convection in Percolation Clusters: NMR Experiments and Numerical FEM/FVM Simulations. MRS Online Proceedings Library 651, 271 (2000). https://doi.org/10.1557/PROC-651-T2.7.1

Download citation