Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-29T09:46:40.517Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  21 March 2011

Rainer Kimmich ,
Andreas Klemm ,
Markus Weber  and
Joseph D. Seymour
Rainer Kimmich
Affiliation:
Sektion Kernresonanzspektroskopie, Universität Ulm, 89069 Ulm, Germany
Andreas Klemm
Affiliation:
Sektion Kernresonanzspektroskopie, Universität Ulm, 89069 Ulm, Germany
Markus Weber
Affiliation:
Sektion Kernresonanzspektroskopie, Universität Ulm, 89069 Ulm, Germany
Get access

Abstract

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.

Type
Research Article
Information
MRS Online Proceedings Library (OPL) , Volume 651: Symposium T – Dynamics in Small Confining Systems V , 2000 , T2.7.1
Copyright
Copyright © Materials Research Society 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Stauffer, D. and Aharony, A., Introduction to Percolation Theory (Taylor Francis, 1992).Google Scholar
2. Bunde, A. and Havlin, S., (Eds.), Fractals and Disordered Systems (Springer-Verlag, 1996).Google Scholar
3. Hermann, H., Stochastic Models of Heterogeneous Materials (Trans Tech Publ., 1991).Google Scholar
4. Feng, S., Halperin, B. I., and Sen, P. N., Phys. Rev. B 35, 197 (1987).Google Scholar
5. Klemm, A., Müller, H.-P., Kimmich, R., Phys. Rev. E 55, 4413 (1997).Google Scholar
6. Kimmich, R., NMR Tomography, Diffusometry, Relaxometry (Springer-Verlag, 1997).Google Scholar
7. Nield, D. A. and Bejan, A., Convection in Porous Media (Springer-Verlag,1992).Google Scholar
8. Müller, H.-P., Weis, J., and Kimmich, R., Phys. Rev. E 52, 5195 (1995).Google Scholar
9. Müller, H.-P., Kimmich, R., and Weis, J., Phys. Rev. E 54, 5278 (1996).Google Scholar
10. Andrade, J. S. Jr, Almeida, M. P., Filho, J. Mendes, Havlin, S., Suki, B., and Stanley, H. E., Phys. Rev. Letters 79, 3901 (1997).Google Scholar
11. Klammler, F. and Kimmich, R., Phys. Med. Biol. 35, 67 (1990).Google Scholar
12. Codd, S. L., Manz, B., Seymour, J. D., and Callaghan, P. T., Phys. Rev. E 60, R3491 (1999).Google Scholar
13. Kapitulnik, A., Aharony, A., Deutscher, G., and Stauffer, D., J. Phys. A: Math. Gen. 16, L269 (1983).Google Scholar
14. Porto, M., Bunde, A., Havlin, S., and Roman, H. E., J. Phys. Rev. E 56, 1667 (1997).Google Scholar
15. Shattuck, M. D., Behringer, R. P., Johnson, G. A., and Georgiadis, J. G., Phys. Rev. Letters 75, 1934 (1995).Google Scholar
16. Alexander, S. and Orbach, R., J. Physique-Lettres (Paris) 43, L625 (1982).Google Scholar
17. Hong, D. C., Havlin, S., Herrmann, H. J., and Stanley, H. E., Phys. Rev. B 30, 4083 (1984).Google Scholar
18. Zabolitzky, J. G., Phys. Rev. B 30, 4077 (1984).Google Scholar
19. Taylor, G. I., Proc. Roy. Soc. Lond. A 219, 186 (1953).Google Scholar
20. Aris, R., Proc. Roy. Soc. Lond. A 235, 67 (1956).Google Scholar
21. Brenner, H., J. Stat. Phys. 62, 1095 (1991).Google Scholar
22. Broeck, C. van den, Physica A 168, 677 (1990).Google Scholar
23. Cushman, J. H, Hu, B.X. and Ginn, T.R., J. Stat. Phys. 75, 859 (1994).Google Scholar
24. Koch, D.L. and Brady, J.F., J. Fluid Mech. 200, 173 (1987).Google Scholar
25. Koplik, J., Redner, S. and Wilkinson, D., Phys. Rev. A 37, 2619 (1988).Google Scholar
26. Sahimi, M., Rev. Mod. Phys, 65, 1393 (1993).Google Scholar
27. Makse, H.A., Andrade, J.S. and Stanley, H.E., Phys. Rev. E 61, 583 (2000).Google Scholar
28. Metzler, R. and Klafter, J., Europhys. Lett. 51, 492 (2000).Google Scholar
29. Schneider, W. R. and Wyss, W., J. Math. Phys. 30, 134 (1989).Google Scholar
30. Seymour, J. D. and Callaghan, P. T., AIChE J. 43, 2096 (1997).Google Scholar
31. Poole, O. J. and Salt, D. W., J. Phys. A; Math. Gen. 29, 7959 (1996).Google Scholar