Letter | Published:

An invasion percolation model of drainage network evolution

Naturevolume 352pages423425 (1991) | Download Citation

Subjects

Abstract

STREAM networks evolve by headward growth and branching away from escarpments such as rift margins. The structure of these networks and their topographic relief are known to be fractal1–3, but no model so far has been able to generate the observed scaling properties. Here I present a statistical model of network growth in which stream heads branch and propagate at a rate that depends only on the local strength of the substrate. This model corresponds to the process of invasion percolation4, with the added requirement of self-avoidance; it is a self-organized critical system5 with properties similar to those of standard percolation models6. A description based on self-avoiding invasion percolation reproduces the known scaling behaviour of stream networks, and may provide a valuable tool for delineation of drainage patterns from digital topographic data sets7,8.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

References

  1. 1

    Mandelbrot, B. B. The Fractal Geometry of Nature (Freeman, New York, 1982).

  2. 2

    Huang, J. & Turcotte, D. L. J. geophys. Res. 94, 7491–7495 (1989).

  3. 3

    Thornes, J. Nature 345, 764–765 (1990).

  4. 4

    Wilkinson, D. & Willemsen, J. F. J. Phys. A 16, 3365–3376 (1983).

  5. 5

    Bak, P., Tang, C. & Wiesenfeld, K. Phys. Rev. A38, 364–374 (1988).

  6. 6

    Stauffer, D. Introduction to Percolation Theory (Taylor & Francis, London, 1985).

  7. 7

    Band, L. E. Water Resour. Res. 12, 15–24 (1986).

  8. 8

    Jenson, S. K. & Domingue, J. O. Photogram. Engng Remote Sens. 54, 1593–1600 (1988).

  9. 9

    Leopold, L. B. & Langbein, W. B. US Geol. Survey Prof. Pap. 500-A, 1–20 (1962).

  10. 10

    Howard, A. D. Geogr. Anal. 3, 29–51 (1971).

  11. 11

    Takayasu, H., Nishikawa, I. & Tasaki, H. Phys. Rev. A 37, 3110–3117 (1988).

  12. 12

    Hjelmfelt, A. T. Jr Water Resour. Bull. 24, 455–459 (1988).

  13. 13

    Rosso, R., Bacchi, B. & La Barbera, P. Water Resour. Res. 27, 381–387 (1991).

  14. 14

    Hack, J. T. US Geol. Survey Prof. Pap. 294-B, 1–97 (1957).

  15. 15

    Gray, D. M. J. geophys. Res. 66, 1215–1223 (1961).

  16. 16

    Weissel, J. K. Pacific Rim Congress 90, Proc. v. III, 63–70 (Austral. Inst. Min. and Metall, Parkville, Vic., 1990).

  17. 17

    Havlin, S. & Ben-Avraham, D. Adv. Phys. 36, 695–798 (1987).

  18. 18

    Havlin, S., Nossal, R. & Trus, B. Phys. Rev. A 32, 3829–3831 (1985).

  19. 19

    Herrmann, H. J. & Stanley, H. E. J. Phys. A 21, L829–L833 (1988).

  20. 20

    Hinrichsen, E. L., Måløy, K. J., Feder, J. & Jøssang, T. J. Phys. A 22, L271–L277 (1989).

  21. 21

    La Barbera, P. & Rosso, R. Water Resour. Res. 25, 735–741 (1989).

  22. 22

    Shreve, R. L. J. Geol. 77, 397–414 (1969).

  23. 23

    Dhar, D. & Ramaswamy, R. Phys. Rev. Lett. 54, 1346–1349 (1985).

  24. 24

    Smart, J. S. Bull. geol. Soc. Am. 80, 1757–1774 (1969).

  25. 25

    Meakin, P. J. Phys. A 20, L1113–L1119 (1987).

  26. 26

    Meakin, P., Majid, I., Havlin, S. & Stanley, H. E. J. Phys. A 17, L975–L981 (1984).

  27. 27

    Mandelbrot, B. B. & Evertsz, C. J. G. Nature 348, 143–145 (1990).

  28. 28

    Strahler, A. N. Handbook of Applied Hydrology (ed. Chow, V. T.) 4–11 (McGraw-Hill. New York, 1964).

Download references

Author information

Affiliations

  1. Department of Earth Sciences, University of Leeds, Leeds, LS2 9JT, UK

    • Colin P. Stark

Authors

  1. Search for Colin P. Stark in:

About this article

Publication history

Received

Accepted

Issue Date

DOI

https://doi.org/10.1038/352423a0

Further reading

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.