Abstract
Based on fault maps, whether or not the fracture geometry of rocks is self-similar, was examined by using a box-counting algorithm. The statistical self-similarity (fractal structure) of the fault fracture systems holds well at the scale of about 2 to 20 km. The fractal dimension in Japan varied from 1.05 to 1.60. The fractal dimension is about 1.5–1.6 at the central part of the Japan Arc, and decreases with distance from the center. At a smaller scale, the fractal structure also holds well in the rock fracture geometry. The fractal dimension of the North Izu Peninsula fault system (branching faults) is 1.49 at the scale of 0.625 to 10 km, the fractal dimension of rock fracture geometry at the scale order of 10−1 to 10−2 meters is about 1.49–1.61. The upper limit of the fractal dimension of rock fracture geometry is about 1.6, judging from the estimation of fractal dimension on actual fracture geometry of rocks. This value may impose a restraint on modeling of faulting and the fracture process of rocks.
Similar content being viewed by others
References
Abe, K. (1978),Dislocations, Source Dimensions and Stresses Associated with Earthquakes in the Izu Peninsula, Japan, J. Phys. Earth26, 253–274.
Aki, K.,A probabilistic synthesis of precursory phenomena, InEarthquake Prediction (eds. Simpson, D. W. and Richards, P. G.) (Maurice Ewing Volume 4, AGU, Washington, D. C. 1981) pp. 566–574.
Allègre, C. J., Le Mouel, J. L., andProvost, A. (1982),Scaling Rules in Rock Fracture and Possible Implications for Earthquake Prediction, Nature297, 47–49.
Aviles, C. A., Scholz, C. H., andBoatwright, J. (1987),Fractal Analysis Applied to Characteristic Segments of the San Andreas Fault, J. Geophys. Res.92, 331–344.
Barton, C. C., andLarsen, E. (1985),Fractal geometry of two-dimensional fracture networks at Yucca Mountain, Southwestern Nevada, InFundamentals of Rock Joints, Proc. Int. Sympo. on Fundamentals of Rock Joints, (ed. Stephansson, Ove), pp. 77–84.
Barton, C. C., andCameron, B. G. (1986),Fractal Scaling of Fracture and Fault Maps at Yucca Mountain, Southern Nevada, EOS (Transactions of the American Geophysical Union)67 (44), 870.
Brown, S. R., andScholz, C. H. (1985),Broad Bandwidth Study of the Topography of Natural Rock Surfaces, J. Geophys. Res.90, 12575–12582.
Davidge, R. W. andGreen, T. J. (1968),The Strength of Two-phase Ceramic/Glass Material, J. Mater. Sci.3, 629–634.
Eykholt, R., andUmberger, D. K. (1986),Characterization of Fat Fractals in Nonlinear Dynamical Systems, Phys. Rev. Lett.57, 2333–2336.
Hirata, T., Satoh, T., andIto, K. (1987),Fractal Structure of Spatial Distribution of Microfracturing in Rock, Geophys. J. R. Astr. Soc.90, 369–374.
Hirata, T. (1987),Omori's Power Law Aftershock Sequence of Microfracturing in the Rock Fracture Experiment, J. Geophys. Res.92, 6215–6221.
Kondratev, V. N., Kulukin, A. M., Ponomarev, V. S., Romashov, A. N., andChubarov, V. M. (1985),Investigation of a Two-layer Model of the Earth's Crust for Two-axis Extension of the Lower Layer, Izvestiya, Earth Physics21, (3), 176–183.
Kagan, Y. Y., andKnopoff, L. (1978),Statistical Study of the Occurrence of Shallow Earthquakes, Geophys. J. R. Astr. Soc.55, 67–86.
Kagan, Y. Y., andKnopoff, L. (1980),Spatial Distribution of Earthquakes: The Two-point Correlation Function, Geophys. J. R. Astr. Soc.62, 303–320.
Kagan, Y. Y., andKnopoff, L. (1981),Stochastic Synthesis of Earthquake Catalogs, J. Geophys. Res.86, 2853–2862.
King, G. (1983),The Accommodation of Large Strains in the Upper Lithosphere of the Earth and Other Solids by Self-similar Fault Systems: The Geometrical Origin of b-value, Pure Appl. Geophys.121, 761–815.
Madden, T. R. (1983),Microcrack Connectivity in Rocks: A Renormalization Group Approach to the Critical Phenomena of Conduction and Failure in Crystalline Rocks, J. Geophys. Res.88, 585–592.
Mandelbrot, B. B.,The Fractal Geometry of Nature (Freeman, New York 1983).
Matsuda, T.,Surface faults associated with Kita-Izu earthquake of 1930 in Izu Peninsula, Japan, inIzu Peninsula (eds. Hoshino, M., and Aoki, H.) (Tokai Univ. Press, Tokyo 1972), pp. 73–93 (in Japanese).
Matsuda, T., Ota, Y., Okada, A., Shimizu, F., andTogo, M. (1977),Aerial Photo-interpretation of Active Faults—The Individual Difference and Examples, Bull, Earthq. Res. Inst. 52, 461–497.
Matsuda, T.,Active faults and damaging earthquakes in Japan—Macroseismic zoning and precaution fault zones, InEarthquake Prediction (eds. Simpson, D. W. and Richards, P.G.) (Maurice Ewing Volume 4, AGU, Washington, D. C. 1981) pp. 279–289.
Mogi, K. (1962),Magnitude-frequency Relation for Elastic Shocks Accompanying Fractures of Various Materials and Some Related Problems in Earthquakes, Bull. Earthq. Res. Inst.40, 831–853.
Nii, Y., Nakamura, H. Ito, K., Fujii, N., andMatsuda, J.,The fractal dimension of fractured fragments in relation with fracture energy, In Proc. the 18th I.S.A.S. Lunar and Planet. Sympo. (eds. H. Mizutani, Oya, H., and Shimizu, M.) (Institute of Space and Astronautical Science, Tokyo 1985) pp. 58–59.
Okubo, P. G., andAki, K. (1987),Fractal Geometry in the San Andreas Fault System, J. Geophys. Res.92, 345–355.
Otsuka, M. (1972),A Chain-reaction-type Source Model as a Tool to Interpret the Magnitude-frequency Relation of Earthquakes, J. Phys. Earth20, 35–45.
Research Group for Active Faults of Japan,Active Faults in Japan, Sheet Maps and Inventories (in Japanese) (Univ. of Tokyo Press, Tokyo 1980).
Sadovskiy, M. A., Golubeva, T. V., Pisarenko, V. F., andShnirman, M.G. (1984),Characteristic Dimensions of Rock and Hierarchial Properties of Seismicity, Izvestiya, Earth Physics20, 87–96.
Saito, M., Kikuchi, M., andKudo, K. (1973),Analytical Solution of “Go-Game Model” of Earthquake, Zisin26, 19–25, (in Japanese).
Scholz, C. H., (1968),Microfractures, Aftershocks, and Seismicity, Bull. Seismol. Soc. Am.58, 1117–1130.
Scholz, C. H., andAviles, C. A. The fractal geometry of faults and faulting InEarthquake Source Mechanics (eds. Das, S., Boatwright, J., and Scholz, C. H.) (Maurice Ewing Volume 6, AGU, Washington, D. C. 1986) pp. 147–155.
Smalley, R. F., Turcotte, D. L., andSolla, S. A. (1985),A Renormalization Group Approach to the Stick-slip Behavior of Faults, J. Geophys. Res.90, 1894–1900.
Stauffer, D.,Introduction to Percolation Theory (Taylor and Francis, London 1985).
Takayasu, H. (1985),A Deterministic Model of Fracture, Prog. Theor. Phys.74, 1343–1345.
Turcotte, D. L., Smalley, R. F., andSolla, S. A. (1985),Collapse of Loaded Fractal Trees, Nature 313, 671–672.
Turcotte, D. L. (1986),Fractals and Fragmentation J. Geophys. Res.91, 1921–1926.
Vere-Jones, D. (1976),A Branching Model for Crack Propagation, Pure Appl. Geophys.114, 711–725.
Watanabe, K. (1986),Stochastic Evaluation of the Two Dimensional Continuity of Fractures in a Rock Mass, Int. J. Rock Mech. Min. Sci. and Geomech. Abstr.23, 431–437.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hirata, T. Fractal dimension of fault systems in Japan: Fractal structure in rock fracture geometry at various scales. PAGEOPH 131, 157–170 (1989). https://doi.org/10.1007/BF00874485
Received:
Revised:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00874485