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Licensed Unlicensed Requires Authentication Published by De Gruyter July 16, 2020

Statistical modeling of three-dimensional fractal point sets with a given spatial probability distribution

  • Dmitriy Kolyukhin ORCID logo EMAIL logo

Abstract

The work is devoted to three-dimensional modeling of fractal sets of points. Additional constraints in the form of probability density caused by the frequency of the generated points’ spatial distribution are considered. The suggested method for statistical simulation allows reproducing both the given probability distribution defining the spatial position of the generated points and the required fractal dimension. Performed numerical computations confirm the accuracy and efficiency of the proposed method for the considered test models.

MSC 2010: 65C05; 65C20

Award Identifier / Grant number: AAAA-A16-116122810045-9

Funding statement: The financial support from IPGG project AAAA-A16-116122810045-9 is gratefully acknowledged.

References

[1] E. Bonnet, O. Bour, N. E. Odling, P. Davy, I. Main, P. Cowie and B. Berkowitz, Scaling of fracture systems in geological media, Rev. Geophys. 39 (2001), no. 3, 347–383. 10.1029/1999RG000074Search in Google Scholar

[2] O. Bou, P. Davy and C. Darcel, A statistical scaling model for fracture network geometry, with validation on multiscale mapping of a joint network (Hornelen Basin, Norway), J. Geophys. Res. 107 (2002), 10.1029/2001JB000176. 10.1029/2001JB000176Search in Google Scholar

[3] A. Cortis, C. E. Puente and B. Sivakumar, Encoding hydrologic information via a fractal geometric approach and its extensions, Stoch. Environ. Res. Risk Assessment 24 (2010), 625–632. 10.1007/s00477-009-0349-4Search in Google Scholar

[4] C. Darcel, O. Bour, P. Davy and J. R. de Dreuzy, Connectivity properties of two-dimensional fracture network with stochastic fractal correlation, Water Resources Res. 39 (2003), no. 20, Article ID 1272. 10.1029/2002WR001628Search in Google Scholar

[5] M. De Stefano, Simulating geophysical models through fractal algorithms, Geophys. Prospecting 66 (2018), no. 1, 26–33. 10.3997/2214-4609.201601160Search in Google Scholar

[6] X. Du Bernard, P. Labaume, C. Darcel, P. Davy and O. Bour, Cataclastic slip band distribution in normal fault damage zones, Nubian sandstones, Suez rift, J. Geophys. Res. 107 (2002), 10.1029/2001JB000493. 10.1029/2001JB000493Search in Google Scholar

[7] P. Grassberger and I. Procaccia, Measuring the strangeness of strange attractors, Phys. D 9 (1983), no. 1–2, 189–208. 10.1007/978-0-387-21830-4_12Search in Google Scholar

[8] D. R. Kolyukhin, Statistical simulation technique for deformation band spatial distribution in the fault damage zone, Sib. Èlektron. Mat. Izv. 12 (2015), 465–479. Search in Google Scholar

[9] B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, San Francisco, 1982. Search in Google Scholar

[10] B. Mazzi and J. C. Vassilicos, Fractal-generated turbulence, J. Fluid Mech. 502 (2004), 65–87. 10.1017/S0022112003007249Search in Google Scholar

[11] R. G. McClarren, Uncertainty Quantification and Predictive Computational Science, Springer, Cham, 2018. 10.1007/978-3-319-99525-0Search in Google Scholar

[12] G. Ouillon and D. Sornette, Unbiased multifractal analysis: Application to fault patterns, Geophys. Res. Lett. 23 (1996), 3409–3412. 10.1029/96GL02379Search in Google Scholar

[13] M. Rouai, Properties of Fracture Network in Sefrou Carbonate Reservoir (Morocco), 78th EAGE Conference and Exhibition, 2016. 10.3997/2214-4609.201601448Search in Google Scholar

[14] A. Saltelli, Global sensitivity analysis: An introduction, Sensitivity Analysis of Model Output, Los Alamos National Laboratory, Loa Alamos (2005), 27–43. Search in Google Scholar

[15] S. Schueller, A. Braathen, H. Fossen and J. Tveranger, Spatial distribution of deformation bands in damage zones of extensional faults in porous sandstones: Statistical analysis of field data, J. Struct. Geol. 52 (2013), 148–162. 10.1016/j.jsg.2013.03.013Search in Google Scholar

[16] L. Seuront, Fractals and Multifractals in Ecology and Aquatic Science, CRC Press, Boca Raton, 2010. 10.1201/9781420004243Search in Google Scholar

[17] K. R. Sreenivasan, Fractals and multifractals in fluid turbulence, Annual Review of Fluid Mechanics. Vol. 23, Annual Reviews, Palo Alto (1991), 539–600. 10.1146/annurev.fl.23.010191.002543Search in Google Scholar

[18] V. V. Uchaikin, If the universe were a Levy–Mandelbrot fractal, Gravitation & Cosmology 10 (2004), no. 1–2, 5–24. Search in Google Scholar

[19] H. Xie, J. A. Wang and M. A. Kwasniewski, Multifractal characterization of rock fracture surfaces, Int. J. Rock Mech. Min. Sci. 36 (1999), 19–27. 10.1016/S0148-9062(98)00172-7Search in Google Scholar

[20] T. Xu, I. D. Moore and J. C. Gallant, Fractals, fractal dimensions and landscapes – a review, Geomorphology 8 (1993), no. 4, 245–262. 10.1016/0169-555X(93)90022-TSearch in Google Scholar

[21] M. Zou, B. Yu, Y. Feng and P. Xu, A Monte Carlo method for simulating fractal surfaces, Phys. A 386 (2007), 176–186. 10.1016/j.physa.2007.07.058Search in Google Scholar

Received: 2020-02-09
Accepted: 2020-06-19
Published Online: 2020-07-16
Published in Print: 2020-09-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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