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General fractal topography: an open mathematical framework to characterize and model mono-scale-invariances

  • Yi JinEmail author
  • Xianhe Liu
  • Huibo Song
  • Junling Zheng
  • Jienan Pan
Original Paper
  • 47 Downloads

Abstract

In this work, we reported there are two kinds of independent complexities in mono-scale-invariance, namely to be behavioral complexity determined by fractal behavior and original one wrapped in scaling object. Quantitative characterization of the complexities is fundamentally important for mechanism exploration and essential understanding of nonlinear systems. Recently, a new concept of fractal topography (\(\mathrm {\Omega }\)) was emerged to define the fractal behavior in self-similarities by scaling lacunarity (P) and scaling coverage (F) as \(\mathrm {\Omega }(P, F)\). It is, however, a “special fractal topography” because fractals are rarely deterministic and always appear stochastic, heterogeneous, and even anisotropic. For that, we reviewed a novel concept of “general fractal topography”, \({\varvec{\Omega }}(\mathrm {{\mathbf {P}}}, \mathrm {{\mathbf {F}}})\). In \({\varvec{\Omega }}\), \(\mathrm {{\mathbf {P}}}\) is generalized to a set accounting for direction-dependent scaling behaviors of the scaling object G, while \(\mathrm {{\mathbf {F}}}\) is extended to consider stochastic and heterogeneous effects. And then, we decomposed G into \(\mathrm {{\mathbf {G}}}\left( G_+, G_-\right) \) to ease type controlling and measurement quantification, with \(G_+\) wrapping the original complexity while \(G_-\) enclosing behavioral complexity. Together with \({\varvec{\Omega }}\) and \(\mathrm {{\mathbf {G}}}\), a mathematical model \(F_\mathrm {3S}\left( {\varvec{\Omega }}, \mathrm {{\mathbf {G}}}\right) \) was then established to unify the definition of deterministic or statistical, self-similar or self-affine, single- or multi-phase properties. For demonstration, algorithms are developed to model natural scale-invariances. Our investigations indicated that the general fractal topography is an open mathematic framework which can admit most complex scaling objects and fractal behaviors.

Keywords

Scale-invariance Fractal dimension Hurst exponent Porous media Fractal networks Fractal topography Self-sameness self-similarity self-affinity (3S) 

List of symbols

D

Space fractal dimension

\(D_x\)

Space fractal dimension in x-direction

\({\mathcal {D}}\)

Property fractal dimension

\({\overline{D}}\)

General Space fractal dimension

\(D_s\)

Mass fractal dimension

H

Hurst exponent

\(H_{xy}\)

General Hurst exponent

P

Scaling lacunarity

\(P_x\)

Scaling lacunarity in x-direction

\(\mathrm {{\mathbf {P}}}\)

Scaling lacunarity set

\({\overline{P}}\)

Effective scaling lacunarity, scalar quantity of \(\mathrm {{\mathbf {P}}}\)

F

Scaling coverage

\(\mathrm {{\mathbf {F}}}\)

Scaling coverage set

\(\langle F\rangle \)

Expectation of scaling coverage

G

Scaling object

\(G_+\)

Regime wrapping original complexity in G

\(G_-\)

Regime enclosing behavioral complexity in G

l

Characteristic linear size of G

\(\mathrm {{\mathbf {l}}}\)

Vector of l

\(\mathrm {{\mathbf {L}}}\)

Range of \(\mathrm {{\mathbf {l}}}\)

\(l_x\)

Characteristic linear size of G in x-direction

\({\overline{l}}\)

Effective characteristic size of G

\({\mathcal {P}}\)

A desired property

Greek symbols

\(\mathrm {\Omega }\)

Special fractal topography

\({\varvec{\Omega }}\)

General fractal topography

\(\varphi \)

Porosity

\(\rho \)

Density of desired property \({\mathcal {P}}\)

\(\epsilon \)

A dimensionless constant

Subscripts

max

Maximum values

min

Minimum values

Notes

Compliance with ethical standards

Conflict of interest

This manuscript has not been published in part or in entirety and is not under consideration by another journal. All authors have been informed and consent to this submission. We declare no competing interests (financial or non-financial) or other interests that might be perceived to influence the results and/or discussion reported in this paper. In this research, all funding sources were correctly included and there are no animals involved. We have read and understood your journal’s policies, and we believe that neither the manuscript nor the study violates any of these.

References

  1. 1.
    Mandelbrot, B.B.: How long is the coast of britain? Statistical self-similarity and fractional dimension. Science 156(3775), 636 (1967).  https://doi.org/10.1126/science.156.3775.636 Google Scholar
  2. 2.
    Stanley, H.E., Meakin, P.: Multifractal phenomena in physics and chemistry. Nature 335(6189), 405 (1988).  https://doi.org/10.1038/335405a0 Google Scholar
  3. 3.
    Zhang, X., Li, N., Gu, G.C., Wang, H., Nieckarz, D., Szabelski, P., He, Y., Wang, Y., Xie, C., Shen, Z.Y., L, J.T., Tang, H., Peng, L.M., Hou, S.M., Wu, K., Wang, Y.F.: Controlling Molecular Growth between Fractals and Crystals on Surfaces. ACS Nano. 9(12), 11909 (2015).  https://doi.org/10.1021/acsnano.5b04427 Google Scholar
  4. 4.
    Mandelbrot, B.B., Passoja, D.E., Paullay, A.J.: Fractal character of fracture surfaces of metals. Nature 308(5961), 721 (1984).  https://doi.org/10.1038/308721a0 Google Scholar
  5. 5.
    Jonkers, A.R.T.: Long-range dependence in the Cenozoic reversal record. Phys. Earth Planet. Inter. 135(4), 253 (2003).  https://doi.org/10.1016/S0031-9201(03)00036-0 Google Scholar
  6. 6.
    Gneiting, T., Schlather, M.: Stochastic models that separate fractal dimension and the Hurst effect. SIAM Rev. 46(2), 269 (2004).  https://doi.org/10.1137/S0036144501394387 MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bejan, A., Lorente, S.: Constructal theory of generation of configuration in nature and engineering. J. Appl. Phys. 100(4), 041301 (2006).  https://doi.org/10.1063/1.2221896 Google Scholar
  8. 8.
    Lanotte, A.S., Benzi, R., Malapaka, S.K., Toschi, F., Biferale, L.: Turbulence on a fractal fourier set. Phys. Rev. Lett. 115, 26 (2015).  https://doi.org/10.1103/PhysRevLett.115.264502 MathSciNetGoogle Scholar
  9. 9.
    Turcotte, D.L.: Fractals and Choas in Geology and Geophysics. Cambridge University Press, New York (1997).  https://doi.org/10.1017/CBO9781139174695 Google Scholar
  10. 10.
    Dubuc, B., Quiniou, J.F., Roquescarmes, C., Tricot, C., Zucker, S.W.: Evaluating the fractal dimension of profiles. Phys. Rev. A 39(3), 1500 (1989).  https://doi.org/10.1103/PhysRevA.39.1500 MathSciNetGoogle Scholar
  11. 11.
    Schlager, W.: Fractal nature of stratigraphic sequences. Geology 32(3), 185 (2004).  https://doi.org/10.1130/G202531.1 Google Scholar
  12. 12.
    Bailey, R.J., Smith, D.G.: Quantitative evidence for the fractal nature of the stratigraphic record: results and implications. P. Geologist Assoc. 116(2), 129 (2005).  https://doi.org/10.1016/S0016-7878(05)80004-5 Google Scholar
  13. 13.
    Koutsoyiannis, D.: Climate change, the Hurst phenomenon, and hydrological statistics. Hydrol. Sci. J. 48(1), 3 (2003).  https://doi.org/10.1623/hysj.48.1.3.43481 Google Scholar
  14. 14.
    Clauset, A., Shalizi, C.R., Newman, M.E.J.: Power-law distributions in empirical data. SIAM Rev. 51(4), 661 (2009).  https://doi.org/10.1137/070710111 MathSciNetzbMATHGoogle Scholar
  15. 15.
    Montanari, A., Rosso, R., Taqqu, M.S.: A seasonal fractional ARIMA model applied to the Nile river monthly flows at Aswan. Water Resour. Res. 36(5), 1249 (2000).  https://doi.org/10.1029/2000WR900012 Google Scholar
  16. 16.
    West, G.B., Brown, J.H., Enquist, B.J.: A general model for the origin of allometric scaling laws in biology. Science 276(5309), 122 (1997).  https://doi.org/10.1126/science.276.5309.122 Google Scholar
  17. 17.
    Ivanov, P.C., Amaral, L.A.N., Goldberger, A.L., Havlin, S., Rosenblum, M.G., Struzik, Z.R., Stanley, H.E.: Multifractality in human heartbeat dynamics. Nature 399(6735), 461 (1999).  https://doi.org/10.1038/20924 Google Scholar
  18. 18.
    Ashkenazy, Y., Ivanov, P.C., Havlin, S., Peng, C.K., Goldberger, A.L., Stanley, H.E.: Magnitude and sign correlations in heartbeat fluctuations. Phys. Rev. Lett. 86(9), 1900 (2001).  https://doi.org/10.1103/PhysRevLett.86.1900 Google Scholar
  19. 19.
    King, R.D., George, A.T., Jeon, T., Hynan, L.S., Youn, T.S., Kennedy, D.N., Dickerson, B.: Characterization of atrophic changes in the cerebral cortex using fractal dimensional analysis. Brain Imaging Behav. 3(2), 154 (2009).  https://doi.org/10.1007/s11682-008-9057-9 Google Scholar
  20. 20.
    Popescu, D.P., Flueraru, C., Mao, Y., Chang, S., Sowa, M.G.: Signal attenuation and box-counting fractal analysis of optical coherence tomography images of arterial tissue. Biomed. Opt. Express 1(1), 268 (2010).  https://doi.org/10.1364/BOE.1.000268 Google Scholar
  21. 21.
    Adler, P.M., Thovert, J.F.: Real porous media: local geometry and macroscopic properties. Appl. Mech. Rev. 51(9), 537 (1998).  https://doi.org/10.1115/1.3099022 Google Scholar
  22. 22.
    Bonnet, E., Bour, O., Odling, N.E., Davy, P., Main, I., Cowie, P., Berkowitz, B.: Scaling of fracture systems in geological media. Rev. Geophys. 39(3), 347 (2001).  https://doi.org/10.1029/1999RG000074 Google Scholar
  23. 23.
    Cheng, Q.M.: Singularity theory and methods for mapping geochemical anomalies caused by buried sources and for predicting undiscovered mineral deposits in covered areas. J. Geochem. Explor. 122, 55 (2012).  https://doi.org/10.1016/j.gexplo.2012.07.007 Google Scholar
  24. 24.
    Thovert, J.F., Wary, F., Adler, P.M.: Thermal conductivity of random media and regular fractals. J. Appl. Phys. 68(8), 3872 (1990).  https://doi.org/10.1063/1.346274 Google Scholar
  25. 25.
    Li, G.X., Moon, F.C.: Fractal basin boundaries in a two-degree-of-freedom nonlinear system. Nonlinear Dyn. 1(3), 209 (1990).  https://doi.org/10.1007/BF01858294 Google Scholar
  26. 26.
    Wu, Y.T., Shyu, K.K., Chen, T.R., Guo, W.Y.: Using three-dimensional fractal dimension to analyze the complexity of fetal cortical surface from magnetic resonance images. Nonlinear Dyn. 58(4), 745 (2009).  https://doi.org/10.1007/s11071-009-9515-y zbMATHGoogle Scholar
  27. 27.
    Anishchenko V.S.: Fractals in nonlinear dynamics. In: Deterministic Nonlinear Systems. Fractals in Nonlinear Dynamics. Deterministic Nonlinear Systems. Springer Series in Synergetics, Springer, Cham (2014)Google Scholar
  28. 28.
    He, Z.: Integer-dimensional fractals of nonlinear dynamics, control mechanisms, and physical implications. Sci. Rep. 8(1), 10324 (2018).  https://doi.org/10.1038/s41598-018-28669-3 Google Scholar
  29. 29.
    El-Nabulsi, R.A.: Path integral formulation of fractionally perturbed lagrangian oscillators on fractal. J. Stat. Phys. 172(6), 1617 (2018).  https://doi.org/10.1007/s10955-018-2116-8 MathSciNetzbMATHGoogle Scholar
  30. 30.
    Yu, B.M., Li, J.H.: Some fractal Characters of porous media. Fractals 9(3), 365 (2001).  https://doi.org/10.1142/S0218348X01000804 Google Scholar
  31. 31.
    Friesen, W.I., Mikula, R.J.: Fractal dimensions of coal particles. J. Colloid Interf. Sci. 120(1), 263 (1987).  https://doi.org/10.1016/0021-9797(87)90348-1 Google Scholar
  32. 32.
    Tyler, S.W., Wheatcraft, S.W.: Fractal processes in soil water retention. Water Resour. Res. 26(5), 1047 (1990).  https://doi.org/10.1029/WR026i005p01047 Google Scholar
  33. 33.
    Sui, L., Ju, Y., Yang, Y., Yang, Y., Li, A.: A quantification method for shale fracability based on analytic hierarchy process. Energy 115, 637 (2016).  https://doi.org/10.1016/j.energy.2016.09.035 Google Scholar
  34. 34.
    Jin, Y., Zhu, Y.B., Li, X., Zheng, J.L., Dong, J.B.: Scaling invariant effects on the permeability of fractal porous media. Transport Porous Med. 109(2), 433 (2015).  https://doi.org/10.1007/s11242-015-0527-4 Google Scholar
  35. 35.
    Yousefi, M., Carranza, E.J.M.: Prediction-area (P-A) plot and C-A fractal analysis to classify and evaluate evidential maps for mineral prospectivity modeling. Comput. Geosci. 79, 69 (2015).  https://doi.org/10.1016/j.cageo.2015.03.007 Google Scholar
  36. 36.
    Zhao, J.N., Chen, S.Y., Zuo, R.G., Carranza, E.J.M.: Mapping complexity of spatial distribution of faults using fractal and multifractal models: vectoring towards exploration targets. Comput. Geosci. 37(12), 1958 (2011).  https://doi.org/10.1016/j.cageo.2011.04.007 Google Scholar
  37. 37.
    Cheng, Q.M., Agterberg, F.P.: Singularity analysis of ore-mineral and toxic trace elements in stream sediments. Comput. Geosci. 35(2), 234 (2009).  https://doi.org/10.1016/j.cageo.2008.02.034 Google Scholar
  38. 38.
    Zuo, R.G., Agterberg, F.P., Cheng, Q.M., Yao, L.Q.: Fractal characterization of the spatial distribution of geological point processes. Int. J. Appl. Earth Obs. 11(6), 394 (2009).  https://doi.org/10.1016/j.jag.2009.07.001 Google Scholar
  39. 39.
    Wheatcraft, S.W., Tyler, S.W.: An explanation of scale-dependent dispersivity in heterogeneous aquifers using concepts of fractal geometry. Water Resour. Res. 24(4), 566 (1988).  https://doi.org/10.1029/WR024i004p00566 Google Scholar
  40. 40.
    Jin, Y., Dong, J.B., Zhang, X.Y., Li, X., Wu, Y.: Scale and size effects on fluid flow through self-affine rough fractures. Int. J. Heat Mass Transf. 105, 443 (2017).  https://doi.org/10.1016/j.ijheatmasstransfer.2016.10.010 Google Scholar
  41. 41.
    Katz, A.J., Thompson, A.H.: Fractal sandstone pores: implications for conductivity and pore formation. Phys. Rev. Lett. 54, 1325 (1985).  https://doi.org/10.1103/PhysRevLett.54.1325 Google Scholar
  42. 42.
    Wheatcraft, S.W., Cushman, J.H.: Hierarchical approaches to transport in heterogeneous porous media. Rev. Geophys. 29(S1), 263 (1991).  https://doi.org/10.1002/rog.1991.29.s1.263 Google Scholar
  43. 43.
    Molz, F.J., Rajaram, H., Lu, S.: Stochastic fractal-based models of heterogeneity in subsurface hydrology: origins, applications, limitations, and future research questions. Rev. Geophys. 42(1), RG1002 (2004).  https://doi.org/10.1029/2003RG000126 Google Scholar
  44. 44.
    Gaci, S.: A new method for characterizing heterogeneities from a core image using local Holder exponents. Arab. J. Geosci. 6(8), 2719 (2013).  https://doi.org/10.1007/s12517-012-0611-9 Google Scholar
  45. 45.
    Pesin, Y.B.: Dimension Theory in Dynamical Systems: Contemporary Views and Applications. University of Chicago Press, Chicago (1997)zbMATHGoogle Scholar
  46. 46.
    Costa, A.: Permeability-porosity relationship: a reexamination of the Kozeny–Carman equation based on a fractal pore-space geometry assumption. Geophys. Res. Lett. 33, L02318 (2006).  https://doi.org/10.1029/2005GL025134 Google Scholar
  47. 47.
    Xu, P., Yu, B.M.: Developing a new form of permeability and Kozeny–Carman constant for homogeneous porous media by means of fractal geometry. Adv. Water Resour. 31(1), 74 (2008).  https://doi.org/10.1016/j.advwatres.2007.06.003 Google Scholar
  48. 48.
    Jin, Y., Song, H.B., Hu, B., Zhu, Y.B., Zheng, J.L.: Lattice Boltzmann simulation of fluid flow through coal reservoir’s fractal pore structure. Sci. China Earth Sci. 56, 1519 (2013).  https://doi.org/10.1007/s11430-013-4643-0 Google Scholar
  49. 49.
    Cai, J.C., Perfect, E., Cheng, C.L., Hu, X.Y.: Generalized modeling of spontaneous imbibition based on Hagen–Poiseuille flow in tortuous capillaries with variably shaped apertures. Langmuir 30(18), 5142 (2014).  https://doi.org/10.1021/la5007204 Google Scholar
  50. 50.
    Turner, M.J., Andrews, P.R., Blackledge, J.M.: Fractal Geometry in Digital Imaging, 1st edn. Academic Press, Orlando (1998)Google Scholar
  51. 51.
    Jin, Y., Wu, Y., Li, H., Zhao, M.Y., Pan, J.N.: Definition of fractal topography to essential understanding of scale-invariance. Sci. Rep. 7, 46672 (2017).  https://doi.org/10.1038/srep46672 Google Scholar
  52. 52.
    Yu, B.M., Li, J.H.: Fractal dimensions for unsaturated porous media. Fractals 12(1), 17 (2004).  https://doi.org/10.1142/S0218348X04002409 zbMATHGoogle Scholar
  53. 53.
    Shokri, N., Sahimi, M.: Structure of drying fronts in three-dimensional porous media. Phys. Rev. E 85, 066312 (2012).  https://doi.org/10.1103/PhysRevE.85.066312 Google Scholar
  54. 54.
    Perrier, E., Bird, N., Rieu, M.: Generalizing the fractal model of soil structure: the pore-solid fractal approach. Geoderma 88(3), 137 (1999).  https://doi.org/10.1016/S0016-7061(98)00102-5 Google Scholar
  55. 55.
    Mandelbrot, B.B.: Self-affine fractals and fractal dimension. Phys. Scr. 32(4), 257 (1985).  https://doi.org/10.1088/0031-8949/32/4/001 MathSciNetzbMATHGoogle Scholar
  56. 56.
    Lovejoy, S., Schertzer, D.: Scaling and multifractal fields in the solid earth and topography. Nonlinear Process. Geophs. 14(4), 465 (2007).  https://doi.org/10.5194/npg-14-465-2007 Google Scholar
  57. 57.
    Turcotte, D.L., Newman, W.I.: Symmetries in geology and geophysics. Proc. Natl. Acad. Sci. 93(25), 14295 (1996).  https://doi.org/10.1073/pnas.93.25.14295 Google Scholar
  58. 58.
    Madadi, M., Sahimi, M.: Lattice Boltzmann simulation of fluid flow in fracture networks with rough, self-affine surfaces. Phys. Rev. E 67(2), 026309 (2003).  https://doi.org/10.1103/PhysRevE.67.026309 Google Scholar
  59. 59.
    Berry, M.V., Lewis, Z.V.: On the Weierstrass–Mandelbrot fractal function. Proc. R. Soc. A. 370(1743), 459 (1980).  https://doi.org/10.1098/rspa.1980.0044 MathSciNetzbMATHGoogle Scholar
  60. 60.
    Yan, W., Komvopoulos, K.: Contact analysis of elastic-plastic fractal surfaces. J. Appl. Phys. 84(7), 3617 (1998).  https://doi.org/10.1063/1.368536 Google Scholar
  61. 61.
    Keller, A.A., Auset, M.: A review of visualization techniques of biocolloid transport processes at the pore scale under saturated and unsaturated conditions. Adv. Water Resour. 30(6–7), 1392 (2007).  https://doi.org/10.1016/j.advwatres.2006.05.013 Google Scholar
  62. 62.
    Joekar-Niasar, V., Doster, F., Armstrong, R.T., Wildenschild, D., Celia, M.A.: Trapping and hysteresis in two-phase flow in porous media: a pore-network study. Water Resour. Res. 49(7), 4244 (2013).  https://doi.org/10.1002/wrcr.20313 Google Scholar
  63. 63.
    Hunt, A.G., Sahimi, M.: Flow, transport, and reaction in porous media: percolation scaling, critical-path analysis, and effective medium approximation. Rev. Geophys. 55(4), 993 (2017).  https://doi.org/10.1002/2017RG000558 Google Scholar
  64. 64.
    Javadpour, F., Fisher, D., Unsworth, M.: Nanoscale gas flow in shale gas sediments. J. Can. Pet. Technol. 46(10), 55 (2007).  https://doi.org/10.2118/07-10-06 Google Scholar
  65. 65.
    Wang, B.Y., Jin, Y., Chen, Q., Zheng, J.L., Zhu, Y.B., Zhang, X.B.: Derivation of permeability-pore relationship for fractal porous reservoirs using series-parallel flow resistance model and lattice Boltzmann method. Fractals 22(3), 1440005 (2014).  https://doi.org/10.1142/S0218348X14400052 Google Scholar
  66. 66.
    Ghanbarian-Alavijeh, B., Hunt, A.G.: Comments on More general capillary pressure and relative permeability models from fractal geometry by Kewen Li. J. Contam. Hydrol. 140–141, 21 (2012).  https://doi.org/10.1016/j.jconhyd.2012.08.004 Google Scholar
  67. 67.
    Liang, Z., RONG, H.E., Chen, Q., Xu, X., SATO, J.: Fractal generation of char pores through random walk. Combust. Sci. Technol. 179(3), 637 (2007)Google Scholar
  68. 68.
    Wang, M.R., Wang, J.K., Pan, N., Chen, S.Y.: Mesoscopic predictions of the effective thermal conductivity for microscale random porous media. Phys. Rev. E 75, 036702 (2007).  https://doi.org/10.1103/PhysRevE.75.036702 Google Scholar
  69. 69.
    Lanning, L.M., Ford, R.M.: Glass micromodel study of bacterial dispersion in spatially periodic porous networks. Biotechnol. Bioeng. 78(5), 556 (2002).  https://doi.org/10.1002/bit.10236 Google Scholar
  70. 70.
    Wu, M., Xiao, F., Johnson-Paben, R.M., Retterer, S.T., Yin, X., Neeves, K.B.: Single- and two-phase flow in microfluidic porous media analogs based on Voronoi tessellation. Lab Chip 12(2), 253 (2012).  https://doi.org/10.1039/c1lc20838a Google Scholar
  71. 71.
    Chatzis, I., Dullien, F.A.L.: Application of the percolation theory for the simulation of penetration into porous media of a non wetting fluid and the prediction of the relative permeability curve. Rev. Inst. Fr. Pét. 37(2), 183 (1982).  https://doi.org/10.2516/ogst:1982011 Google Scholar
  72. 72.
    Dullien, F.A.L.: Porous Media: Fluid Transport and Pore Structure, 2nd edn. Academic Press, New York (1992).  https://doi.org/10.1016/C2009-0-26184-8 Google Scholar
  73. 73.
    Ayón, A.A., Braff, R., Lin, C.C., Sawin, H.H., Schmidt, M.A.: Characterization of a time multiplexed inductively coupled plasma etcher. J. Electrochem. Soc. 146(1), 339 (1999).  https://doi.org/10.1149/1.1391611 Google Scholar
  74. 74.
    Gostick, J.T.: Random pore network modeling of fibrous PEMFC gas diffusion media using voronoi and delaunay tessellations. J. Electrochem. Soc. 160(8), F731 (2013).  https://doi.org/10.1149/2.009308jes Google Scholar
  75. 75.
    Jin, Y., Li, X., Zhao, M.Y., Liu, X.H., Li, H.: A mathematical model of fluid flow in tight porous media based on fractal assumptions. Int. J. Heat Mass Transf. 108(Part A), 1078 (2017).  https://doi.org/10.1016/j.ijheatmasstransfer.2016.12.096 Google Scholar
  76. 76.
    Moon, F.C.M.: Chaotic and Fractal Dynamics: An Introduction for Applied Scientists and Engineers. Wiley, New York (1992)Google Scholar
  77. 77.
    Hilborn, R.: Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers, 2nd edn. Oxford University Press, Oxford (2004)zbMATHGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Resources and EnvironmentHenan Polytechnic UniversityJiaozuoChina
  2. 2.Collaborative Innovation Center of Coalbed Methane and Shale Gas for Central Plains Economic RegionJiaozuoChina

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