Scaling of Fracture Patterns in Three Deep Boreholes and Implications for Constraining Fractal Discrete Fracture Network Models

  • Mohammad Javad Afshari MoeinEmail author
  • Benoît Valley
  • Keith F. Evans
Original Paper


We present a methodology for generating fractal fracture networks in one, two and three dimensions that respects the dual power-law model, in which the scaling characteristics are set by the two independent parameters: (1) the correlation dimension that pertains to separation of fracture centers, and (2) the length exponent that governs the distribution of fracture lengths. Synthetic fracture distributions were generated to evaluate the stereological relationships between the scaling parameters of 2D and 3D networks, and the scaling of fracture intersection points along a scanline through the network. The results showed that it is not possible to estimate the 2D and 3D fractal scaling parameters of the correlation dimension from the 1D correlation dimension of fracture spacing from scanlines through the network, even if the length exponent is known a priori. Synthetic 1D distributions of fracture spacing of known correlation dimension were used as a benchmark to test the consistency of estimates of fractal dimension derived from box-counting, two-point correlation, and power-law fitting. The results showed that the correlation dimension obtained from the two-point correlation method provided the most stable and reliable estimate of the fractal dimension of fractures on 1D scanlines or boreholes. Application of the two-point correlation function to the observed fracture distributions along three deep boreholes in crystalline rock at Basel, Switzerland and Soultz-sous-Forêts, France showed that the distribution was fractal over more than two orders of magnitude in scale, and in all cases the fractal dimensions was in the range 0.86–0.88. Similar results were obtained for fracture sets of common orientation within the wells, although the fractal dimension ranged between 0.65 and 0.75. This constitutes strong evidence that fracturing in rock masses penetrated by the wells follows a fractal organization.


Discrete fracture networks (DFN) Fracture spacing Power-law scaling Deep boreholes Fractal geometry 

List of symbols


Correlation dimension


Fracture length exponent


Distance between fracture centers (m)


Fracture length (m)


Correlation function

\({N_{\text{p}}}\left( r \right)\)

Number of pairs of fractures whose center-to-center distance is less than r


Total number of fractures


Domain length in 1D, 2D and 3D (m)


Number of equal-sized sub-domains in a Multiplicative Cascade process


Minimum fracture length (m) in DFN generation


The ratio of sub-domain side length to the domain side length in Multiplicative Cascade process


Probability of having a fracture in a sub-domain in Multiplicative Cascade process


Fracture assignment vector


Cumulative probability density vector


Ruler length in 1D box-counting technique


Number of rulers of length t containing at least one fracture




Fracture spacing


Number of spacings greater than or equal to a specific spacing s


Power-law exponent of fracture spacing


Fischer coefficient



The research leading to these results received funding from the European Community’s Seventh Framework Program under Grant agreement no. 608553 (Project IMAGE). We would like thank Simon Löw for his continuous support and constructive comments during this project. We are also thankful to the anonymous reviewers, associate editor and the editor for their valuable suggestions that led to improvements of the manuscript.


  1. Afshari Moein MJ (2018) Linkage between fracture network, stress heterogeneities and induced seismicity in deep geothermal reservoirs (Doctoral Dissertation). Zurich: ETH Zurich.
  2. Afshari Moein MJ, Somogyvári M, Valley B, Jalali M, Loew S, Bayer P (2018a) Fracture network characterization using stress-based tomography. J Geophys Res Solid Earth,123.
  3. Afshari Moein MJ, Tormann T, Valley B, Wiemer S (2018b) Maximum magnitude forecast in hydraulic stimulation based on clustering and size distribution of early microseismicity. Geophys Res Lett 45:6907–6917. CrossRefGoogle Scholar
  4. Alghalandis YF, Dowd PA, Xu C (2013) The RANSAC method for generating fracture networks from micro-seismic event data. Math Geosci 45:207–224CrossRefGoogle Scholar
  5. Alghalandis YF, Dowd PA, Xu C (2015) connectivity field: a measure for characterising fracture networks. Math Geosci 47:63–83. CrossRefGoogle Scholar
  6. Allegre CJ, Lemouel JL, Provost A (1982) Scaling rules in rock fracture and possible implications for earthquake prediction. Nature 297:47–49. doi: CrossRefGoogle Scholar
  7. Baecher GB, Lanney NA (1978) Trace length biases in joint surveys. In: Proceedings of the 19th U. S. Symposium on Rock Mechanics, vol 1, pp 56–65Google Scholar
  8. Baghbanan A, Jing LR (2007) Hydraulic properties of fractured rock masses with correlated fracture length and aperture. Int J Rock Mech Min Sci 44:704–719. CrossRefGoogle Scholar
  9. Bak P, Tang C, Wiesenfeld K (1988) Self-organized criticality. Phys Rev A Gen Phys 38:364–374CrossRefGoogle Scholar
  10. Barton CC (1995) Fractal analysis of scaling and spatial clustering of fractures. Fractals in the earth sciences. Springer, Berlin, pp 141–178Google Scholar
  11. Barton CA, Zoback MD (1992) Self-similar distribution and properties of macroscopic fractures at depth in crystalline rock in the Cajon Pass Scientific Drill Hole. J Geophys Res Solid Earth (1978–2012) 97:5181–5200CrossRefGoogle Scholar
  12. Berkowitz B, Hadad A (1997) Fractal and multifractal measures of natural and synthetic fracture networks. J Geophys Res Solid Earth (1978–2012) 102(B6):12205–12218. CrossRefGoogle Scholar
  13. Boadu FK, Long LT (1994) The fractal character of fracture spacing and RQD. Int J Rock Mech Min Sci Geomech Abstr 31(2):127 ± 134CrossRefGoogle Scholar
  14. Bonnet E, Bour O, Odling NE, Davy P, Main I, Cowie P, Berkowitz B (2001) Scaling of fracture systems in geological media. Rev Geophys 39:347–383. CrossRefGoogle Scholar
  15. Bour O, Davy P (1997) Connectivity of random fault networks following a power law fault length distribution. Water Resour Res 33:1567–1583CrossRefGoogle Scholar
  16. Bour O, Davy P (1999) Clustering and size distributions of fault patterns: Theory and measurements. Geophys Res Lett 26(13):2001–2004CrossRefGoogle Scholar
  17. Bour O, Davy P, Darcel C, Odling N (2002) A statistical scaling model for fracture network geometry, with validation on a multiscale mapping of a joint network (Hornelen Basin, Norway). J Geophys Res Solid Earth 107Google Scholar
  18. Chilès J (1988) Fractal and geostatistical methods for modeling of a fracture network. Math Geol 20:631–654CrossRefGoogle Scholar
  19. Clemo T, Smith L (1997) A hierarchical model for solute transport in fractured media. Water Resour Res 33:1763–1783. CrossRefGoogle Scholar
  20. Cowie P, Knipe R, Main I (1996) Scaling laws for fault and fracture populations-Analyses and applications-Introduction. J Struct Geol 18:R5–R11Google Scholar
  21. Darcel C (2002) Corrélations dans les réseaux de fractures: caractérisation et conséquences sur les propriétés hydrauliques. Doctoral Dissertation. Université Rennes 1. RennesGoogle Scholar
  22. Darcel C, Bour O, Davy P (2003a) Cross-correlation between length and position in real fracture networks. Geophys Res Lett 30(12):1650. CrossRefGoogle Scholar
  23. Darcel C, Bour O, Davy P (2003b) Stereological analysis of fractal fracture networks. J Geophys Res. CrossRefGoogle Scholar
  24. Darcel C, Bour O, Davy P, De Dreuzy JR (2003c) Connectivity properties of two-dimensional fracture networks with stochastic fractal correlation. Water Resour Res 39(10)Google Scholar
  25. Davy P (1993) On the frequency-length distribution of the San Andreas fault system. J Geophys Res Solid Earth 98:12141–12151CrossRefGoogle Scholar
  26. Davy P, Sornette A, Sornette D (1990) Some consequences of a proposed fractal nature of continental faulting. Nature 348:56–58CrossRefGoogle Scholar
  27. Davy, Le Goc R, Darcel C, Bour O, De Dreuzy J-R, Munier R (2010) A likely universal model of fracture scaling and its consequence for crustal hydromechanics. J Geophys Res Solid Earth 115Google Scholar
  28. Davy P, Darcel C, Le Goc R, Munier R, Selroos JO, Ivars MD (2018) DFN, why, how and what for, concepts, theories and issues. In: 2nd international discrete fracture network engineering conference. American Rock Mechanics AssociationGoogle Scholar
  29. Day-Lewis AD (2008) Characterization and modeling of in situ stress heterogeneity. (PhD thesis), Stanford University, California, USAGoogle Scholar
  30. de Dreuzy J-R, Davy P, Bour O (2001) Hydraulic properties of two-dimensional random fracture networks following a power law length distribution: 1. Effective connectivity. Water Resour Res 37:2065–2078CrossRefGoogle Scholar
  31. de Dreuzy JR, Davy P, Bour O (2002) Hydraulic properties of two-dimensional random fracture networks following power law distributions of length and aperture. Water Resour Res 38Google Scholar
  32. Deere DU, Deere DW (1988) The rock quality designation (RQD) index in practice. In: Kirakaldie L (ed) Rock classification systems for engineering purposes. ASTM special publication 984. American Society for Testing Materials, Philadelphia, pp 91–101CrossRefGoogle Scholar
  33. Dershowitz WS, Einstein HH (1988) Characterizing rock joint geometry with joint system models. Rock Mech Rock Eng 21:21–51. CrossRefGoogle Scholar
  34. Dershowitz W, Lee G, Geier J, Hitchcock S, La Pointe P (1993) FracMan user documentation. Golder Associates Inc, SeattleGoogle Scholar
  35. Dezayes C, Genter A, Valley B (2010) Structure of the low permeable naturally fractured geothermal reservoir at Soultz. CR Geosci 342(7–8):517–530CrossRefGoogle Scholar
  36. Dèzes P, Schmid SM, Ziegler PA (2004) Evolution of the European Cenozoic Rift System: interaction of the Alpine and Pyrenean orogens with their foreland lithosphere. Tectonophysics 389(1–2):1–33CrossRefGoogle Scholar
  37. Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc Lond A 241(1226):376–396CrossRefGoogle Scholar
  38. Evans KF (2005) Permeability creation and damage due to massive fluid injections into granite at 3.5 km at Soultz: 2. Critical stress and fracture strength. J Geophys Res Solid Earth 110:B4Google Scholar
  39. Evans KF, Genter A, Sausse J (2005) Permeability creation and damage due to massive fluid injections into granite at 3.5 km at Soultz: 1. Borehole observations. J Geophys Res Solid Earth 110:B4Google Scholar
  40. Genter A, Evans KF, Cuenot N, Fritsch D, Sanjuan B (2010) Contribution of the exploration of deep crystalline fractured reservoir of Soultz to the knowledge of enhanced geothermal systems (EGS). CR Geosci 342:502–516. CrossRefGoogle Scholar
  41. Gillespie P, Howard C, Walsh J, Watterson J (1993) Measurement and characterisation of spatial distributions of fractures. Tectonophysics 226:113–141CrossRefGoogle Scholar
  42. Häring MO, Schanz U, Ladner F, Dyer BC (2008) Characterisation of the Basel 1 enhanced geothermal system. Geothermics 37(5):469–495CrossRefGoogle Scholar
  43. Harthong B, Scholtès L, Donzé F-V (2012) Strength characterization of rock masses, using a coupled DEM–DFN model. Geophys J Int 191:467–480CrossRefGoogle Scholar
  44. Hentschel H, Procaccia I (1983) The infinite number of generalized dimensions of fractals and strange attractors. Physica D 8:435–444CrossRefGoogle Scholar
  45. Hirata T, Satoh T, Ito K (1987) Fractal structure of spatial distribution of microfracturing in rock. Geophys J Int 90:369–374. CrossRefGoogle Scholar
  46. Kim TH (2007) Fracture characterization and estimation of fracture porosity of naturally fractured reservoirs with no matrix porosity using stochastic fractal models. PhD Thesis. Texas A&M University, TexasGoogle Scholar
  47. La Pointe P (1988) A method to characterize fracture density and connectivity through fractal geometry. Int J Rock Mech Min Sci Geomech, vol 6. Elsevier, pp 421–429Google Scholar
  48. Ledésert B, Dubois J, Genter A, Meunier A (1993) Fractal analysis of fractures applied to Soultz-sous-Forêts hot dry rock geothermal program. J Volcanol Geoth Res 57:1–17CrossRefGoogle Scholar
  49. Lei LQ, Gao K (2018) Correlation between fracture network properties and stress variability in geological media. Geophys Res Lett 45(9):3994–4006CrossRefGoogle Scholar
  50. Lei Q, Latham JP, Tsang CF, Xiang J, Lang P (2015) A new approach to upscaling fracture network models while preserving geostatistical and geomechanical characteristics. J Geophys Res Solid Earth 120:4784–4807CrossRefGoogle Scholar
  51. Lei Q, Latham JP, Tsang CF (2017) The use of discrete fracture networks for modelling coupled geomechanical and hydrological behaviour of fractured rocks. Comput Geotech 85:151–176CrossRefGoogle Scholar
  52. Lovejoy S, Schertzer D (1986) Scale invariance, symmetries, fractals, and stochastic simulations of atmospheric phenomena. Bull Am Meteorol Soc 67:21–32CrossRefGoogle Scholar
  53. Manning CE (1994) Fractal clustering of metamorphic veins. Geology 22:335–338CrossRefGoogle Scholar
  54. Meakin P (1991) Fractal aggregates in geophysics. Rev Geophys 29:317–354CrossRefGoogle Scholar
  55. Merceron T, Velde B (1991) Application of Cantor’s method for fractal analysis of fractures in the Toyoha Mine, Hokkaido, Japan. J Geophys Res Solid Earth 96:16641–16650CrossRefGoogle Scholar
  56. Moein MJA, Valley B, Ziegler M (2016) Preliminary fractal analysis of fracture spacing inferred from an acoustic televiewer log run in the Basel-1 geothermal well (Switzerland). Rock mechanics and rock engineering: from the past to the future. CRC Press, London, pp 103–1107Google Scholar
  57. Moller J, Waagepetersen RP (2003) Statistical inference and simulation for spatial point processes. Monographs on statistics and applied probability, vol 100. Chapman and Hall, UKGoogle Scholar
  58. Odling (1992) Network properties of a two-dimensional natural fracture pattern. Pure Appl Geophys 138:95–114CrossRefGoogle Scholar
  59. Odling NE, Gillespie P, Bourgine B, Castaing C, Chiles JP, Christensen NP, Fillion E, Genter A, Olsen C, Thrane L, Trice R (1999) Variations in fracture system geometry and their implications for fluid flow in fractured hydrocarbon reservoirs. Pet Geosci 5:373–384CrossRefGoogle Scholar
  60. Pollard D, Segall P (1987) Theoretical displacements and stresses near fractures in rock: with applications to faults, joints, veins, dikes, and solution surfaces. Fract Mech Rock 277:277–349CrossRefGoogle Scholar
  61. Power WL, Tullis TE (1991) Euclidean and fractal models for the description of rock surface roughness. J Geophys Res Solid Earth 96:415–424CrossRefGoogle Scholar
  62. Renshaw CE (1999) Connectivity of joint networks with power law length distributions. Water Resour Res 35:2661–2670CrossRefGoogle Scholar
  63. Roy A, Perfect E, Dunne WM, McKay LD (2014) A technique for revealing scale-dependent patterns in fracture spacing data. J Geophys Res Solid Earth 119:5979–5986CrossRefGoogle Scholar
  64. Scholz CH (2002) The mechanics of earthquakes and faulting. Cambridge University press, New YorkCrossRefGoogle Scholar
  65. Sornette D (2006) Critical phenomena in natural sciences: chaos, fractals, selforganization and disorder: concepts and tools. Springer, BerlinGoogle Scholar
  66. Sornette D, Davy P, Sornette A (1990) Structuration of the lithosphere in plate tectonics as a self-organized critical phenomenon. J Geophys Res Solid Earth 95:17353–17361CrossRefGoogle Scholar
  67. Spyropoulos C, Scholz CH, Shaw BE (2002) Transition regimes for growing crack populations. Phys Rev E 65:056105CrossRefGoogle Scholar
  68. Valley B (2007) The relation between natural fracturing and stress heterogeneities in deep-seated crystalline rocks at Soultz-sous-Forêts (France). Doctoral Dissertation. ETH Zürich, ZürichGoogle Scholar
  69. Valley B, Evans K (2014) Preliminary assessment of the scaling relationships of in-situ stress orientation variations indicated by wellbore failure data. In: The 2014 ISRM European Rock Mechanics Symposium (EUROCK 2014). rock engineering and rock mechanics: structure in and on rock masses. CRC Press, Vigo, pp 463–468Google Scholar
  70. Valley B, Dezayes C, Genter A (2007) Multiscale fracturing in the Soultz-sous-Forêts basement from borehole image analyses. In: Proceedings EHDRA scientific conference, 28. Soultz-sous-Forêts, FranceGoogle Scholar
  71. Valley B, Jalali MR, Ziegler M, Evans KF (2014) Constraining DFN characteristics for deep geothermal projects by considering the effects of fractures on stress variability. In: International discrete fracture network engineering conference DFNE 2014. Vancouver, BC, CanadaGoogle Scholar
  72. Velde B, Dubois J, Touchard G, Badri A (1990) Fractal analysis of fractures in rocks: the Cantor’s Dust method. Tectonophysics 179:345–352CrossRefGoogle Scholar
  73. Verscheure M, Fourno A, Chilès J-P (2012) Joint inversion of fracture model properties for CO2 storage monitoring or oil recovery history matching. Oil Gas Sci Technol Revue d’IFP Energies nouvelles 67:221–235CrossRefGoogle Scholar
  74. Watanabe K, Takahashi H (1993) Fractal characterization of subsurface fracture network for geothermal energy extraction system. In: Proceedings, eighteenth workshop on geothermal reservoir engineering, Stanford University, Stanford, CA. Report No. SGP-TR-145-17Google Scholar
  75. Ziegler M, Valley B, Evans F (2015) Characterisation of nature fractures and fracture zones of the Basel EGS reservoir inferred from geophysical logging of the Basel-1 well. In: Proceedings world geothermal congress 2015, Melbourne, AustraliaGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Engineering Geology, Department of Earth Sciences, Geological InstituteETH ZürichZurichSwitzerland
  2. 2.Center for Hydrogeology and GeothermicsUniversity of NeuchâtelNeuchâtelSwitzerland
  3. 3.Geothermal Energy and Geofluids, Department of Earth Sciences, Institute of GeophysicsETH ZürichZurichSwitzerland

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