An Experimental Study on Scale-Dependent Velocity and Anisotropy in Fractured Media Based on Artificial Rocks with Controlled Fracture Geometries


The scale of the fractures can vary, making the seismic velocity and anisotropy substantially scale dependent. Two mechanisms of the scale-dependent phenomenon may be considered: scattering and wave-induced fluid flow. In this study, we measure the scale-dependent velocity and anisotropy effects through laboratory experiments on porous and non-porous artificial rocks containing aligned fractures. This allows us to isolate the effects of these two mechanisms for the first time, yielding some insights into the scale-dependent phenomenon. For short-wavelength waves, scattering dominates with less wave-induced fluid flow effects. For intermediate- and long-wavelength waves, the P-wave is strongly scale dependent mainly due to wave-induced fluid flow mechanism, and the slow shear-wave is also strongly scale dependent but due to both scattering and wave-induced fluid flow. However, the fast shear-wave is almost scale independent. Moreover, a multi-scale equivalent medium theory can model the P-wave propagation accurately.

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\({\uplambda }_{{\text{P}}}\). \({\uplambda }_{{\text{s}}}\) :

Wavelength of P- and S-waves

d :

Fracture scale length

\({\uptau }_{{\text{m}}}\) :

Relaxation time at grain scale

\(\uptau_{{\text{f}}}\) :

Relaxation time at fracture scale

\(\eta\) :

Fluid viscosity

\(k\) :


\(c_{{\text{v}}}\) :

Volume of the individual cracks

\(K_{{\text{c}}}\) :

Inverse of the crack space compressibility

\(c_{1}\) :

Number of connections to other elements of the pore space

\(\sigma_{{\text{c}}}\) :

Critical stress

\({\upmu }\) :

Shear-wave modulus

r :

Aspect ratio of the crack


Poisson’s ratio of the isotropic rock matrix

\(\varsigma\),\({ }a_{{\text{f}}}\) :

Grain size and fracture size

\(\phi_{p}\), \(\epsilon_{{\text{c}}}\), \(\epsilon_{f}\) :

Porosity, crack density and fracture deity

\(C_{{{\text{ijkl}}}}\) :

Stiffness tensor of fractured rock

\(C_{{{\text{ijkl}}}}^{0}\) :

Elastic properties of the unfractured porous rock

\({\text{C}}_{{{\text{ijkl}}}}^{1}\), \({\text{C}}_{{{\text{ijkl}}}}^{2}\), \({\text{C}}_{{{\text{ijkl}}}}^{3}\) :

Contributions from the pores, micro-scale cracks and meso-scale fractures, respectively

\(\lambda^{0}\), \(\mu^{0}\) :

Lame parameters of porous background matrix


  1. Al-Harrasi OH, Kendall J-M, Chapman M (2011) Fracture characterization using frequency-dependent shear wave anisotropy analysis of microseismic data. Geophys J Int 185(2):1059–1070

    Google Scholar 

  2. Backus GE (1962) Long-wave elastic anisotropy produced by horizontal layering. J Geophys Res 67(11):4427–4440

    Google Scholar 

  3. Baird A, Kendall J, Angus D (2013) Frequency-dependent seismic anisotropy due to fractures: fluid flow versus scattering. Geophysics 78(2):WA111–WA122

    Google Scholar 

  4. Barbier M, Hamon Y, Callot J-P, Floquet M, Daniel J-M (2012) Sedimentary and diagenetic controls on the multiscale fracturing pattern of a carbonate reservoir: The Madison Formation (Sheep Mountain, Wyoming, USA). Mar Petrol Geol 29(1):50–67

    Google Scholar 

  5. Berryman JG (1980) Long-wavelength propagation in composite elastic media II. Ellipsoidal inclusions J Acoust Soc Am 68(6):1820–1831

    Google Scholar 

  6. Berryman JG (1992) Single-scattering approximations for coefficients in Biot’s equations of poroelasticity. J Acoust Soc Am 91(2):551–571

    Google Scholar 

  7. Boness NL, Zoback MD (2004) Stress-induced seismic velocity anisotropy and physical properties in the SAFOD Pilot Hole in Parkfield. CA Geophys Res Lett 31(15):L15S17

    Google Scholar 

  8. Carcione J, Kosloff D, Behle A (1991) Long-wave anisotropy in stratified media: a numerical test. Geophysics 56(2):245–254

    Google Scholar 

  9. Chapman M (2003) Frequency-dependent anisotropy due to meso-scale fractures in the presence of equant porosity. Geophys Prospect 51(5):369–379

    Google Scholar 

  10. Chapman M (2009) Modeling the effect of multiple sets of mesoscale fractures in porous rock on frequency-dependent anisotropy. Geophysics 74(6):D97–D103

    Google Scholar 

  11. Chapman M, Maultzsch S, Liu E, Li XY (2003) The effect of fluid saturation in an anisotropic multi-scale equant porosity model. J Appl Geophys 54(3):191–202

    Google Scholar 

  12. Chapman M, Zatsepin SV, Crampin S (2002) Derivation of a microstructural poroelastic model. Geophys J Int 151(2):427–451

    Google Scholar 

  13. Cheadle SP, Brown RJ, Lawton DC (1991) Orthorhombic anisotropy: A physical seismic modeling study. Geophysics 56(10):1603–1613

    Google Scholar 

  14. Cosentino L et al (2002) Integrated study of a fractured Middle East reservoir with stratiform super-K intervals-part 2: upscaling and dual-media simulation. Spe Reserv Eval Eng 5(01):24–32

    Google Scholar 

  15. 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(8):2065–2078

    Google Scholar 

  16. Dellinger J, Vernik L (1994) Do travel times in pulse-transmission experiments yield anisotropic group or phase velocities? Geophysics 59(11):1774–1779

    Google Scholar 

  17. Ding P, Di B, Wang D, Wei J, Li X (2014) P and S wave anisotropy in fractured media: experimental research using synthetic samples. J Appl Geophys 109:1–6

    Google Scholar 

  18. Ding P, Di B, Wang D, Wei J, Li X (2017) Measurements of seismic anisotropy in synthetic rocks with controlled crack geometry and different crack densities. Pure Appl Geophys 174(5):1907–1922

    Google Scholar 

  19. Ding P, Wang D, Di G, Li X (2019) Investigation of the effects of fracture orientation and saturation on the Vp/Vs ratio and their implications. Rock Mech Rock Eng 52(9):3293–3304

    Google Scholar 

  20. Ekanem AM, Wei J, Li X-Y, Chapman M, Main IG (2013) P-wave attenuation anisotropy in fractured media: a seismic physical modelling study. Geophys Prospect 61(s1):420–433

    Google Scholar 

  21. Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc Lond A 241(1226):376–396

    Google Scholar 

  22. Fan LF, Gao JW, Wu ZJ, Yang SQ, Ma GW (2018) An investigation of thermal effects on micro-properties of granite by X-ray CT technique. Appl Therm Eng 140:505–519

    Google Scholar 

  23. Fan LF, Yi XW, Ma GW (2013) Numerical manifold method (NMM) simulation of stress wave propagation through fractured rock mass. Int J Appl Mech 05(02):1350022

    Google Scholar 

  24. Hall SA, Kendall J-M, Maddock J, Fisher Q (2008) Crack density tensor inversion for analysis of changes in rock frame architecture. Geophys J Int 173(2):577–592

    Google Scholar 

  25. Hudson JA (1981) Wave speeds and attenuation of elastic waves in material containing cracks. Geophys J R Astr Soc 64(1):133–150

    Google Scholar 

  26. Hudson JA, Liu E, Crampin S (1996) The mechanical properties of materials with interconnected cracks and pores. Geophys J Int 124(1):105–112

    Google Scholar 

  27. Hudson JA, Pointer T, Liu E (2001) Effective-medium theories for fluid-saturated materials with aligned cracks. Geophys Prospect 49(5):509–522

    Google Scholar 

  28. Jiang H-Y, Chen Z-B, Zeng X-X, Lv H, Liu X (2016) Velocity calibration for microseismic event location using surface data. Pet Sci 13(2):225–236

    Google Scholar 

  29. Li JC, Li HB, Zhao J (2015a) An improved equivalent viscoelastic medium method for wave propagation across layered rock masses. Int J Rock Mech Min Sci 73:62–69

    Google Scholar 

  30. Li JC, Liu TT, Li HB, Liu YQ, Liu B, Xia X (2015b) Shear wave propagation across filled joints with the effect of interfacial shear strength. Rock Mech Rock Eng 48(4):1547–1557

    Google Scholar 

  31. Li JC, Rong LF, Li HB, Hong SN (2019) An SHPB test study on stress wave energy attenuation in jointed rock masses. Rock Mech Rock Eng 52:403–420

    Google Scholar 

  32. Liu RC, Jiang YJ, Huang N, Sugimoto S (2018) Hydraulic properties of 3D crossed rock fractures by considering anisotropic aperture distributions. Adv Geo-Energy Res 2(2):113–121

    Google Scholar 

  33. Mavko G, Mukerji T, Dvorkin J (2009) The rock physics handbook: tools for seismic analysis of porous media, 2nd edn. Cambridge University Press, California

    Google Scholar 

  34. Nicolas A, Christensen NI (1987) Formation of anisotropy in upper mantle peridotites—a review. In: Fuchs K, Froidevaux C (eds) Composition, structure and dynamics of the lithosphere-asthenosphere system. American Geophysical Union, pp 111–123. doi: 10.1029/GD016p0111

  35. Pan B-Z, Yuan M-X, Fang C-H, Liu W-B, Guo Y-H, Zhang L-H (2017) Experiments on acoustic measurement of fractured rocks and application of acoustic logging data to evaluation of fractures. Pet Sci 14(3):520–528

    Google Scholar 

  36. Pyrak-Nolte LJ, Nolte DD (1992) Frequency dependence of fracture stiffness. Geophys Res Lett 19(3):325–328

    Google Scholar 

  37. Schoenberg M, Sayers CM (1995) Seismic anisotropy of fractured rock. Geophysics 60(1):204–211

    Google Scholar 

  38. Stephenson BJ, Koopman A, Hillgartner H, McQuillan H, Bourne S, Noad JJ, Rawnsley K (2007) Structural and stratigraphic controls on fold-related fracturing in the Zagros Mountains, Iran: Implications for reservoir development. In: Lonergan L, Jolly RJH, Rawnsley K, Sanderson DJ (eds) Fractured Reservoirs, vol 270. vol 1. Geological Society, London, pp 1–21. doi: 10.1144/gsl.sp.2007.270.01.01

  39. Thomsen L (1986) Weak elastic anisotropy. Geophysics 51(10):1954–1966

    Google Scholar 

  40. Tillotson P, Chapman M, Sothcott J, Best AI, Li X-Y (2014) Pore fluid viscosity effects on P- and S-wave anisotropy in synthetic silica-cemented sandstone with aligned fractures. Geophys Prospect 62(6):1238–1252

    Google Scholar 

  41. Tran NH, Chen Z, Rahman SS (2006) Integrated conditional global optimisation for discrete fracture network modelling. Comput Geosci-UK 32(1):17–27

    Google Scholar 

  42. Verdon JP, Kendall JM (2011) Detection of multiple fracture sets using observations of shear-wave splitting in microseismic data. Geophys Prospect 59(4):593–608

    Google Scholar 

  43. Wang D, Qu SL, Zhao Q, Yin XY, Zhou F (2017) Laboratory studies of ultrasonic wave response of fractures with different lengths: Anisotropy characteristics and coda analysis. Ultrasonics 80(Supplement C):101–112

    Google Scholar 

  44. Wang Y (2011) Seismic anisotropy estimated from P-wave arrival times in crosshole measurements. Geophys J Int 184(3):1311–1316

    Google Scholar 

  45. Yousef BM, Angus DA (2016) When do fractured media become seismically anisotropic? Some implications on quantifying fracture properties. Earth Planet Sc Lett 444:150–159

    Google Scholar 

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This work is supported by the National Natural Science Fund Projects (41804105, U19B6003), the Fundamental Research Funds for the Central Universities (2462018YJRC008) and the National Science and Technology Major Project (2017ZX05018005).

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Correspondence to Xiang-Yang Li.

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Ding, P., Wang, D. & Li, X. An Experimental Study on Scale-Dependent Velocity and Anisotropy in Fractured Media Based on Artificial Rocks with Controlled Fracture Geometries. Rock Mech Rock Eng 53, 3149–3159 (2020).

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  • Dispersion
  • Fluid flow
  • Velocity anisotropy
  • Fracture