Transport in Porous Media

, Volume 118, Issue 2, pp 301–326 | Cite as

Fluid Flow Through Single Fractures With Directional Shear Dislocations

  • Long Cheng
  • Guan RongEmail author
  • Jie Yang
  • Chuangbing Zhou


This paper numerically investigates the fluid flow behavior through single fractures with directional shear dislocations. Synthetic fractures are generated with directional shear dislocations, and the lattice Boltzmann method is used to simulate the fracture flow. With an ignorance of tortuosity effect, a notable overestimation of hydraulic conductivity is observed when the simplified local cubic law is used. During the closure process, the decreasing rate of conductivity is found to be highly related to the roughness of fractures. The conductivity of smoother fractures decreases faster than that of rougher fractures. By conducting simulations on fractures with a constant shear displacement, the effective conductivity is found to vary with the shear directions. The results show that the conductivity of rougher fractures is less sensitive to the shear directions than that of smoother fractures. As fracture surfaces come into contact, a sharp decrease in effective conductivity is observed and the decreasing trend flattens as the contact ratio continues to increase. A new model is proposed based on the bottleneck model to predict the conductivity of sheared fractures. By integrating the tortuosity and channeling effects into the original model, the proposed new model shows a better performance in predicting the conductivity, especially for fractures with rougher surfaces.


Fracture Flow Roughness Lattice Boltzmann method (LBM) Tortuosity 

List of symbols


Fractal dimension


Conductivity in x-axis direction



\(\nabla p\)

Pressure gradient


Volumetric flow rate


Reynolds number

\(\mathbf u \)

Velocity vector

\({\mu }\)

Dynamic viscosity


Constant separation between parallel plates

\({\rho }\)

Density of fluid

\({{\beta }}\)

Inclination angle between the macroscopic flow direction and shear direction


Width of the fracture

Pin, Pout

Constant pressure set on the inlet and outlet of the fracture, respectively

\(\Delta {P}\)

Driving pressure


Local aperture at point (xy)

\(\langle \,\,\,\rangle \)

Arithmetic mean

\(\left\langle {h(x,y)} \right\rangle _0\)

Initial aperture with only a single contact point between the opposing surfaces

\(\left\langle {h(x,y)^{3}} \right\rangle \)

Arithmetic mean local conductivity

\(\left\langle {h(x)^{-3}} \right\rangle ^{-1}\)

Nominal conductivity in x-axis direction

\({R}_{0}\), \({{\delta }}\) and \({{\lambda }}\)

Geometrical parameters of sinusoidal constricted pipes


\({{n}}_{{c}}\) is the number of contact points, and \({{N}}_{{c}}\) is the total number of data points

\({\eta }\)

Contact ratio


Real conductivity of fracture

\(\left\langle {\tau _x^2} \right\rangle \)

Hydraulic tortuosity parameter in x-axis

\(\chi _{\ell _1 }^2 \)

In-plane tortuosity parameter of the most constrictive path \(\ell _1 \)

\(\left\langle {\tau _{\ell _2 }^2 } \right\rangle \)

Averaged mean geometrical tortuosity for preferential channels

\(\Delta \)

Width of the bottleneck region

\(\mathrm{d}\vec {l}, \vec {e}_y \)

Direction vector of the most constrictive path and unit vector in y-axis direction, respectively

\(\delta x\), \(\delta y\)

Lattice unit length in x-axis and y-axis directions


The contribution fraction of bottleneck structure to the pressure drop.



The authors are grateful for the National Natural Science Foundation of China (Grant No. 51579189), the National Basic Research Program of China (“973” Program, Grant No. 2011CB013501), and the Fundamental Research Funds for the Central Universities (2042016kf0171) for providing financial support.


  1. Auradou, H., et al.: Permeability anisotropy induced by the shear displacement of rough fracture walls. Water Resour. Res. 41(9), W09423 (2005)CrossRefGoogle Scholar
  2. Brown, S.R.: Fluid flow through rock joints: the effect of surface roughness. J. Geophys. Res. Solid Earth 92(B2), 1337–1347 (1987)CrossRefGoogle Scholar
  3. Brown, S.R.: Simple mathematical model of a rough fracture. J. Geophys. Res. Solid Earth 100(B4), 5941–5952 (1995)CrossRefGoogle Scholar
  4. Crandall, D., et al.: Numerical simulations examining the relationship between wall-roughness and fluid flow in rock fractures. Int. J. Rock Mech. Min. Sci. 47(5), 784–796 (2010)CrossRefGoogle Scholar
  5. Deiber, J.A., Peirotti, M.B., Bortolozzi, R.A., Durelli, R.J.: Flow of Newtonian fluids though sinusoidally constricted tubes. Numerical and experimental results. Chem. Eng. Commun. 117, 241–262 (1992)CrossRefGoogle Scholar
  6. Develi, K., Babadagli, T.: Experimental and visual analysis of single-phase flow through rough fracture replicas. Int. J. Rock Mech. Min. Sci. 73, 139–155 (2015)Google Scholar
  7. Durham, W.B., Bonner, B.P.: Self-propping and fluid flow in slightly offset joints at high effective pressures. J. Geophys. Res. Solid Earth 99(B5), 9391–9399 (1994)CrossRefGoogle Scholar
  8. Gavrilenko, P., Gueguen, Y.: Pressure dependence of permeability: a model for cracked rocks. Geophys. J. Int. 98(1), 159–172 (1989)CrossRefGoogle Scholar
  9. Glover, P.W.J., Matsuki, K., Hikima, R., Hayashi, K.: Synthetic rough fractures in rocks. J. Geophys. Res. Solid Earth 103(B5), 9609–9620 (1998)CrossRefGoogle Scholar
  10. He, X., Luo, L.-S.: Lattice Boltzmann model for the incompressible Navier–Stokes equation. J. Stat. Phys. 88(3), 927–944 (1997)CrossRefGoogle Scholar
  11. Kim, I., Lindquist, W.B., Durham, W.B.: Fracture flow simulation using a finite-difference lattice Boltzmann method. Phys. Rev. E 67(4), 046708 (2003)CrossRefGoogle Scholar
  12. Kishida, K., et al.: Estimation of fracture flow considering the inhomogeneous structure of single rock fractures. Soils Found. 53(1), 105–116 (2013)CrossRefGoogle Scholar
  13. Koyama, T., et al.: Shear induced anisotropy and heterogeneity of fluid flow in a single rock fracture by translational and rotary shear displacements—a numerical study. Int. J. Rock Mech. Min. Sci. 41(3), 360–365 (2004)CrossRefGoogle Scholar
  14. Koyama, T., et al.: Numerical simulation of shear-induced flow anisotropy and scale-dependent aperture and transmissivity evolution of rock fracture replicas. Int. J. Rock Mech. Min. Sci. 43(1), 89–106 (2006)CrossRefGoogle Scholar
  15. Lee, S.H., Lee, K.K., Yeo, I.W.: Assessment of the validity of Stokes and Reynolds equations for fluid flow rough walled fracture with flow imaging. Geophys. Res. Lett. 41(13), 4578–4585 (2014)CrossRefGoogle Scholar
  16. Llewellin, E.W.: LBflow: an extensible lattice Boltzmann framework for the simulation of geophysical flows. Part I: theory and implementation. Comput. Geosci. 36(2), 115–122 (2010)CrossRefGoogle Scholar
  17. Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, New York (1982)Google Scholar
  18. Matsuki, K., et al.: Size effect on aperture and permeability of a fracture as estimated in large synthetic fractures. Int. J. Rock Mech. Min. Sci. 43, 726–755 (2006)CrossRefGoogle Scholar
  19. Méheust, Y., Schmittbuhl, J.: Geometrical heterogeneities and permeability anisotropy of rough fractures. J. Geophys. Res. Solid Earth 106(B2), 2089–2102 (2001)CrossRefGoogle Scholar
  20. Murata, S., Saito, T.: Estimation of tortuosity of fluid flow through a single fracture. J. Can. Pet. Technol. 42, 12 (2003)CrossRefGoogle Scholar
  21. Ogilvie, S.R., et al.: Fluid flow through rough fractures in rocks. II: a new matching model for rough rock fractures. Earth Planet. Sci. Lett. 241(3–4), 454–465 (2006)CrossRefGoogle Scholar
  22. Oron, A.P., Berkowitz, B.: Flow in rock fractures: the local cubic law assumption reexamined. Water Resour. Res. 34(11), 2811–2825 (1998)CrossRefGoogle Scholar
  23. Pan, P.-Z., et al.: Modelling fluid flow through a single fracture with different contacts using cellular automata. Comput. Geotech. 38(8), 959–969 (2011)CrossRefGoogle Scholar
  24. Qian, Y.H., et al.: Lattice BGK models for Navier–Stokes equation. EPL (Europhys. Lett.) 17(6), 479 (1992)CrossRefGoogle Scholar
  25. Qian, J., et al.: Experimental evidence of scale-dependent hydraulic conductivity for fully developed turbulent flow in a single fracture. J. Hydrol. 339, 206–215 (2007)CrossRefGoogle Scholar
  26. Rong, G., et al.: Experimental study of flow characteristics in non-mated rock fractures considering 3D definition of fracture surfaces. Eng. Geol. 220, 152–163 (2017)CrossRefGoogle Scholar
  27. Scesi, L., Gattinoni, P.: Roughness control on hydraulic conductivity in fractured rocks. Hydrogeol. J. 15(2), 201–211 (2007)CrossRefGoogle Scholar
  28. Skjetne, E., Hansen, A., Gudmundsson, J.S.: High-velocity flow in a rough fracture. J. Fluid Mech. 383, 1–28 (1999)CrossRefGoogle Scholar
  29. Sisavath, S., et al.: Creeping flow through a pipe of varying radius. Phys. Fluids 13(10), 2762–2772 (2001)CrossRefGoogle Scholar
  30. Talon, L., et al.: Permeability estimates of self-affine fracture faults based on generalization of the bottleneck concept. Water Resour. Res. 46(7), W07601 (2010)CrossRefGoogle Scholar
  31. Tsang, Y.W.: The effect of tortuosity on fluid flow through a single fracture. Water Resour. Res. 20(9), 1209–1215 (1984)Google Scholar
  32. Tzelepis, V., et al.: Experimental investigation of flow behavior in smooth and rough artificial fractures. J. Hydrol. 521, 108–118 (2015)CrossRefGoogle Scholar
  33. Walsh, J.B., Brace, W.F.: The effect of pressure on porosity and the transport properties of rock. J. Geophys. Res. Solid Earth 89(B11), 9425–9431 (1984)CrossRefGoogle Scholar
  34. Wang, M., et al.: Influence of surface roughness on nonlinear flow behaviors in 3D self-affine rough fractures: Lattice Boltzmann simulations. Adv. Water Resour. 96, 373–388 (2016)CrossRefGoogle Scholar
  35. Watanabe, N., et al.: Determination of aperture structure and fluid flow in a rock fracture by high-resolution numerical modeling on the basis of a flow-through experiment under confining pressure. Water Resour. Res. 44(6), 1–11 (2008)CrossRefGoogle Scholar
  36. Xiao, W., Xia, C., Wei, W., Bian, Y.: Combined effect of tortuosity and surface roughness on estimation of flow rate through a single rough joint. J. Geophys. Eng. 10(4), 045015 (2013)CrossRefGoogle Scholar
  37. Xie, H., Chen, Z.: Fractal geometry and fracture of rock. Acta Mech. Sin. 4(3), 255–264 (1988)CrossRefGoogle Scholar
  38. Yeo, I.W., et al.: Effect of shear displacement on the aperture and permeability of a rock fracture. Int. J. Rock Mech. Min. Sci. 35(8), 1051–1070 (1998)CrossRefGoogle Scholar
  39. Zimmerman, R.W., et al.: The effect of contact area on the permeability of fractures. J. Hydrol. 139(1), 79–96 (1992)CrossRefGoogle Scholar
  40. Zimmerman, R., Bodvarsson, G.: Hydraulic conductivity of rock fractures. Transp. Porous Media 23(1), 1–30 (1996)CrossRefGoogle Scholar
  41. Zou, L., Jing, L., Cvetkovic, V.: Roughness decomposition and nonlinear fluid flow in a single rock fracture. Int. J. Rock Mech. Min. Sci. 75, 102–118 (2015)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  • Long Cheng
    • 1
    • 2
  • Guan Rong
    • 1
    • 2
    Email author
  • Jie Yang
    • 1
    • 2
  • Chuangbing Zhou
    • 1
    • 2
  1. 1.State Key Laboratory of Water Resources and Hydropower Engineering ScienceWuhan UniversityWuhanChina
  2. 2.Key Laboratory of Rock Mechanics in Hydraulic Structural Engineering, Ministry of EducationWuhan UniversityWuhanChina

Personalised recommendations