A Stochastic Study of Flow Anisotropy and Channelling in Open Rough Fractures


The quantification of fluid flow in rough fractures is of high interest for reservoir engineering, especially for deep geothermal applications. Herein, rough self-affine fractures are stochastically generated with incremental shear displacement and geometrically described by two aperture definitions, the vertical aperture \( a_{\text{vert}} \) and the effective aperture \( a_{\text{eff}} \). In order to compare their effect on fracture flow, such as anisotropy and channelling, Local Cubic Law (LCL) model-based 2D fluid flow is simulated. The particularity of this approach is the combination of a stochastic generation of self-affine fractures with a statistical analysis (560 individual realizations) of the impact of the LCL’s aperture constraint on fracture flow. The results show that aperture definition affects the quantitative interpretation of flow anisotropy and channeling as well as the aperture distribution of the fractures with shearing. Higher values of mean aperture for a given fracture are found using \( a_{\text{vert}} \), whereas the aperture standard deviation is larger with \( a_{\text{eff}} \). In addition, flow anisotropy is significantly sensitive to aperture definition for small shear displacements and shows a relative higher dispersion with \( a_{\text{eff}} \). Thus, LCL prediction models based on \( a_{\text{vert}} \) are expected to lead to higher dispersion of anisotropy results with a higher uncertainty (factor ~ 2). Realizations based on \( a_{\text{vert}} \) lead to an enhanced clustering of high flow rates for higher shearing displacements. This channeling development results in higher total flow rates for these simulations. These findings support the direct calibration of pre-existing LCL anisotropy simulations based on \( a_{\text{vert}} \) towards more representative results using \( a_{\text{eff}} \).

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\( \alpha \) :

Slope of the linear trend lines for \( I_{2} \) in terms of \( I_{1} \) (unitless)

\( \beta \) :

Intercept of the linear trend lines for \( I_{2} \) in terms of \( I_{1} \) (unitless)

\( \Delta h \) :

Height difference used to define \( p \) (in m)

\( \Delta \varvec{r} \) :

Vector distance used to define \( p \) (in m)

\( \Delta x \) :

Discretisation element length along the \( x \)-axis (in mm)

\( \Delta y \) :

Discretisation element length along the \( y \)-axis (in mm)

\( \lambda \) :

Arbitrary scaling factor used to define \( p \) (unitless)

\( \nu \) :

Kinematic viscosity of the fluid (in m2 s−1)

\( \mu \) :

Dynamic viscosity of the fluid (in kg m−1 s−1)

\( \rho \) :

Fluid density (in kg m−3)

\( \sigma^{a} \) :

Standard deviation of the aperture within a fracture (in mm)

\( \sigma_{\text{eff}}^{a} \) :

Standard deviation of the aperture within a fracture based on the effective aperture definition (in mm)

\( \sigma_{\text{vert}}^{a} \) :

Standard deviation of the aperture within a fracture based on the vertical aperture definition (in mm)

\( \sigma_{{N_{\text{offset}} }}^{I} \) :

Standard deviation of the indicator \( I \) associated with the shearing \( N_{\text{offset}} \) (unitless)

\( \tilde{\sigma }_{{N_{\text{offset}} }}^{I} \) :

Relative standard deviation for the indicator \( I \) at a given \( N_{\text{offset}} \) (unitless)

\( \tilde{\sigma }_{{N_{\text{offset}} }}^{{I_{1} }} \) :

Relative standard deviation for the indicator \( I_{1} \) at a given \( N_{\text{offset}} \) (unitless)

\( \tilde{\sigma }_{{N_{\text{offset}} }}^{{I_{2} }} \) :

Relative standard deviation for the indicator \( I_{2} \) at a given \( N_{\text{offset}} \) (unitless)

\( {\text{AF}} \) :

Anisotropy factor (unitless)

\( {\text{AF}}_{\text{eff}} \) :

Anisotropy factor based on the effective aperture definition (unitless)

\( {\text{AF}}_{\text{vert}} \) :

Anisotropy factor based on the vertical aperture definition (unitless)

\( a \) :

Arithmetic mean aperture (in m)

\( \bar{a} \) :

Mean aperture within a fracture (in mm)

\( \bar{a}_{\text{eff}} \) :

Mean aperture within a fracture based on the effective aperture definition (in mm)

\( \bar{a}_{\text{vert}} \) :

Mean aperture within a fracture based on the vertical aperture definition (in mm)

\( a_{\text{e}} \) :

Equivalent fracture aperture in the parallel-plate model (in mm)

\( a_{\text{eff}} \) :

Effective aperture (in mm)

\( a_{ {\max} } \) :

Maximum local aperture observed among the fractures (in mm)

\( a_{i} \) :

Aperture value of the \( i^{th} \) element of a considered fracture (in mm)

\( a_{\text{vert}} \) :

Vertical aperture (in mm)

\( a_{x,y} \) :

Local aperture at \( \left( {x,y} \right) \) (in m)

\( {\mathcal{C}} \) :

Set of elements belonging to one of the \( n/2 + 1 \) channels (unitless)

\( C_{{N_{\text{offset}} }}^{I} \) :

Centroid value for all indicators \( I \) of one offset (unitless)

\( {\mathbf{f}} \) :

Body force vector acting on the fluid (in kg m−2 s−2)

\( {\mathbf{g}} \) :

Acceleration of gravity vector (in kg m−2 s−2)

\( {\mathbf{grad}}\left( \varvec{m} \right) \) :

Gradient of the median \( m\left( {x,y,N_{\text{offset}} } \right) \) (unitless)

\( H \) :

Hurst roughness exponent (unitless)

\( h_{\text{bot}} \) :

Height of a fracture element on its bottom surface (in mm)

\( h_{\text{top}}^{0} \) :

Height of a fracture element on its initial top surface before shearing (in mm)

\( h_{\text{top}}^{{N_{\text{offset}} }} \) :

Height of a fracture element on its top surface after shearing of \( N_{\text{offset}} \) (in mm)

\( I_{1} \) :

Relative proportion of channel area in a fracture (unitless)

\( I_{2} \) :

Maximum channel length in a given fracture normalized by \( n \) (unitless)

\( L \) :

Scaling for the total length of the fractures (in m)

\( L_{X} \) :

Fracture width along the \( x \)-axis (in m)

\( L_{Y} \) :

Fracture width along the \( y \)-axis (in m)

\( l \) :

Scaling of the maximum surface amplitude of the fractures (in mm)

\( l_{w} \) :

Channel width (in mm)

\( m\left( {x,y,N_{\text{offset}} } \right) \) :

Height of the median between the top and bottom surface passing by the point \( \left( {x,y,N_{\text{offset}} } \right) \) (in mm)

\( \dot{m}_{x} \) :

Finite derivative of \( m\left( {x,y,N_{\text{offset}} } \right) \) along the \( x \)-axis (unitless)

\( \dot{m}_{y} \) :

Finite derivative of \( m\left( {x,y,N_{\text{offset}} } \right) \) along the \( y \)-axis (unitless)

\( N \) :

Total number of elements constituting a fracture (unitless)

\( N_{\text{offset}} \) :

Value of the shear displacement along the \( x \)-axis (in mm or in number of element)

\( n \) :

Number of elements constituting the side length of a fracture (in number of element)

\( \varvec{n}\left( \varvec{m} \right) \) :

Normal vector to the middle plane (unitless)

\( {\text{OF}} \) :

Projection of the outgoing flow perpendicular to the pressure gradient (in m3 s−1)

\( {\text{OF}}_{\parallel } \) :

Projection of the outgoing flow perpendicular to the pressure gradient parallel to the shearing direction (in m3 s−1)

\( {\text{OF}}_{ \bot } \) :

Projection of the outgoing flow perpendicular to the pressure gradient perpendicular to the shearing direction (in m3 s−1)

\( p_{\text{d}} \) :

Probability density function of the roughness distribution (in m−1)

\( p \) :

Hydrodynamic pressure (in kg m−1 s−2)

\( Q \) :

Volumetric flow rate (in m3 s−1)

\( Q1 \) :

Lower quartile value (units of the variable considered)

\( Q3 \) :

Upper quartile value (units of the variable considered)

\( \varvec{Q}_{\varvec{i}} \) :

Volumetric flow rate vector at the ith element of the fracture (in m3 s−1)

\( Q_{X} \) :

Projected volumetric flow rate on the \( x \)-axis (in m3 s−1)

\( Q_{j}^{x,y} \) :

Total volumetric flow rate for each element of the fracture identified by the coordinates \( \left( {x,y} \right) \) in the direction \( j \) where \( j = \left\{ {X,Y} \right\} \) (in m3 s−1)

\( Re \) :

Reynolds number (unitless)

\( Re_{ {\max} }^{\text{local}} \) :

Maximum local Reynolds number observed (unitless) element surface of a considered fracture (in m2)

\( U \) :

Flow velocity (in m s−1)

\( \varvec{u} \) :

Flow velocity vector (in m s−1)

\( x \) :

Fracture coordinate along the \( x \)-axis (in mm)

\( y \) :

Fracture coordinate along the \( y \)-axis (in mm.)


  1. Amitrano D, Schmittbuhl J (2002) Fracture roughness and gouge distribution of a granite shear band. J Geophys Res 107:2375. https://doi.org/10.1029/2002JB001761

    Article  Google Scholar 

  2. Auradou H, Hulin JP, Roux S (2001) Experimental study of miscible displacement fronts in rough self-affine fractures. Phys Rev E Stat Nonlin Soft Matter Phys 63:066306. https://doi.org/10.1103/PhysRevE.63.066306

    Article  Google Scholar 

  3. Auradou H, Drazer G, Hulin JP, Koplik J (2005) Permeability anisotropy induced by the shear displacement of rough fracture walls. Water Resour Res. https://doi.org/10.1029/2005WR003938

    Article  Google Scholar 

  4. Auradou H, Drazer G, Boschan A, Hulin J-P, Koplik J (2006) Flow channeling in a single fracture induced by shear displacement. Geothermics 35:576–588. https://doi.org/10.1016/j.geothermics.2006.11.004

    Article  Google Scholar 

  5. Bear J, Tsang CF, De Marsily G (1993) Flow and contaminant transport in fractured rock. Academic Press, London

    Google Scholar 

  6. Berkowitz B (2002) Characterizing flow and transport in fractured geological media: a review. Adv Water Resour 25:861–884. https://doi.org/10.1016/S0309-1708(02)00042-8

    Article  Google Scholar 

  7. Bouchaud E (1997) Scaling properties of cracks. J Phys Condens Matter 9:4319

    Article  Google Scholar 

  8. Brown SR (1987) Fluid flow through rock joints: the effect of surface roughness. J Geophys Res 92:1337–1347. https://doi.org/10.1029/JB092iB02p01337

    Article  Google Scholar 

  9. Brown S (1989) Transport of fluid and electric current in a single fracture. J Geophys Res 94:9429–9438. https://doi.org/10.1029/JB094iB07p09429

    Article  Google Scholar 

  10. Brown S, Stockman HW, Reeves SJ (1995) Applicability of the Reynolds Equation for modeling fluid flow between rough surfaces. Geophys Res Lett 22:2537–2540. https://doi.org/10.1029/95GL02666

    Article  Google Scholar 

  11. Brown S, Arvind C, Robert H (1998) Experimental observation of fluid flow channels in a single fracture. J Geophys Res 103:5125–5132. https://doi.org/10.1029/97JB03542

    Article  Google Scholar 

  12. Brush DJ, Thomson NR (2003) Fluid flow in synthetic rough-walled fractures: Navier-Stokes, Stokes, and local cubic law simulations. Water Resour Res. https://doi.org/10.1029/2002WR001346

    Article  Google Scholar 

  13. Evans DD, Rasmussen TC, Nicholson TJ (2013) Flow and transport through unsaturated fractured rock, vol 42. American Geophysical Union, New York

    Google Scholar 

  14. Foias C, Manley O, Rosa R, Temam R (2001) Navier-Stokes equations and turbulence. Cambridge University Press, Cambridge

    Google Scholar 

  15. Ge S (1997) A governing equation for fluid flow in rough fractures. Water Resour Res 33:53–61. https://doi.org/10.1029/96WR02588

    Article  Google Scholar 

  16. Gentier S, Lamontagne E, Archambault G, Riss J (1997) Anisotropy of flow in a fracture undergoing shear and its relationship to the direction of shearing and injection pressure. Int J Rock Mech Min 34:94.e1–94.e12. https://doi.org/10.1016/S1365-1609(97)00085-3

    Article  Google Scholar 

  17. Hötzer J, Reiter A, Hierl H, Steinmetz P, Selzer M, Nestler B (2018) The parallel multi-physics phase-field framework Pace3D. J Comput Sci Neth 26:1–12. https://doi.org/10.1016/j.jocs.2018.02.011

    Article  Google Scholar 

  18. Knudby C, Ramírez J (2005) On the relationship between indicators of geostatistical, flow and transport connectivity. Adv Water Resour 28:405–421. https://doi.org/10.1016/j.advwatres.2004.09.001

    Article  Google Scholar 

  19. Koltermann CE, Gorelick SM (1996) Heterogeneity in sedimentary deposits: a review of structure-imitating, process-imitating, and descriptive approaches. Water Resour Res 32:2617–2658. https://doi.org/10.1029/96WR00025

    Article  Google Scholar 

  20. Konzuk JS, Kueper BH (2004) Evaluation of cubic law based models describing single-phase flow through a rough-walled fracture. Water Resour Res. https://doi.org/10.1029/2003WR002356

    Article  Google Scholar 

  21. Kumar Singh K, Narain Singh D, Pathegama Gamage R (2016) Effect of sample size on the fluid flow through a single fractured granitoid. J. Rock Mech Geo Eng 8–3:329–340. https://doi.org/10.1016/j.jrmge.2015.12.004

    Article  Google Scholar 

  22. Le Goc R, de Dreuzy J-R, Davy P (2010) Statistical characteristics of flow as indicators of channeling in heterogeneous porous and fractured media. Adv Water Resour 33:257–269. https://doi.org/10.1016/j.advwatres.2009.12.002

    Article  Google Scholar 

  23. Lichun W, Bayani CM, Slottke DT, Ketcham RA, Sharp JM (2015) Modification of the Local Cubic Law of fracture flow for weak inertia, tortuosity, and roughness. Water Resour Res 51:2064–2080. https://doi.org/10.1002/2014WR015815

    Article  Google Scholar 

  24. Méheust Y, Schmittbuhl J (2001) Geometrical heterogeneities and permeability anisotropy of rough fractures. J Geophys Res 106:2089–2102. https://doi.org/10.1029/2000JB900306

    Article  Google Scholar 

  25. Méheust Y, Schmittbuhl J (2003) Scale Effects Related To Flow In Rough Fractures. In: Kümpel H-J (ed) Thermo-hydro-mechanical coupling in fractured rock. Basel, Birkhäuser Basel, pp 1023–1050

    Google Scholar 

  26. Moreno L, Tsang C-F (1994) Flow channeling in strongly heterogeneous porous media: a numerical study. Water Resour Res 30:1421–1430. https://doi.org/10.1029/93wr02978

    Article  Google Scholar 

  27. Mourzenko V, Thovert J-F, Adler P (1995) Permeability of a single fracture; validity of the Reynolds equation. J Phys Paris II 5:465–482. https://doi.org/10.1051/jp2:1995133

    Article  Google Scholar 

  28. Oron AP, Berkowitz B (1998) Flow in rock fractures. Water Resour Res 34:2811–2825. https://doi.org/10.1029/98WR02285

    Article  Google Scholar 

  29. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in C: the art of scientific computing. Cambridge University Press, Cambridge

    Google Scholar 

  30. Rasmuson A, Neretnieks I (1986) Radionuclide transport in fast channels in crystalline rock. Water Resour Res 22:1247–1256. https://doi.org/10.1029/WR022i008p01247

    Article  Google Scholar 

  31. Schmittbuhl J, Gentier S, Roux S (1993) Field measurements of the roughness of fault surfaces. Geophys Res Lett 20:639–641. https://doi.org/10.1029/93GL00170

    Article  Google Scholar 

  32. Schmittbuhl J, Schmitt F, Scholz C (1995) Scale invariance of crack surfaces. J Geophys Res 100:5953–5973. https://doi.org/10.1029/94JB02885

    Article  Google Scholar 

  33. Schmittbuhl J, Steyer A, Jouniaux L, Toussaint R (2008) Fracture morphology and viscous transport. Int J Rock Mech Min 45:422–430. https://doi.org/10.1016/j.ijrmms.2007.07.007

    Article  Google Scholar 

  34. Selzer M (2014) Mechanische und Strömungsmechanische Topologieoptimierung mit der Phasenfeldmethode. Dissertation. Karlsruhe Institute of Technology

  35. Silliman SE (1989) An interpretation of the difference between aperture estimates derived from hydraulic and tracer tests in a single fracture. Water Resour Res 25:2275–2283. https://doi.org/10.1029/WR025i010p02275

    Article  Google Scholar 

  36. Talon L, Auradou H, Hansen A (2010) Permeability of self-affine aperture fields. Phys Rev E Stat Nonlinear Soft Matter Phys 82:046108. https://doi.org/10.1103/PhysRevE.82.046108

    Article  Google Scholar 

  37. Tsang YW, Tsang CF (1989) Flow channeling in a single fracture as a two-dimensional strongly heterogeneous permeable medium. Water Resour Res 25:2076–2080. https://doi.org/10.1029/WR025i009p02076

    Article  Google Scholar 

  38. Watanabe N, Ishibashi T, Tsuchiya N (2015) Predicting the channeling flows through fractures at various scales. In: Proceedings world geothermal congress 2015

  39. Wells DL, Coppersmith KJ (1994) New empirical relationships among magnitude, rupture length, rupture width, rupture area, and surface displacement. B Seismol Soc Am 84:974–1002

    Google Scholar 

  40. Witherspoon P, Wang J, Iwai K, Gale JE (1980) Validity of Cubic Law for fluid flow in a deformable rock fracture. Water Resour Res 16:1016–1024. https://doi.org/10.1029/WR016i006p01016

    Article  Google Scholar 

  41. Zimmerman RW, Bodvarsson GS (1996) Hydraulic conductivity of rock fractures. Transp Porous Med 23:1–30. https://doi.org/10.1007/BF00145263

    Article  Google Scholar 

  42. Zimmerman RW, Kumar S, Bodvarsson GS (1991) Lubrication theory analysis of the permeability of rough-walled fractures. Int J Rock Mech Min 28:325–331. https://doi.org/10.1016/0148-9062(91)90597-F

    Article  Google Scholar 

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The study is part of the Helmholtz portfolio project “Geoenergy”. The support from the program “Renewable Energies”, under the topic “Geothermal Energy Systems”, is gratefully acknowledged. We also thank the EnBW Energie Baden-Württemberg AG for supporting geothermal research at KIT.

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Correspondence to Sophie Marchand.

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Marchand, S., Mersch, O., Selzer, M. et al. A Stochastic Study of Flow Anisotropy and Channelling in Open Rough Fractures. Rock Mech Rock Eng 53, 233–249 (2020). https://doi.org/10.1007/s00603-019-01907-4

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  • Fracture
  • Roughness
  • Aperture
  • Stochastic
  • Anisotropy
  • Channelling