Critical Reynolds number for nonlinear flow through rough-walled fractures: The role of shear processes

Water Resources Research

Volumn 50, Issue 2, 2014, Pages 1789-1804

  Javadi, Morteza ^a   Sharifzadeh, Mostafa ^a,b   Shahriar, Kourosh ^a   Mitani, Yasuhiro ^c  

^a AMIRKABIR UNIVERSITY OF TECHNOLOGY   (Iran)

^b CURTIN UNIVERSITY   (Australia)

^c KYUSHU UNIVERSITY   (Japan)

Author keywords

critical Reynolds number; Forchheimer's Law; nonlinear flow; rough walled fracture; shear flow; shear processes

Indexed keywords

FLOW OF FLUIDS; FRACTURE; REYNOLDS NUMBER; ROCKS;

CRITICAL REYNOLDS NUMBER; FORCHHEIMER'S LAW; NONLINEAR FLOW; ROUGH-WALLED FRACTURES; SHEAR PROCESS;

SHEAR FLOW;

FLUID FLOW; FRACTURE; GRADIENT ANALYSIS; NONLINEARITY; QUANTITATIVE ANALYSIS; REYNOLDS NUMBER; SHEAR FLOW;

EID: 84894575925     PISSN: 00431397     EISSN: 19447973     Source Type: Journal    
DOI: 10.1002/2013WR014610     Document Type: Article

References (116)

1. Al-Yaarubi, A. H., C. C. Pain, C. A. Grattoni, and, R. W. Zimmerman, (2005), Navier-Stokes simulations of fluid flow through a rock fracture, in Dynamics of Fluids and Transport in Fractured Rocks, edited by, B. Faybishenko, et al., AGU, Washington, D. C., doi: 10.1029/162GM07.
2. Archambault, G., S. Gentier, J. Riss, and, R. Flamand, (1997), The evolution of void spaces (permeability) in relation with rock joint shear behavior, Int. J. Rock Mech. Min. Sci., 3 (4), 14.e1-14.e15, doi: 10.1016/S1365-1609(97)00046-4.
3. Auradou, H., (2009), Influence of wall roughness on the geometrical, mechanical and transport properties of single fractures, J. Phys. D Appl. Phys., 42 (21), 214015, doi: 10.1088/0022-3727/42/21/214015.
4. Auradou, H., G. Drazer, J. P. Hulin, and, J. Koplik, (2005), Permeability anisotropy induced by the shear displacement of rough fracture walls, Water Resour. Res., 41, W09423, doi: 10.1029/2005WR003938.
5. Auradou, H., G. Drazer, A. Boschan, J. P. Hulin, and, J. Koplik, (2006), Flow channeling in a single fracture induced by shear displacement, Geothermics, 35 (5-6), 576-588, doi: 10.1016/j.geothermics.2006.11.004. (Pubitemid 46158296)
6. Babanouri, N., S. Karimi Nasab, A. Baghbanan, and, H. R. Mohamadi, (2011), Over-consolidation effect on shear behavior of rock joints, Int. J. Rock Mech. Min. Sci., 48 (8), 1283-1291, doi: 10.1016/j.ijrmms.2011.09.010.
7. Bandis, S., A. C. Lumsden, and, N. R. Barton, (1981), Experimental studies of scale effects on the Shear behaviour of rock joints, Int. J. Rock Mech. Min. Sci., 18 (1), 1-21, doi: 10.1016/0148-9062(81)90262-X. (Pubitemid 11487984)
8. Bandis, S. C., A. C. Lumsden, and, N. R. Barton, (1983), Fundamentals of rock joint deformation, Int. J. Rock Mech. Min. Sci., 20 (6), 249-268, doi: 10.1016/0148-9062(83)90595-8.
9. Barton, N., (1973), Review of a new shear-strength criterion for rock joints, Eng. Geol., 7 (4), 287-332, doi: 10.1016/0013-7952(73)90013-6.
10. Barton, N., (1976), The shear strength of rock and rock joints, Int. J. Rock Mech. Min. Sci., 13 (9), 255-279, doi: 10.1016/0148-9062(76)90003-6.
11. Barton, N., and, S. Bandis, (1980), Some effects of scale on the shear strength of joints, Int. J. Rock Mech. Min. Sci., 17, 69-73.
12. Barton, N., S. Bandis, and, K. Bakhtar, (1985), Strength, deformation and conductivity coupling of rock joints, Int. J. Rock Mech. Min. Sci., 22 (3), 121-140, doi: 10.1016/0148-9062(85)93227-9. (Pubitemid 15545494)
13. Bear, J., (1972), Dynamics of Fluids in Porous Media, Am. Elsevier, New York.
14. Belem, T., M. Souley, and, F. Homand, (2007), Modeling surface roughness degradation of rock joint wall during monotonic and cyclic shearing, Acta Geotech., 2 (4), 227-248, doi: 10.1007/s11440-007-0039-7.
15. Boulon, M., G. Armand, N. Hoteit, and, P. Divoux, (2002), Experimental investigations and modelling of shearing of calcite healed discontinuities of granodiorite under typical stresses, Eng. Geol., 64 (2-3), 117-133, doi: 10.1016/S0013-7952(01)00112-0. (Pubitemid 34580421)
16. Boulon, M. J., A. P. S. Selvadurai, H. Benjelloun, and, B. Feuga, (1993), Influence of rock joint degradation on hydraulic conductivity, Int. J. Rock Mech. Min. Sci., 30 (7), 1311-1317, doi: 10.1016/0148-9062(93)90115-T.
17. Brown, S. R., (1987), Fluid flow through rock joints: The effect of surface roughness, J. Geophys. Res., 92 (B2), 1337-1347, doi: 10.1029/JB092iB02p01337.
18. Brown, S. R., H. W. Stockman, and, S. J. Reeves, (1995), Applicability of the Reynolds equation for modeling fluid flow between rough surfaces, Geophys. Res. Lett., 22 (18), 2537-2540, doi: 10.1029/95GL02666. (Pubitemid 26458692)
19. Brush, D., and, N. R. Thomson, (2003), Fluid flow in synthetic rough-walled fractures: Navier-Stokes, Stokes, and local cubic law simulations, Water Resour. Res., 39 (4), 1085, doi: 10.1029/2002WR001346.
20. Bués, M., M. Panfilov, S. Crosnier, and, C. Oltean, (2004), Macroscale model and viscous-inertia effects for Navier-Stokes flow in a radial fracture with corrugated walls, J. Fluid Mech., 504, 41-60, doi: 10.1017/S002211200400816X. (Pubitemid 38554622)
21. Chern, S.-G., T.-C. Cheng, and, W.-Y. Chen, (2012), Behavior of regular triangular joints under cyclic shearing, J. Mar. Sci. Technol., 20 (5), 508-513, doi: 10.6119/JMST-011-0428-1.
22. Cherubini, C., C. I. Giasi, and, N. Pastore, (2012), Bench scale laboratory tests to analyze non-linear flow in fractured media, Hydrol. Earth Syst. Sci. Discuss., 9, 5575-5609, doi: 10.5194/hessd-9-5575-2012.
23. Cook, N. G. W., (1992), Natural joints in rock: Mechanical, hydraulic and seismic behaviour and properties under normal stress, Int. J. Rock Mech. Min. Sci., 29 (3), 198-223, doi: 10.1016/0148-9062(92)93656-5.
24. Cooke, C. E., (1973), Conductivity of fracture proppants in multiple layers, J. Pet. Technol., 25 (9), 1101-1107, doi: 10.2118/4117-PA.
25. Cornwell, D. K., and, H. D. Murphy, (1985), Experiments with non-Darcy flow in joints with large scale roughness, paper presented at International Symposium on Fundamentals of Rock Joints, Centek, Lulea.
26. Davies, S. J., and, C. M. White, (1928), An experimental study of the flow of water in pipes of rectangular section, Proc. R. Soc. London, Ser. A, 119, 92-107, doi: 10.1098/rspa.1928.0086.
27. Dieterich, J. H., (1972), Time-dependent friction in rocks, J. Geophys. Res., 77 (20), 3690-3697, doi: 10.1029/JB077i020p03690.
28. Dowding, C. H., A. Zubelewicz, K. M. O'Connor, and, T. B. Belytschko, (1991), Explicit modeling of dilation, asperity degradation and cyclic seating of rock joints, Comput. Geotech., 11 (3), 209-227, doi: 10.1016/0266-352X(91) 90020-G. (Pubitemid 21720111)
29. Elsworth, D., and, T. W. Doe, (1986), Application of non-linear flow laws in determining rock fissure geometry from single borehole pumping tests, Int. J. Rock Mech. Min. Sci., 23 (3), 245-254, doi: 10.1016/0148-9062(86)90970-8. (Pubitemid 16603833)
30. Elsworth, D., and, R. E. Goodman, (1986), Characterization of rock fissure hydraulic conductivity using idealized wall roughness profiles, Int. J. Rock Mech. Min. Sci., 23 (3), 233-243, doi: 10.1016/0148-9062(86)90969-1. (Pubitemid 16603832)
31. Esaki, T., H. Hojo, T. Kimura, and, N. Kameda, (1991), Shear-flow coupling test on rock joints, paper presented at 7th International Congress on Rock Mechanics, 4 pp., Aachen, Germany, 16-20 Sep.
32. Esaki, T., S. Du, Y. Mitani, K. Ikusada, and, L. Jing, (1999), Development of a shear-flow test apparatus and determination of coupled properties for a single rock joint, Int. J. Rock Mech. Min. Sci., 36 (5), 641-650, doi: 10.1016/S0148-9062(99)00044-3. (Pubitemid 29452298)
33. Garrouch, A. A., and, L. Ali, (2001), Predicting the onset of inertial effects in sandstone rocks, Transp. Porous Media, 44 (3), 487-505, doi: 10.1023/A:1010671012287. (Pubitemid 32832405)
34. Ge, S., (1997), A governing equation for fluid flow in rough fractures, Water Resour. Res., 33 (1), 53-61, doi: 10.1029/96WR02588. (Pubitemid 27028010)
35. Gentier, S., D. Hopkins, and, J. Riss, (2000), Role of fracture geometry in the evolution of flow paths under stress, in Dynamic of Fluids in Fractured Rock, edited by, B. Faybishenko, et al., AGU, Washington, D. C., doi: 10.1029/GM122p0169.
36. Giger, S. B., M. B. Clennell, C. Harbers, P. Clark, M. Ricchetti, J. H. Ter Heege, B. B. T. Wassing, and, B. Orlic, (2011), Design, operation and validation of a new fluid-sealed direct shear apparatus capable of monitoring fault-related fluid flow to large displacements, Int. J. Rock Mech. Min. Sci., 48 (7), 1160-1172, doi: 10.1016/j.ijrmms.2011.09.005.
37. Goodman, R. E., (1969), The deformability of joints, paper presented at Determination of the in-situ modulus of deformation of rock, Amer. Soc. Test & Mats., STP 477, Denver, Colo.
38. Grasselli, G., J. Wirth, and, P. Egger, (2002), Quantitative three-dimensional description of a rough surface and parameter evolution with shearing, Int. J. Rock Mech. Min. Sci., 39 (6), 789-800, doi: 10.1016/S1365-1609(02)00070-9.
39. Haberfield, C. M., and, I. W. Johnston, (1994), A mechanistically-based model for rough rock joints, Int. J. Rock Mech. Min. Sci., 31 (4), 279-292, doi: 10.1016/0148-9062(94)90898-2.
40. Haberfield, C. M., and, J. P. Seidel, (1999), Some recent advances in the modelling of soft rock joints in direct shear, Geotech. Geol. Eng., 17 (3-4), 177-195, doi: 10.1023/A:1008900905076. (Pubitemid 30168427)
41. Hans, J., and, M. Boulon, (2003), A new device for investigating the hydro-mechanical properties of rock joints, Int. J. Numer. Anal. Methods Geomech., 27 (6), 513-548. doi: 10.1002/nag.285. (Pubitemid 37078449)
42. Haque, A., (1999), Shear behaviour of rock joints under constant normal stiffness, PhD thesis, Univ. of Wollongong, New South Wales, Australia.
43. Holditch, S. A., and, R. A. Morse, (1976), The effects of non-Darcy flow on the behavior of hydraulically fractured gas wells, J. Pet. Technol., 28 (10), 1169-1179, doi: 10.2118/5586-PA.
44. Holland, M., H. van Gent, L. Bazalgette, N. Yassir, E. H. Hoogerduijn Strating, and, J. L. Urai, (2011), Evolution of dilatant fracture networks in a normal fault-Evidence from 4D model experiments, Earth Planet. Sci. Lett., 304 (3-4), 399-406, doi: 10.1016/j.epsl.2011.02.017.
45. Huang, T. H., C. S. Chang, and, C. Y. Chao, (2002), Experimental and mathematical modeling for fracture of rock joint with regular asperities, Eng. Fract. Mech., 69 (17), 1977-1996, doi: 10.1016/S0013-7944(02)00072-3.
46. Huang, X., B. C. Haimson, M. E. Plesha, and, X. Qiu, (1993), An investigation of the mechanics of rock joints part I. Laboratory investigation, Int. J. Rock Mech. Min. Sci., 30 (3), 257-269, doi: 10.1016/0148-9062(93)92729- A.
47. Huitt, J. J., (1956), Fluid flow in simulated fractures, AICHE J., 2 (2), 259-264, doi: 10.1002/aic.690020224.
48. Isakov, E., S. R. Ogilvie, C. W. Taylor, and, P. W. J. Glover, (2001), Fluid flow through rough fractures in rocks I: High resolution aperture determinations, Earth Planet. Sci. Lett., 191 (3-4), 267-282, doi: 10.1016/S0012-821X(01)00424-1. (Pubitemid 32974632)
49. Javadi, M., M. Sharifzadeh, and, K. Shahriar, (2010), A new geometrical model for non-linear fluid flow through rough fractures, J. Hydrol., 389 (1-2), 18-30, doi: 10.1016/j.jhydrol.2010.05.010.
50. Ji, S.-H., H.-B. Lee, I. W. Yeo, and, K.-K. Lee, (2008), Effect of nonlinear flow on DNAPL migration in a rough-walled fracture, Water Resour. Res., 44, W11431, doi: 10.1029/2007WR006712.
51. Jiang, Y., J. Xiao, Y. Tanabashi, and, T. Mizokami, (2004), Development of an automated servo-controlled direct shear apparatus applying a constant normal stiffness condition, Int. J. Rock Mech. Min. Sci., 41 (2), 275-286, doi: 10.1016/j.ijrmms.2003.08.004. (Pubitemid 38149744)
52. Jing, L., E. Nordlund, and, O. Stephansson, (1992), An experimental study on the anisotropy and stress-dependency of the strength and deformability of rock joints, Int. J. Rock Mech. Min. Sci., 29 (6), 535-542, doi: 10.1016/0148-9062(92)91611-8.
53. Jing, L., E. Nordlund, and, O. Stephansson, (1994), A 3-D constitutive model for rock joints with anisotropic friction and stress dependency in shear stiffness, Int. J. Rock Mech. Min. Sci., 31 (2), 173-178, doi: 10.1016/0148-9062(94)92808-8. (Pubitemid 24427237)
54. Jung, R., (1989), Hydraulic in situ investigations of an artificial fracture in the Falkenberg granite, Int. J. Rock Mech. Min. Sci., 26 (3-4), 301-308, doi: 10.1016/0148-9062(89)91978-5. (Pubitemid 20621805)
55. Karami, A., and, D. Stead, (2008), Asperity degradation and damage in the direct shear test: A hybrid FEM/DEM approach, Rock Mech. Rock Eng., 41 (2), 229-266, doi: 10.1007/s00603-007-0139-6. (Pubitemid 351614307)
56. Kohl, T., K. F. Evans, R. J. Hopkirk, R. Jung, and, L. Rybach, (1997), Observation and simulation of non-Darcian flow transients in fractured rock, Water Resour. Res., 33 (3), 407-418, doi: 10.1029/96WR03495.
57. Kolditz, O., (2001), Non-linear flow in fractured rock, Int. J. Numer. Methods Heat Fluid Flow, 11 (6), 547-575, doi: 10.1108/EUM0000000005668. (Pubitemid 32828490)
58. Konzu, J. S., and, B. H. Kueper, (2004), Evaluation of cubic law based models describing single-phase flow through a rough-walled fracture, Water Resour. Res., 40, W02402, doi: 10.1029/2003WR002356.
59. Koyama, T., N. Fardin, O. Stephansson, and, L. Jing, (2006), 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, doi: 10.1016/j.ijrmms.2005.04.006. (Pubitemid 41671407)
60. Koyama, T., B. Li, Y. Jiang, and, L. Jing, (2009), Numerical modelling of fluid flow tests in a rock fracture with a special algorithm for contact areas, Comput. Geotech., 36 (1-2), 291-303, doi: 10.1016/j.compgeo.2008.02.010.
61. Kwafniewski, M. A., and, J. A. Wang, (1997), Surface roughness evolution and mechanical behavior of rock joints under shear, Int. J. Rock Mech. Min. Sci., 34 (3-4), 157.e1-157.e14, doi: 10.1016/S1365-1609(97)00042-7.
62. Leichnitz, W., (1985), Mechanical properties of rock joints, Int. J. Rock Mech. Min. Sci., 22 (5), 313-321, doi: 10.1016/0148-9062(85)92063-7.
63. Li, B., Y. Jiang, T. Koyama, L. Jing, and, Y. Tanabashi, (2008), Experimental study of the hydro-mechanical behavior of rock joints using a parallel-plate model containing contact areas and artificial fractures, Int. J. Rock Mech. Min. Sci., 45 (3), 362-375, doi: 10.1016/j.ijrmms.2007.06.004.
64. Lomize, G. M., (1951), Filtratsia v treshchinovatykh porodakh (Seepage in Jointed Rocks), Gosudarstvennoe Energetitcheskoe Izdatel'stvo, Moskva-Leningrad.
65. Lopez, P., J. Riss, and, G. Archambault, (2003), An experimental method to link morphological properties of rock fracture surfaces to their mechanical properties, Int. J. Rock Mech. Min. Sci., 40 (6), 947-954, doi: 10.1016/S1365-1609(03)00052-2. (Pubitemid 37111239)
66. Louis, C., (1969), A study of groundwater flow in jointed rock and its influence on the stability of rock masses, Rock Mech. Res. Rep. 10, Imp. Coll., London.
67. Matsuki, K., Y. Chida, and, K. Sakaguchi, (2005), Anisotropic and heterogenous water flow in a sheared fracture as estimated in large synthetic fractures, Trans. Geotherm. Resour. Counc., 29, 401-406. (Pubitemid 43907426)
68. Matsuki, K., Y. Chida, K. Sakaguchi, and, P. W. J. Glover, (2006), Size effect on aperture and permeability of a fracture as estimated in large synthetic fractures, Int. J. Rock Mech. Min. Sci., 43, 726-755, doi: 10.1016/j.ijrmms.2005.12.001.
69. Matsuki, K., Y. Kimura, K. Sakaguchi, A. Kizaki, and, A. A. Giwelli, (2010), Effect of shear displacement on the hydraulic conductivity of a fracture, Int. J. Rock Mech. Min. Sci., 47 (3), 436-449, doi: 10.1016/j.ijrmms.2009.10.002.
70. Mitani, Y., M. Sharifzadeh, T. Esaki, and, F. Urakawa, (2005), Development of shear-flow test apparatus and determination of coupled properties of rock joint, in Proceeding of the Eurock 2005-Impact of Human Activity on the Geological Environment, pp. 397-403, A. A. Balkema, Leiden, Netherlands.
71. Moradian, Z. A., G. Ballivy, P. Rivard, C. Gravel, and, B. Rousseau, (2010), Evaluating damage during shear tests of rock joints using acoustic emissions, Int. J. Rock Mech. Min. Sci., 47, 590-598, doi: 10.1016/j.ijrmms. 2010.01.004.
72. Mourzenko, V. V., J.-F. Thovert, and, P. M. Adler, (1995), Permeability of a single fracture: Validity of the Reynolds equation, J. Phys. II, 5 (3), 465-482, doi: 10.1051/jp2:1995133.
73. Moutsopoulos, K. N., (2009), Exact and approximate analytical solutions for unsteady fully developed turbulent flow in porous media and fractures for time dependent boundary conditions, J. Hydrol., 369 (1-2), 78-89, doi: 10.1016/j.jhydrol.2009.02.025.
74. Neuzil, C. E., and, J. V. Tracy, (1981), Flow through fractures, Water Resour. Res., 17 (1), 191-199, doi: 10.1029/WR017i001p00191.
75. Nicholl, M. J., H. Rajaram, R. J. Glass, and, R. L. Detwiler, (1999), Saturated flow in a single fracture: Evaluation of the Reynolds equation in measured aperture fields, Water Resour. Res., 35 (11), 3361-3373, doi: 10.1029/1999WR900241.
76. Nowamooz, A., G. Radilla, and, M. Fourar, (2009), Non-Darcian two-phase flow in a transparent replica of a rough-walled rock fracture, Water Resour. Res., 45, W07406, doi: 10.1029/2008WR007315.
77. O'Brien, G. S., C. J. Bean, and, F. McDermott, (2003), Numerical investigations of passive and reactive flow through generic single fractures with heterogeneous permeability, Earth Planet. Sci. Lett., 213 (3-4), 271-284, doi: 10.1016/S0012-821X(03)00342-X. (Pubitemid 37061271)
78. Ogilvie, S. R., E. Isakov, and, P. W. J. Glover, (2006), 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, doi: 10.1016/j.epsl.2005.11.041. (Pubitemid 43092539)
79. Olsson, R., and, N. Barton, (2001), An improved model for hydromechanical coupling during shearing of rock joints, Int. J. Rock Mech. Min. Sci., 38 (3), 317-329, doi: 10.1016/S1365-1609(00)00079-4. (Pubitemid 32596110)
80. Olsson, W. A., (1992), The effect of slip on the flow of fluid through a fracture, Geophys. Res. Lett., 19 (6), 541-543, doi: 10.1029/92GL00197.
81. Olsson, W. A., and, S. R. Brown, (1993), Hydromechanical response of a fracture undergoing compression and shear, Int. J. Rock Mech. Min. Sci., 30 (7), 845-851, doi: 10.1016/0148-9062(93)90034-B.
82. Oron, A. P., and, B. Berkowitz, (1998), Flow in rock fractures: The local cubic law assumption reexamined, Water Resour. Res., 34 (11), 2811-2825, doi: 10.1029/98WR02285. (Pubitemid 28494915)
83. Park, J.-W., and, J.-J. Song, (2009), Numerical simulation of a direct shear test on a rock joint using a bonded-particle model, Int. J. Rock Mech. Min. Sci., 46 (7), 1315-1328, doi: 10.1016/j.ijrmms.2009.03.007.
84. Parrish, D. R., (1963), Fluid flow in rough fractures, paper presented at Production Research Symposium, Joint SPE-AIME, Univ. of Okla., Norman, Okla.
85. Pereira, J. P., and, M. H. de Freitas, (1993), Mechanisms of shear failure in artificial fractures of sandstone and their implication for models of hydromechanical coupling, Rock Mech. Rock Eng., 26 (3), 195-214, doi: 10.1007/BF01040115.
86. Pyrak-Nolte, L. J., N. G. W. Cook, and, D. D. Nolte, (1988), Fluid percolation through single fractures, Geophys. Res. Lett., 15 (11), 1247-1250, doi: 10.1029/GL015i011p01247.
87. Qian, J., H. Zhan, W. Zhao, and, F. Sun, (2005), Experimental study of turbulent unconfined groundwater flow in a single fracture, J. Hydrol., 311 (1-4), 134-142, doi: 10.1016/j.jhydrol.2005.01.013. (Pubitemid 41261603)
88. Qian, J., H. Zhan, Z. Chen, and, H. Ye, (2011), Experimental study of solute transport under non-Darcian flow in a single fracture, J. Hydrol., 399 (3), 246-254, doi: 10.1016/j.jhydrol.2011.01.003.
89. Quinn, P. M., J. A. Cherry, and, B. L. Parker, (2011), Quantification of non-Darcian flow observed during packer testing in fractured sedimentary rock, Water Resour. Res., 47, W09533, doi: 10.1029/2010WR009681.
90. Ranjith, P. G., and, W. Darlington, (2007), Nonlinear single-phase flow in real rock joints, Water Resour. Res., 43, W09502, doi: 10.1029/2006WR005457.
91. Ranjith, P. G., and, D. R. Viete, (2011), Applicability of the 'cubic law' for non-Darcian fracture flow, J. Pet. Sci. Eng., 78 (2), 321-327, doi: 10.1016/j.petrol.2011.07.015.
92. Renshaw, C. E., (1995), On the relationship between mechanical and hydraulic apertures in rough-walled fractures, J. Geophys. Res., 100 (B12), 629-636, doi: 10.1029/95JB02159.
93. Ruina, A., (1983), Slip instability and state variable friction laws, J. Geophys. Res., 88 (B12), 10,359-10,370, doi: 10.1029/JB088iB12p10359.
94. Ruth, D., and, H. Ma, (1992), On the derivation of the Forchheimer equation by means of the averaging theorem, Transp. Porous Media, 7 (3), 255-264, doi: 10.1007/BF01063962. (Pubitemid 23568602)
95. Saeb, S., and, B. Amadei, (1992), Modelling rock joints under shear and normal loading, Int. J. Rock Mech. Min. Sci., 29 (3), 267-278, doi: 10.1016/0148-9062(92)93660-C.
96. Schrauf, T. W., and, D. D. Evans, (1986), Laboratory studies of gas flow through a single natural fracture, Water Resour. Res., 22 (7), 1038-1050, doi: 10.1029/WR022i007p01038. (Pubitemid 16590982)
97. Seidel, J. P., and, C. M. Haberfield, (2002), A theoretical model for rock joints subjected to constant normal stiffness direct shear, Int. J. Rock Mech. Min. Sci., 39 (5), 539-553, doi: 10.1016/S1365-1609(02)00056-4.
98. Sharifzadeh, M., (2005), Experimental and theoretical research on hydro-mechanical coupling properties of rock joint, PhD thesis, Dep. of Civ. and Struct. Eng., Kyushu Univ., Fukuoka, Japan.
99. Sharifzadeh, M., Y. Mitani, and, T. Esaki, (2006), Rock joint surfaces measurement and analysis of aperture distribution under different normal and shear loading using GIS, Rock Mech. Rock Eng., 41 (2), 299-323, doi: 10.1007/s00603-006-0115-6. (Pubitemid 351614309)
100. Skjetne, E., A. Hansen, and, J. S. Gudmundsson, (1999), High-velocity flow in a rough fracture, J. Fluid Mech., 383, 1-28, doi: 10.1017/ S0022112098002444.
101. Sun, Z., C. Gerrard, and, O. Stephansson, (1985), Rock joint compliance tests compression for and shear loads, Int. J. Rock Mech. Min. Sci., 22 (4), 197-213, doi: 10.1016/0148-9062(85)92948-1. (Pubitemid 15541099)
102. Talon, L., H. Auradou, and, A. Hansen, (2010), Permeability of self-affine aperture fields, Phys. Rev. E, 82 (4), 046108, doi: 10.1103/PhysRevE.82.046108.
103. Tsang, Y. W., (1984), The effect of tortuosity on fluid-flow through a single fracture, Water Resour. Res., 20 (9), 1209-1215, doi: 10.1029/WR020i009p01209.
104. Tsang, Y. W., and, C. F. Tsang, (1987), Channel model of flow through fractured media, Water Resour. Res., 23 (3), 467-479, doi: 10.1029/ WR023i003p00467.
105. Tsang, Y. W., and, P. A. Witherspoon, (1981), Hydromechanical behavior of a deformable rock fracture subject to normal stress, J. Geophys. Res., 86 (B10), 9287-9298, doi: 10.1029/JB086iB10p09287.
106. Watanabe, N., N. Hirano, and, N. Tsuchiya, (2008), 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, W06412, doi: 10.1029/2006WR005411.
107. Wen, Z., G. Huang, and, H. Zhan, (2006), Non-Darcian flow in a single confined vertical fracture toward a well, J. Hydrol., 330 (3-4), 698-708, doi: 10.1016/j.jhydrol.2006.05.001. (Pubitemid 44573203)
108. Witherspoon, P. A., J. S. Y. Wang, K. Iwai, and, J. E. Gale, (1980), Validity of cubic law for fluid flow in a deformable rock fracture, Water Resour. Res., 16 (6), 1016-1024, doi: 10.1029/WR016i006p01016.
109. Xiong, X., B. Li, Y. Jiang, T. Koyama, and, C. Zhang, (2011), Experimental and numerical study of the geometrical and hydraulic characteristics of a single rock fracture during shear, Int. J. Rock Mech. Min. Sci., 48 (8), 1292-1302, doi: 10.1016/j.ijrmms.2011.09.009.
110. Yeo, I. W., and, S. Ge, (2001), Solute dispersion in rock fractures by non-Darcian flow, Geophys. Res. Lett., 28 (20), 3983-3986, doi: 10.1029/2001GL013274. (Pubitemid 33056437)
111. Yeo, I. W., M. H. de Freitas, and, R. W. Zimmerman, (1998), Effect of shear displacement on the aperture and permeability of a rock fracture, Int. J. Rock Mech. Min. Sci., 35 (8), 1051-1070, doi: 10.1016/S0148-9062(98)00165-X. (Pubitemid 29007445)
112. Zeng, Z., and, R. Grigg, (2006), A criterion for non-Darcy flow in porous media, Transp. Porous Media, 63 (1), 57-69, doi: 10.1007/s11242-005-2720-3.
113. Zhang, H. Q., Z. Y. Zhao, C. A. Tang, and, L. Song, (2006), Numerical study of shear behavior of intermittent rock joints with different geometrical parameters, Int. J. Rock Mech. Min. Sci., 43 (5), 802-816, doi: 10.1016/j.ijrmms.2005.12.006.
114. Zhao, J., (1997), Joint surface matching and shear strength part B: JRC-JMC shear strength criterion, Int. J. Rock Mech. Min. Sci., 34 (2), 179-185, doi: 10.1016/S0148-9062(96)00063-0. (Pubitemid 27271174)
115. Zimmerman, R. W., and, G. S. Bodvarsson, (1996), Hydraulic conductivity of rock fractures, Transp. Porous Media, 23 (1), 1-30, doi: 10.1007/BF00145263.
116. Zimmerman, R. W., A. H. Al-Yaarubi, C. C. Pain, and, C. A. Grattoni, (2004), Non-linear regimes of fluid flow in rock fractures, Int. J. Rock Mech. Min. Sci., 41 (3), 163-169, doi: 10.1016/j.ijrmms.2003.12.045.