Water flow into tunnels in discontinuous rock: a short critical review of the analytical solution of the art

  • Hadi FarhadianEmail author
  • Arash Nikvar-Hassani
Original Paper


Indeed, determination of groundwater inflow into tunnels, especially in discontinuous rock masses, is still one of the most challenging issues faced by tunneling experts, which may have destructive consequences and result in impediments to tunneling operations. Hence, it is necessary to predict the location and amount of inflow in the tunneling designation phase. In order to quantify the groundwater inflow rate into tunnels, various analytical, empirical, and numerical methods exist in the literature. Analytical solutions are of great interest since they provide desirable approximations quickly and without the necessity for advanced computers. However, there are no comprehensive and unique databases concerning analytical methods. Therefore, the attempt of this study was to present the various analytical methods that exist in the literature, their application domain, and their validation.


Water inflow Tunnel Analytical equation Rock media 




Cross-section area (m2)


Constant parameter


Distance from the axis of the tunnel to the bottom of the aquifer (m)

aF( J)

Aperture of joints in the field (mm)


Corner of normal direction of fracture and coordinate axis (°)


Aperture surface area of joints (m2)


Aquifer thickness (m)


Constant parameter


A coefficient only related to the shape and depth


Inter-connectivity coefficient


Obtained circumference of shape from intersection of joint strike with tunnel axis (m)


Median of the specific capacity of the rock mass (m2/s)


Hydraulic load above land surface (m)


Discontinuity (ith discontinuity for jth set) of diameter (discontinuity with circular shape) (m)


Average aperture of the i-set of discontinuities (m)


Average frequency of the i-set of discontinuities (m − 1)


Earth gravity (m/s2)


Hydraulic head into the tunnel (m)


Depth of the tunnel center from the water table (m)


Water depth above the ground (m)


Energy head (m)


Distance of the tunnel center to the ground surface (m)


Distance of the tunnel center to the ground surface or sea floor (m)


Distance of the tunnel center to the water table (m)


Hydraulic gradient (m)


Identity matrix


Modified Bessel function of the first kind of order zero


Cluster number of fracture


Permeability coefficient (m/s)


Equivalent hydraulic conductivity (m/s)


Maximum components of the hydraulic conductivity (m/s)


Minimum components of the hydraulic conductivity (m/s)


Empirical hydraulic conductivity (m/s)


Modified Bessel function of the second kind of order zero


Permeability coefficient of grouting circle (m/s)


Permeability coefficient of lining (m/s)


Permeability coefficient of surrounding rocks (m/s)


Permeability coefficient of secondary lining (m/s)


Tunnel length (m)


The number of joint sets


Total number of discontinuities sets


Crude seepage field force (kg/m.s2)


Excess pressure at opening (kg/m2)


Inflow to tunnel water (m2/s)


Approximated water inflow rate (m2/s)


The average flow rate (m2/s)


Leakage into a certain section of a tunnel quantified according to the seismic velocity (m3/s)


Influence radius of the flow (m)


Radius of the objective volume of the rock mass around the tunnel (m)


The horizontal influence distance of groundwater level drawdown from the center of the tunnel (m)


The vertical influence distance of groundwater level drawdown from the initial groundwater level (m)


Tunnel radius (m)


Borehole radius for which the capacity is estimated (m)


External tunnel radius with lining (m)


Radius of grouting circle (m)


Internal tunnel radius with lining (m)


Radius of lining (m)


Radius of secondary lining (m)


Radius of surrounding rock (m)


Influence radius of borehole (m)


A coefficient only related to the tunnel’s shape and depth


Drawdown (m)


Spacing of joints (m)


Mean value of seismic velocity (m/s) of the whole tunnel line


Depth of the tunnel’s centerline (m)


Gradient operator


Unit weight of water


Water specific weight


Acute angle between the axis of the tunnel and the normal vector of the ith discontinuity of the jth set (°)


Minimum joint dip (°)


Dynamic viscosity of water (kg/m/s)


Kinematic viscosity of water (3.20 ·10− 6 m2/s)

\( \overline{v} \)

Mean value of the seismic velocity (m/s) of the rock mass which has been penetrated by the boreholes


A mathematical constant, commonly approximated as 3.14159


Angle between direction minimum components of the hydraulic conductivity and the horizontal plane (°)


Hydraulic head at the tunnel perimeter (m)

Operator and symbol

\( \overrightarrow{n_i} \) ⌊n1; n2; n3

Dimensionless unitary vector normal to the average plane of the discontinuity set



The authors would like to thank the enthusiastic researchers in the field of groundwater in underground spaces and related topics. Moreover, the valuable and constructive comments of the anonymous reviewers are greatly appreciated.


  1. Airaksinen JU (1978) Soil and groundwater hydrology. Pohjoinen, Oulu, Finland (in Finnish)Google Scholar
  2. Anagnostou G (1995a) Seepage flow around tunnels in swelling rock. Int J Numer Anal Methods Geomech 19(10):705–724Google Scholar
  3. Anagnostou G (1995b) The influence of tunnel excavation on the hydraulic head. Int J Numer Anal Methods Geomech 19(10):725–746Google Scholar
  4. Bai T, Pollard DD, Gross MR (2000) Mechanical prediction of fracture aperture in layered rocks. J Geophys Res Solid Earth 105(B1):707–721Google Scholar
  5. Bandis SC, Lumsden AC, Barton NR (1983) Fundamentals of rock joint deformation. Int J Rock Mech Min Sci Geomech Abstr 20(6):249–268Google Scholar
  6. Barton N, Harvik L, Christianson M, Vik G (1986) Estimation of joint deformations, potential leakage and lining stresses for a planned urban road tunnel. In: Proceedings of the international symposium on large rock caverns, Helsinki, Finland, August 1986. Pergamon Press, vol 2, pp 1171–1182Google Scholar
  7. Bear J, Zalavsky D, Irmay S (1968) Chapter 13. Hydraulics of wells. In: Physical principles of water percolation and seepage. UNESCO, Paris, pp 395–434Google Scholar
  8. Bello-Maldonado AA (1974) Seepage towards tunnels. In: Proceedings of the 2nd international congress of the International Association of Engineering Geology, São Paulo, Brazil, August 1974, vol 7, pp 15.1–15.10Google Scholar
  9. Bello-Moldonado AA (1983) Post construction seepage towards tunnels in variable head aquifers. In: Proceedings of the 5th ISRM congress, Melbourne, Australia, April 1983, vol B, pp Bll1–Bll7Google Scholar
  10. Bouvard M, Pinto N (1969) Aménagement Capivari-Cachoeira: étude du puits en charge. La Houille Blanche, Paris 7:747–760Google Scholar
  11. Brantberger M, Dalmalm T, Eriksson M, Stille H (1998) Steering factors of tightness around a pregrouted tunnel. Royal Institute of Technology, Stockholm (in Swedish)Google Scholar
  12. Briggs S, Karney BW, Sleep BE (2017) Numerical modeling of the effects of roughness on flow and eddy formation in fractures. J Rock Mech Geotech Eng 9(1):105–115Google Scholar
  13. Brown SR, Kranz RL, Bonner BP (1986) Correlation between the surfaces of natural rock joints. Geophys Res Lett 13:1430–1433Google Scholar
  14. Butscher C (2012) Steady-state groundwater inflow into a circular tunnel. Tunn Undergr Space Technol 32:158–167Google Scholar
  15. Butscher C, Huggenberger P, Zechner E, Einstein HH (2011a) Relation between hydrogeological setting and swelling potential of clay-sulfate rocks in tunneling. Eng Geol 122(3–4):204–214Google Scholar
  16. Butscher C, Huggenberger P, Zechner E (2011b) Impact of tunneling on regional groundwater flow and implications for swelling of clay–sulfate rocks. Eng Geol 117(3–4):198–206Google Scholar
  17. Butscher C, Scheidler S, Farhadian H, Dresmann H, Huggenberger P (2017) Swelling potential of clay-sulfate rocks in tunneling in complex geological settings and impact of hydraulic measures assessed by 3D groundwater modeling. Eng Geol 221:143–153Google Scholar
  18. Cao YB, Feng XT, Yan EC, Chen G, Lü FF, Ji HB, Song KY (2016) Calculation method and distribution characteristics of fracture hydraulic aperture from field experiments in fractured granite area. Rock Mech Rock Eng 49(5):1629–1647Google Scholar
  19. Carlsson A, Olsson T (1977) Water leakage in the Forsmark Tunnel, Uppland, Sweden. Swedish State Power Board, Stockholm, Sweden (in Swedish)Google Scholar
  20. Carlsson A, Olsson T (1993) The analysis of fractures, stress and water flow for rock engineering projects. Compr Rock Eng 2:126–133Google Scholar
  21. Ceryan Ş (2008) New chemical weathering indices for estimating the mechanical properties of rocks: a case study from the Kürtün granodiorite, NE Turkey. Turk J Earth Sci 17(1):187–207Google Scholar
  22. Cesano D (1999) Methods for prediction of groundwater flows into underground constructions in hard rocks. Royal Institute of Technology, StockholmGoogle Scholar
  23. Cesano D, Olofsson B, Bagtzoglou AC (2000) Parameters regulating groundwater inflows into hard rock tunnels—a statistical study of the Bolmen tunnel in southern Sweden. Tunn Undergr Space Technol 15(2):153–165Google Scholar
  24. Chen Y, Zhou C, Sheng Y (2007) Formulation of strain-dependent hydraulic conductivity for a fractured rock mass. Int J Rock Mech Min Sci 44(7):981–996Google Scholar
  25. Cherubini C, Giasi CI, Pastore N (2012) Bench scale laboratory tests to analyze non-linear flow in fractured media. Hydrol Earth Syst Sci 16(8):2511–2522Google Scholar
  26. Chu H, Xu G, Yasufuku N, Yu Z, Liu P, Wang J (2017) Risk assessment of water inrush in karst tunnels based on two-class fuzzy comprehensive evaluation method. Arab J Geosci 10(7):179Google Scholar
  27. Coli N, Pranzini G, Alfi A, Boerio V (2008) Evaluation of rock-mass permeability tensor and prediction of tunnel inflows by means of geostructural surveys and finite element seepage analysis. Eng Geol 101(3–4):174–184Google Scholar
  28. Custodio E (1983) Hidráulica de captaciones de agua subterránea. In: Custodio E, Llamas MR (eds) Hidrología Subterránea. Ediciones Omega, Barcelona, pp 614–695 (in Spanish)Google Scholar
  29. El Tani M (1999) Water inflow into tunnels. In: Proceedings of the World Tunnel Congress ITA-AITES 1999, Oslo, Norway, May/June 1999. Balkema, Rotterdam, pp 61–70Google Scholar
  30. El Tani M (2003) Circular tunnel in a semi-infinite aquifer. Tunn Undergr Space Technol 18:49–55Google Scholar
  31. El Tani M (2010) Helmholtz evolution of a semi-infinite aquifer drained by a circular tunnel. Tunn Undergr Space Technol 25:54–62Google Scholar
  32. Farhadian H, Katibeh H (2015a) Groundwater seepage estimation into Amirkabir tunnel using analytical methods and DEM and SGR method. Int J Civ Environ Eng 9(3):296–301Google Scholar
  33. Farhadian H, Katibeh H (2015b) Effect of model dimension in numerical simulation on assessment of water inflow to tunnel in discontinues rock. Int J Civ Environ Eng 9(4):358–361Google Scholar
  34. Farhadian H, Katibeh H (2017) New empirical model to evaluate groundwater flow into circular tunnel using multiple regression analysis. Int J Min Sci Technol 27(3):415–421Google Scholar
  35. Farhadian H, Aalianvari A, Katibeh H (2012) Optimization of analytical equations of groundwater seepage into tunnels: a case study of Amirkabir tunnel. Geolog Soc Ind 80:96–100Google Scholar
  36. Farhadian H, Katibeh H, Huggenberger P, Butscher C (2016a) Optimum model extent for numerical simulation of tunnel inflow in fractured rock. Tunn Undergr Space Technol 60:21–29Google Scholar
  37. Farhadian H, Katibeh H, Huggenberger P (2016b) Empirical model for estimating groundwater flow into tunnel in discontinuous rock masses. Environ Earth Sci 75(6):471Google Scholar
  38. Farhadian H, Hassani AN, Katibeh H (2017) Groundwater inflow assessment to Karaj Water Conveyance tunnel, northern Iran. KSCE J Civ Eng 21(6):2429–2438Google Scholar
  39. Fernandez G, Moon J (2010) Excavation-induced hydraulic conductivity reduction around a tunnel—part 1: guideline for estimate of ground water inflow rate. Tunn Undergr Space Technol 25(5):560–566Google Scholar
  40. Freeze RA, Cherry JA (1979) Groundwater. Prentice-Hall, Englewood Cliffs, New Jersey, TIC 217571Google Scholar
  41. Gale JE (1982) The effects of fracture type (induced versus natural) on the stress-fracture closure-fracture permeability relationships. In: Proceedings of the 23rd US symposium on rock mechanics, Berkeley, California, August 1982, pp 209–217Google Scholar
  42. Gargini A, Bernini A, Castaldelli G, Fano EA, Franchini M, Pontin A (2009). Minimum flow estimation tools validated through hydrological and ecological monitoring in alpine rivers. In Water engineering for a sustainable environment (pp. 6692–6699). International Association of Hydraulic Engineering and Research (IAHR)Google Scholar
  43. Gattinoni P, Scesi L (2010) An empirical equation for tunnel inflow assessment: application to sedimentary rock masses. Hydrogeol J 18(8):1797–1810Google Scholar
  44. Gentier S, Billaux D (1989) Caracterisation en laboratoire de l’espace fissural d’une fracture. In: Proceedings of the international symposium on rock at great depth, Pau, France, August 1989. Balkema, RotterdamGoogle Scholar
  45. Goodman RE, Moye DG, Van Schalkwyk A, Javandel I (1965) Ground water inflows during tunnel driving. Eng Geol 2:39–56Google Scholar
  46. Gustafson G (1986) Hydrogeological preinvestigations in rock. Theoretical basis and applications. Swedish Rock Engineering Research Foundation, report BeFo 84:1/86Google Scholar
  47. Harr ME (1962) Groundwater and seepage. McGraw-Hill, New YorkGoogle Scholar
  48. Heuer RE (1995) Estimating rock-tunnel water inflow. In: Proceedings of the rapid excavation and tunneling conference, San Francisco, California, June 1995, pp 18–21Google Scholar
  49. Hosseinian A, Rasouli V, Utikar R (2010) Fluid flow response of JRC exemplar profiles. In: Proceedings of EUROCK 2010—rock mechanics in civil and environmental engineering, Lausanne, Switzerland, June 2010Google Scholar
  50. Huangfu M, Wang MS, Tan ZS, Wang XY (2010) Analytical solutions for steady seepage into an underwater circular tunnel. Tunn Undergr Space Technol 25(4):391–396Google Scholar
  51. Hwang J-H, Lu C-C (2007) A semi-analytical method for analyzing the tunnel water inflow. Tunn Undergr Space Technol 22:39–46Google Scholar
  52. Iwai K (1976) Fundamental studies of fluid flow through a single fracture. PhD thesis, University of California, Berkeley, CA, USA, 208 ppGoogle Scholar
  53. Jacob CE, Lohman SW (1952) Nonsteady flow to a well of constant drawdown in an extensive aquifer. Trans Am Geophys Union 33(4):559–569Google Scholar
  54. Jin X, Li Y, Luo Y, Liu H (2016) Prediction of city tunnel water inflow and its influence on overlain lakes in karst valley. Environ Earth Sci 75(16):1162Google Scholar
  55. Karlsrud K (2001) Water control when tunneling under urban areas in the Olso region. NFF Publ 12(4):27–33Google Scholar
  56. Kelsall PC, Case JB, Chabannes CR (1984) Evaluation of excavation-induced changes in rock permeability. Int J Rock Mech Min Sci Geomech Abstr 21(3):123–135Google Scholar
  57. Király L (1969) Anisotropie et hétérogénéité de la perméabilité dans les calcaires fissurés (Anisotropy and heterogeneity of permeability in fractured limestones). Eclogae Geol Helv 62(2):613–619Google Scholar
  58. Király L (1978) La notion d’unité hydrogéologique. Essai de définition (Definition of the hydrogeological unit). Bull Cent Hydrogéol 2:83–216Google Scholar
  59. Kolditz O (2001) Non-linear flow in fractured rock. Int J Numer Methods Heat Fluid Flow 11(6):547–575Google Scholar
  60. Kolymbas D, Wagner P (2007) Groundwater ingress to tunnels—the exact analytical solution. Tunn Undergr Space Technol 22(1):23–27Google Scholar
  61. Kranz RL, Frankel AD, Engelder T, Scholz CH (1979) Permeability of whole and jointed Barre granite. Int J Rock Mech Min Sci 16:225–234Google Scholar
  62. Lapcevic PA, Novakowski KS, Cherry JA (1990) The characterization of two discrete horizontal fractures in shale. In: Proceedings of the technology transfer conference, Ontario Ministry of the Environment, Toronto, Canada, November 1990Google Scholar
  63. Lee SH, Lee KK, Yeo IW (2014) Assessment of the validity of Stokes and Reynolds equations for fluid flow through a rough-walled fracture with flow imaging. Geophys Res Lett 41(13):4578–4585Google Scholar
  64. Lei S (1999) An analytical solution for steady flow into a tunnel. Groundwater 37:23–26Google Scholar
  65. Lin L, Xu Y (2006) A tensor approach to the estimation of hydraulic conductivities in Table Mountain Group aquifers of South Africa. Water SA 32(3):371–378Google Scholar
  66. Lin BS, Lee CH, Yu JL (2000) Analysis of groundwater seepage of tunnels in fractured rock. J Chin Inst Eng 23(2):155–160Google Scholar
  67. Liu F, Xu G, Huang W, Hu S, Hu M (2012) The effect of grouting reinforcement on groundwater seepage in deep tunnels. Blucher Mech Eng Proc 1(1):4727–4737Google Scholar
  68. Loew S (2001) Natural groundwater pathways and models for regional groundwater flow in crystalline rocks. In: Seiler KP, Wohnlich S (eds) New approaches characterizing groundwater flow: proceedings of the XXXI IAH congress, Munich, Germany, September 2001Google Scholar
  69. Lombardi G (2002) Private communicationGoogle Scholar
  70. Long JCS (1983) Investigation of equivalent porous medium permeability in networks of discontinuous fractures. PhD thesis, University of California, Berkeley, CA, USAGoogle Scholar
  71. Long JCS, Witherspoon PA (1985) The relationship of the degree of interconnection to permeability in fracture networks. J Geophys Res Solid Earth 90:3087–3098Google Scholar
  72. Long JCS, Karasaki K, Davey A, Peterson J, Landsfeld M, Kemeny J, Martel S (1991) An inverse approach to the construction of fracture hydrology models conditioned by geophysical data: an example from the validation exercises at the Stripa Mine. Int J Rock Mech Min Sci Geomech Abstr 28:121–142Google Scholar
  73. Louis CA (1969) A study of groundwater flow in jointed rock and its influence on the stability of rock masses. Imperial College of Science and Technology, LondonGoogle Scholar
  74. Louis C (1974) Introduction a l’hydraulique des roches [Introduction to rock hydraulics]. Bur Rech Géol Min 4(III):283–356Google Scholar
  75. Luo S, Zhao Z, Peng H, Pu H (2016) The role of fracture surface roughness in macroscopic fluid flow and heat transfer in fractured rocks. Int J Rock Mech Min Sci 87:29–38Google Scholar
  76. Maleki MR (2018) Groundwater seepage rate (GSR); a new method for prediction of groundwater inflow into jointed rock tunnels. Tunn Undergr Space Technol 71:505–517Google Scholar
  77. Maréchal JC (1998) Les circulations d eau dans les massifs cristallins alpins et leurs relations avec les ouvrages souterrains. Thèse Ecole Polytechnique Fédérale de Lausanne, Switzerland, 296 pGoogle Scholar
  78. Maréchal J-C, Perrochet P (2003) New analytical solution for the study of hydraulic interaction between Alpine tunnels and groundwater. Bull Soc Géol Fr 174(5):441–448Google Scholar
  79. Mas Ivars D (2006) Water inflow into excavations in fractured rock—a three-dimensional hydro-mechanical numerical study. Int J Rock Mech Min Sci 43(5):705–725Google Scholar
  80. Masset O (2011) Transient tunnel inflow and hydraulic conductivity of fractured crystalline rocks in the Central Alps (Switzerland). Dissertation, ETH ZürichGoogle Scholar
  81. Meng Z, Zhang J, Wang R (2011) In-situ stress, pore pressure and stress-dependent permeability in the southern Qinshui Basin. Int J Rock Mech Min Sci 48:122–131Google Scholar
  82. Meyer T, Einstein HH (2002) Geologic stochastic modeling and connectivity assessment of fracture systems in the Boston area. Rock Mech Rock Eng 35(1):23–44Google Scholar
  83. Milanovic P (2002) The environmental impacts of human activities and engineering constructions in karst regions. Episodes 25:13–21Google Scholar
  84. Min KB, Jing L, Stephansson O (2004) Determining the equivalent permeability tensor for fractured rock masses using a stochastic REV approach: method and application to the field data from Sellafield, UK. Hydrogeol J 12(5):497–510Google Scholar
  85. Moeini H, Farhadian H, Nikvar-Hassani A (2018) Determination of the optimum sealing method for Azad pumped storage dam considering seepage analysis. Arab J Geosci 11(14):389Google Scholar
  86. Molinero J, Samper J, Juanes R (2002) Numerical modeling of the transient hydrogeological response produced by tunnel construction in fractured bedrocks. Eng Geol 64(4):369–386Google Scholar
  87. Moon J, Fernandez G (2010) Effect of excavation-induced groundwater level drawdown on tunnel inflow in a jointed rock mass. Eng Geol 110(3–4):33–42Google Scholar
  88. Moon J, Jeong S (2011) Effect of highly pervious geological features on ground-water flow into a tunnel. Eng Geol 117(3–4):207–216Google Scholar
  89. Muskat M (1937) The flow of homogeneous fluids through porous media. McGraw-Hill, New York, pp 175–181Google Scholar
  90. Nikvar-Hassani A, Katibeh H, Farhadian H (2016) Numerical analysis of steady-state groundwater inflow into Tabriz line 2 metro tunnel, northwestern Iran, with special consideration of model dimensions. Bull Eng Geol Environ 75(4):1617–1627Google Scholar
  91. Nikvar-Hassani A, Farhadian H, Katibeh H (2018) A comparative study on evaluation of steady-state groundwater inflow into a circular shallow tunnel. Tunn Undergr Space Technol 73:15–25Google Scholar
  92. Oda M (1985) Permeability tensor for discontinuous rock masses. Geotechnique 35(4):483–495Google Scholar
  93. Olofsson B (1991) Impact on groundwater conditions by tunnelling in hard crystalline rock. Doctoral dissertation, Royal Institute of Technology, StockholmGoogle Scholar
  94. Palma B, Ruocco A, Lollino P, Parise M (2012) Analysis of the behaviour of a carbonate rock mass due to tunneling in a karst setting. In: Han KC, Park C, Kim JD, Jeon S, Song JJ (eds) The present and future of rock engineering, proceedings of the 7th Asian rock mechanics symposium, Seoul, Korea, 15–19 October 2012, pp 772–781Google Scholar
  95. Palmstrom A, Stille H (2007) Ground behaviour and rock engineering tools for underground excavations. Tunn Undergr Space Technol 22(4):363–376Google Scholar
  96. Park KH, Owatsiriwong A, Lee JG (2008) Analytical solution for steady-state groundwater inflow into a drained circular tunnel in a semi-infinite aquifer: a revisit. J Tunn Undergr Space Technol 23:206–209Google Scholar
  97. Perello P, Baietto A, Burger U, Skuk S (2014) Excavation of the Aica-Mules pilot tunnel for the Brenner base tunnel: information gained on water inflows in tunnels in granitic massifs. Rock Mech Rock Eng 47(3):1049–1071Google Scholar
  98. Perrochet P (2005) Confined flow into a tunnel during progressive drilling: an analytical solution. Groundwater 43(6):943–946Google Scholar
  99. Perrochet P, Dematteis A (2007) Modeling transient discharge into a tunnel drilled in a heterogeneous formation. Groundwater 45(6):786–790Google Scholar
  100. Piggott AR (1990) Analytical and experimental studies of rock fracture hydraulics. PhD thesis, Pennsylvania State University, University Park, PA, USAGoogle Scholar
  101. Polla J, Ritola J (1989) Large rock caverns. Drainage and sealing of rock caverns. Technical Research Centre of Finland, Research notes 1000. Espoo, Finland (in Finnish)Google Scholar
  102. Polubarinova-Kochina PA (1962) Theory of ground water movement. Translated by De Wiest RJM. Princeton University Press, Princeton, New JerseyGoogle Scholar
  103. Qian JZ, Wang M, Zhang Y, Yan XS, Zhao WD (2015) Experimental study of the transition from non-Darcian to Darcy behavior for flow through a single fracture. J Hydrodynam 27(5):679–688Google Scholar
  104. Quinn PM, Cherry JA, Parker BL (2011) Quantification of non-Darcian flow observed during packer testing in fractured sedimentary rock. Water Resour Res 47(9):178–187Google Scholar
  105. Rat M (1973) Ecoulement et répartition des pressions interstitielles autour des tunnels. Bull Liaison Lab Ponts Chauss 68:109–124Google Scholar
  106. Ribacchi R, Graziani A, Boldini D (2002) Previsione degli afflussi d’acqua in galleria ed influenza sull’ambiente. Meccanica e Ingegneria delle rocce, pp 143–199Google Scholar
  107. Rouleau A, Gale JE (1987) Stochastic discrete fracture simulation of groundwater flow into an underground excavation in granite. Int J Rock Mech Min Sci Geomech Abstr 24(2):99–112Google Scholar
  108. Rutqvist J (2015) Fractured rock stress–permeability relationships from in situ data and effects of temperature and chemical–mechanical couplings. Geofluids 15:48–66Google Scholar
  109. Scesi L, Gattinoni P (2009) Water circulation in rocks. Springer, London, 165 pp. ISBN: 9789048124169Google Scholar
  110. Sievanen U (2001) Leakage and groutability. Working Report 2001-06Google Scholar
  111. Snow DT (1969) Anisotropie permeability of fractured media. Water Resour Res 5(6):1273–1289Google Scholar
  112. Su K, Zhou Y, Wu H, Shi C, Zhou L (2017) An analytical method for groundwater inflow into a drained circular tunnel. Groundwater 55(5):712–721Google Scholar
  113. Tolppanen P (1997) Water Leakage amounts in excavations, literature study. Memorandum T-2000-18/97 (based on a memorandum of 1996). Consulting Engineers Saanio & Riekkola Oy, Helsinki, Finland (in Finnish)Google Scholar
  114. Ündül Ö, Tuğrul A (2012) The influence of weathering on the engineering properties of dunites. Rock Mech Rock Eng 45(2):225–239Google Scholar
  115. Vigna B, D’Angeli IM, De Waele J (2017) Hydrogeological flow in gypsum karst areas: some examples from northern Italy and main circulation models. Int J Speleol 46(2):205–217Google Scholar
  116. Walsh JB (1981) Effect of pore pressure and confining pressure on fracture permeability. Int J Rock Mech Min Sci Geomech Abstr 18:429–435Google Scholar
  117. Wang M, Kulatilake PHSW (2008) Understanding of hydraulic properties from configurations of stochastically distributed fracture networks. Hydrol Process 22(8):1125–1135Google Scholar
  118. Wang XY, Wang MS, Zhang M (2004) A simple method to calculate tunnel discharge and external water pressure on lining. J Northern Jiaotong Univ 28:8–10Google Scholar
  119. Xu Z, Zhao Z, Sun J (2013) Determination of water flow rate into subsea deep rock cavern with horseshoe cross-section. In: Wu F, Qi S (eds) Global view of engineering geology and the environment. CRC Press, Beijing, China, pp 345–349Google Scholar
  120. Yang G, Wang X, Wang X, Cao Y (2016) Analyses of seepage problems in a subsea tunnel considering effects of grouting and lining structure. Mar Georesour Geotechnol 34(1):65–70Google Scholar
  121. Zarei HR, Uromeihy A, Sharifzadeh M (2011) Evaluation of high local groundwater inflow to a rock tunnel by characterization of geological features. Tunn Undergr Space Technol 26(2):364–373Google Scholar
  122. Zhan SS, Teng ZR, Lin JH, Wang TT, Huang TH (2014) Determination of aperture characteristics and hydraulic conductivity of fracture set using core laboratory tests. In: Proceedings of the ISRM international symposium—8th Asian rock mechanics symposium, Sapporo, Japan, October 2014Google Scholar
  123. Zhang L, Franklin JA (1993) Prediction of water flow into rock tunnels: an analytical solution assuming an hydraulic conductivity gradient. Int J Rock Mech Min Sci Geomech Abstr 30(1):37–46Google Scholar
  124. Zhang W, Dai B, Liu Z, Zhou C (2017) On the non-Darcian seepage flow field around a deeply buried tunnel after excavation. Bull Eng Geol Environ 1–13Google Scholar
  125. Zhao J (1998) Rock mass hydraulic conductivity of the Bukit Timah granite. Sing Eng Geol 50:211–216Google Scholar
  126. Zhou CB, Sharma RS, Chen YF, Rong G (2008) Flow–stress coupled permeability tensor for fractured rock masses. Int J Numer Anal Methods Geomech 32(11):1289–1309Google Scholar
  127. Zimmerman RW, Bodvarsson GS (1995) Effective transmissivity of two-dimensional fracture networks. Report number: LBL-37332Google Scholar
  128. Zoorabadi M, Indraratna B, Nemcik JA (2012a) A new equation for the equivalent hydraulic conductivity of rock mass around a tunnel. Int J Rock Mech Min Sci 54:125–128Google Scholar
  129. Zoorabadi M, Saydam S, Timms W, Hebblewhite B (2012b) Sensitivity analysis of effective parameters on the permeability of rock mass around a tunnel. In: Proceedings of the ISRM regional symposium—7th Asian rock mechanics symposium, Seoul, Korea, 15–19 October 2012Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mining Engineering, Faculty of EngineeringUniversity of BirjandBirjandIran
  2. 2.Department of Civil Engineering and Engineering MechanicsUniversity of ArizonaTucsonUSA

Personalised recommendations