Rock Mechanics and Rock Engineering

, Volume 52, Issue 11, pp 4731–4746 | Cite as

Experimental and Numerical Studies of the Hydraulic Properties of Three-Dimensional Fracture Networks with Spatially Distributed Apertures

  • Na Huang
  • Yujing JiangEmail author
  • Richeng Liu
  • Bo Li
Original Paper


Fluid flow tests on three-dimensional (3D) transparent fracture networks were performed to clarify the effect of aperture heterogeneity on the hydraulic properties of rough fractures. The five-axis machining combined with the 3D construction technique was applied to create the transparent sample, which ensures the direct visualization and quantification of flow behavior through the fractures. The effects of the anisotropic aperture on flow channeling and permeability of the fracture networks were evaluated using a developed numerical code, the validity of which has been verified by the flow test results. The numerical investigation was extended to complex 3D discrete fracture networks with large fracture densities. The results show that five-axis machining can reproduce a 3D fracture network sample with spatially distributed apertures by first machining the fracture blocks and subsequently assembling them. An obvious channeling flow in the sample was observed during the flow tests. With decreasing mean and/or increasing deviation of the aperture fields following a normal distribution, the localization of fluid flow is increasingly obvious; the main flow channels account for approximately 26–67% of the fracture planes. The ratio of permeability of the model with apertures following truncated lognormal distributions to those following complete lognormal distributions first showed remarkably increase and subsequently approached 1.0 with the increasing truncation threshold of the aperture distribution. A mathematical expression is proposed for predicting the critical truncation threshold, which is defined when the normalized permeability is equal to 0.9. The suitability of upscaling the proposed expression to complex 3D discrete fracture networks is verified, which can help to determine the proper truncation threshold when simulating a fracture aperture with lognormal distribution functions in fractured rock masses.


Fracture network Flow test Aperture Channeling flow Truncation threshold 

List of Symbols


Coefficient in the Forchheimer’s law representing viscous effect


Cross-sectional area of the fracture


Coefficient in the Forchheimer’s law representing inertial effect


The power law exponent


Fracture aperture


Mean value of normal distribution of aperture


Truncation threshold of aperture


Critical truncation threshold of aperture


Body force


Gravitational acceleration


Hydraulic head


The trace of the hydraulic head on Sk in fracture plane f


Hydraulic gradient


The equivalent permeability


The equivalent permeability of the model with parallel plates


The equivalent permeability calculated with aperture following truncated lognormal distribution


The equivalent permeability calculated with aperture following complete lognormal distribution


The equivalent permeability obtained by laboratorial experiment


The equivalent permeability obtained by numerical simulation


Fracture length


Fracture number


Hydraulic pressure


Flow rate


The ratios of the area of preferential flow paths to that of the total fracture planes


The relative error of permeability between experimental and numerical results


Fracture intersection line




The logarithm of the mean value of lognormal distribution of aperture


Flow velocity tensor


Fracture width


Fluid density


The ratio of permeability the complete lognormal distribution


Dynamic viscosity


The deviation of normal distribution of aperture


The deviation of lognormal distribution of aperture



This study has been partially funded by National Natural Science Foundation of China, China (Grant nos. 51709260, 51609136), Natural Science Foundation of Jiangsu Province, China (no. BK20170276), Japan Society for the Promotion of Science (JSPS) (no. P17382), and JSPS Grant-in-Aid for Scientific Research (no. 17H03506). These supports are gratefully acknowledged.


  1. Ahmad R, Soberi MSF (2018) Changeover process improvement based on modified SMED method and other process improvement tools application: an improvement project of 5-axis CNC machine operation in advanced composite manufacturing industry. Int J Adv Manuf Technol 94(1–4):433–450Google Scholar
  2. Anderson MP (1984) Movement of contaminants in groundwater: groundwater transport–advection and dispersion. Groundw Contam 1984:37–45Google Scholar
  3. Ando K, Kostner A, Neuman SP (2003) Stochastic continuum modeling of flow and transport in a crystalline rock mass: Fanay-Augeres, France, revisited. Hydrogeol J 11(5):521–535Google Scholar
  4. Baghbanan A, Jing L (2007) Hydraulic properties of fractured rock masses with correlated fracture length and aperture. Int J Rock Mech Min Sci 44(5):704–719Google Scholar
  5. Barton N, Choubey V (1977) The shear strength of rock joints in theory and practice. Rock Mech 10(1–2):1–54Google Scholar
  6. Barton N, Bandis S, Bakhtar K (1985) Strength, deformation and conductivity coupling of rock joints. Int J Rock Mech Min Sci Geomech Abstr 22(3):121–140 (Pergamon) Google Scholar
  7. Bear J (2013) Dynamics of fluids in porous media. Courier Corporation, North ChelmsfordGoogle Scholar
  8. Berrone S, Fidelibus C, Pieraccini S, Scialo S, Vicini F (2018) Unsteady advection-diffusion simulations in complex discrete fracture networks with an optimization approach. J Hydrol 566:322–345Google Scholar
  9. Black JH, Woodman ND, Barker JA (2017) Groundwater flow into underground openings in fractured crystalline rocks: an interpretation based on long channels. Hydrogeol J 25(2):445–463Google Scholar
  10. Bodvarsson GS, Boyle W, Patterson R, Williams D (1999) Overview of scientific investigations at Yucca Mountain—the potential repository for high-level nuclear waste. J Contam Hydrol 38(1–3):3–24Google Scholar
  11. Bouchaud E, Lapasset G, Planes J (1990) Fractal dimension of fractured surfaces: a universal value? EPL (Europhys Lett) 13(1):73Google Scholar
  12. Brown SR, Kranz RL, Bonner BP (1986) Correlation between the surfaces of natural rock joints. Geophys Res Lett 13(13):1430–1433Google Scholar
  13. de Dreuzy JR, Méheust Y, Pichot G (2012) Influence of fracture scale heterogeneity on the flow properties of three-dimensional discrete fracture networks (DFN). J Geophys Res Solid Earth 117:B11Google Scholar
  14. Dershowitz WS, Einstein HH (1988) Characterizing rock joint geometry with joint system models. Rock Mech Rock Eng 21(1):21–51Google Scholar
  15. Dessirier B, Tsang CF, Niemi A (2018) A new scripting library for modeling flow and transport in fractured rock with channel networks. Comput Geosci 111:181–189Google Scholar
  16. Dijk P, Berkowitz B, Bendel P (1999) Investigation of flow in water-saturated rock fractures using nuclear magnetic resonance imaging (NMRI). Water Resour Res 35(2):347–360Google Scholar
  17. Ebigbo A, Lang PS, Paluszny A, Zimmerman RW (2016) Inclusion-based effective medium models for the permeability of a 3D fractured rock mass. Transp Porous Media 113(1):137–158Google Scholar
  18. Einstein HH, Baecher GB (1983) Probabilistic and statistical methods in engineering geology. Rock Mech Rock Eng 16(1):39–72Google Scholar
  19. El-Kadi AI (1986) A computer program for generating two-dimensional fields of autocorrelated parameters. Groundwater 24(5):663–667Google Scholar
  20. Foias C, Manley O, Rosa R, Temam R (2001) Navier-Stokes equations and turbulence. Cambridge University Press, CambridgeGoogle Scholar
  21. Hakami E, Larsson E (1996) Aperture measurements and flow experiments on a single natural fracture. Int J Rock Mech Min Sci 33(4):395–404Google Scholar
  22. Head D, Vanorio T (2016) Effects of changes in rock microstructures on permeability: 3-D printing investigation. Geophys Res Lett 43(14):7494–7502Google Scholar
  23. Huang N, Jiang Y, Li B, Liu R (2016) A numerical method for simulating fluid flow through 3-D fracture networks[J]. J Natural Gas Sci Eng 33:1271–1281Google Scholar
  24. Huang N, Liu R, Jiang Y, Li B, Yu L (2018) Effects of fracture surface roughness and shear displacement on geometrical and hydraulic properties of three-dimensional crossed rock fracture models. Adv Water Resour 113:30–41Google Scholar
  25. Huang N, Liu R, Jiang Y, Cheng Y, Li B (2019) Shear-flow coupling characteristics of a three-dimensional discrete fracture network-fault model considering stress-induced aperture variations. J Hydrol 571:416–424Google Scholar
  26. Hudson JA, Priest SD (1983) Discontinuity frequency in rock masses. Int J Rock Mech Min Sci Geomech Abstr. 20(2):73–89 (Pergamon) Google Scholar
  27. Hyman JD, Karra S, Makedonska N, Gable CW, Painter SL, Viswanathan HS (2015) dfnWorks: a discrete fracture network framework for modeling subsurface flow and transport. Comput Geosci 84:10–19Google Scholar
  28. Hyman JD, Aldrich G, Viswanathan H, Makedonska N, Karra S (2016) Fracture size and transmissivity correlations: implications for transport simulations in sparse three-dimensional discrete fracture networks following a truncated power law distribution of fracture size. Water Resour Res 52(8):6472–6489Google Scholar
  29. Illman WA, Liu X, Takeuchi S, Yeh TCJ, Ando K, Saegusa H (2009) Hydraulic tomography in fractured granite: Mizunami Underground Research site, Japan. Water Resour Res 45:1Google Scholar
  30. Ishibashi T, Watanabe N, Hirano N, Okamoto A, Tsuchiya N (2012) GeoFlow: a novel model simulator for prediction of the 3-D channeling flow in a rock fracture network. Water Resour Res 48:7Google Scholar
  31. Jing Y, Armstrong RT, Mostaghimi P (2017) Rough-walled discrete fracture network modelling for coal characterisation. Fuel 191:442–453Google Scholar
  32. Johns RA, Steude JS, Castanier LM, Roberts PV (1993) Nondestructive measurements of fracture aperture in crystalline rock cores using X ray computed tomography. J Geophys Res Solid Earth 98(B2):1889–1900Google Scholar
  33. Ju Y, Xie H, Zheng Z, Lu J, Mao L, Gao F, Peng R (2014) Visualization of the complex structure and stress field inside rock by means of 3D printing technology. Chin Sci Bull 59(36):5354–5365Google Scholar
  34. Jun CS, Cha K, Lee YS (2003) Optimizing tool orientations for 5-axis machining by configuration-space search method. Comput Aided Des 35(6):549–566Google Scholar
  35. Keller A (1998) High resolution, non-destructive measurement and characterization of fracture apertures. Int J Rock Mech Min Sci 35(8):1037–1050Google Scholar
  36. Keller AA, Roberts PV, Blunt MJ (1999) Effect of fracture aperture variations on the dispersion of contaminants. Water Resour Res 35(1):55–63Google Scholar
  37. Ketcham RA, Carlson WD (2001) Acquisition, optimization and interpretation of X ray computed tomographic imagery: applications to the geosciences. Comput Geosci 27(4):381–400Google Scholar
  38. Kilmer NH, Morrow NR, Pitman JK (1987) Pressure sensitivity of low permeability sandstones. J Petrol Sci Eng 1(1):65–81Google Scholar
  39. Lang PS, Paluszny A, Zimmerman RW (2014) Permeability tensor of three-dimensional fractured porous rock and a comparison to trace map predictions. J Geophys Res Solid Earth 119(8):6288–6307Google Scholar
  40. Lapcevic PA, Novakowski KS, Sudicky EA (1999) The interpretation of a tracer experiment conducted in a single fracture under conditions of natural groundwater flow. Water Resour Res 35(8):2301–2312Google Scholar
  41. Lee YS (1997) Admissible tool orientation control of gouging avoidance for 5-axis complex surface machining. Comput Aided Des 29(7):507–521Google Scholar
  42. Lee HS, Cho TF (2002) Hydraulic characteristics of rough fractures in linear flow under normal and shear load. Rock Mech Rock Eng 35(4):299–318Google Scholar
  43. Lee J, Kang JM, Choe J (2003) Experimental analysis on the effects of variable apertures on tracer transport. Water Resour Res 39:1Google Scholar
  44. Li Y, Sun S, Tang C (2019) Analytical prediction of the shear behaviour of rock joints with quantified waviness and unevenness through wavelet analysis. Rock Mech and Rock Eng. CrossRefGoogle Scholar
  45. Liu HH, Bodvarsson GS, Lu S, Molz FJ (2004) A corrected and generalized successive random additions algorithm for simulating fractional Levy motions. Math Geol 36(3):361–378Google Scholar
  46. Liu R, Li B, Jiang Y, Yu L (2018) A numerical approach for assessing effects of shear on equivalent permeability and nonlinear flow characteristics of 2-D fracture networks. Adv Water Resour 111:289–300Google Scholar
  47. Long JCS, Remer JS, Wilson CR, Witherspoon PA (1982) Porous media equivalents for networks of discontinuous fractures. Water Resour Res 18(3):645–658Google Scholar
  48. Madadi M, Sahimi M (2003) Lattice Boltzmann simulation of fluid flow in fracture networks with rough, self-affine surfaces. Phys Rev E 67(2):026309Google Scholar
  49. Masciopinto C (2005) Pumping-well data for conditioning the realization of the fracture aperture field in groundwater flow models. J Hydrol 309(1–4):210–228Google Scholar
  50. Min KB, Rutqvist J, Tsang CF, Jing L (2004) Stress-dependent permeability of fractured rock masses: a numerical study. Int J Rock Mech Min Sci 41(7):1191–1210Google Scholar
  51. Moreno L, Tsang YW, Tsang CF et al (1988) Flow and tracer transport in a single fracture: a stochastic model and its relation to some field observations. Water Resour Res 24(12):2033–2048Google Scholar
  52. Murphy HD, Tester JW, Grigsby CO, Potter RM (1981) Energy extraction from fractured geothermal reservoirs in low-permeability crystalline rock. J Geophys Res Solid Earth 86(B8):7145–7158Google Scholar
  53. Ngo TD, Fourno A, Noetinger B (2017) Modeling of transport processes through large-scale discrete fracture networks using conforming meshes and open-source software. J Hydrol 554:66–79Google Scholar
  54. Noetinger B, Jarrige N (2012) A quasi steady state method for solving transient Darcy flow in complex 3D fractured networks. J Comput Phys 231(1):23–38Google Scholar
  55. Power WL, Tullis TE (1991) Euclidean and fractal models for the description of rock surface roughness. J Geophys Res Solid Earth 96(B1):415–424Google Scholar
  56. Pruess K (2006) Enhanced geothermal systems (EGS) using CO2 as working fluid—a novel approach for generating renewable energy with simultaneous sequestration of carbon. Geothermics 35(4):351–367Google Scholar
  57. Quinn PM, Cherry JA, Parker BL (2011) Quantification of non-Darcian flow observed during packer testing in fractured sedimentary rock. Water Resour Res 47:9Google Scholar
  58. Renshaw CE (1995) On the relationship between mechanical and hydraulic apertures in rough-walled fractures. J Geophys Res Solid Earth 100(B12):24629–24636Google Scholar
  59. Suzuki A, Watanabe N, Li K, Horne RN (2017) Fracture network created by 3-D printer and its validation using CT images. Water Resour Res 53(7):6330–6339Google Scholar
  60. Tse R, Cruden DM (1979) Estimating joint roughness coefficients. Int J Rock Mech Min Sci Geomech Abstr 16(5):303–307Google Scholar
  61. Warkentin A, Hoskins P, Ismail F, Bedi S (2019) Computer aided 5-axis machining. In: Computer-aided design, engineering, and manufacturing. CRC Press, pp 118–151Google Scholar
  62. Witherspoon PA, Wang JSY, Iwai K, Gale JE (1980) Validity of cubic law for fluid flow in a deformable rock fracture. Water Resour Res 16(6):1016–1024Google Scholar
  63. Xiong X, Li B, Jiang Y, Koyama T, Zhang C (2011) Experimental and numerical study of the geometrical and hydraulic characteristics of a single rock fracture during shear. Int J Rock Mech Min Sci 48:1292–1302Google Scholar
  64. Yang Z, Niemi A, Fagerlund F, Illangasekare T (2012) Effects of single-fracture aperture statistics on entrapment, dissolution and source depletion behavior of dense non-aqueous phase liquids[J]. J Contam Hydrol 133:1–16Google Scholar
  65. Zębala W, Plaza M (2014) Comparative study of 3-and 5-axis CNC centers for free-form machining of difficult-to-cut material. Int J Prod Econ 158:345–358Google Scholar
  66. Zhao Z, Jing L, Neretnieks I (2010) Evaluation of hydrodynamic dispersion parameters in fractured rocks. J Rock Mech Geotech Eng 2(3):243–254Google Scholar
  67. Zhao Z, Li B, Jiang Y (2014) Effects of fracture surface roughness on macroscopic fluid flow and solute transport in fracture networks. Rock Mech Rock Eng 47(6):2279–2286Google Scholar
  68. Zhou JQ, Wang M, Wang L, Chen YF, Zhou CB (2018) Emergence of nonlinear laminar flow in fractures during shear. Rock Mech Rock Eng 51:1–9Google Scholar
  69. Zhu JB, Zhou T, Liao ZY, Sun L, Chen R (2018) Replication of internal defects and investigation of mechanical and fracture behaviour of rock using 3D printing and 3D numerical methods in combination with X ray computerized tomography. Int J Rock Mech Min Sci 106:198–212Google Scholar
  70. Zimmerman RW, Bodvarsson GS (1996) Hydraulic conductivity of rock fractures. Transp Porous Media 23(1):1–30Google Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Petroleum EngineeringChina University of Petroleum (East China)QingdaoChina
  2. 2.School of EngineeringNagasaki UniversityNagasakiJapan
  3. 3.State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and TechnologyShandong University of Science and TechnologyQingdaoPeople’s Republic of China
  4. 4.State Key Laboratory for Geomechanics and Deep Underground EngineeringChina University of Mining and TechnologyXuzhouPeople’s Republic of China
  5. 5.Center of Rock Mechanics and GeohazardsShaoxing UniversityShaoxingChina

Personalised recommendations