Abstract
The permeability evolution of fractal-based two-dimensional discrete fracture networks (DFNs) during shearing under constant normal stiffness (CNS) boundary conditions is numerically modeled and analyzed based on a fully coupled hydromechanical (HM) model. The effects of fractal dimension, boundary normal stiffness and hydraulic pressure on the evolutions of mechanical behaviors, aperture distributions and permeability are quantitatively investigated. The results show that with increasing confining pressure from 0 to 30 MPa, the permeability decreases from the magnitude of 10−13 m2 to 10−16 m2, which is generally consistent with previous models reported in the literature. With the increment of shear displacement from 0 to 500 mm, the variations in shear stress, normal stress and normal displacement exhibit the same patterns with the conceptual model. As shear advances, the permeability evolution exhibits a three-stage behavior. In the first stage, the permeability decreases due to the compaction of fractures induced by the increasing shear stress from 0 to the peak value. In the second stage, the permeability holds almost constant values under constant normal load (CNL) boundary conditions, whereas that under CNS boundary conditions decreases by approximately one order of magnitude. Under CNS boundary conditions, although the aperture of shearing fracture increases enhancing its own permeability, the apertures of surrounding fractures are compacted due to the simultaneous increases in the normal and shear stresses, which result in the decrease in the total permeability of DFNs. When the fractal dimension increases from 1.4 to 1.5, the permeability increases following exponential functions in the early stage of shear, which fail to characterize the permeability in the residual stage due to the complex flow path distributions. At the start of shear, the ratio of permeability perpendicular to the shear direction to that parallel to shear decreases approximately from 1.0 to 0.5 and then gradually decreases from 0.5 to 0.3 in the residual stage. The hydraulic pressure tends to open up the fractures and enhances the permeability. The magnitude in permeability enhancement is of approximately the same order with the increase in the hydraulic pressure.
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References
Anderson JG, Wesnousky SG, Stirling MW (1996) Earthquake size as a function of fault slip rate. Bull Seismol Soc Am 86(3):683–690
Andrade JS Jr, Oliveira EA, Moreira AA, Herrmann HJ (2009) Fracturing the optimal paths. Phys Rev Lett 103(22):225503
Baghbanan A, Jing L (2008) Stress effects on permeability in a fractured rock mass with correlated fracture length and aperture. Int J Rock Mech Min Sci 45(8):1320–1334
Balberg I, Anderson CH, Alexander S, Wagner N (1984) Excluded volume and its relation to the onset of percolation. Phys Rev B 30(7):3933
Barton CC (1995) Fractal analysis of scaling and spatial clustering of fracture. In: Barton CC, LaPointe PR (eds) Fractals in the earth sciences. Plenum, New York, pp 141–178
Barton CA, Zoback MD (1992) Self-similar distribution and properties of macro-scopic fractures at depth in crystalline rock in the Cajon Pass Scientific Drill Hole. J Geophys Res Solid Earth 97(B4):5181–5200
Bisdom K, Bertotti G, Nick HM (2016) The impact of different aperture distribution models and critical stress criteria on equivalent permeability in fractured rocks. J Geophys Res 121(5):4045–4063
Bisdom K, Nick HM, Bertotti G (2017) An integrated workflow for stress and flow modelling using outcrop-derived discrete fracture networks. Comput Geosci 103:21–35
Bour O, Davy P (1998) On the connectivity of three-dimensional fault networks. Water Resour Res 34(10):2611–2622
Brune JN (1968) Seismic moment, seismicity, and rate of slip along major fault zones. J Geophys Res 73(2):777
Chelidze T, Gueguen Y (1990) Evidence of fractal fracture. Int J Rock Mech Min Sci 27(3):223–225
Chen D, Pan Z, Ye Z (2015) Dependence of gas shale fracture permeability on effective stress and reservoir pressure: model match and insights. Fuel 139:383–392
Cvetkovic V, Painter S, Outters N, Selroos JO (2004) Stochastic simulation of radionuclide migration in discretely fractured rock near the Äspö Hard Rock Laboratory. Water Resources Res 40(2):2004
Davy P (1993) On the frequency-length distribution of the San Andreas fault system. J Geophys Res 98(B7):12141–12151
De Dreuzy JR, Davy P, Bour O (2001) Hydraulic properties of two-dimensional random fracture networks following a power law length distribution: 1. Effective connectivity. Water Resources Res 37(8):2065–2078
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. https://doi.org/10.1029/2012JB009461
Dverstorp B, Andersson J (1989) Application of the discrete fracture network concept with field data: possibilities of model calibration and validation. Water Resour Res 25(3):540–550
Englman R, Gur Y, Jaeger Z (1983) Fluid flow through a crack network in rocks. J Appl Mech 50(4a):707–711
Esaki T, Du S, Mitani Y, Ikusada Jing L (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
Fang Y, Elsworth D, Wang C, Ishibashi T, Fitts JP (2017) Frictional stability-permeability relationships for fractures in shales. J Geophys Res 122(3):1760–1776
Fryer B, Siddiqi G, Laloui L (2020) Injection-induced seismicity: strategies for reducing risk using high stress path reservoirs and temperature-induced stress preconditioning. Geophys J Int 220(2):1436–1446
Grant MA, Donaldson IG, Bixley PF (1983) Geotherm Reservoir Eng. Academic Press, New York
Gueguen Y, Dienes J (1989) Transport properties of rocks from statistics and percolation. Math Geol 21(1):1–13
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 571:416–424
Indraratna B, Haque A, Aziz N (1999) Shear behaviour of idealized infilled joints under constant normal stiffness. Géotechnique 49(3):331–355
Indraratna B, Thirukumaran S, Brown ET, Zhu SP (2015) Modelling the shear behaviour of rock joints with asperity damage under constant normal stiffness. Rock Mech Rock Eng 48(1):179–195
Itasca Consulting Group Inc. (2014) UDEC User’s Guide, ver 4.0. Minneapolis, Minnesota
Jafari A, Babadagli T (2012) Estimation of equivalent fracture network permeability using fractal and statistical network properties. J Petrol Sci Eng 92:110–123
Jiang Y, Xiao J, Tanabashi Y, Mizokami T (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
Jiang Y, Li B, Tanabashi Y (2006) Estimating the relation between surface roughness and mechanical properties of rock joints. Int J Rock Mech Min Sci 43(6):837–846
Johnston IW, Lam TSK, Williams AF (1987) Constant normal stiffness direct shear testing for socketed pile design in weak rock. Geotechnique 37(1):83–89
Juanes R, Spiteri EJ, Orr FM Jr, Blunt MJ (2006) Impact of relative permeability hysteresis on geological CO2 storage. Water Resources Res 42(12):W12418
Kolyukhin D, Torabi A (2013) Power-law testing for fault attributes distributions. Pure Appl Geophys 170(12):2173–2183
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 119(8):6288–6307
Lang PS, Paluszny A, Nejati M, Zimmerman RW (2018) Relationship between the orientation of maximum permeability and intermediate principal stress in fractured rocks. Water Resour Res 54(11):8734–8755
Latham JP, Xiang J, Belayneh M, Nick HM, Tsang CF, Blunt MJ (2013) Modelling stress-dependent permeability in fractured rock including effects of propagating and bending fractures. Int J Rock Mech Min Sci 57:100–112
Lee YK, Park JW, Song JJ (2014) Model for the shear behavior of rock joints under CNL and CNS conditions. Int J Rock Mech Min Sci 70:252–263
Lei Q, Latham JP, Xiang J, Tsang CF, Lang P, Guo L (2014) Effects of geomechanical changes on the validity of a discrete fracture network representation of a realistic two-dimensional fractured rock. Int J Rock Mech Min Sci 70:507–523
Lei Q, Latham JP, Xiang J, Tsang CF (2015) Polyaxial stress-induced variable aperture model for persistent 3D fracture networks. Geomech Energy Environ 1:34–47
Leung CTO, Zimmerman RW (2012) Estimating the hydraulic conductivity of two-dimensional fracture networks using network geometric properties. Transp Porous Media 93(3):777–797
Li B, Jiang YJ, Koyama T, Jing LR (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
Liu R, Jiang Y, Li B, Wang X (2015) A fractal model for characterizing fluid flow in fractured rock masses based on randomly distributed rock fracture networks. Comput Geotech 65:45–55
Liu R, Li B, Jiang Y (2016) A fractal model based on a new governing equation of fluid flow in fractures for characterizing hydraulic properties of rock fracture networks. Comput Geotech 75:57–68
Liu R, Li B, Yu L, Jiang Y, Jing H (2018) A discrete-fracture-network fault model revealing permeability and aperture evolutions of a fault after earthquakes. Int J Rock Mech Min Sci 107:19–24
Liu R, Wang C, Li B, Jiang Y, Jing H (2020) Modeling linear and nonlinear fluid flow through sheared rough-walled joints taking into account boundary stiffness. Comput Geotech 120:103452
Long JCS, Remer JS, Wilson CR, Witherspoon PA (1982) Porous media equivalents for networks of discontinuous fractures[. Water Resour Res 18(3):645–658
Ma W, Wang Y, Wu X, Liu G (2020) Hot dry rock (HDR) hydraulic fracturing propagation and impact factors assessment via sensitivity indicator. Renewable Energy 146:2716–2723
MacMinn CW, Szulczewski ML, Juanes R (2010) CO2 migration in saline aquifers. Part 1. Capillary trapping under slope and groundwater flow. J Fluid Mech 662:329–351
Mandelbrot BB (1982) The fractal geometry of nature. San Francisco, CA
Matsuki K, Lee JJ, Sakaguchi K (1999) Size effect in flow conductance of a closed small-scale hydraulic fracture in granite. Geotherm Sci Technol 6(1–4):113–138
Miao T, Yu B, Duan Y, Fang Q (2015) A fractal analysis of permeability for fractured rocks. Int J Heat Mass Transf 81:75–80
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):1191–1210
Mora P, Wang Y, Alonso-Marroquin F (2015) Lattice solid/Boltzmann microscopic model to simulate solid/fluid systems—a tool to study creation of fluid flow networks for viable deep geothermal energy. J Earth Sci 26(1):11–19
Nolen-Hoeksema RC, Gordon RB (1987) Optical detection of crack patterns in the opening-mode fracture of marble. Int J Rock Mech Min Sci 24(2):135–144
Oliveira DAF, Indraratna B (2010) Comparison between models of rock discontinuity strength and deformation. J Geotech Geoenviron Eng 136(6):864–874
Olsson R, Barton N (2001) An improved model for hydromechanical coupling during shearing of rock joints. Int J Rock Mech Min Sci 38(3):317–329
Pan JB, Lee CC, Lee CH, Yeh HF, Lin HI (2010) Application of fracture network model with crack permeability tensor on flow and transport in fractured rock. Eng Geol 116(1–2):166–177
Peng Z, Gomberg J (2010) An integrated perspective of the continuum between earthquakes and slow-slip phenomena. Nat Geosci 3:599–607
Prassetyo SH, Gutierrez M, Barton N (2017) Nonlinear shear behavior of rock joints using a linearized implementation of the Barton-Bandis model. J Rock Mech Geotech Eng 9(4):671–682
Ramsay JG (1980) Shear zone geometry: a review. J Struct Geol 2(1–2):83–99
Rasouli V, Hosseinian A (2011) Correlations developed for estimation of hydraulic parameters of rough fractures through the simulation of JRC flow channels. Rock Mech Rock Eng 44(4):447–461
Reeves DM, Parashar R, Pohll G, Carroll R, Badger T, Willoughby K (2013) The use of discrete fracture network simulations in the design of horizontal hillslope drainage networks in fractured rock. Eng Geol 163:132–143
Robinson PC (1984) Numerical calculations of critical densities for lines and planes. J Phys A: Math Gen 17(14):2823
Saeb S, Amadei B (1990) Modelling joint response under constant or variable normal stiffness boundary conditions. Int J Rock Mech Min Sci 27(3):213–217
Sahimi M (1993) Flow phenomena in rocks: from continuum models to fractals, percolation, cellular automata, and simulated annealing. Rev Mod Phys 65(4):1393
Schmidt DA, Burgmann R, Nadeau RM, d’Alessio M (2005) Distribution of aseismic slip rate on the Hayward fault inferred from seismic and geodetic data. J Geophys Res 110:B08406
Shrivastava AK, Rao KS (2015) Shear behaviour of rock joints under CNL and CNS boundary conditions. Geotech Geol Eng 33(5):1205–1220
Thirukumaran S, Indraratna B (2016) A review of shear strength models for rock joints subjected to constant normal stiffness. J Rock Mech Geotech Eng 8(3):405–414
Tsang CF (2005) Is current hydrogeologic research addressing long-term predictions? Groundwater 43(3):296–300
Tsang YW, Tsang CF, Hale FV, Dverstorp B (1996) Tracer transport in a stochastic continuum model of fractured media. Water Resour Res 32(10):3077–3092
Tsang CF, Neretnieks I, Tsang Y (2015) Hydrologic issues associated with nuclear waste repositories. Water Resour Res 51(9):6923–6972
Vignes-Adler M, Le Page A, Adler PM (1991) Fractal analysis of fracturing in two African regions, from satellite imagery to ground scale. Tectonophysics 196(1):69–86
Wang Y, Li X, Zhou RQ, Tang CA (2016) Numerical evaluation of the shear stimulation effect in naturally fractured formations. Sci China Earth Sci 59(2):371–383
Xie LZ, Gao C, Ren L et al (2015) Numerical investigation of geometrical and hydraulic properties in a single rock fracture during shear displacement with the Navier-Stokes equations. Environ Earth Sci 73(11):7061–7074
Xiong X, Li B, Jiang Y, 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(8):1292–1302
Xue L, Li HB, Brodsky EE, Xu Z, Kano Y, Wang H, Mori JJ, Si J, Pei J, Zhang W, Yang G, Sun Z, Huang Y (2013) Continuous permeability measurements record healing inside the Wenchuan earthquake fault zone. Science 340(6140):1555–1559
Yang H, Yang T, Zhang S, Zhao S, Hu K, Jiang Y (2020) Rainfall-induced landslides and debris flows in Mengdong Town, Yunnan Province, China. Landslides. https://doi.org/10.1007/s10346-019-01336-y
Yeo IW, De Freitas MH, Zimmerman RW (1998) Effect of shear displacement on the aperture and permeability of a rock fracture. Int J Rock Mech Min Sci 35(8):1051–1070
Zhang X, Sanderson DJ (1996) Effects of stress on the two-dimensional permeability tensor of natural fracture networks. Geophys J Int 125(3):912–924
Zhou CB, Sharma RS, Chen YF, Rong G (2008) Flow–stress coupled permeability tensor for fractured rock masses. Int J Numer Anal Meth Geomech 32(11):1289–1309
Zhuang L, Kim KY, Jung SG, Diaz M, Min KB (2019) Effect of water infiltration, injection rate and anisotropy on hydraulic fracturing behavior of granite. Rock Mech Rock Eng 52(2):575–589
Zimmerman RW, Bodvarsson GS (1996) Hydraulic conductivity of rock fractures. Transp Porous Media 23(1):1–30
Zimmerman RW, Al-Yaarubi A, Pain CC et al (2004) Non-linear regimes of fluid flow in rock fractures. Int J Rock Mech Min Sci 41:163–169
Acknowledgements
This study has been partially funded by Natural Science Foundation of Zhejiang Province, China (Grant No. LR19E090001) and Natural Science Foundation of China, China (Grant Nos. 51609136, 51979272, 51709260). Miss Larissa Kamseu helped in the data processing. These supports are gratefully acknowledged.
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Li, B., Bao, R., Wang, Y. et al. Permeability Evolution of Two-Dimensional Fracture Networks During Shear Under Constant Normal Stiffness Boundary Conditions. Rock Mech Rock Eng 54, 409–428 (2021). https://doi.org/10.1007/s00603-020-02273-2
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DOI: https://doi.org/10.1007/s00603-020-02273-2