Abstract
Fluid flow through rock media is highly significant in underground water management, the geothermal recovery process, and various underground engineering applications. Fractures have a critical effect on the fluid flow through low-permeability rocks since they serve as the major flow channels in the rock formation. Consequently, the evaluation of fracture permeability is crucial to many engineering applications, such as the geothermal recovery process, exploitation of hydrocarbon resources, and various underground engineering applications. However, fluid flow through rough fractures is complex under the effects of rough profiles and variable apertures. The variation of fracture apertures causes the nonlinear distribution of the pressure along the fracture and thus intensifies the difficulties of studying the flow in fractures. In this study, variable-aperture fractures were simplified as axisymmetric fractures using the Weierstrass–Mandelbrot function. To address the nonlinear distribution of pressure caused by aperture variations, a method is proposed to segment fractures with variable lengths by considering the weights of fracture apertures. With the segmented results, we evaluated the permeability of rough fractures using a modified local law. The evaluation results aligned with the lattice Boltzmann simulation results. Finally, combining the analytical solution of flow through asymmetric fractures with sinusoidal profiles, the proposed methods were thereby validated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
The data used to support the findings of this study are available through the following link: https://doi.org/10.6084/m9.figshare.12083190.
References
An FH, Cheng YP (2014) An explanation of large-scale coal and gas outbursts in underground coal mines: the effect of low-permeability zones on abnormally abundant gas. Nat Hazards Earth Syst Sci 14:2125–2132. https://doi.org/10.5194/nhess-14-2125-2014
Bennour Z, Watanabe S, Chen Y et al (2018) Evaluation of stimulated reservoir volume in laboratory hydraulic fracturing with oil, water and liquid carbon dioxide under microscopy using the fluorescence method. Geomech Geophys Geo-Energy Geo-Resour 4:39–50. https://doi.org/10.1007/s40948-017-0073-3
Berry MV, Lewis ZV, Nye JF (1980) On the Weierstrass-Mandelbrot fractal function. Proc R Soc A Math Phys Eng Sci 370:459–484. https://doi.org/10.1098/rspa.1980.0044
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
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
Cheung NJ, Ding X, Shen H (2017) A nonhomogeneous cuckoo search algorithm based on quantum mechanism for real parameter optimization. IEEE Trans Cybern 47:391–402. https://doi.org/10.1109/TCYB.2016.2517140
D’Humières DD, Lallemand P, Frisch U (1986) Lattice gas models for 3D hydrodynamics. EPL 2:291–297. https://doi.org/10.1209/0295-5075/2/4/006
Develi K, Babadagli T (2015) Experimental and visual analysis of single-phase flow through rough fracture replicas. Int J Rock Mech Min Sci 73:139–155. https://doi.org/10.1016/j.ijrmms.2014.11.002
Drazer G, Auradou H, Koplik J, Hulin JP (2004) Self-affine fronts in self-affine fractures: large and small-scale structure. Phys Rev Lett 92:014501. https://doi.org/10.1103/PhysRevLett.92.014501
Ge S (1997) A governing equation for fluid flow in rough fractures. Water Resour Res 33:53–61. https://doi.org/10.1029/96WR02588
Gutfraind R, Hansen A (1995) Study of fracture permeability using lattice gas automata. Transp Porous Media 18:131–149. https://doi.org/10.1007/BF01064675
Hajjar A, Scholtès L, Oltéan C, Buès MA (2018) Effects of the geometry of two-dimensional fractures on their hydraulic aperture and on the validity of the local cubic law. Hydrol Process 32:2510–2525. https://doi.org/10.1002/hyp.13181
Hariri-Ardebili MA, Seyed-Kolbadi SM, Mirzabozorg H (2013) A smeared crack model for seismic failure analysis of concrete gravity dams considering fracture energy effects. Struct Eng Mech 48:17–39. https://doi.org/10.12989/sem.2013.48.1.017
Hyman JD, Jiménez-Martínez J (2018) Dispersion and mixing in three-dimensional discrete fracture networks: nonlinear interplay between structural and hydraulic heterogeneity. Water Resour Res 54:3243–3258. https://doi.org/10.1029/2018WR022585
Ishibashi T, Elsworth D, Fang Y et al (2018) Friction-stability-permeability evolution of a fracture in granite. Water Resour Res 54:9901–9918. https://doi.org/10.1029/2018WR022598
Jin Y, Dong JB, Zhang XY et al (2017) Scale and size effects on fluid flow through self-affine rough fractures. Int J Heat Mass Transf 105:443–451. https://doi.org/10.1016/j.ijheatmasstransfer.2016.10.010
Ju Y, Dong J, Gao F, Wang J (2019) Evaluation of water permeability of rough fractures based on a self-affine fractal model and optimized segmentation algorithm. Adv Water Resour 129:99–111. https://doi.org/10.1016/j.advwatres.2019.05.007
Komvopoulos K, Yan W (1997) A fractal analysis of stiction in microelectromechanical systems. J Tribol 119:391–400. https://doi.org/10.1115/1.2833500
Konzuk JS, Kueper BH (2004) Evaluation of cubic law based models describing single-phase flow through a rough-walled fracture. Water Resour Res 40:W02402. https://doi.org/10.1029/2003WR002356
Krüger T, Kusumaatmaja H, Kuzmin A et al (2016) The lattice boltzmann method: principles and practice. Springer, Berlin
Kumar H, Mishra MK, Mishra S (2019) Experimental and numerical evaluation of CBM potential in Jharia coalfield India. Geomech Geophys Geo-Energy Geo-Resour 5:289–314. https://doi.org/10.1007/s40948-019-00114-3
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 54:8734–8755. https://doi.org/10.1029/2018WR023189
Lee I-H, Ni C-F, Lin F-P et al (2019) Stochastic modeling of flow and conservative transport in three-dimensional discrete fracture networks. Hydrol Earth Syst Sci 23:19–34. https://doi.org/10.5194/hess-23-19-2019
Li K, Ma M, Wang X (2011) Experimental study of water flow behaviour in narrow fractures of cementitious materials. Cem Concr Compos 33:1009–1013. https://doi.org/10.1016/j.cemconcomp.2011.08.005
Li Q, Luo KH, Kang QJ et al (2016) Lattice Boltzmann methods for multiphase flow and phase-change heat transfer. Prog Energy Combust Sci 52:62–105. https://doi.org/10.1016/j.pecs.2015.10.001
Liu R, Li B, Jiang Y (2016) Critical hydraulic gradient for nonlinear flow through rock fracture networks: the roles of aperture, surface roughness, and number of intersections. Adv Water Resour 88:53–65. https://doi.org/10.1016/j.advwatres.2015.12.002
Lomize GM (1951) Flow in fractured rocks. Gosenergoizdat, Moscow (in Russian)
Louis C (1969) A study of groundwater flow in jointed rock and its influence on the stability of rock masses. Imperial College of Science and Technology, London
Lu Y, Wang L (2015) Numerical simulation of mining-induced fracture evolution and water flow in coal seam floor above a confined aquifer. Comput Geotech 67:157–171. https://doi.org/10.1016/j.compgeo.2015.03.007
Mandelbrot BB, Van Ness JW (1968) Fractional Brownian motions, fractional noises and applications. SIAM Rev 10:422–437. https://doi.org/10.1137/1010093
McClure JE, Prins JF, Miller CT (2014) A novel heterogeneous algorithm to simulate multiphase flow in porous media on multicore CPU–GPU systems. Comput Phys Commun 185:1865–1874. https://doi.org/10.1016/j.cpc.2014.03.012
Mourzenko VV, Thovert J-F, Adler PM (1995) Permeability of a single fracture; validity of the reynolds equation. J Phys II France 5:465–482. https://doi.org/10.1051/jp2:1995133
Mourzenko VV, Thovert J-F, Adler PM (1996) Geometry of simulated fractures. Phys Rev E 53:5606–5626. https://doi.org/10.1103/PhysRevE.53.5606
Mourzenko VV, Thovert J-F, Adler PM (1999) Percolation and conductivity of self-affine fractures. Phys Rev E 59:4265–4284. https://doi.org/10.1103/PhysRevE.59.4265
Nazridoust K, Ahmadi G, Smith DH (2006) A new friction factor correlation for laminar, single-phase flows through rock fractures. J Hydrol 329:315–328. https://doi.org/10.1016/j.jhydrol.2006.02.032
Oron AP, Berkowitz B (1998) Flow in rock fractures: the local cubic law assumption reexamined. Water Resour Res 34:2811–2825. https://doi.org/10.1029/98WR02285
Paramanathan P, Uthayakumar R (2010) Fractal interpolation on the Koch curve. Comput Math Appl 59:3229–3233. https://doi.org/10.1016/j.camwa.2010.03.008
Qian YH, D’Humieres D, Lallemand P (1992) Lattice BGK models for Navier–Stokes equation. EPL (Europhys Lett) 17:479–484. https://doi.org/10.1006/jcph.2000.6616
Renshaw CE (1995) On the relationship between mechanical and hydraulic apertures in rough-walled fractures. J Geophys Res 100:24629–24636. https://doi.org/10.1029/95JB02159
Renshaw CE, Dadakis JS, Brown SR (2000) Measuring fracture apertures: a comparison of methods. Geophys Res Lett 27:289–292. https://doi.org/10.1029/1999GL008384
Sahimi M, Mukhopadhyay S (1996) Scaling properties of a percolation model with long-range correlations. Phys Rev E 54:3870–3880. https://doi.org/10.1103/PhysRevE.54.3870
Sisavath S, Al-Yaaruby A, Pain CC, Zimmerman RW (2003) A Simple model for deviations from the cubic law for a fracture undergoing dilation or closure. Pure appl Geophys 160:1009–1022. https://doi.org/10.1007/PL00012558
Talon L, Auradou H, Hansen A (2010a) Permeability of self-affine aperture fields. Phys Rev E 82:046108. https://doi.org/10.1103/PhysRevE.82.046108
Talon L, Auradou H, Hansen A (2010b) Permeability estimates of self-affine fracture faults based on generalization of the bottleneck concept. Water Resour Res 46:W07601. https://doi.org/10.1029/2009WR008404
Tang Y, Ma T, Chen P, Ranjith PG (2020) An analytical model for heat extraction through multi-link fractures of the enhanced geothermal system. Geomech Geophys Geo-Energy Geo-Resour. https://doi.org/10.1007/s40948-019-00123-2
Tian ZW, Wang JY (2017) Lattice Boltzmann simulation of CO2 reactive transport in network fractured media. Water Resour Res 53:7366–7381. https://doi.org/10.1002/2017WR021063
Wang S, Komvopoulos K (1994) A fractal theory of the interfacial temperature distribution in the slow sliding regime: Part II—multiple domains, elastoplastic contacts and applications. J Tribol 116:824–832. https://doi.org/10.1115/1.2927341
Wang LC, Cardenas MB, Slottke DT et al (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
Wang M, Chen YF, Ma GW et al (2016) Influence of surface roughness on nonlinear flow behaviors in 3D self-affine rough fractures: lattice Boltzmann simulations. Adv Water Resour 96:373–388. https://doi.org/10.1016/j.advwatres.2016.08.006
Wang Z, Xu C, Dowd P (2018) A modified cubic law for single-phase saturated laminar flow in rough rock fractures. Int J Rock Mech Min Sci 103:107–115. https://doi.org/10.1016/j.ijrmms.2017.12.002
Watanabe N, Hirano N, Tsuchiya N (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. https://doi.org/10.1029/2006WR005411
Witherspoon PA, Wang JSY, Iwai K, Gale JE (1980) Validity of Cubic law for fluid flow in a deformable rock fracture. Water Resour Res 16:1016. https://doi.org/10.1029/WR016i006p01016
Wu J-J (2000) Characterization of fractal surfaces. Wear 239:36–47. https://doi.org/10.1016/S0043-1648(99)00362-2
Yang XS, Deb S (2010) Engineering optimisation by Cuckoo search. Int J Math Model Numer Optim 1:330–343. https://doi.org/10.1504/IJMMNO.2010.035430
Zhang CP, Cheng P, Lu YY et al (2020) Experimental evaluation of gas flow characteristics in fractured siltstone under various reservoir and injection conditions: an application to CO2-based fracturing. Geomech Geophys Geo-Energy Geo-Resour 6:23. https://doi.org/10.1007/s40948-020-00145-1
Zhu HG, Xie HP, Yi C et al (2013) Analysis of properties of fluid flow in rock fractures. Chin J Rock Mech Eng 32:657–663 (in Chinese)
Zimmerman RW, Bodvarsson GS (1996) Hydraulic conductivity of rock fractures. Transp Porous Media 23:1–30. https://doi.org/10.1007/BF00145263
Zou Q, He X (1997) On pressure and velocity boundary conditions for the lattice Boltzmann BGK model. Phys Fluids 9:1591–1598. https://doi.org/10.1063/1.869307
Acknowledgements
The authors gratefully acknowledge the financial support of the Fundamental Research Funds for the Central Universities (Grant No. 2018BSCXC37) and the Postgraduate Research and Practice Innovation Program of Jiangsu Province (Grant No. KYCX18_1974).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest. No conflict of interest exists in the submission of this manuscript, and the manuscript is approved by all authors for publication. I would like to declare on behalf of my co-authors that the work described was original research that has not been published previously, and is not under consideration for publication elsewhere in whole or in part.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Dong, J., Ju, Y. Quantitative characterization of single-phase flow through rough-walled fractures with variable apertures. Geomech. Geophys. Geo-energ. Geo-resour. 6, 42 (2020). https://doi.org/10.1007/s40948-020-00166-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40948-020-00166-w