Abstract
Determining effective hydraulic conductivity is critical to understand groundwater flow behavior in fractured rock masses and is required for large-scale numerical simulations. A new approach is developed to estimate the effective hydraulic conductivity of a discrete fracture network with aperture and trace length being stochastic and correlated. The aperture and length are assumed to follow a joint lognormal distribution. The average flowrate through the network is determined by integrating flowrate in the fracture ensemble with the laminar flow and non-linear flow being separately considered based on the Reynolds number. The effective hydraulic conductivity of the discrete fracture network is then determined from the idea that Darcy’s law still applies to the overall flow behavior through the network. For comparison, the simple average hydraulic conductivity is also developed based on the assumption that flow in all fractures is laminar and aperture and trace length are uncorrelated. Based on the developed approach, the effects of aperture and trace length uncertainty, their correlation degree, and non-linear flow characteristics on the ratio of effective hydraulic conductivity over simple average hydraulic conductivity are examined. The results demonstrate that the effective hydraulic conductivity can be either larger than or smaller than the simple average hydraulic conductivity depending on correlation level between aperture and trace length and the degree of non-linear effect in the network. Non-linear flow reduces the effective hydraulic conductivity of the fracture network. Correlation between aperture and trace length increases the effective hydraulic conductivity. Aperture uncertainty tends to reduce the effective hydraulic conductivity.
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References
Aitchison, J. and Brown, J.A.C., 1957, The Lognormal Distribution. Cambridge University Press, New York, 176 p.
Baecher, G.B., 1983, Statistical analysis of rock mass fracturing. Mathematical Geology, 15, 329–348. https://doi.org/10.1007/BF01036074
Baghbanan, A. and Jing, L., 2007, Hydraulic properties of fracture rock masses with correlated fracture length and aperture. International Journal of Rock Mechanics and Mining Sciences, 44, 704–19. https://doi.org/10.1016/j.ijrmms.2006.11.001
Baghbanan, A. and Jing, L., 2008, Stress effects on permeability in a fractured rock mass with correlated fracture length and aperture. International Journal of Rock Mechanics and Mining Sciences, 45, 1320–1334. https://doi.org/10.1016/j.ijrmms.2008.01.015
Barton, C.M., 1978, Analysis of joint traces. Proceedings of the 19th U.S. Symposium on Rock Mechanics, Reno, USA, May 1–3, p. 38–42.
Bear, J., 1972, Dynamics of Fluids in Porous Media. Courier Corporation, North Chelmsford, 764 p.
Bonnet, E., Bour, O., Odling, N.E., Davy, P., Main, I., Cowie, P., and Berkowitz, B., 2001, Scaling of fracture systems in geological media. Review of Geophysics, 39, 347–383. https://doi.org/10.1029/1999RG000074
Bridges, M.C., 1975, Presentation of fracture data for rock mechanics. Proceedings of the 2nd Australia-New Zealand Conference on Geomechanics, Brisbane, Australia, Jul. 21–25, p. 144–148.
Chelidze, T. and Guguen, Y., 1990, Evidence of fractal fracture. International Journal of Rock Mechanics and Mining Sciences, 27, 223–225. https://doi.org/10.1016/0148-9062(90)94332-N
Chen, Y.F., Zhou, J.Q., Hu, S.H., Hu, R., and Zhou, C.B., 2015, Evaluation of Forchheimer equation coefficients for non-Darcy flow in deformable rough-walled fractures. Journal of Hydrology, 529, 993–1006. https://doi.org/10.1016/j.jhydrol.2015.09.021
Crow, E.L. and Shimizu, K., 1988, Lognormal Distributions: Theory and Applications. Marcel Dekker, Inc., New York, 387 p.
Earle, S., 2015, Physical Geology. BCcampus Open Education, Victoria, B.C., 717 p. https://opentextbc.ca/geology/
Einstein, H.H. and Baecher, G.B., 1983, Probabilistic and statistical methods in engineering geology. Rock Mechanics and Rock Engineering, 16, 39–72. https://doi.org/10.1007/BF01030217
Gale, J.F.W., Reed, R.M., and Holder, J., 2007, Natural fractures in the Barnett Shale and their importance for hydraulic fracture treatments. American Association of Petroleum Geologists Bulletin, 91, 603–622. https://doi.org/10.1306/11010606061
Ghanbarian, B., Perfect, E., and Liu, H.-H., 2019, A geometrical aperture-width relationship for rock fractures. Fractals, 27, 1940002. https://doi.org/10.1142/S0218348X19400024
Hatton, C.G., Main, I.G., and Meredith, P.G., 1994. Non-universal of fracture length and opening displacement. Nature, 367, 160–162. https://doi.org/10.1038/367160a0
Hooker, J.N., Gale, J.F.W., Gomez, L.A., Laubach, S.E., Marrett, R., and Reed, R.M., 2009, Aperture-size scaling variations in a low-strain opening-mode fracture set, Cozzette Sandstone, Colorado. Journal of Structural Geology, 31, 707–718. https://doi.org/10.1016/j.jsg.2009.04.001
Javadi, M., Sharifzadeh, M., Shahriar, K., and Mitani, Y., 2014, Critical Reynolds number for nonlinear flow through rough-walled fractures: the role of shear processes. Water Resources Research, 50, 1789–1804. https://doi.org/10.1002/2013WR014610
Klimczak, C., Schultz, R.A., Parashar, R., and Reeves, D.M., 2010, Cubic law with aperture-length correlation: implications for network scale fluid flow. Hydrogeology Journal, 18, 851–862. https://doi.org/10.1007/s10040-009-0572-6
Koyama, T., Neretnieks, I., and Jing, L.A., 2007, A numerical study on differences in using Navier-Stokes and Reynolds equations for modeling the fluid flow and particle transport in single rock fractures with shear. International Journal of Rock Mechanics and Mining Sciences, 45, 1082–1101. https://doi.org/10.1016/j.ijrmms.2007.11.006
Laubach, S.E., Lamarche, J., Gauthier, B.D.M., Dunne, W.M., and Sanderson, D.J., 2018, Spatial arrangement of faults and opening-mode fractures. Journal of Structural Geology, 108, 2–15. https://doi.org/10.1016/j.jsg.2017.08.008
Liu, R., Yu, L., Jiang, Y., Wang, Y., and Li, B., 2017, Recent developments on relationships between the equivalent permeability and fractal dimension of two-dimensional rock fracture networks. Journal of Natural Gas Science and Engineering, 45, 771–785. https://doi.org/10.1016/j.jngse.2017.06.013
Ma, G.W., Wang, H.D., Fan, L.F., and Chen, Y., 2018, Segmented two-phase flow analysis in fractured geological medium based on the numerical manifold method. Advances in Water Resources, 121, 112–129. https://doi.org/10.1016/j.advwatres.2018.08.012
Manzocchi, T., 2002, The connectivity of two-dimensional networks of spatially correlated fractures. Water Resources Research, 38, 1162. https://doi.org/10.1029/2000WR000180
McMahon, B., 1974, Design of rock slopes against sliding on pre-existing fractures. Proceedings of the 3rd Congress of the International Society for Rock Mechanics, Denver, Sep. 1–7, p. 803–808.
Odling, N.E., 1997, Scaling and connectivity of joint systems in sandstone from western Norway. Journal of Structural Geology, 19, 1257–1271. https://doi.org/10.1016/S0191-8141(97)00041-2
Odling, N.E., Gillespie, P., Bourgine, B., Castaing, C., Chiles, J.P., Christensen, N.P., Fillion, E., Genter, A., Olsen, C., Thrane, L., Trice, R., Aarseth, E., Walsh, J.J., and Watterson, J., 1999, Variations in fracture system geometry and their implications for fluid flow in fractured hydrocarbon reservoirs. Petroleum Geoscience, 5, 373–384. https://doi.org/10.1144/petgeo.5.4.373
Pina, A., Donado, L.D., and Blessent, D., 2019, Analysis of the scale-dependence of the hydraulic conductivity in complex fractured media. Journal of Hydrology, 569, 556–572. https://doi.org/10.1016/j.jhydrol.2018.12.006
Priest, S.D. and Hudson, J.A., 1981, Estimation of discontinuity spacing and trace length using scanline surveys. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 18, 183–197. https://doi.org/10.1016/0148-9062(81)90973-6
Renshaw, C.E. and Park, J.C., 1997, Effect of mechanical interactions on the scaling of fracture length and aperture. Nature, 386, 482–684. https://doi.org/10.1038/386482a0
Rouleau, A. and Gale, J.E., 1985, Statistical characterization of the fracture system in the Stripa granite, Sweden. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 22, 353–367. https://doi.org/10.1016/0148-9062(85)90001-4
Sahimi, M., 1993, Flow phenomena in rocks: from continuum models to fractals, percolation, cellular automata, and simulated annealing. Review of Modern Physics, 65, 1393–1534. https://doi.org/10.1103/RevModPhys.65.1393
Scholz, C.H. and Cowie, P.A., 1990, Determination of total strain from faulting using slip measurements. Nature, 346, 837–839. https://doi.org/10.1038/346837a0
Schrauf, T.W. and Evans, D.D., 1986, Laboratory studies of gas flow through a single natural fracture. Water Resources Research, 22, 1038–1050. https://doi.org/10.1029/WR022i007p01038
Stone, D., 1984, Sub-surface fracture maps predicted from borehole data: an example from the Eye-Dashwa Pluton, Atikolan, Canada. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 21, 183–194. https://doi.org/10.1016/0148-9062(84)90795-2
Torabi, A. and Berg, S.S., 2011, Scaling of fault attributes: a review. Marine and Petroleum Geology, 28, 1444–1460. https://doi.org/10.1016/j.marpetgeo.2011.04.003
Vermilye, J.M. and Scholz, C.H., 1995, Relation between vein length and aperture. Journal of Structural Geology, 17, 423–434. https://doi.org/10.1016/0191-8141(94)00058-8
Ward, J.C., 1964, Turbulent flow in porous media. Journal of the Hydraulics Division, 90, 1–12.
Witherspoon, P.A., Wang, J.S.Y., Iwai, K., and Gale, J.E., 1980, Validity of cubic law for fluid flow in a deformable rock fracture. Water Resources Research, 16, 1016–1024. https://doi.org/10.1029/WR016i006p01016
Yan, C. and Zheng, H., 2016, Three-dimensional hydromechanical model of hydraulic fracturing with arbitrarily discrete fracture networks using finite-discrete element method. International Journal of Geomechanics, 17, 04016133. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000819
Zhu, J., 2019, Impact of scaling break and fracture orientation on effective permeability of fractal fracture network. SN Applied Sciences, 1, 4. https://doi.org/10.1007/s42452-018-0002-2
Zhu, J. and Cheng, Y., 2018, Effective permeability of fractal fracture rocks: significance of turbulent flow and fractal scaling. International Journal of Heat and Mass Transfer, 116, 549–556. https://doi.org/10.1016/j.ijheatmasstransfer.2017.09.026
Zimmerman, R.W. and Bodvarsson, G.S., 1996, Hydraulic conductivity of rock fractures. Transport in Porous Media, 23, 1–30. https://doi.org/10.1007/BF00145263
Zoorabadi, M., Saydam, S., Timms, W., and Hebblewhite, B., 2015, Non-linear flow behavior of rough fractures having standard JRC profiles. International Journal of Rock Mechanics & Mining Sciences, 76, 192–199. https://doi.org/10.1016/j.ijrmms.2015.03.004
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Zhu, J. Effective hydraulic conductivity of discrete fracture network with aperture-length correlation. Geosci J 24, 329–338 (2020). https://doi.org/10.1007/s12303-019-0025-8
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DOI: https://doi.org/10.1007/s12303-019-0025-8