Abstract
The fracture transmissivity characteristics curve (Witherspoon et al. in Water Resour. Res. 16(6):1016–1024, 1980) is found to deviate from cubic law as aperture decreases and still have residual transmissivity when aperture is very small. The existing models can partly explain the deviation from cubic law (e.g., Sisavath et al. in PAGEOPH 160:1009–1022, 2003), or the residual transmissivity due to irreducible flow (e.g., Nolte et al. in PAGEOPH 131(1/2):111–138, 1989). In order to predict the transmissivity curve with both the above characteristics, in this study, a simple statistical model is employed with the following assumptions: (1) fracture boundaries are assumed parallel flat at global scale, but with normally distributed aperture variations at local scale (like frosted glasses); and (2) in this case, the flow field is assumed regular with straight head-contours and flow-lines. Then the equivalent transmissivity can be approximated as a series of parallel-connected local transmissivities. The transmissivity curve can be fitted very well with both the above characteristics. It is suggested that the reason for the deviation from cubic low is possibly due to the variations of local apertures which induce redistribution of hydraulic gradients, and the residual foot is because of residual open apertures or micro-fractures in the fracture surfaces.
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References
Brown, S.R.: Fluid flow through rock joints: the effect of surface roughness. J. Geophys. Res. 92(B2), 1337–1347 (1987)
Brush, D.J., Thomson, N.R.: Fluid flow in synthetic rough-walled fractures: Navier–Stokes, Stokes, and local cubic law simulations. Water Resour. Res. 39(4), 1085 (2003)
Curtis, J.B.: Fractured shale-gas systems. AAPG Bull. 86(11), 1921–1938 (2002)
Dagan, G.: Models of groundwater flow in statistically homogeneous porous formations. Water Resour. Res. 15, 47–63 (1979)
Dagan, G.: Higher-order correction of effective permeability of heterogeneous isotropic formations of lognormal conductivity distribution. Transp. Porous Media 12, 279–290 (1993)
de Marsily, G.: Quantitative Hydrogeology. Academic Press, San Diego (1986)
Dixon, N.A., Durham, W.B., Suzuki, A.M., Mei, S.: Measurements of grain size sensitivity in olivine deformed at high pressure in the deformation DIA. Resented at AGU 2009 Fall Meeting, San Francisco. MR41A-1849 Poster (2009)
Garven, G.: Continental-scale groundwater flow and geologic processes. Ann. Rev. Earth Planet. Sci. 23, 89–118 (1995)
Koyama, T., Li, B., Jiang, Y., Jing, L.: Numerical modelling of fluid flow tests in a rock fracture with a special algorithm for contact areas. Comput. Geotech. 36, 291–303 (2009)
Koyama, T., Neretnieks, I., Jing, L.: 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. Int J. Rock Mech. Mining Sci. 45, 1082–1101 (2008)
Li, B., Wong, R.C.K., Milnes, T.: Anisotropy in capillary invasion and flow fluid through induced sandstone and shale fractures. Int. J. Rock Mech. Mining Sci. 65, 129–140 (2014)
Long, J.C.S., Ewing, R.C.: Yucca mountain: earth-science issues at a geologic repository for high-level nuclear waste. Ann. Rev. Earth Planet. Sci. 32, 363–401 (2004)
Mourzenko, V.V., Thovert, J.F., Adler, P.M.: Permeability of a single fracture: validity of the Reynolds equation. J. Phys. II Paris 5(3), 465–482 (1995)
Myers, T.: Potential contaminant pathways from hydraulically fractured shale to aquifers. Groundwater 50(6), 872–882 (2012)
Neuzil, C.E., Tracy, J.V.: Flow through fractures. Water Resour. Res. 17, 191–199 (1981)
Nolte, D.D., Pyrak-Nolte, L.J., Cook, N.G.W.: The fractal geometry of flow paths in natural fractures in rock and the approach to percolation. PAGEOPH 131(1/2), 111–138 (1989)
Pyrak-Nolte, L.J., Cook, N.G.W., Nolte, D.D.: Fluid percolation through single fractures. Geophys. Res. Lett. 15(11), 1247–1250 (1988)
Pyrak-Nolte, L.J., Myer, L.R., Cook, G.W., Witherspoon, P.A.: Hydraulic and mechanical properties of natural fractures in low permeability rock. In: Proceedings of the Sixth International Congress on Rock Mechanics, Montreal, Canada, August 1987, G. HergetS. Vongpaisal225-231, Pubs. A.A. Balkema, Rotterdam (1987)
Silliman, S.E.: An interpretation of the difference between aperture estimates derived from hydraulic and tracer tests in a single fracture. Water Resour. Res. 25, 2275–2283 (1989)
Sisavath, S., Al-Yaaruby, A., Pain, C.C., Zimmerman, W.R.: A simple model for deviations from the cubic law for a fracture undergoing dilation or closure. PAGEOPH 160, 1009–1022 (2003)
Tsang, Y.W.: The effect of tortuosity on fluid flow through a single fracture. Water Resour. Res. 20(9), 1209–1215 (1984)
Tsang, Y.W., Tsang, C.F.: Channel model of flow through fractured media. Water Resour. Res. 23(3), 467–479 (1987)
Velde, B., Dubois, J., Touchard, G., Badri, A.: Fractal analysis of fractures in rocks: the Cantor’s Dust method. Tectonophysics 179(3–4), 345–352 (1990)
Wintsch, P., Christoffersen, R., Kronenberg, A.K.: Fluid-rock reaction weakening of fault zones. J. Geophys. Res. Solid Earth (1978–2012) 100(B7), 13021–13032 (1995)
Witherspoon, P.A., Wang, J.S.Y., Iwai, K., Gale, J.E.: Validity of cubic law for fluid flow in a deformable rock fracture. Water Resour. Res. 16(6), 1016–1024 (1980)
Zimmerman, R.W., Bodvarsson, G.S.: Hydraulic conductivity of rock fractures. Transp. Porous Media 23, 1–30 (1996)
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This study is financially supported by National Natural Science Foundation of China (NSFC) (No. 51409028).
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Yu, C. A Simple Statistical Model for Transmissivity Characteristics Curve for Fluid Flow Through Rough-Walled Fractures. Transp Porous Med 108, 649–657 (2015). https://doi.org/10.1007/s11242-015-0493-x
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DOI: https://doi.org/10.1007/s11242-015-0493-x