Abstract
Darcy’s law only applying to the flow domain is extended to the entire fracture network domain including the dry domain. The partial differential equation (PDE) formulation for unconfined seepage flow problems for discrete fracture network is established, in which a boundary condition of Signorini’s type is prescribed over the potential seepage surfaces. In order to reduce the difficulty in selecting trial functions, a new variational inequality formulation is presented and mathematically proved to be equivalent to the PDE formulation. The numerical procedure based on the VI formulation is proposed and the corresponding algorithm has been developed. Since a continuous penalized Heaviside function is introduced to replace a jump function in finite element analysis, oscillation of numerical integration for facture elements cut by the free surface is eliminated and stability of numerical solution is assured. The numerical results from two typical examples demonstrate, on the one hand the effectiveness and robustness of the proposed method, and on the other hand the capability of predicting main seepage pathways in fractured rocks and flow rates out of the drainage system, which is very important for performance assessments and design optimization of complex drainage system.
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References
Long J C S, Remer J S, Wilson C R, et al. Porous media equivalents for networks of discontinuous fractures. Water Resour Res, 1982, 18(3): 645–658
Oda M. An equivalent continuum model for coupled stress and fluid flow analysis in jointed rock masses. Water Resour Res, 1986, 22(13): 1845–1856
Wang Y, Su B Y, Xu Z Y. C Coupling seepage model of 3D fractured rock masses and finite element simulation (in Chinese). Hydrogeology Eng Geology, 19 1995, (3): 1–4
Tian K M, Wan L. Research and Evaluation of the Permeability of Anisotropic Fractured Media (in Chinese). Beijing: Academy Press, 1989
Zhang Q F, Wu Z R. The improved cut-off negative pressure method for unsteady seepage flow with free surface (in Chinese). Chinese J Geotech Eng, 2005, 127(1): 48–54
Zhang Y Z, Zhang W Q. 3D thermo-hydro-mechanical-migratory coupling model and FEM analyses for dual-porosity medium. Sci China Tech Sci, 2010, 53(8): 2172–2182
Hu R, Chen Y F, Zhou C B. Modeling of coupled deformation, water flow and gas transport in soil slopes subjected to rain infiltration. Sci China Tech Sci, 2011, 54(10): 2561–2575
Wittke W. Rock Mechanics-Theory and Applications with Case Histories. Berlin: Springer, 1990. 377–386
Wilson C R, Witherspoon P A. Steady state flow in rigid networks of fractures. Water Resour Res, 1974, 10(2): 328–335
Mao C X, Chen P, Li Z Y, et a1. Research on fractured rock mass seepage calculation method (in Chinese). Chinese J Geotech Eng, 1991, 13(6): 1–10
Wang E Z. Network analysis and seepage flow model of fractured rockmass (in Chinese). Chinese J Rock Mech Eng, 1993, 12(3): 214–221
Chai J R, Xu W S. Coupling analysis of unsteady seepage and stress fields in discrete fractures network of rock mass in dam foundation. Sci China Tech Sci, 2011, 54(Suppl. 1): 133–139
Long J C S, Gilmour P, Witherspoon P A. A method for steady fluid flow in random three-dimensional networks of disc-shaped fractures. Water Resour Res, 1985, 21(8): 1105–1115
Dershowitz W S, Einstein H H. Three dimensional flow modeling in jointed rock masses. In: Proceedings of the sixth international congress on rock mechanics, Rotterdam, Balkema, 1987. 87–92
Cacas M C, Ledoux E, Marsily de G, et al. Modeling fracture flow with a stochastic discrete fracture network: Calibration and validation 1. The flow model. Water Resour Res, 1990, 26(3): 479–489
Wang E Z, Sun Y, Huang Y Z, et al. 3-D seepage flow model for discrete fracture network and verification experiment (in Chinese). Shuili Xuebao, 2002, (5): 37–40
Wang E Z. Seepage calculation method in fissure networks on vertical section (in Chinese). Hydrogeology Eng Geology, 1993, 20(4): 27–29
Chai J R, Wu Y Q. The method for determination of the position of free surface in fractured rock masses (in Chinese). Geotech Investigation Surveying, 2000, 1: 23–24
Zheng H, Liu D F, Lee C F, et al. A new formulation of Signorini’s type for seepage problems with free surfaces. Int J Numer Meth Eng, 2005, 64: 1–16
Bathe K J, Khoshgoftaar M R. Finite element free surface seepage analysis without mesh iteration. Int J Numerical Analytical Methods Geomech, 1979, 3 (l): 13–22
Harr M E. Groundwater and Seepage. New York: Dover Publications, 1990
Shi G H. Discontinuous Deformation Analysis-a New Numerical Model for the Statics and Dynamics of Block System. Dissertation of Doctoral Degree. Berkeley: University of California, 1988
Zhang Y T, Zhou W Y. Deformation and Stability of High Rock Slope. Beijing: China Water Power Press, 1999
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Jiang, Q., Ye, Z., Yao, C. et al. A new variational inequality formulation for unconfined seepage flow through fracture networks. Sci. China Technol. Sci. 55, 3090–3101 (2012). https://doi.org/10.1007/s11431-012-4904-8
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DOI: https://doi.org/10.1007/s11431-012-4904-8