Abstract
The space block search technology is used to determine a connected three-dimensional fracture network in polygonal shapes, i.e., seepage paths. After triangulation on these polygons, a finite element mesh for 3D fracture network seepage is obtained. Through introduction of the generalized Darcy’s law, conservative equations for both fracture surface and fracture interactions are established. Combined with the boundary condition of Signorini’s type, a partial differential equation (PDE) formulation is presented for the whole domain concerned. To solve this problem efficiently, an equivalent variational inequality (VI) formulation is given. With the penalized Heaviside function, a finite element procedure for unconfined seepage problem in 3D fracture network is developed. Through an example in a homogeneous rectangular dam, validity of the algorithm is verified. The analysis of an unconfined seepage problem in a complex fracture network shows that the proposed algorithm is very applicable to complex three-dimensional problems, and is effective in describing some interesting phenomenon usually encountered in practice, such as “preferential flow”.
Similar content being viewed by others
References
Berkowitz B. Characterizing flow and transport in fractured geological media: A review. Adv Water Resour, 2002, 25 (8):3861–3884
Neretnieks I, Eriksen T, Tahtinen P. Tracer movement in a single fissure in granitic rock: Some experimental results and their interpretation. Water Resour Res, 1982, 18(4):849–858
Neretnieks I. Solute transport in fracture rock—Applications toradionuclide waste repositories. In: Bear J, Tsang C F, de Marsily G, ed. Flow and Contaminant Transport in Fractured Rock. San Diego: Academic Press, Inc; 1993. 39–127
Oda M. Permeability tensor for discontinuous rock masses. Geotech, 1985, 35(4):483–495
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
Neuman S P. Saturated-unsaturated seepage by finite elements. J Hydraulics Div, 1973, 99(12):2233–2250
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
Adler P M, Thovert J F. Fractures and Fracture Networks. Netherlands: Kluwer Academic Publishers, 1999
Cacas M C, Ledoux E, Marsily G de, et al. Modeling fracture flow with a stochastic discrete fracture network: Calibration and validation: 2. The transport model. Water Resources Res, 1990, 26(3): 491–500
Nordqvist A W, Tsang Y W, Tsang C F, et al. A variable aperture fracture network model for flow and transport in fractured rocks. Water Resources Res, 1992, 28(6):1703–1713
Rouleau A, Gale J E. Stochastic discrete fracture simulation of groundwater flow into an underground excavation in granite. Int J Rock Mech Mining Sci Geomech Abstracts, 1987, 24(2):99–112
Andersson J, Dverstorp B. Conditional simulations of fluid flow in three-dimensional networks of discrete fractures. Water Resources Res, 1987, 23(10):1876–1886
Koudina N, Gonzalez G R, Thovert J F, et al. Permeability of three-dimensional fracture networks. Phys Rev E, 1998, 57(4):4466–4479
Khamforoush M, Shams M, Thovert J F, et al. Permeability and percolation of anisotropic three-dimensional fracture networks. Phys Rev E, 2008, 77(5):056307
Mourzenko V V, Thovert J F, Adler P M. Permeability of isotropic and anisotropic fracture networks, from the percolation threshold to very large densities. Phys Rev E, 2011, 84(3):036307
Zhang Y T, Chen P, Wang L. Initial flow method for seepage analysis with free surface (in Chinese). J Hydraulic Eng, 1988, 8(1):18–26
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
Chen Y F, Zhou C B, Zheng H. A numerical solution to seepage problems with complex drainage systems. Comput Geotech, 2008, 35(3):383–393
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
Jiang Q H, Ye Z Y, Yao C, et al. A new variational inequality formulation for unconfined seepage flow through fracture networks. Sci China Tech Sci, 2012, 55(11):3090–3101
Yao C, Jiang Q H, Ye Z Y, et al. Initial flow method for unconfined seepage problems of fracture networks (in Chinese). Rock Soil Mech, 2012, 33(6):1896–1903
Jiang Q H, Yao C, Ye Z Y, et al. Seepage flow with free surface in fracture networks. Water Resour Res, 2013, 49, doi: 10.1029/2012-WR011991
Liu Z, Zhang Y T. Analysis of free surface seepage problems in three dimensional network of fracture (in Chinese). Shuili Xuebao, 1996, 6:34–38
Zhang Q H, Wu A Q. Three-dimensional arbitrary fracture network seepage model and its solution (in Chinese). Chinese J Rock Mech Eng, 2010, 29 (4):720–730
Zhang Q H, Wu A Q. General methodology of spatial block topological identification with stochastic discontinuities cutting (in Chinese). Chinese J Rock Mech Eng, 2007, 26 (10):2044–2048
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yao, C., Jiang, Q., Wei, W. et al. The variational inequality formulation for unconfined seepage through three-dimensional dense fracture networks. Sci. China Technol. Sci. 56, 1241–1247 (2013). https://doi.org/10.1007/s11431-013-5191-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11431-013-5191-8