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Geofluids
Volume 2018, Article ID 9062569, 16 pages
https://doi.org/10.1155/2018/9062569
Research Article

The Parabolic Variational Inequalities for Variably Saturated Water Flow in Heterogeneous Fracture Networks

1Hubei Key Laboratory for Efficient Utilization and Agglomeration of Metallurgical Mineral Resources, Wuhan University of Science and Technology, Wuhan 430081, China
2Rock Mechanics in Hydraulic Structural Engineering, Ministry of Education, Wuhan University, Wuhan 430072, China
3School of Civil Engineering, Wuhan University, Wuhan 430072, China
4School of Civil Engineering and Architecture, Nanchang University, Nanchang 330031, China
5Department of Mathematical and Statistical Sciences, University of Colorado Denver, Denver, CO 80204, USA

Correspondence should be addressed to Qinghui Jiang; nc.ude.uhw@2791hqj

Received 18 August 2017; Revised 14 November 2017; Accepted 13 December 2017; Published 11 January 2018

Academic Editor: Daniele Pedretti

Copyright © 2018 Zuyang Ye et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. S. Y. Wang and T. N. Narasimhan, “Hydrologic mechanismsgoverning fluid flow in a partially saturated, fractured,porous medium,” Water Resources Research, vol. 21, no. 12, pp. 1861–1874, 1985. View at Publisher · View at Google Scholar · View at Scopus
  2. R. R. Peters and E. A. Klavetter, “A continuum model for water movement in an unsaturated fractured rock mass,” Water Resources Research, vol. 24, no. 3, pp. 416–430, 1988. View at Publisher · View at Google Scholar · View at Scopus
  3. J. J. Nitao and T. A. Buscheck, “Infiltration of a liquid front in an unsaturated, fractured porous medium,” Water Resources Research, vol. 27, no. 8, pp. 2099–2112, 1991. View at Publisher · View at Google Scholar · View at Scopus
  4. R. Therrien and E. A. Sudicky, “Three-dimensional analysis of variably-saturated flow and solute transport in discretely-fractured porous media,” Journal of Contaminant Hydrology, vol. 23, no. 1-2, pp. 1–44, 1996. View at Publisher · View at Google Scholar · View at Scopus
  5. A. Abdel-Salam and C. V. Chrysikopoulos, “Unsaturated flow in a quasi-three-dimensional fractured medium with spatially variable aperture,” Water Resources Research, vol. 32, no. 6, pp. 1531–1540, 1996. View at Publisher · View at Google Scholar · View at Scopus
  6. C. Masciopinto and M. C. Caputo, “Modeling unsaturated-saturated flow and nickel transport in fractured rocks,” Vadose Zone Journal, vol. 10, no. 3, pp. 1045–1057, 2011. View at Publisher · View at Google Scholar · View at Scopus
  7. D. Pedretti, A. Russian, X. Sanchez-Vila, and M. Dentz, “Scale dependence of the hydraulic properties of a fractured aquifer estimated using transfer functions,” Water Resources Research, vol. 52, no. 7, pp. 5008–5024, 2016. View at Publisher · View at Google Scholar · View at Scopus
  8. J. T. Fabryka-Martin, A. V. Wolfsberg, P. R. Dixon, S. Levy, J. Musgrave, and H. J. Turin, Summary report of chlorine-36 studies: sampling, analysis and simulation of chlorine-36 in the Exploratory Studies Facility. Rep. LA-CST-TIP-96-002, Los Alamos Natl Lab, Los Alamos, 1996.
  9. I. Yang, G. Rattray, and P. Yu, “Interpretation of chemical and isotopic data from boreholes in the unsaturated zone at Yucca Mountain, Nevada,” Tech. Rep. USGS/WRIR--96-4058, 1996. View at Publisher · View at Google Scholar
  10. R. Nativ, E. Adar, O. Dahan, and M. Geyh, “Water Recharge and Solute Transport Through the Vadose Zone of Fractured Chalk Under Desert Conditions,” Water Resources Research, vol. 31, no. 2, pp. 253–261, 1995. View at Publisher · View at Google Scholar · View at Scopus
  11. J. T. Geller and K. Pruess, “On water infiltration in rough-walled fractures,” in Proceedings of the Sixth Annual International High-Level Radioactive Waste Management Conference, Las Vegas, pp. 23–25, La Grange Park, IL, USA, 1995.
  12. K. Pruess, “On water seepage and fast preferential flow in heterogeneous, unsaturated rock fractures,” Journal of Contaminant Hydrology, vol. 30, no. 3-4, pp. 333–362, 1998. View at Publisher · View at Google Scholar · View at Scopus
  13. R. Hu, Y.-F. Chen, C.-B. Zhou, and H.-H. Liu, “A numerical formulation with unified unilateral boundary condition for unsaturated flow problems in porous media,” Acta Geotechnica, vol. 12, no. 2, pp. 277–291, 2017. View at Publisher · View at Google Scholar · View at Scopus
  14. R. Hu, J. Hong, Y. Chen, and C. Zhou, “Hydraulic hysteresis effects on the coupled flow–deformation processes in unsaturated soils: Numerical formulation and slope stability analysis,” Applied Mathematical Modelling, vol. 54, pp. 221–245, 2017. View at Publisher · View at Google Scholar
  15. Q. Jiang, Z. Ye, and C. Zhou, “A numerical procedure for transient free surface seepage through fracture networks,” Journal of Hydrology, vol. 519, pp. 881–891, 2014. View at Publisher · View at Google Scholar · View at Scopus
  16. Z. Ye, Q. Jiang, C. Zhou, and Y. Liu, “Numerical analysis of unsaturated seepage flow in two-dimensional fracture networks,” International Journal of Geomechanics, vol. 17, no. 5, Article ID 04016118, 2017. View at Publisher · View at Google Scholar · View at Scopus
  17. L. A. Richards, “Capillary conduction of liquids through porous mediums,” Journal of Applied Physics, vol. 1, no. 5, pp. 318–333, 1931. View at Publisher · View at Google Scholar · View at Scopus
  18. P. A. Witherspoon, J. S. Y. Wang, K. Iwai, and J. E. Gale, “Validity of cubic law for fluid flow in a deformable rock fracture,” Water Resources Research, vol. 16, no. 6, pp. 1016–1024, 1980. View at Publisher · View at Google Scholar · View at Scopus
  19. K. G. Raven and J. E. Gale, “Water flow in a natural rock fracture as a function of stress and sample size,” International Journal of Rock Mechanics and Mining Sciences, vol. 22, no. 4, pp. 251–261, 1985. View at Publisher · View at Google Scholar · View at Scopus
  20. S. R. Brown, “Fluid flow through rock joints: The effect of surface roughness,” Journal of Geophysical Research: Atmospheres, vol. 92, no. B2, p. 1337, 1987. View at Publisher · View at Google Scholar
  21. R. Zimmerman and G. Bodvarsson, “Hydraulic conductivity of rock fractures,” Transport in Porous Media, vol. 23, no. 1, 1996. View at Publisher · View at Google Scholar
  22. L. Jing, “Flow equations of connected fracture system,” in Fundamentals of Discrete Element Methods for Rock Engineering: Theory and Applications: Theory and Application, p. 126, Elsevier, Oxford, UK, 2007. View at Google Scholar
  23. S. Reitsma and B. H. Kueper, “Laboratory measurement of capillary pressure-saturation relationships in a rock fracture,” Water Resources Research, vol. 30, no. 4, pp. 865–878, 1994. View at Publisher · View at Google Scholar · View at Scopus
  24. H. Liu and G. S. Bodvarsson, “Constitutive relations for unsaturated flow in a fracture network,” Journal of Hydrology, vol. 252, no. 1-4, pp. 116–125, 2001. View at Publisher · View at Google Scholar
  25. M. T. van Genuchten, “A closed-form equation for predicting the hydraulic conductivity of unsaturated soils,” Soil Science Society of America Journal, vol. 44, no. 5, pp. 892–898, 1980. View at Publisher · View at Google Scholar · View at Scopus
  26. Y. Mualem, “A new model for predicting the hydraulic conductivity of unsaturated porous media,” Water Resources Research, vol. 12, no. 3, pp. 513–522, 1976. View at Publisher · View at Google Scholar · View at Scopus
  27. H. Liu, M. Wei, and J. Rutqvist, “Normal-stress dependence of fracture hydraulic properties including two-phase flow properties,” Hydrogeology Journal, vol. 21, no. 2, pp. 371–382, 2013. View at Publisher · View at Google Scholar
  28. K. Pruess and Y. W. Tsang, “On two‐phase relative permeability and capillary pressure of rough‐walled rock fractures,” Water Resources Research, vol. 26, no. 9, pp. 1915–1926, 1990. View at Publisher · View at Google Scholar · View at Scopus
  29. Y. Li, Y.-F. Chen, and C.-B. Zhou, “Hydraulic properties of partially saturated rock fractures subjected to mechanical loading,” Engineering Geology, vol. 179, pp. 24–31, 2014. View at Publisher · View at Google Scholar · View at Scopus
  30. Z. Ye, H.-H. Liu, Q. Jiang, and C. Zhou, “Two-phase flow properties of a horizontal fracture: The effect of aperture distribution,” Advances in Water Resources, vol. 76, pp. 43–54, 2015. View at Publisher · View at Google Scholar · View at Scopus
  31. Z. Ye, H.-H. Liu, Q. Jiang, Y. Liu, and A. Cheng, “Two-phase flow properties in aperture-based fractures under normal deformation conditions: Analytical approach and numerical simulation,” Journal of Hydrology, vol. 545, pp. 72–87, 2017. View at Publisher · View at Google Scholar · View at Scopus
  32. C. Chen and R. N. Horne, “Two-phase flow in rough-walled fractures: Experiments and a flow structure model,” Water Resources Research, vol. 42, no. 3, 2006. View at Publisher · View at Google Scholar
  33. H. Zheng, D. F. Liu, C. F. Lee, and L. G. Tham, “A new formulation of Signorini's type for seepage problems with free surfaces,” International Journal for Numerical Methods in Engineering, vol. 64, no. 1, pp. 1–16, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  34. I. Borsi, A. Farina, and M. Primicerio, “A rain water infiltration model with unilateral boundary condition: qualitative analysis and numerical simulations,” Mathematical Methods in the Applied Sciences, vol. 29, no. 17, pp. 2047–2077, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  35. Y. Chen, R. Hu, C. Zhou, D. Li, and G. Rong, “A new parabolic variational inequality formulation of Signorini's condition for non-steady seepage problems with complex seepage control systems,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 35, no. 9, pp. 1034–1058, 2011. View at Publisher · View at Google Scholar · View at Scopus
  36. Q. Jiang, C. Yao, Z. Ye, and C. Zhou, “Seepage flow with free surface in fracture networks,” Water Resources Research, vol. 49, no. 1, pp. 176–186, 2013. View at Publisher · View at Google Scholar
  37. A. W. Warrick, D. O. Lomen, and S. R. Yates, “A generalized solution to infiltration.,” Soil Science Society of America Journal, vol. 49, no. 1, pp. 34–38, 1985. View at Publisher · View at Google Scholar · View at Scopus
  38. S. J. Lacy and J. H. Prevost, “Flow through porous media: A procedure for locating the free surface,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 11, no. 6, pp. 585–601, 1987. View at Publisher · View at Google Scholar · View at Scopus
  39. R. I. Borja and S. S. Kishnani, “On the solution of elliptic free-boundary problems via Newton's method,” Computer Methods Applied Mechanics and Engineering, vol. 88, no. 3, pp. 341–361, 1991. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  40. J-P. Chiles and G. de Marsily, “Stochastic models of fracture systems and their use in flow and transport modeling,” in Flow And Contaminant Transport in Fractured Rock, J. Bear, C. F. Tsang, and G. de Marsily, Eds., pp. 169–236, Academic Press, New York, NY, USA, 1993. View at Google Scholar
  41. W. Press, S. Teukolsky, W. Vetterling, and et al., Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, UK, 2nd edition, 1992.
  42. L. Smith and F. W. Schwartz, “Solute transport through fracture networks,” in Flow and contaminant transport in fractured rock, J. Bear, C. F. Tsang, and G. de Marsily, Eds., Academic Press, New York, NY, USA, 1993. View at Google Scholar
  43. M. Bianchi and D. Pedretti, “Geological entropy and solute transport in heterogeneous porous media,” Water Resources Research, vol. 53, no. 6, pp. 4691–4708, 2017. View at Publisher · View at Google Scholar