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含随机裂纹网络孔隙材料渗透率的逾渗模型研究

李乐 李克非

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含随机裂纹网络孔隙材料渗透率的逾渗模型研究

李乐, 李克非

Permeability of cracked porous solids through percolation approach

Li Le, Li Ke-Fei
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  • 采用逾渗理论对含随机裂纹网络的孔隙材料渗透性进行研究. 开裂孔隙材料渗透率的影响因素包括裂纹网络的几何特征、孔隙材料本体渗透率以及裂纹开度, 本文使用连续区逾渗理论模型建立了渗透率的标度律. 对于裂纹网络的几何特征, 本文基于连续区逾渗理论并考虑裂纹网络的分形特征提出了有限区域内二维随机裂纹网络的连通度定义; 对随机裂纹网络的几何分析表明, 随机裂纹局部团簇效应会降低裂纹网络的整体连通性, 随机裂纹网络的标度指数并非经典逾渗理论给出的固定值, 而是随着网络的分形维数的减小而增大. 本文在网络连通度和主裂纹团的曲折度的基础上, 提出了开裂孔隙材料渗透率标度律的解析表达, K=K0(Km,b)(-c), 分别考虑了裂纹网络的几何逾渗特征 (-c)、孔隙材料渗透率Km 以及裂纹开度比b; 对有限区域含有随机裂纹网络的孔隙材料渗透过程的有限元模拟表明, K0 在裂纹逾渗阈值附近与b呈指数关系, 但当裂纹的局部渗透率与Km比值高于106 后, 开度比b对渗透率不再有影响.
    This paper investigates the permeability of microcracked porous solids incorporating random crack networks in terms of continuum percolation theory. Main factors of permeability include the geometry of crack networks, permeability of porous matrix, and crack opening. For the two-dimensional random crack networks, a new connectivity factor is defined to take into consideration the spanning cluster of cracks, fractal dimension of networks, and the size of a finite domain. For an infinite domain, the connectivity factor around a percolation threshold observes the scaling law, so this definition of connectivity is proved to be consistent with the percolation concepts. Geometric analysis reveals that the local clustering will not necessarily contribute to the global connectivity of networks. It is also found that too strong a local clustering of cracks will decrease the probability of the global percolation, and this adverse aspect of the local clustering effect has never been reported in the literature. The percolation threshold changes with the crack pattern of networks and the scaling exponents of percolation are not constant but depend on the fractal dimension of the crack networks. On the basis of connectivity and tortuosity of crack networks, the scaling law for permeability is established, K=K0(Km,b)(-c), taking into consideration the geometris characteristics through (-c), the permeability of porous matrix Km, and the crack opening aperture b. Then the permeability of a solid incorporating random crack networks is solved by finite element methods: all the cracks are idealized as 2-node elements and the matrix is divided into 6-node triangle elements. The fluid is assumed to be incompressible and Newtonian. With these assumptions the effective permeability of numerical samples is evaluated through Darcy's law. The scaling exponents of the permeability obtained numerically are very near to the theoretical values, and the impact of crack opening is less important as the crack density is far below the percolation threshold and the effect of crack opening becomes significant only as the crack density approaches the percolation threshold. Influence of crack opening on the permeability is strongly dependent on the opening aperture of the cracks. Finite element simulation results show that K0 depends on b through a power law near the percolation threshold and this dependence disappears as the ratio between the local permeability of crack and the matrix permeability exceeds 106.
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    Jensen A D, Chatterji S 1996 Mater. Struct. 29 3

    [4]

    Guéguen Y, Chelidze T, Le Ravalec M 1997 Tectonophys. 279 23

    [5]

    Broadbent S R, Hammersley J M 1957 Math. Proc. Cambridge Philos. Soc. 53 629

    [6]

    Liu Z F, Lai Y T, Zhao G, Zhang Y W, Liu Z F, Wang X H 2008 Acta Phys. Sin. 57 2011 (in Chinese) [刘志峰, 赖远庭, 赵刚, 张有为, 刘正锋, 王晓宏 2008 物理学报 57 2011]

    [7]

    Feng Z C, Zhao Y S, Lu Z X 2007 Acta Phys. Sin. 56 2796 (in Chinese) [冯增朝, 赵阳升, 吕兆兴 2007 物理学报 56 2796]

    [8]

    Hestir K, Long J 1990 J. Geophys. Res. 95 21565

    [9]

    Leung C T O, Zimmerman R W 2012 Transp. Porous Med. 93 777

    [10]

    Bour O, Davy P 1997 Water Resour. Res. 33 1567

    [11]

    Robinson P C 1983 J. Phys. A: Math. Gen. 16 605

    [12]

    Berkowitz B 1995 Math. Geol. 27 467

    [13]

    Balberg I, Anderson C H, Alexander S, Wagner N 1984 Phys. Rev. B: Condens. Matter 30 3933

    [14]

    Masihi M, King P R 2007 Water Resour. Res. 43 W07439

    [15]

    Robinson P C 1984 J. Phys. A: Math. Gen. 17 2823

    [16]

    Zhou C, Li K, Pang X 2011 Mech. Mater. 43 969

    [17]

    Li J H, Zhang L M 2011 Comput. Geotech. 38 217

    [18]

    Stauffer D 1979 Phys. Reports 54 1

    [19]

    Stauffer D, Aharony A 2003 Introduction to percolation theory 2nd edition (London: Taylor & Francis) pp15-19

    [20]

    Zhou C, Li K, Pang X 2012 Cem. Concr. Res. 42 1261

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    Bonnet E, Bour O, Odling N E, Davy P, Main I, Cowie P, Berkowitz B 2001 Rev. Geophys. 39 347

    [22]

    Sheppard A P, Knackstedt M A, Pinczewski W V, Sahimi M 1999 J. Phys. A: Math. Gen. 32 L521

  • [1]

    Mehta P K 1991 ACI Spec. Publ. 126 1

    [2]

    Feldman R F 1986 Proceedings of the Eighth International Congress on the Chemistry of Cement (Rio de Janeiro: FINEP) p336

    [3]

    Jensen A D, Chatterji S 1996 Mater. Struct. 29 3

    [4]

    Guéguen Y, Chelidze T, Le Ravalec M 1997 Tectonophys. 279 23

    [5]

    Broadbent S R, Hammersley J M 1957 Math. Proc. Cambridge Philos. Soc. 53 629

    [6]

    Liu Z F, Lai Y T, Zhao G, Zhang Y W, Liu Z F, Wang X H 2008 Acta Phys. Sin. 57 2011 (in Chinese) [刘志峰, 赖远庭, 赵刚, 张有为, 刘正锋, 王晓宏 2008 物理学报 57 2011]

    [7]

    Feng Z C, Zhao Y S, Lu Z X 2007 Acta Phys. Sin. 56 2796 (in Chinese) [冯增朝, 赵阳升, 吕兆兴 2007 物理学报 56 2796]

    [8]

    Hestir K, Long J 1990 J. Geophys. Res. 95 21565

    [9]

    Leung C T O, Zimmerman R W 2012 Transp. Porous Med. 93 777

    [10]

    Bour O, Davy P 1997 Water Resour. Res. 33 1567

    [11]

    Robinson P C 1983 J. Phys. A: Math. Gen. 16 605

    [12]

    Berkowitz B 1995 Math. Geol. 27 467

    [13]

    Balberg I, Anderson C H, Alexander S, Wagner N 1984 Phys. Rev. B: Condens. Matter 30 3933

    [14]

    Masihi M, King P R 2007 Water Resour. Res. 43 W07439

    [15]

    Robinson P C 1984 J. Phys. A: Math. Gen. 17 2823

    [16]

    Zhou C, Li K, Pang X 2011 Mech. Mater. 43 969

    [17]

    Li J H, Zhang L M 2011 Comput. Geotech. 38 217

    [18]

    Stauffer D 1979 Phys. Reports 54 1

    [19]

    Stauffer D, Aharony A 2003 Introduction to percolation theory 2nd edition (London: Taylor & Francis) pp15-19

    [20]

    Zhou C, Li K, Pang X 2012 Cem. Concr. Res. 42 1261

    [21]

    Bonnet E, Bour O, Odling N E, Davy P, Main I, Cowie P, Berkowitz B 2001 Rev. Geophys. 39 347

    [22]

    Sheppard A P, Knackstedt M A, Pinczewski W V, Sahimi M 1999 J. Phys. A: Math. Gen. 32 L521

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出版历程
  • 收稿日期:  2014-11-19
  • 修回日期:  2015-02-04
  • 刊出日期:  2015-07-05

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