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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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Keywords:
- crack network /
- permeability /
- percolation threshold /
- connectivity
[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
期刊类型引用(9)
1. 周建，廖星川，刘福深，尚肖楠，沈君逸. 应用卷积近场动力学快速模拟随机裂纹扩展. 岩土力学. 2025(02): 625-639 . 
百度学术2. 侯昭飞，唐发满，张杨，杨佺忠，尚立涛，孙逊，赵福垚. 冲击波致裂在青海一里坪卤水开采中的增渗效果（英文）. 盐湖研究. 2024(03): 78-86 . 百度学术3. 魏德葆，纪佑军，王泽根，蒋国斌. 微裂缝对灰岩地层固体废弃物回注能力的影响. 吉林大学学报(地球科学版). 2024(04): 1339-1349 . 百度学术4. 寻之朋，郝大鹏. 含复杂近邻的二维正方格子键渗流的蒙特卡罗模拟. 物理学报. 2022(06): 365-370 . 百度学术5. 李滔，李骞，胡勇，彭先，冯曦，朱占美，赵梓寒. 不规则微裂缝网络定量表征及其对多孔介质渗流能力的影响. 石油勘探与开发. 2021(02): 368-378 . 百度学术6. LI Tao，LI Qian，HU Yong，PENG Xian，FENG Xi，ZHU Zhanmei，ZHAO Zihan. Quantitative characterization of irregular microfracture network and its effect on the permeability of porous media. Petroleum Exploration and Development. 2021(02): 430-441 . 必应学术7. 董少群，王涛，曾联波，刘凯，梁锋，尹启航，曹东升. 地下空间逾渗与裂缝属性的关系分析. 地学前缘. 2019(03): 140-146 . 百度学术8. 李乐. 开裂孔隙材料渗透率的细观力学模型研究. 力学学报. 2018(05): 1032-1040 . 百度学术9. 钱鹏，徐千军. 基于单元嵌入技术和弹性比拟的含裂纹混凝土三维渗流模拟方法. 工程力学. 2017(04): 125-133 . 百度学术其他类型引用(8)
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[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
期刊类型引用(9)
1. 周建，廖星川，刘福深，尚肖楠，沈君逸. 应用卷积近场动力学快速模拟随机裂纹扩展. 岩土力学. 2025(02): 625-639 . 百度学术2. 侯昭飞，唐发满，张杨，杨佺忠，尚立涛，孙逊，赵福垚. 冲击波致裂在青海一里坪卤水开采中的增渗效果（英文）. 盐湖研究. 2024(03): 78-86 . 百度学术3. 魏德葆，纪佑军，王泽根，蒋国斌. 微裂缝对灰岩地层固体废弃物回注能力的影响. 吉林大学学报(地球科学版). 2024(04): 1339-1349 . 百度学术4. 寻之朋，郝大鹏. 含复杂近邻的二维正方格子键渗流的蒙特卡罗模拟. 物理学报. 2022(06): 365-370 . 百度学术5. 李滔，李骞，胡勇，彭先，冯曦，朱占美，赵梓寒. 不规则微裂缝网络定量表征及其对多孔介质渗流能力的影响. 石油勘探与开发. 2021(02): 368-378 . 百度学术6. LI Tao，LI Qian，HU Yong，PENG Xian，FENG Xi，ZHU Zhanmei，ZHAO Zihan. Quantitative characterization of irregular microfracture network and its effect on the permeability of porous media. Petroleum Exploration and Development. 2021(02): 430-441 . 必应学术7. 董少群，王涛，曾联波，刘凯，梁锋，尹启航，曹东升. 地下空间逾渗与裂缝属性的关系分析. 地学前缘. 2019(03): 140-146 . 百度学术8. 李乐. 开裂孔隙材料渗透率的细观力学模型研究. 力学学报. 2018(05): 1032-1040 . 百度学术9. 钱鹏，徐千军. 基于单元嵌入技术和弹性比拟的含裂纹混凝土三维渗流模拟方法. 工程力学. 2017(04): 125-133 . 百度学术其他类型引用(8)
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