STUDY ON CONNECTIVITY AND ENTROPY SCALE OF THREE-DIMENSIONAL FRACTURE NETWORK
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摘要: 裂隙网络是岩体地下水的主要流动通道, 而工程岩体中裂隙网络错综复杂, 裂隙网络的几何特征和连通性对其渗透性有着重要影响. 为了综合量化裂隙迹长、间距、倾角、开度对裂隙网络连通性和渗透性的影响, 基于信息熵原理, 提出了三维裂隙网络地质熵理论和连通性指标−熵尺度, 对比熵尺度与其他传统三维裂隙网络连通性指标, 验证了熵尺度评价三维裂隙网络连通性和渗透性的合理性. 结合锦屏一级水电站左岸边坡裂隙统计分布, 建立三维裂隙网络渗流数值计算方法, 分析不同裂隙迹长、倾角、间距、开度条件下三维裂隙面密度、无量纲逾渗密度、熵尺度和渗透系数的变化关系. 结果表明: 当体积率一定, 考虑开度影响时, 三维裂隙面密度和无量纲逾渗密度无法定量表征迹长和间距对裂隙网络连通性的影响; 裂隙迹长与熵尺度和渗透系数呈负相关关系, 裂隙间距和开度与熵尺度和渗透系数呈正相关关系, 裂隙倾角变化对熵尺度和渗透系数影响较小; 熵尺度与渗透系数的非线性关系近似满足二次多项式.Abstract: The fracture network is complex in engineering rock mass, both the geometric characteristics and connectivity have an important influence on its permeability. In order to comprehensively quantify the influence of fracture trace length, dip angle, spacing and aperture on the connectivity and permeability of fracture network, basing on the principle of information entropy, the geological entropy theory and connectivity index entropy scale of three-dimensional fracture network is proposed. Compared the entropy scale with other traditional connectivity indexes, the rationality of entropy scale in evaluating the connectivity and permeability of three-dimensional fracture network is verified. The results show that the trace length of the fracture is negatively correlated with the entropy scale and the permeability coefficient. The fracture spacing and aperture are positively correlated with the entropy scale and the permeability coefficient. The dip angle of the fracture has little effect on the entropy scale and the permeability coefficient. The nonlinear relationship between entropy scale and permeability coefficient approximately satisfies quadratic polynomial. Basing on the statistical distribution of the fractures on the Left Bank Slope Jinping Hydropower Station, a numerical calculation method of three-dimensional fracture network seepage is established. By analyzing the relationship between the three-dimensional fracture network geometric characteristics and fracture areal intensity, dimensionless percolation density, entropy scale and permeability coefficient, The following conclusions are obtained: when the volume ratio is constant and the influence of aperture is considered, the fracture areal intensity and dimensionless percolation density cannot quantitatively characterize the influence of fracture network geometric characteristics. The length of fracture is negatively correlated with entropy scale and permeability coefficient. The fracture spacing and aperture are positively correlated with entropy scale and permeability coefficient. The dip angle of fracture have little influence on entropy scale and permeability coefficient. The nonlinear relationship between entropy scale and permeability coefficient approximately satisfies the quadratic polynomial.
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Key words:
- fracture network /
- geometric features /
- connectivity /
- entropy scale
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图 1 (a) 三维裂隙网络与子域划分和(b) 子域与裂隙相交关系
Figure 1. (a) The 3D fracture network and subdomain mesh and (b) the intersection between the subdomain and fractures
图 3 裂隙三角形单元与六面体子域的切割关系
Figure 3. Relationship between triangular element of fracture and subdomain
图 6 连通性指标与渗透系数的关系
Figure 6. Relationship between connectivity index and permeability coefficient
图 7 不同几何特征裂隙网络模型
Figure 7. Fracture network model of different geometric characteristics
图 8 不同迹长裂隙网络的熵尺度变化
Figure 8. Variation of fracture entropy scale with different lengths
图 9 不同间距裂隙网络的熵尺度变化
Figure 9. Variation of fracture entropy scale with different spacings
图 10 不同倾角裂隙网络的熵尺度变化
Figure 10. Variation of fracture entropy scale with different dips
图 11 不同开度均值裂隙网络的熵尺度变化
Figure 11. Variation of fracture entropy scale with different apertures
图 12 不同开度标准差裂隙网络的熵尺度变化
Figure 12. Variation of fracture entropy scale with different aperture standard deviations
图 17 不同迹长裂隙网络的熵尺度和渗透系数关系
Figure 17. Variation of fracture entropy scale and permeability coefficient with different lengths
图 20 不同间距裂隙网络的熵尺度和渗透系数关系
Figure 20. Variation of fracture entropy scale and permeability coefficient with different spacings
图 23 不同倾角裂隙网络的熵尺度和渗透系数关系
Figure 23. Variation of fracture entropy scale and permeability coefficient with different dip angles
表 1 不同网格尺度下的熵尺度
Table 1. Entropy scale with different grid scales
Grid scale HR HS 10 1 5 5 0.992527 2.537365 3.333333 0.986026 1.726512 2.5 0.981387 1.323399 2 0.973093 1.08842 1.666667 0.968802 0.930529 1.428571 0.963176 0.819883 1.25 0.958964 0.736894 1.111111 0.95343 0.673758 1 0.949158 0.623163 0.909091 0.94488 0.582159 0.833333 0.940987 0.548137 0.769231 0.935639 0.520206 0.714286 0.932679 0.49564 0.666667 0.92932 0.474633 0.625 0.923088 0.457349 0.588235 0.920473 0.441198 0.555556 0.91755 0.427018 0.526316 0.914108 0.414559 0.5 0.910686 0.403432 
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表 2 传统三维连通性指标
Table 2. Traditional 3D connectivity index
Symbol Expression Reference Description $ {\rho _N} $ ${\rho _N} = \dfrac{ {\displaystyle\sum {\tau (i,j)} } }{V}$ Ref. [22] V is the volume of rockmass $ \overline L $ $\overline L = \dfrac{ {\displaystyle\sum\limits_{n = 1}^N { {l_i} } } }{V}$ Ref. [23] N is the number of fracture
intersecting lines,
li is the length of fracture intersection line$ {P_{32}} $ ${P_{32} } = \dfrac{ {\displaystyle\sum\nolimits_i^N S } }{V}$ Ref. [10] S is the area of the fracture surface,
N is the number of fracture surface${\rho '_{3 {\rm{D}}} }$ ${\rho '_{3 {\rm{D}}} } = {\rho _V}{V_{ex} }$ Ref. [15] $ {\rho _V} $is the fracture density,
$ {V_{ex}} $is the excluded volume
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表 3 裂隙模型几何参数
Table 3. Geometric parameters of fracture model
Group Fracture set Mean length/m Dip direction/(°) Dip angle/(°) Spacing/m Aperture/μm Mean e Standard deviation A 1 [2,3] 309 60 2 73.2 0 2 156 77 B 1 2.5 309 60 [1,3] 72.6 0 2 156 77 C 1 2.5 309 60 2 78.1 0 2 156 [60,80] D 1 2.5 309 60 2 [20,100] [0,10] 2 156 77 E 1 [2,3] 309 60 [1,3] [20,300] 0 2 156 [60,80]
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