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Permeability coefficient investigation based on fractal characteristics of porous media soil
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摘要: 堤坝工程渗流计算中确定土体渗透系数尤为重要。利用分形维数不同尺度域，分析渗透破坏试验土样无标度区，指出土体细颗粒含量是决定土体分形维数的主要因素。基于多孔介质毛管束模型，推导了渗透系数和孔隙率与分形维数之间分形关系解析式，阐释了多孔介质土体渗透系数影响因子包括分形系数、孔径大小、分形维数及流体黏滞系数。利用土体渗透破坏试验结果，进一步论证了渗透系数和孔隙率与分形维数之间的非线性关系。结果表明：当分形维数大于2.83时，孔隙率随着分形维数的增大而减小，但在颗粒吸着水和薄膜水形成的黏聚力影响下，渗透系数随着分形维数增大而减小的规律不明显。研究结果可为渗透破坏形成机制及发展过程分析提供理论依据，减少堤坝渗透破坏致灾隐患。Abstract: Permeability coefficient of soil is extremely important to dike and dam engineering. Scale-invariant space of seepage failure testing soils were statistically analyzed by multifractal dimensions, which shows that fine particle content is the major factor of mass fractal dimension. Based on a pipe bundle model of porous medium, the theoretical relationship between permeability coefficient and porosity was deduced, indicating that the influence factors of permeability coefficient include fractal coefficient, particle size, fractal dimension and fluid viscosity coefficient. The nonlinear relationship of permeability coefficient, porosity and fractal dimension was verified for further studies based on seepage failure experimental results. The results show that when the fractal dimension value is greater than 2.83, porosity decreases obviously with the increase of the fractal dimension, while permeability coefficient decreases insignificantly under the cohesive force of hydroscopic water and film water. The results provide a theoretical base for seepage formation mechanism and evolution process, which can decrease the seepage disaster risk of dams.
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Key words:
- porous media /
- seepage failure /
- permeability coefficient /
- scale-invariant space /
- fractal dimensions
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图 5 土体分形维数和孔隙率理论值与试验值对比
Figure 5. Comparison between theoretical values and experimental results of soil fractal dimension and porosity
图 6 土体分形维数和渗透系数理论值与试验值对比
Figure 6. Comparison between theoretical values and experimental results of soil fractal dimension and permeability coefficient
表 1 不同试验土样颗粒级配
Table 1. Particle size distribution of different experimental soils
土样 不同粒径区间质量百分比/% >20 mm 20~10 mm 10~5 mm 5~2 mm 2~1 mm 1~0.5 mm 0.5~0.25 mm 0.25~0.10 mm 0.10~0.075 mm 0.075~0.025 mm <0.025 mm 1 5.37 19.98 26.38 16.68 8.96 8.61 8.22 4.80 0.70 0.21 0.09 2 8.32 17.92 29.97 15.85 9.12 8.55 8.34 1.80 0.08 0.04 0.01 3 7.32 20.32 30.73 14.97 11.45 10.23 3.23 1.47 0.20 0.06 0.02 4 6.85 19.42 26.56 18.54 13.22 9.43 4.18 1.70 0.07 0.02 0.01 5 2.58 10.80 15.42 15.94 17.32 20.30 12.64 4.70 0.20 0.08 0.02 6 1.16 5.30 9.70 9.93 11.22 19.37 18.32 14.55 8.45 1.98 0.02 7 / / / 0.30 1.80 6.80 22.12 35.30 20.38 8.21 5.09 8 / / / / 0.10 2.34 11.23 32.21 33.28 15.20 5.64 9 / / / / / / / / 0.87 2.11 97.02 10 / / / / / / / / 4.91 10.67 84.42 注：编号1~4为圆砾，5~8为砂，9~10为黏土。 
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表 2 不同试验土样质量分形维数与无标度区统计结果
Table 2. Statistical results of mass fractal dimension and scale-invariant space of different experimental soils
土样编号 土样类别 无标度区/mm 无标度区土体
颗粒含量/%分形维数 相关系数 下限 上限 1 圆砾 0.250 20.000 88.83 2.652 6 0.954 3 2 圆砾 0.250 20.000 89.75 2.734 6 0.881 2 3 圆砾 0.500 20.000 87.70 2.613 0 0.993 7 4 圆砾 0.500 20.000 87.17 2.567 8 0.984 1 5 粗砂 0.500 10.000 68.98 2.870 9 0.996 1 6 中砂 0.250 10.000 68.54 2.839 5 0.936 9 7 细砂 0.100 0.500 57.42 2.898 2 0.789 6 8 粉砂 0.075 0.250 65.49 2.999 0 0.906 6 9 黏土 0.002 0.005 97.02 2.995 1 0.903 9 10 黏土 0.002 0.005 84.42 2.966 0 0.882 1
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