Analytical calculation of steady seepage in unsaturated layered soils
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摘要:
广泛存在的成层土非饱和稳态渗流的解析计算研究相对薄弱。基于达西渗流定律和土层界面的连续性条件，构建了描述非饱和成层土稳态渗流过程的数学模型。使用分离变量技术和数学归纳思想，获得了成层土同一剖面的基质吸力、有效饱和度与吸应力沿高程分布的解析表达式。基于COMSOL数值分析平台对解析算法进行了验证计算，从而实现了非饱和成层土稳态渗流过程的解析求解。而后，探讨了土层界面的存在对渗流过程的影响并开展了参数敏感度分析。分析表明：（1）相同入渗条件下相同高程处砂土的基质吸力最大，黏土最小；地表渗流速率的不同对粉土层有效饱和度分布影响最大，砂土最小。（2）黏土层中吸应力近乎线性增长，砂土的吸应力则沿高程先增大后减小；土层界面的存在会影响基质吸力沿高程的增长速率，可使有效饱和度和吸应力沿高程分布发生突变。（3）Gardner模型参数α值越小，相同高程处基质吸力值越大，饱和土渗透系数（ks值）越小，基质吸力增长速率越慢；α与ks取值越小，有效饱和度降低速率越慢；α取值越小或ks取值越大，地表处的吸应力值越大。研究成果可为诸如边坡稳定等工程地质问题提供一定的理论支撑。
Abstract:The analytical calculations of unsaturated steady-state seepage in the widespread layered soils wewre relatively rare. Based on the Darcy seepage theorem and the continuity condition of soil layer interface, a mathematical model describing the steady seepage process of unsaturated layered soil was constructed. The analytical expressions of matric suction, effective saturation, and suction stress along the elevation for the same profile of layered soil were obtained by using the separation variable technique and mathematical induction. The analytical algorithm was validated based on the COMSOL numerical analysis platform, which enabled the analytical solution of the steady-state seepage process in unsaturated layered soils. Then, the influence of soil layer interface on seepage process was discussed and the sensitivity analysis of parameters was carried out. The analysis shows that: (1) Under the same infiltration conditions, the matric suction of sand at the same elevation is the highest, while that of clay is the lowest. The difference in surface seepage rate has the greatest impact on the distribution of effective saturation in the silt layer, while it has the smallest impact in the sand layer. (2) The suction stress in the clay layer increases almost linearly, while the suction stress in sand increases first and then decreases along the elevation. The presence of soil interface can affect the growth rate of matric suction along the elevation, leading to a sudden change in the distribution of effective saturation and suction stress along the elevation. (3) The smaller Gardner model parameter α, the greater the matric suction value at the same elevation, while the smaller the ks value of saturated soil, the slower the growth rate of matric suction. The smaller the values of ks and α, the slower the rate of effective saturation reduction. The smaller the value of α or the larger the value of ks, the greater the suction stress value at the surface. The research results can provide theoretical support for engineering geological problems such as slope stability.
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表 1 砂土、粉土与黏土的典型参数
Table 1. Typical parameters of sandy, powdery and clayey soils
土的类型 n α/kPa−1 ks/（m·s−1） 砂土 5.0 0.100 3×10−5 粉土 4.0 0.010 1×10−7 黏土 2.0 0.005 5×10−8 
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表 2 解析方法的拟合优度R2
Table 2. Goodness of fit R2 of analytical method
拟合优度 渗流速率/（m·s−1） R2 砂土 粉土 黏土 计算值 −3.14×10−8 1.000 0.977 1.000 −3.14×10−9 1.000 0.996 1.000 0 1.000 1.000 1.000 1.15×10−9 1.000 0.999 0.997 1.15×10−8 1.000 0.987 0.977
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