Numerical Analysis of Time Fractional-Order Unsaturated Flow and Its Application
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摘要:
非饱和渗流过程的数值模拟对土质边坡稳定性分析、地下污染物迁移模拟等众多领域有着重要的意义。Richards方程由于其普遍适用性被广泛地应用，然而Richards方程所描述的渗流过程并未考虑在自然环境和实验中存在的反常扩散现象。针对这一问题，该文结合Caputo导数得到了具有更广泛渗流意义的时间分数阶Richards方程，采用有限差分法得到其离散格式并采用Picard法迭代求解，以及对分数阶参数和土水特征曲线进行了敏感性分析。最后，结合土柱入渗实验数据，比较了不同土水特征曲线下时间分数阶Richards方程得到的数值解。结果表明，VGM模型的时间分数阶Richards方程与实测数据具有更好的拟合效果，能够更好地描述地下水在非饱和土中的渗流过程。
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关键词:
- 非饱和渗流 /
- Richards方程 /
- 时间分数阶 /
- 有限差分 /
- 土柱实验
Abstract:Numerical simulation of the unsaturated flow process is of great significance to many fields such as soil slope stability analysis and migration simulation of underground pollutants. Generally, it is widely used due to the universal applicability of the Richards equation, but the seepage process described by the Richards equation does not involve the anomalous diffusion phenomenon in natural environment and experiments. To address this problem, the Caputo derivative was applied to obtain the time fractional-order Richards equation with broader seepage significance. Then the finite difference method was used to get the discretization scheme and the Picard method was chosen to solve it iteratively, and the sensitivity analysis of the fractional parameters and soil-water characteristic curves was carried out. Finally, combined with the experimental data of soil column infiltration, the numerical solutions obtained from the time fractional-order Richards equation under different soil-water characteristic curves were compared. The results show that, the time fractional-order Richards equation of the VGM model has better fitting effects for the measured data and can better describe the seepage process of groundwater in unsaturated soil.
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图 1 均质非饱和土的一维入渗模型
Figure 1. The 1D infiltration model for homogeneous unsaturated soil
图 3 不同模型和阶次
$\gamma $ 下获得的压力水头曲线： （a） Gardner模型；（b） Fredlund-Xing模型；（c） van Genuchten-Mualem模型Figure 3. Pressure head curves obtained under different models and orders
$\gamma $ : （a） Gardner model; （b） Fredlund-Xing model; （c） van Genuchten-Mualem model
图 5 水分测定装置：（a） EC-5；（b） MPS-6；（c） EM50
Figure 5. Moisture measuring devices: (a) EC-5; (b) MPS-6; (c) EM50
图 6 不同模型拟合得到的土壤水分特征曲线
Figure 6. Soil moisture characteristic curves fitted by different models
图 7 比较不同SWCC下不同阶次
$\gamma $ 获得的数值解： （a） VGM模型，γ =1；（b） VGM模型，γ = 0.97； （c） FX模型，γ =1；（d） FX模型，γ = 0.97Figure 7. Comparison of the numerical solutions obtained by different orders
$\gamma $ under different SWCC: （a） VGM model, γ =1; （b） VGM model, γ = 0.97; （c） FX model, γ =1;（d） FX model, γ = 0.97表 1 拟合参数
Table 1. Fitting parameters
model ${\theta _{\rm{s}}}$ ${\theta _{\rm{r}}}$ $\alpha $ n m Gardner 0.50 0.11 0.016 – – VGM 0.46 0.10 0.028 2.24 – FX 0.48 0.11 2513 1.17 181 
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表 2 不同模型的拟合参数
Table 2. Fitting parameters of different models
model ${\theta _{\rm{s}}}$ ${\theta _{\rm{r}}}$ $\alpha $ n m VGM 0.43 0.023 0.0142 1.74 – FX 0.43 0.023 33.7 1.62 1.60
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表 3 t =24 h的数值精度
Table 3. Numerical accuracy at t = 24 h
model VGM FX δRSE δRE/% δRSE δRE/% $\gamma $= 1 0.11 7.41 0.51 43.3 $\gamma $= 0.97 2.43E–2 1.93 0.48 41.8
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