Effect of Coefficient of Variation of Particle Size of Porous Media on Contaminant Transport
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摘要: 多孔介质污染物运移对于明晰地下水污染很重要，但在多孔介质中粒径变异系数（coefficient of variation，COV）对内部微观孔隙结构中污染物运移过程影响的研究还存在不足.为此，基于随机算法，提出了一种不同COV且孔隙度一致的多孔介质几何模型构建方法，通过对Navier-Stokes（N-S）方程和对流‒扩散方程（advection-diffusion equation，简称ADE）进行耦合求解得到多孔介质地下水流场及污染物浓度场，引入克里斯琴森均匀系数，定量评价流场流速分布的均匀性，基于MIM（mobile and immobile）模型和ADE模型分析耦合求解得到的穿透曲线特征.结果表明：随着粒径COV的增大，流场流速分布的不均性增强，MIM模型中的溶质流动区域占比$ \beta $、无量纲传质率$ {\alpha }^{*} $均增大；MIM模型拟合优度高于ADE模型，且随COV增大，ADE模型的拟合全局误差$ {E}_{\mathrm{i}} $逐步增大.总体上，粒径COV控制了溶质流动区域和非流动区域的大小及其之间的溶质交换强度，造成了多孔介质内部溶质运移的“非费克”行为，使得ADE模型的误差逐步增大，对于较大COV的多孔介质，MIM模型具有更好的适用性.Abstract: The contaminant transport in porous media is crucial for a comprehensive understanding of groundwater contamination. However, there are still inadequacies in the research concerning the effect of the coefficient of variation (COV) of particle size of porous media on the contaminant transport in the internal microscopic pore structure. A method is proposed to construct a geometric model of porous media with different coefficients of variation and consistent porosity based on a random algorithm. The groundwater flow field and contaminant concentration field in porous media are obtained by coupling the Navier-Stokes equation and the advection-diffusion equation. The uniformity of flow velocity distribution is evaluated quantitatively by introducing Christiansen uniformity coefficient. Based on the MIM (mobile and immobile) model and the ADE (advection-dispersion equation) model, the characteristics of breakthrough curves are analyzed. The results show that as the COV of particle size increases, the non-uniformity of the flow velocity distribution is enhanced. The COV of particle size is positively correlated with the ratio of solute mobile domain $ \beta $ and the dimensionless mass transfer rate $ {\alpha }^{*} $; the goodness-of-fitting of MIM model is better than that of ADE model, and as the COV increases, the fitting global error $ {E}_{\mathrm{i}} $ of ADE model gradually increases. Overall, the COV of particle size controls the size of mobile and immobile domain and solute exchange intensity between two domains, resulting in the non-Fickian behavior of solute transport in porous media. This also causes the error of ADE model gradually increases. MIM model has better applicability for the porous media with larger COV.
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Key words:
- porous media /
- contaminant transport /
- coefficient of variation /
- breakthrough curve /
- inverse model /
- groundwater
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图 4 孔隙结构内部优先流现象（COV=0.64）
Fig. 4. Preferential flow phenomenon in pore structure(COV=0.64)
图 6 (a) 水流场及(b)浓度场中流动区域与非流动区域分布
Fig. 6. Mobile and immobile domain distribution in (a) flow field and (b) concentration field
图 7 不同COV的多孔介质内部污染物运移穿透曲线
Fig. 7. Contaminant transport breakthrough curves in porous media with different coefficients of variation
图 9 标准化流速概率分布函数曲线
Fig. 9. Probability distribution functions of the normalized flow velocity
图 10 不同COV的MIM模型及ADE模型拟合穿透曲线
Fig. 10. Fitted breakthrough curves with different coefficients of variation based on MIM model and ADE model
表 1 不同COV多孔介质流速均匀性评价指标
Table 1. Evaluation indices of flow velocity uniformity of the porous media with different coefficients of variation
COV CV $ {\gamma }_{\mathrm{\upsilon }} $ CU 0.18 1.253 4 0.546 4 0.092 8 0.27 1.108 1 0.580 4 0.160 8 0.45 1.231 8 0.537 4 0.074 9 0.64 1.161 5 0.564 3 0.128 5 
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表 2 不同COV多孔介质的MIM模型和ADE模型拟合优度统计结果
Table 2. Statistical results from the goodness-of-fitting of the MIM model and ADE model for porous media with different coefficients of variation
COV MIM模型 ADE模型 $ {{R}^{2}}_{\mathrm{M}\mathrm{I}\mathrm{M}} $ $ {E}_{\mathrm{M}\mathrm{I}\mathrm{M}} $ $ {{R}^{2}}_{\mathrm{A}\mathrm{D}\mathrm{E}} $ $ {E}_{\mathrm{A}\mathrm{D}\mathrm{E}} $ 0.18 0.999 9 0.005 925 0.999 8 0.005 458 0.27 1.000 0 0.001 071 0.999 9 0.004 751 0.45 1.000 0 0.001 614 0.999 8 0.006 208 0.64 1.000 0 0.000 863 0.999 6 0.009 094
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表 3 不同COV多孔介质的MIM模型和ADE模型参数估计值
Table 3. Estimated values of parameters from the MIM model and ADE model for porous media with different coefficients of variation
COV MIM模型 ADE模型 $ \beta $ $ {\alpha }^{*} $a $ {D}_{\mathrm{L},\mathrm{M}\mathrm{I}\mathrm{M}} $
(m2·s‒1)$ {\lambda }_{\mathrm{M}\mathrm{I}\mathrm{M}} $b(m) $ {\stackrel{-}{u}}_{\mathrm{f}\mathrm{i}\mathrm{t},\mathrm{M}\mathrm{I}\mathrm{M}}/\stackrel{-}{u} $ $ {D}_{\mathrm{L},\mathrm{A}\mathrm{D}\mathrm{E}} $
(m2·s-1)$ {\lambda }_{\mathrm{A}\mathrm{D}\mathrm{E}} $c
(m)$ {\stackrel{-}{u}}_{\mathrm{f}\mathrm{i}\mathrm{t},\mathrm{A}\mathrm{D}\mathrm{E}}/\stackrel{-}{u} $ 0.18 0.71 0.003 0.134 843.032 0.54 0.193 1214.218 0.77 0.27 0.98 0.128 0.076 478.138 0.80 0.089 559.925 0.80 0.45 0.98 0.132 0.091 572.507 0.81 0.111 698.333 0.82 0.64 0.96 0.201 0.104 654.294 0.79 0.140 880.780 0.80 注：$ {\alpha }^{*} $a表示无量纲传质率；$ {\lambda }_{\mathrm{M}\mathrm{I}\mathrm{M}} $b表示MIM模型的拟合弥散度，$ {\lambda }_{\mathrm{M}\mathrm{I}\mathrm{M}}={D}_{\mathrm{L},\mathrm{M}\mathrm{I}\mathrm{M}}/\stackrel{-}{u} $；$ {\lambda }_{\mathrm{A}\mathrm{D}\mathrm{E}} $c表示ADE模型的拟合弥散度，$ {\lambda }_{\mathrm{A}\mathrm{D}\mathrm{E}}={D}_{\mathrm{L},\mathrm{A}\mathrm{D}\mathrm{E}}/\stackrel{-}{u} $.
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