基于负表皮层影响的径向溶质运移模型构建与新求解方法

doi:10.11867/j.issn.1001-8166.2023.010

地球科学进展 ›› 2023, Vol. 38 ›› Issue (4): 429 -440. doi: 10.11867/j.issn.1001-8166.2023.010

张开鑫 ¹(

), 黄璟胜 ¹ ^, ²(

), 王晨 ¹, 童晨晨 ¹, 王子成 ¹

^1.河海大学水文水资源学院，江苏南京 210098
^2.河海大学长江保护与绿色发展研究院，江苏南京 210098

收稿日期:2022-10-05 修回日期:2023-01-07 出版日期:2023-04-04
通讯作者: 黄璟胜 E-mail:zkx_0111@qq.com;cshuang0318@hhu.edu.cn
基金资助:
国家自然科学基金项目“耦合大—小尺度介质地下水流的滞后理论”(52079042);“淮河流域遥感土壤水分数据产品验证及暴雨洪水数据同化系统构建”(41830752)

A New Analytical Method for Modeling Radially Divergent Solute Transport in Two-zone Confined Aquifers with Negative Skin Effects

Kaixin ZHANG ¹( ), Ching-Sheng HUANG ¹ ^, ²( ), Chen WANG ¹, Chenchen TONG ¹, Zicheng WANG ¹

^1.College of Hydrology and Water Resources, Hohai University, Nanjing, 210098, China
^2.Yangtze Institution for Conservation and Development, Hohai University, Nanjing 210098, China

Received:2022-10-05 Revised:2023-01-07 Online:2023-04-04 Published:2023-04-18
Contact: Ching-Sheng HUANG E-mail:zkx_0111@qq.com;cshuang0318@hhu.edu.cn
About author:ZHANG Kaixin (1998-), female, Lianyungang City, Jiangsu Province, Master student. Research areas include groundwater mechanism and numerical simulation. E-mail: zkx_0111@qq.com
Supported by:
the National Natural Science Foundation of China “The lagging theory coupling groundwater flows in large- and small-scale media”(52079042);“Verification of remote sensing soil moisture data products and construction of rainstorm and flood data assimilation system in the huaihe river basin”(41830752)

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对于模拟示踪试验的溶质运移，传统方法采用细网格离散小尺度表皮层，存在网格数量多和计算时间长等问题。由于表皮层外缘的参数变化剧烈，即使采用细网格，仍然存在显著误差。因此发展基于完整井的新示踪试验溶质运移模型，提出新的瞬态Robin边界条件反映表皮层吸附/解吸溶质的影响，实现表皮层的无网格。应用拉普拉斯变换导出模型的解析解，通过有限元法建立基于非完整井的数值解。结果显示新瞬态Robin边界条件精准反映负表皮层的影响。对于求解反演问题，表皮层宽度的估计范围由0.45~0.54 m缩至0.47~0.48 m，纵向弥散度由0.6~10.0 m缩至6.4~7.7 m，参数估计的可靠度显著提升。表皮层的无网格节省97%的数值解计算时间。

关键词: 示踪试验, 溶质吸附/解吸, 瞬态Robin边界条件, 解析解, 有限元解

Various models have been developed for radially divergent tracer tests in two-zone confined aquifers of the skin and formation zones. However, existing numerical solutions require considerable computing time because of the fine spatial discretization of skins. The abrupt change in parameters near the skin-formation interface produces significant errors while predicting the spatiotemporal concentration near the interface, despite fine spatial discretization. In this study, a new model was developed for conducting radially divergent tracer tests in a partially penetrating well in a two-zone-confined aquifer. The skin was treated as a new transient Robin boundary condition specified at the skin-formation interface to reflect the effect of solute adsorption/release in the skin and achieve no skin discretization. A finite element solution for the model was developed. The analytical solution of the model modified for full penetration of the well was developed using the Laplace transform. These results suggest that the transient Robin boundary condition leads to accurate concentration predictions affected by negative skin. The analytical solution predicts reliable ranges of 0.47~0.48 m for the skin width w and 6.4~7.7 m for the longitudinal dispersivity $α l'$ , whereas a traditional solution exhibits a range of 0.45 m≤w≤0.54 m and 0.6 m≤ $α l'$ ≤10 m. The finite element solution required only 3% of the computing time for obtaining a finite element solution based on fine skin discretization. In conclusion, this study provides implications not only for theoretical advances but also for useful numerical methods.

Key words: Tracer test, Solute adsorption/release, Transient Robin boundary condition, Analytical solution, Finite element solution

中图分类号:

P641.2

图1 基于小尺度表皮层和大尺度含水层的承压含水层示踪试验及网格离散的示意图

Fig. 1 Schematic diagram of radially divergent tracer tests at a partially-penetrating well in a two-zone confined aquifer of small-scale skin and large-scale formation zones

图2 不同无量纲负表皮层宽度影响下的无量纲浓度时空分布

Fig. 2 Spatiotemporal distributions of the dimensionless concentration for different dimensionless widths of negative skins

图3 穿透曲线对每个参数变化的标准化敏感系数时域分布
（a）新单区解析解；（b）双区解析解

Fig. 3 Temporal normalized sensitivity coefficient for breakthrough curve to the change in each parameter plotted
（a） The single-zone analytical solution；（b） The two-zone analytical solution

图4 不同负表皮层迟滞因子影响下的无量纲浓度时空分布

Fig. 4 Spatiotemporal distributions of the dimensionless concentration for different skin retardation factors

图5 穿透曲线对每个参数变化的标准化敏感系数时域分布
（a）新单区解析解；（b）双区解析解

Fig. 5 Temporal normalized sensitivity coefficient for breakthrough curve to the change in each parameter plotted
（a） The single-zone analytical solution；（b） The two-zone analytical solution

图6 网格分辨率对新单区数值解和双区数值解的精度和效率的影响

Fig. 6 Influence of grid resolutions on the accuracy and efficiency of the single-zone and two-zone numerical solutions

图7 完整井和非完整井案例的无量纲浓度时空分布

Fig. 7 Spatiotemporal distributions of the dimensionless concentration for the cases of the fully-penetrating and partially-penetrating wells

图8 穿透曲线对每个参数变化的标准化敏感系数时域分布
（a）修改的解析解；（b）新单区数值解

Fig. 8 Temporal normalized sensitivity coefficient for breakthrough curve to the change in each parameter plotted
（a） The modified single-zone analytical solution；（b） The single-zone numerical solution

图9 新单区解析解与穿透曲线噪声数据的拟合
（a）~（b）穿透曲线标准化敏感系数的时域分布；（c）~（d）表皮层参数的标准误差等值线

Fig. 9 Curve fitting between the single-zone analytical solution and noisy data of breakthrough curve
（a）~（b） Temporal normalized sensitivity coefficients for breakthrough curves to each parameter；（c）~（d） Contour of standard error of estimate for skin parameters

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A New Analytical Method for Modeling Radially Divergent Solute Transport in Two-zone Confined Aquifers with Negative Skin Effects