喀斯特表层岩溶带岩−土界面优势流数值模拟研究

吴怡; 陈喜; 张志才; 刘维翰; 高峰钧; 刘秀强; 彭韬

doi:10.11932/karst2026y021

喀斯特表层岩溶带岩−土界面优势流数值模拟研究

doi: 10.11932/karst2026y021

吴怡^1,,
陈喜^{1, 2, ,},
张志才³,
刘维翰^{1, 2},
高峰钧³,
刘秀强^{1, 2},
彭韬^{4, 5}

1.
天津大学地球系统科学学院表层地球系统科学研究院, 天津 300072
2.
天津市环渤海关键带科学与可持续发展重点实验室, 天津 300072
3.
河海大学水文水资源学院, 江苏南京 210098
4.
中国科学院地球化学研究所环境地球化学国家重点实验室, 贵州贵阳 550081
5.
中国科学院普定喀斯特生态系统观测研究站, 贵州普定 562100

基金项目: 国家自然科学基金重点支持项目（42030506）, 国家自然科学基金面上项目（42571044），国家自然科学基金国际(地区)合作与交流项目，喀斯特社会-生态系统动力学与可持续发展（42561144297）

详细信息

作者简介:
吴怡（2000－），女，硕士研究生，研究方向为水文地质数值模拟。E-mail：wuyi001@tju.edu.cn

通讯作者:
陈喜（1964－），男，教授，研究方向为地下水及水文数值模拟。E-mail：xi_chen@tju.edu.cn。

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出版历程
- 收稿日期: 2025-09-29
- 录用日期: 2026-05-22
- 修回日期: 2026-05-07
- 网络出版日期: 2026-06-30

Numerical Simulation Study of Preferential Flow at the Rock-Soil Interface in the Epikarst Zone

WU Yi^1
,,
CHEN Xi^{1, 2
, ,},
ZHANG Zhicai³,
LIU Weihan^{1, 2},
GAO Fengjun³,
LIU Xiuqiang^{1, 2},
PENG Tao^{4, 5}

1.
Institute of Surface-Earth System Science, School of Earth System Science, Tianjin University, Tianjin 300072, China
2.
Tianjin Key Laboratory of Earth Critical Zone Science and Sustainable Development in Bohai Rim, Tianjin University, Tianjin 300072, China
3.
College of Hydrology and Water Resources, Hohai University, Nanjing 210098, China
4.
State Key Laboratory of Environment Geochemistry, Institute of Geochemistry, Chinese Academy of Sciences, Guiyang 550081, China
5.
Puding Karst Ecosystem Research Station, Chinese Academy of Sciences; Puding, Guizhou 562100, China

摘要

摘要: 我国西南喀斯特表层岩溶带具有土壤充填的溶槽以及交错发育的裂隙，降雨入渗过程受溶槽土壤、岩−土界面优势通道以及裂隙水流共同作用。本文基于原位溶槽剖面染色示踪试验与土壤、裂隙水分长期观测资料，构建水分运移数值模型，定量揭示岩−土界面优势流与周边倾斜裂隙对水分运移过程及其出流通量的贡献。结果表明：岩−土界面优势流为溶槽中水分垂向运移提供了高效优先通道，实际情况（考虑优势流和裂隙影响）最大流速可达0.027 m∙s⁻¹，未考虑优势流时水分运移呈现典型活塞流特征，未考虑裂隙影响时流速约为考虑裂隙时的1.3倍；溶槽下边界优势流区水分通量随降雨强度增大而显著增加，非优势流区仅在大雨和暴雨事件后呈缓慢上升；在土壤优势流区，强降雨条件下倾斜裂隙对优势流流出具有削弱作用，在非优势流区，大雨和暴雨事件后忽略周边裂隙的下边界出流增加量总体大于有裂隙情形。研究明确了岩−土界面优势流是降水高效补给地下水的主导路径，为喀斯特地区水资源管理与生态保护提供了协同调控的科学依据。
- 表层岩溶带 /
- 优势流 /
- 裂隙 /
- 岩−土界面
Abstract: In the karst region of southwest China, the epikarst zone is characterized by soil-filled solution grooves and an intricate network of fractures. Rainfall infiltration is jointly controlled by the groove soil, preferential flow pathways at the rock-soil interface, and the surrounding fractures. However, the dynamic interactions among the groove soil, rock-soil interface, and fracture network, especially their response mechanisms under different rainfall intensities, remain poorly quantified. Based on in-situ dye tracer tests on a solution groove profile, combined with long-term monitoring of soil moisture and fracture water content, this study developed a numerical model of water flow to quantitatively reveal the contributions of preferential flow at the rock-soil interface and adjacent inclined fractures to water movement and the outlet flux at the lower boundary. The study was conducted on a typical epikarst profile in Puding, Guizhou Province. The profile is 10 m long and 2.24-2.5 m high, containing three solution grooves with widths of 0.22-0.82 m, four sub-horizontal fractures (dip angle~5.4°) and three sub-vertical fractures (dip angle~83.2°). A dye tracer test using Brilliant Blue FCF was carried out on the middle solution groove at a rainfall intensity of 60 mm·h⁻¹ for one hour, followed by excavation to reveal the spatial distribution of preferential flow. Five soil moisture sensors were installed at depths of 5, 20, 50, 100 and 180 cm in the groove soil, and five fracture moisture sensors (F 1-F 5) were placed in adjacent fractures. Meteorological data, including rainfall, were recorded every 30 minutes from September 2022 to September 2024. Based on the soil and fracture moisture responses, four representative rainfall events were selected for detailed analysis: light rain (4.3 mm), moderate rain (12.3 mm), heavy rain (34.2 mm), and storm (62.1 mm).A numerical model of water flow was constructed using COMSOL Multiphysics. Unsaturated flow in the groove soil and fractures was described by the Richards equation, and the soil water retention and hydraulic conductivity were described by the van Genuchten model. Solute transport in the dye tracer experiment was simulated using the convection-diffusion equation with Langmuir isothermal adsorption. The effective aperture of the open rock-soil interface was inversely calibrated against the dye distribution and soil moisture dynamics, yielding an aperture of 4.22 mm in the upper zone (A) and 1.66 mm in the lower zone (B). Hydraulic parameters for the soil and fractures were optimized using the BOBYQA iterative algorithm. Model validation against two years of soil and fracture moisture data gave root-mean-square errors (RMSE) mostly below 0.05 m³·m⁻³ and mean absolute percentage errors (MAPE) below 5%, indicating that the calibrated parameters represent the average hydraulic conditions over the entire monitoring period.Three simulation scenarios were compared under a storm event: (A) actual conditions with an open rock-soil interface (preferential flow) and surrounding fractures; (B) closed rock-soil interface (no preferential flow); and (C) no surrounding fractures. The results show that preferential flow at the rock-soil interface provides an efficient pathway for vertical water movement. Under actual conditions (scenario A), the maximum flow velocity reaches 0.027 m·s⁻¹. When preferential flow is ignored (scenario B), the flow exhibits typical piston-flow characteristics with a uniform flow field and a maximum velocity of only 7.32$ \times $10⁻⁵ m·s⁻¹. When surrounding fractures are absent (scenario C), the flow velocity in the groove is about 1.3 times that of scenario A, indicating that inclined fractures retard vertical flow under heavy rainfall. For the four representative rainfall events, the outlet water flux at the lower boundary of the solution groove was analyzed. In the preferential flow zone of the rock-soil interface, the flux increased significantly with rainfall intensity. In the non-preferential flow zone (soil matrix), the flux was very small or even negative during light and moderate rain events, indicating that infiltrated water did not reach the lower boundary; only after heavy rain and storm events did a slow increase occur, reflecting slow matrix flow. The cumulative outflow volumes over 24 hours were 0.19 L (light rain), 18.25 L (moderate rain), 29.62 L (heavy rain), and 132.33 L (storm rain) in the preferential flow zone, whereas the non-preferential zone contributed only 0.12-0.23 L for all events. This demonstrates that the preferential flow zone is one to two orders of magnitude more effective in recharging the deep epikarst than the non-preferential zone. Furthermore, the presence of inclined fractures significantly influences the outflow process. In the preferential flow zone, after heavy rain and storm events, ignoring the surrounding fractures led to a faster rise and a larger flux, indicating that the inclined fractures weaken the preferential outflow. In the non-preferential flow zone, after heavy rain and storm events, the increase in outlet flux when neglecting the surrounding fractures was generally greater than when fractures were present, suggesting that fractures also modulate matrix flow, but with a more persistent effect. In conclusion, this study confirms that preferential flow at the rock-soil interface is the dominant pathway for rapid groundwater recharge from precipitation. The combination of dye tracer tests, long-term in-situ monitoring, and physically-based numerical modeling provides a reliable approach for quantifying the roles of rock-soil interface aperture and fractures under different rainfall intensities. The findings offer a scientific basis for the coordinated management of water resources and ecological protection in karst regions.
- epikarst /
- preferential flow /
- fissure /
- rock-soil interface

HTML

图 1 剖面实景图及模型概化图

Figure 1. Profile photograph and conceptual model

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图 2 染色试验剖面概化图（a）、静置24 h后实景图（b）及局部放大图（c）

Figure 2. (a) Generalized diagram of the dye staining test profile, (b) actual view after standing for 24 hours, and (c) partially enlarged detail

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图 3 土壤及裂隙含水率动态变化

Figure 3. Dynamic changes in soil and fissure moisture content

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图 4 不同降雨事件土壤和裂隙含水率响应

Figure 4. Response of soil and fissure moisture content to rainfall events

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图 5 染色试验土壤水分监测与模拟对比

Figure 5. Comparison of monitored and simulated soil moisture in the dye staining experiment

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图 6 染色试验与溶质运移模拟的染色剂影响范围对比

Figure 6. Comparison of dye tracer influence range between the dye test and solute transport simulation

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图 7 溶槽中土壤和裂隙2023-2024年含水率实测与模拟值对比

Figure 7. Comparison of measured and simulated moisture content in soil and fractures of the solution groove (2023-2024)

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图 8 溶槽中岩−土界面不闭合（A）和闭合（B）以及未考虑周边裂隙（C）情景下水流流速和流向对比

Figure 8. Comparison of flow velocity and direction in the solution groove under scenarios of (A) open rock-soil interface, (B) closed rock-soil interface, and (C) excluding adjacent fractures

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图 9 不同降雨事件溶槽下边界（a）优势流区和（b）非优势流区出流通量

Figure 9. Outlet flux at the lower boundary of the solution groove under different rainfall events: (a) preferential flow zone and (b) non-preferential flow zone

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图 10 不同边界条件（a）优势流区和（b）非优势流区下边界出流通量

Figure 10. Outflow flux at the lower boundary under different boundary conditions: (a) preferential flow zone and (b) non-preferential flow zone

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表 1 土壤及裂隙特征值

Table 1. Soil and Fissure Characteristics

土壤编号	土壤质地	黏土 (<0.002 mm)	粉砂 (0.002-0.05 mm)	沙土 (>0.05 mm)	容重/ g∙cm⁻³	裂隙编号	隙宽 /mm	渗透系数/ $ \times $10⁻⁵
土壤编号	土壤质地	%			容重/ g∙cm⁻³	裂隙编号	隙宽 /mm	m∙s⁻¹
S 1	粉质黏壤土	36.99	55.49	7.52	1.31	F 1	0.470	0.46
S 2	粉质黏壤土	30.51	65.49	4.00	1.58	F 2	0.204	0.39
S 3	粉质黏土	56.60	41.08	2.32	1.51	F 3	1.110	4.94
S 4		56.61	42.62	0.77	1.64	F 4	15.000	9.05
S 5		57.23	41.89	0.88	1.60	F 5	21.000	28.93
S 5		57.23	41.89	0.88	1.60	F _┴	2.000	/

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表 2 选取的典型降雨场次

Table 2. Rainfall Events Selection

降雨类型	降雨日期	总雨量/mm	历时/h	降雨强度/mm∙h⁻¹
小雨	2023/9/27	4.3	5	0.86
中雨	2024/5/26	12.3	1	12.30
大雨	2024/8/10	34.2	5	6.84
暴雨	2024/5/15	62.1	14	4.44

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表 3 模型率定的土壤和裂隙水力学参数

Table 3. Hydraulic parameters of soil and fractures calibrated by the model

介质类型	埋深/cm/编号	K/m∙s⁻¹	$ {\theta }_{s} $	$ {\theta }_{r} $	α/m⁻¹	n
土壤	0-14.5	$ 7.04\times {10}^{-4} $	0.587	0.033	0.264	7.818
	14.5-35	$ 8.51\times {10}^{-5} $	0.488	0.016	0.236	7.789
	35-75	$ 6.21\times {10}^{-5} $	0.464	0.057	0.272	5.589
	75-140	$ 7.32\times {10}^{-5} $	0.471	0.005	0.479	3.201
	140-200	$ 5.84\times {10}^{-6} $	0.469	0.004	0.400	6.602
	200-220	$ 3.12\times {10}^{-6} $	0.466	0.004	0.788	4.012
水平裂隙垂向裂隙	F 1	0.0047	1	0.060	0.419	4.496
	F 2	0.0050	1	0.026	0.493	3.549
	F 3	0.0234	0.625	0.041	0.367	7.809
	F 4	0.0839	0.450	0.040	0.463	4.229
	F 5	0.0334	0.327	0.160	3.585	4.645
	F _┴	0.0001	0.833	0.001	0.810	1.420
岩−土间隙	A	$ 3.30\times {10}^{-2} $	1	0.001	0.464	6.124
	B	$ 8.71\times {10}^{-2} $	1	0.001	0.556	4.333
注：为简化模型，三条垂向裂隙的水力参数赋值相同。

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表 4 2023-2024年土壤和裂隙含水率模拟误差统计结果

Table 4. Statistical results of simulation errors for soil and fracture moisture content (2023-2024)

土壤测点	RMSE/m³·m⁻³	MAPE/%	裂隙测点	RMSE/m³·m⁻³	MAPE/%
S 1	0.039	3.407	F 1	0.042	3.325
S 2	0.044	3.510	F 2	0.044	2.500
S 3	0.027	1.023	F 3	0.050	3.541
S 4	0.021	2.760	F 4	0.027	1.400
S 5	0.051	5.008	F 5	0.006	0.155

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表 5 不同降雨场次下边界累计出流体积

Table 5. Cumulative outflow volume at the lower boundary for different rainfall events

降雨类型	降雨日期	总雨量/mm	优势流区出流体积/L	非优势流区出流体积/L
小雨	2023/9/27	4.3	0.19	0.12
中雨	2024/5/26	12.3	18.25	0.15
大雨	2024/8/10	34.2	29.62	0.14
暴雨	2024/5/15	62.1	132.33	0.23
注：出流体积(L)由下边界通量（kg·m⁻¹·s⁻¹）在24小时内积分得到累计出流量（kg∙m⁻¹），除以水密度（1000 kg∙m⁻³）再乘以模型纵深（视为1 m）换算得出。

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参考文献(33)

[1]	凡非得, 王克林, 熊鹰, 宣勇, 张伟, 岳跃民. 西南喀斯特区域水土流失敏感性评价及其空间分异特征 [J]. 生态学报, 31(21): 6353-6362. Fan F D, Wang K L, Xiong Y, Xuan Y, Zhang W, Yue Y M. Assessment and spatial distribution of water and soil loss in karst regions, southwest China [J]. Acta Ecologica Sinica, 2011, 31(21): 6353-6362.
[2]	邹小阳, 周旺, 孙文俊, 肖克飚, 胡封兵. 桂西北岩溶区土壤理化性质及相关性分析 [J]. 东北农业科学, 2021, 46(3): 41-7+129. Zou Xiaoyang, Zhou Wang, Sun Wenjun, Xiao Kebiao, Hu Fengbing. Soil physical and chemical properties and correlation analysis in karst area of Northwest Guangxi [J]. Journal of Northeast Agricultural Sciences, 2021, 46(3): 41-47+129.
[3]	覃小群, 蒋忠诚. 表层岩溶带及其水循环的研究进展与发展方向[J]. 中国岩溶, 2005, 24(3): 250-254. Qin Xiaoqun, Jiang Zhongcheng. A review on recent advances and perspective in epikarst water study[J]. Carsologica Sinica, 2005, 24(3): 250-254.
[4]	杨继翔, 张志才, 陈喜, 刘秀强, 谢永玉, 彭韬, 陈波. 喀斯特关键带裂隙土壤水分时空变化特征及其水流路径[J]. 中国岩溶, 2025, 44(5): 1063-1073. doi: 10.11932/karst20250510 Yang Jixiang, Zhang Zhicai, Chen Xi, Liu Xiuqiang, Xie Yongyu, Peng Tao, Chen Bo. Spatiotemporal variation characteristics and water flow path of soil moisture of the critical zones fissures in karst area[J]. Carsologica Sinica, 2025, 44(5): 1063-1073. doi: 10.11932/karst20250510
[5]	Zhou J, Tang Y Q, Yang P, Zhang X H, Zhou N Q, Wang J X. Inference of creep mechanism in underground soil loss of karst conduits I. Conceptual model[J]. Natural Hazards, 2012, 62(3): 1191-1215. doi: 10.1007/s11069-012-0143-3
[6]	Zhang Z C, Chen X, Ghadouani A, Shi P, Xue X W. Modeling hydrological processes influenced by soil, rock and vegetation in a small karst basin of southwest China[J]. Hydrology and Earth System Sciences Discussions, 2010, 7: 561-592.
[7]	冷德明, 史文兵, 李华, 梁风. 表层喀斯特带溶蚀岩体单轴压缩力学特性模拟研究[J]. 中国岩溶, 2023, 42(6): 1258-1269. Leng Deming, Shi Wenbing, Li Hua, Liang Feng. Numerical simulation on uniaxial compressive mechanical properties of karstified rock mass in epikarst zone[J]. Carsologica Sinica, 2023, 42(6): 1258-1269.
[8]	Liu X Q, Liu W H, Chen X, Wang L C, Zhang Z C, Peng T. Estimating fracture characteristics and hydraulic conductivity from slug tests in epikarst of southwest China[J]. Journal of Hydrology: Regional Studies, 2024, 53: 101777. doi: 10.1016/j.ejrh.2024.101777
[9]	Sohrt J, Ries F, Sauter M, Lange J. Significance of preferential flow at the rock soil interface in a semi-arid karst environment[J]. Catena, 2014, 123: 1-10. doi: 10.1016/j.catena.2014.07.003
[10]	Zhao Z M, Wang Q H. Effect of Rock Film Mulching on Preferential Flow at Rock−Soil Interfaces in Rocky Karst Areas [J/OL]. Water, 2023, 15(9): 1775
[11]	Liu J X, Deng Z M, Chen L D, Huang Y, Peng X Y. Quantifying the influence of soil-rock interfaces on water infiltration rate in karst landscapes[J]. Geoderma, 2025, 460: 117432. doi: 10.1016/j.geoderma.2025.117432
[12]	王甲荣, 陈喜, 张志才, 张润润, 朱彪, 龚轶芳, 刘皓, 袁瞬飞. 喀斯特溶槽岩-土界面优势流及其对土壤水分动态的影响[J]. 中国岩溶, 2019, 38(1): 109-116. Wang Jiarong, Chen Xi, Zhang Zhicai, Zhang Runrun, Zhu Biao, Gong Yifang, Liu Hao, Yuan Shunfei. Preference flow at rock-soil interface and its influence on soil water dynamics in the karst troughs[J]. Carsologica Sinica, 2019, 38(1): 109-116.
[13]	许胜兵, 彭旭东, 戴全厚, 刘婷婷, 岑龙沛. 喀斯特高原石漠化区露石岩-土界面与非岩-土界面土壤入渗特性差异[J]. 水土保持学报, 2022, 36(5): 136-143. Xu Shengbing, Peng Xudong, Dai Quanhou, Liu Tingting, Cen Longpei. Differences of soil infiltration characteristics between exposed rock-soil interface and non-rock-soil interface in rocky desertification area of karst plateau[J]. Journal of Soil and Water Conservation, 2022, 36(5): 136-143.
[14]	豆红强, 谢森华, 简文彬, 王浩, 郭朝旭. 球状风化花岗岩类土质边坡土-岩界面优势流潜蚀特性研究[J]. 岩土力学, 2024, 45(4): 950-960. doi: 10.16285/j.rsm.2023.0644 Dou Hongqiang, Xie Senhua, Jian Wenbin, Wang Hao, Guo Chaoxu. Characteristics of preferential flow suffosion of soil-rock interface in spherical weathered granite slopes[J]. Rock and Soil Mechanics, 2024, 45(4): 950-960. doi: 10.16285/j.rsm.2023.0644
[15]	徐子凡, 陈喜, 刘维翰, 刘皓, 张志才. 表层岩溶带土岩结构对降雨−径流响应特征的影响[J]. 中国岩溶, 2024, 43(4): 863-875. doi: 10.11932/karst20240406 Xu Zifan, Chen Xi, Liu Weihan, Liu Hao, Zhang Zhicai. Effect of soil-rock structures on the characteristics of rainfall-runoff responses in epikarst zones[J]. Carsologica Sinica, 2024, 43(4): 863-875. doi: 10.11932/karst20240406
[16]	Zhang J, Chen H S, Fu Z Y, Luo Z D, Wang F, Wang K L. Effect of soil thickness on rainfall infiltration and runoff generation from karst hillslopes during rainstorms[J]. European Journal of Soil Science, 2022, 73(4): e13288. doi: 10.1111/ejss.13288
[17]	曾庥, 彭旭东, 戴全厚, 刘婷婷, 许胜兵, 岑龙沛. 石漠化区露石岩-土界面流形成过程模拟试验[J]. 农业工程学报, 2022, 38(17): 110-117. doi: 10.11975/j.issn.1002-6819.2022.17.012 Zeng Xiu, Peng Xudong, Dai Quanhou, Liu Tingting, Xu Shengbing, Cen Longpei. Simulation test on the formation of rock-soil interface flow from outcrops in the rocky desertification areas[J]. Transactions of the Chinese Society of Agricultural Engineering, 2022, 38(17): 110-117. doi: 10.11975/j.issn.1002-6819.2022.17.012
[18]	沈蔚, 王建力, 王家录, 蒋先淑, 毛庆亚, 陈挚秋, 刘肖. 贵州纳朵洞洞穴水水文地球化学变化特征及其环境意义[J]. 环境科学, 2015, 36(12): 4455-4463. doi: 10.13227/j.hjkx.2015.12.017 Shen Wei, Wang Jianli, Wang Jialu, Jiang Xianshu, Mao Qingya, Chen Zhiqiu, Liu Xiao. Variation characteristics of cave water hydrogeochemistry in Naduo Cave of Guizhou and its implications for environment research[J]. Environmental Science, 2015, 36(12): 4455-4463. doi: 10.13227/j.hjkx.2015.12.017
[19]	Liu X Q, Chen X, Zhang Z C, Liu W H, Gao F J, Cheng Q B, Chen J, Peng T. The role of rock fractures as a water source for trees growing in karst[J]. Water Resources Research, 2025, 61(11): e2024WR039588. doi: 10.1029/2024WR039588
[20]	Indraratna B, Premadasa W, Brown E, Gens A, Heitor A. Shear strength of rock joints influenced by compacted infill[J]. International Journal of Rock Mechanics and Mining Sciences, 2014, 70: 296-307. doi: 10.1016/j.ijrmms.2014.04.019
[21]	Khosravi A, Mousavi S, Serej Ali D. Hydraulic behavior of infilled fractured rocks under unsaturated conditions [C]// Proceedings of the Geotechnical and Structural Engineering Congress 2016. Reston, Virginia: American Society of Civil Engineers, 2016: 1708-1718.
[22]	Richards L A. Capillary Conduction of Liquids through Porous Mediums[J]. Physics, 1931, 1(5): 318-333. doi: 10.1063/1.1745010
[23]	Celia M A, Bouloutas E T, Zarba R L. A general mass-conservative numerical solution for the unsaturated flow equation[J]. Water Resources Research, 1990, 26(7): 1483-1496. doi: 10.1029/WR026i007p01483
[24]	Liu W H, Chen X, Wang L C, Zhang Z C, Liu X Q, Peng T, Chen J, Dong J Z. The role of karstic soil-rock structures on subsurface stormflow dynamics in southwest China[J]. Journal of Hydrology, 2025, 657: 133126. doi: 10.1016/j.jhydrol.2025.133126
[25]	Van Genuchten M T. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils[J]. Soil Science Society of America Journal, 1980, 44(5): 892-898.
[26]	Camillo P, Gurney R. A resistance parameter for bare-soil evaporation models[J]. Soil Science, 1986, 141(2): 95-105. doi: 10.1097/00010694-198602000-00001
[27]	Wei Y Q, Cao X D. A COMSOL-PHREEQC Coupled Python Framework for Reactive Transport Modeling in Soil and Groundwater[J]. Groundwater, 2022, 60(2): 284-294. doi: 10.1111/gwat.13144
[28]	Li J, Werth C J. Modeling sorption isotherms of volatile organic chemical mixtures in model and natural solids[J]. Environmental Toxicology and Chemistry, 2002, 21(7): 1377-1383. doi: 10.1002/etc.5620210707
[29]	Clancy A, Chen D, Bruns J, Nadella J, Stealey S, Zhang Y, Timperman A, Zustiak S P. Hydrogel-based microfluidic device with multiplexed 3D in vitro cell culture[J]. Scientific Reports, 2022, 12(1): 17781. doi: 10.1038/s41598-022-22439-y
[30]	Ketelsen H, Meyer-Windel S. Adsorption of brilliant blue FCF by soils[J]. Geoderma, 1999, 90(1): 131-145.
[31]	Zhang X X, Qi X B, Zhou X G, Pang H B. An in situ method to measure the longitudinal and transverse dispersion coefficients of solute transport in soil[J]. Journal of Hydrology, 2006, 328(3): 614-619. doi: 10.1016/j.jhydrol.2006.01.004
[32]	Powell M J D. The BOBYQA algorithm for bound constrained optimization without derivatives [R]. Cambridge, UK: Department of Applied Mathematics and Theoretical Physics, University of Cambridge, 2009.
[33]	Yang W, Peng X D, Dai Q H, Li C L, Xu S B, Liu T T. Storage infiltration of rock-soil interface soil on rock surface flow in the rocky desertification area[J]. Geoderma, 2023, 435: 116512. doi: 10.1016/j.geoderma.2023.116512