2022 , Vol. 24 >Issue 11: 2071 - 2088

• 王信雷 , 1 ,
• 芮小平 , 1, * ,
• 谢宜霖 1 ,
• 朱益虎 2 ,
• 杨蕴 1

• 1.河海大学地球科学与工程学院，南京 211000
• 2.江苏省地质测绘院，南京 210008
*芮小平（1975-），男，江苏苏州人，博士，教授，主要从事地理信息科学方面的研究。E-mail:
 王信雷（1997-），男，江西上饶人，硕士生，研究方向为地理信息系统。E-mail: xinleical@163.com

修回日期: 2022-04-12

网络出版日期: 2023-01-25

A Cellular Automata Simulation Model of Site Surface Pollution Diffusion with Adaptive Time Step

• WANG Xinlei , 1 ,
• RUI Xiaoping , 1 ,
• XIE Yilin 1 ,
• ZHU Yihu 2 ,
• YANG Yun 1
Expand
• 1. School of Earth Sciences and Engineering, Hohai University, Nanjing 211000, China
• 2. Jiangsu Geologic Surveying and Mapping Institute, Nanjing 210008, China
*RUI Xiaoping, E-mail:

Revised date: 2022-04-12

Online published: 2023-01-25

Supported by

National Key Research and Development Program of China(2019YFC1804304)

National Natural Science Foundation of China(41771478)

Fundamental Research Funds for the Central Universities(2019B02514)

### 2 研究方法

#### 2.1.2 元胞空间划分

##### 图2 AABB包围盒

Fig. 2 AABB bounding box

##### 图3 元胞空间划分方式

Fig. 3 Cellular space division

#### 2.2 地表污染元胞演变规则的建立

$\begin{array}{*{35}{l}} M_{i,j}^{t+\Delta t}=M_{i,j}^{t}+ \\ m\left[ \begin{array}{*{35}{l}} \left( M_{i-1,j}^{t}-M_{i,j}^{t} \right)+\left( M_{i,j-1}^{t}-M_{i,j}^{t} \right)+ \\ \left( M_{i,j+1}^{t}-M_{i,j}^{t} \right)+\left( M_{i+1,j}^{t}-M_{i,j}^{t} \right) \\ \end{array} \right]+ \\ md\left[ \begin{array}{*{35}{l}} \left( M_{i-1,j-1}^{t}-M_{i,j}^{t} \right)+\left( M_{i-1,j+1}^{t}-M_{i,j}^{t} \right)+ \\ \left( M_{i+1,j-1}^{t}-M_{i,j}^{t} \right)+\left( M_{i+1,j+1}^{t}-M_{i,j}^{t} \right) \\ \end{array} \right] \\ \end{array}$

$\begin{matrix} & M_{i,j}^{t+\Delta t}=M_{i,j}^{t}+m\left[ \begin{array}{*{35}{l}} {{K}_{i-1,j}}\left( M_{i-1,j}^{t}-M_{i,j}^{t} \right)+ \\ {{K}_{i,j-1}}\left( M_{i,j-1}^{t}-M_{i,j}^{t} \right)+ \\ {{K}_{i,j+1}}\left( M_{i,j+1}^{t}-M_{i,j}^{t} \right)+ \\ {{K}_{i+1,j}}\left( M_{i+1,j}^{t}-M_{i,j}^{t} \right) \\ \end{array} \right]+ \\ & md\left[ \begin{array}{*{35}{l}} {{K}_{i-1,j-1}}\left( M_{i-1,j-1}^{t}-M_{i,j}^{t} \right)+ \\ {{K}_{i-1,j+1}}\left( M_{i-1,j+1}^{t}-M_{i,j}^{t} \right)+ \\ {{K}_{i+1,j-1}}\left( M_{i+1,j-1}^{t}-M_{i,j}^{t} \right)+ \\ {{K}_{i+1,j+1}}\left( M_{i+1,j+1}^{t}-M_{i,j}^{t} \right) \\ \end{array} \right]-us \Delta t \\ \end{matrix}$

$K={{K}_{d}}\times {{K}_{r}}+1$

#### 2.2.1 高差影响系数的计算

${{K}_{{{d}_{i}}}}=\frac{{{H}_{i}}-{{H}_{0}}}{\sum ({{H}_{j}}-{{H}_{0}})} \ \ {{H}_{i}}>{{H}_{0}},{{H}_{j}}>{{H}_{0}}$

#### 2.2.2 地表粗糙度影响系数的计算

${{K}_{{{r}_{i}}}}=\frac{1/{{r}_{i}}}{\sum \left( 1/{{r}_{i}} \right)},{{r}_{j}}>0$

${{r}_{i}}=1/\text{tan}\left( slope\times \text{ }\!\!\Pi\!\!\text{ }/180 \right)$

#### 2.3.1 地表污染物平均流速计算

$v=\frac{k}{n}R_{h}^{2/3}{{s}^{1/2}}$

$Re=\rho vd/\mu$

$v={{f}_{re}}\frac{k}{n}R_{h}^{2/3}{{s}^{1/2}}$

${{f}_{re}}=R_{e}^{t}/R_{e}^{o}$

#### 2.3.2 非降雨条件下平均流速的计算

${{R}_{h}}=\frac{M}{\rho \times s}$

$K=p\rho g/\eta$

${{K}_{T}}={{f}_{re}}K$

${{R}_{h}}=\frac{M}{\rho s}-{{K}_{T}}\text{ }\!\!\Delta\!\!\text{ }t$

#### 2.3.3 降雨条件下平均流速的计算

$R=P-I$

$f\left( t \right)={{f}_{c}}+\left( {{f}_{0}}-{{f}_{c}} \right){{e}^{-ut}}$

Horton方程在实际应用中有很多局限性，当其他因素（如灌溉、蒸散）影响土壤水分时，方程不能准确地描述雨水的下渗量，Yang[28]等针对这种现象在Horton方程的基础上提出了一种改进方程，实验表明其能更精确地计算下渗量，因此，本文采用这种改进的Horton方程计算雨水在某个时刻的下渗量，计算公式如下。
$f\left( \theta \right)={{f}_{c}}+\left( {{f}_{0}}-{{f}_{c}} \right){{e}^{-k\left( \frac{\theta -{{\theta }_{0}}}{{{\theta }_{c}}-\theta } \right)}}$

$K={{K}_{s}}S_{e}^{l}{{\left[ 1-{{(1-S_{e}^{\left( 1/m \right)})}^{m}} \right]}^{2}}$

${{S}_{e}}=\frac{\theta -{{\theta }_{r}}}{{{\theta }_{s}}-{{\theta }_{r}}}$

$\theta =f\left( K \right)$

$u\sim {{R}^{1-m}}$

$v={{f}_{re}}{{R}^{1-n}}R_{h}^{2/3}{{s}^{1/2}}$

#### 2.3.4 自适应时间步长演变算法

##### 图4 一次演变示意

Fig. 4 A schematic diagram of evolution

##### 图5 元胞自适应时间演变算法流程

Fig. 5 Flow chart of cellular adaptive time evolution algorithm

$\begin{array}{*{35}{l}} \Delta{{t}_{x,y}}=\frac{l}{{{v}_{x,y}}} \\ x\in \left( i-1,i+1 \right),y\in \left( j-1,j+1 \right),\left( x\ne i,y\ne j \right) \\ \end{array}$

$\begin{array}{*{35}{l}}\Delta{{t}_{i,j}}=Max\left( \Delta{{t}_{i-1,j-1}},\Delta{{t}_{i-1,j}},\ldots,\Delta{{t}_{i+1,j+1}} \right) \\ \left(\Delta{{t}_{i,j}}=0 \right) \\ \end{array}$

$\begin{array}{*{35}{r}} S_{i,j}^{t+\Delta t}=S_{i,j}^{t}+\frac{\left( v_{i-1,j-1}^{t}\Delta{{t}_{i-1,j-1}}+\ldots +v_{i+1,j+1}^{t}\Delta{{t}_{i+1,j+1}} \right)}{l} \\ \left( \Delta{{t}_{i,j}}=0 \right) \\ \end{array}$

### 3 实验区概况与预处理

#### 3.1 实验区

##### 图6 实验区地理位置

Fig. 6 Geographical location of the study area

#### 3.3 实验区划分

##### 图7 元胞空间划分

Fig. 7 Cellular space division

### 4 仿真结果及分析

#### 4.1.1 下垫面为一般性土壤时的污染扩散仿真与分析

##### 图8 污染物在土壤地面扩散1 d后的范围

Fig. 8 The range of pollutants diffused on the soil surface for 1 day

##### 图9 污染物在土壤地面扩散时间随距离的变化

Fig. 9 Variation of pollutant diffusion time on the soil surface with distance

#### 4.1.2 下垫面为水泥路面时的污染扩散仿真与分析

##### 图10 污染物在水泥地面扩散1 d后的范围

Fig. 10 The range of pollutants diffused on the cement floor for 1 day

##### 图11 污染物在水泥地面扩散时间随距离的变化

Fig. 11 Variation of pollutant diffusion time on the cement floor with distance

#### 4.2.1 降雨时长对污染物扩散的影响

##### 图12 降雨不同时间后污染物5 min的扩散范围

Fig. 12 Diffusion range of pollutants in 5 min utes after rainfall at different times

#### 4.2.2 降雨强度随时间变化下污染物的扩散模拟

##### 图13 降雨强度随时间变化时半小时内污染物的扩散范围

Fig. 13 Diffusion range of pollutants with time-varying rainfall intensity within half an hour

##### 图14 降雨半小时内污染物扩散随时间变化曲线

Fig. 14 Variation curve of pollutant diffusion with time within half an hour of rainfall

#### 4.2.3 不同地点污染物浓度随时间的变化

##### 图15 稳定降雨下不同位置污染物浓度随时间变化

Fig. 15 Concentrations of pollutants at different locations with time under stable rainfall

#### 4.2.4 污染扩散结果分析与验证

##### 图16 坡度图

Fig. 16 Slope map

##### 图17 污染扩散结果与坡度叠加示意

Fig. 17 Schematic diagram of pollution diffusion results and slope overlay

#### 4.3 一次演变中污染物不同方向上的扩散时间差异仿真与分析

##### 图18 第1次演变时不同邻域方向上的扩散时间差异

Fig. 18 Diffusion time difference in different neighborhood directions during the first evolution

##### 图19 第6次演变时不同邻域方向上的扩散时间差异

Fig. 19 Diffusion time difference in different neighborhood directions during the 6th evolution

##### 图20 第10次演变时不同邻域方向上的扩散时间差异

Fig. 20 Diffusion time difference in different neighborhood directions during the 10th evolution

### 5 结论

（1）对于不规则场地边界，基于最小包围盒的思想确定了元胞空间，考虑地表高度时影响污染物扩散的主要因素，根据地形最高分辨率设定了元胞大小，并划分了元胞空间，能满足小范围场地的元胞空间划分需求。
（2）地表坡度越大、粗糙度越小时，液态污染物扩散速率越快，基于元胞自动机模型只受分子扩散影响下的推演规则，根据坡度、粗糙度倒数之间的比值，区分了一次推演过程中的污染物扩散通量，建立了污染物受坡度和粗糙度影响的推演规则，实验结果表明苯在坡度较大的方向迁移更为迅速。
（3）本文基于坡面流流速计算方法，提出了一种更普适的液态坡面流流速计算方法，对降雨和非降雨条件的污染扩散进行了模拟。无降雨条件下的污染物扩散速率十分缓慢，在坡度较缓的情况下，苯的最大直线扩散范围保持在2.50 m左右；不同下垫面时苯的扩散速率有较大的差异，下垫面为水泥路面时苯的平均扩散速率约为下垫面为一般性土壤的2.7倍左右。降雨条件下，苯的扩散主要受地表径流影响，降雨对其有促进作用；当降雨强度随时间变化时，苯的扩散速率也是一个不均匀的曲线，最大速率在雨强峰值附近；实验模拟了降雨条件下，同一位置的苯浓度随时间的变化过程，当苯随着水流迁移，高浓度区间范围慢慢变小，低浓度区间范围慢慢变大，浓度差异渐渐 变小。
（4）针对污染物在不同环境下的流速不同，本文提出了一种元胞自适应时间步长动态演变算法，解决了地表污染物扩散元胞自动机模型的扩散时间难以确定的问题。实验对降雨条件下苯的扩散过程进行了分析，由于外界环境的不同，第1次推演所需0.16 min左右，不过在0.04 min和0.06 min时刻污染物扩散状态存在一定的差异，第6次和第10次推演也是如此，在0.1 min的时间内，存在多个时刻扩散特征较为明显的差异。实验符合污染物扩散的实际规律，具有更强的适应性，可更好地模拟时间非均匀情况下污染扩散的时空演变过程。

### 参考文献

 [1] 中华人民共和国环境保护部. 场地环境调查技术导则:HJ 25.1—2014[S]. 北京: 中国环境科学出版社, 2014. [ Ministry of Environmental Protection of the People's Republic of China. Technical guidelines for site environmental site investigation: HJ 25.1—2014[S]. Beijing: China Environmental Science Press, 2014. ]
 [2] 苑克帅. 我国污染场地再开发风险管控法律规制研究[D]. 重庆: 西南政法大学, 2016. [ Yuan K S. Brownfield redevelopment risk control legal regulation research in China[D]. Chongqing: Southwest University of Political Science and Law, 2016. ]
 [3] 张亦弛. 工业搬迁遗留场地环境风险管理体系研究[D]. 西安: 长安大学, 2012. [ Zhang Y C. The system of environmental risk management of former relocation industrial sites[D]. Xi'an: Chang'an University, 2012. ]
 [4] Li J K, Zhang B, Li Y J, et al. Simulation of rain garden effects in urbanized area based on mike flood[J]. Water, 2018, 10(7):860-882. DOI:10.3390/w10070860
 [5] Prodanovic V, Jamali B, Kuller M, et al. Calibration and sensitivity analysis of a novel water flow and pollution model for future city planning: Future Urban Stormwater Simulation (FUSS)[J]. Water Science and Technology: A Journal of the International Association on Water Pollution Research, 2022, 85(4):961-969. DOI: 10.2166/wst.2022.046
 [6] Lye X Y, Nakayama A, Nizamani Z. Development of smoothed particle hydrodynamics for simulation of flow and contaminant transport on natural urban terrain and streams[J]. IOP Conference Series: Earth and Environmental Science, 2021, 945(1): 012009. DOI:10.1088/1755-1315/945/1/012009
 [7] 赵莉, 杨俊, 李闯, 等. 地理元胞自动机模型研究进展[J]. 地理科学, 2016, 36(8):1190-1196. [ Zhao L, Yang J, Li C, et al. Progress on geographic cellular automata model[J]. Scientia Geographica Sinica, 2016, 36(8):1190-1196. ] DOI:10.13249/j.cnki.sgs.2016.08.009
 [8] 谢志文, 王海军, 张彬, 等. 城市扩展元胞自动机多结构卷积神经网络模型[J]. 测绘学报, 2020, 49(3):375-385. [ Xie Z W, Wang H J, Zhang B, et al. Urban expansion cellular automata model based on multi-structures convolutional neural networks[J]. Acta Geodaetica et Cartographca Sinica, 2020, 49(3):375-385. ] DOI:10.11947/j.AGCS.2020.20190147
 [9] Tong X H, Feng Y J. A review of assessment methods for cellular automata models of land-use change and urban growth[J]. International Journal of Geographical Information Science, 2020, 34(5):866-898. DOI:10.1080/13658816.2019.1684499
 [10] Guidolin M, Chen A S, Ghimire B, et al. A weighted cellular automata 2D inundation model for rapid flood analysis[J]. Environmental Modelling & Software, 2016, 84:378-394. DOI:10.1016/j.envsoft.2016.07.008
 [11] 惠珊, 芮小平, 李尧. 一种耦合元胞自动机的改进林火蔓延仿真算法[J]. 武汉大学学报·信息科学版, 2016, 41(10):1326-1332. [ Hui S, Rui X P, Li Y. An improved forest fire spread simulation algorithm coupled with cellular automata[J]. Geomatics and Information Science of Wuhan University, 2016, 41(10):1326-1332. ] DOI:10.13203/j.whugis20140811
 [12] Lin M L, Yao Y P. Simulation of water pollution accident based on cellular automata[C]// CMSS 2018: Proceedings of the 2018 2nd International Conference on Management Engineering, Software Engineering and Service Sciences. 2018:270-274. DOI: 10.1145/3180374.3180380
 [13] Dai Y, Chen L, Zhang P, et al. Construction of a cellular automata-based model for rainfall-runoff and NPS pollution simulation in an urban catchment[J]. Journal of Hydrology, 2019, 568:929-942. DOI:10.1016/j.jhydrol.2018.11.029
 [14] 高华, 代侦勇. 基于改进元胞自动机的水污染扩散模拟[J]. 测绘地理信息, 2020, 45(6):138-140. [ Gao H, Dai Z Y. Simulation of water pollution dispersion based on improved cellular automata[J]. Journal of Geomatics, 2020, 45(6):138-140. ] DOI: 10.14188/j.2095-6045.2019431
 [15] 王璐, 谢能刚, 李锐, 等. 基于元胞自动机的水体污染带扩散漂移仿真[J]. 水利学报, 2009(4):481-485. [ Wang L, Xie N G, Li R, et al. Simulation of drift-diffusion of water pollution zone based on cellular automata[J]. Journal of Hydraulic Engineering, 2009, 40(4):481-485. ] DOI:10.3321/j.issn:0559-9350.2009.04.014
 [16] 周成虎, 欧阳, 马廷, 等. 地理系统模拟的CA模型理论探讨[J]. 地理科学进展, 2009, 28(6):833-838. [ Zhou C H, Ou Y, Ma T, et al. Theoretical perspectives of CA-based geographical system modeling[J]. Progress in Geography, 2009, 28(6):833-838. ] DOI:10.11820/dlkxjz.2009.06.001
 [17] Dahal K R, Chow T E. Characterization of neighborhood sensitivity of an irregular cellular automata model of urban growth[J]. International Journal of Geographical Information Science, 2015, 29(3):475-497. DOI:10.1080/13658816.2014.987779
 [18] 于瑞云, 赵金龙, 余龙, 等. 结合轴对齐包围盒和空间划分的碰撞检测算法[J]. 中国图象图形学报, 2018, 23(12):1925-1937. [ Yu R Y, Zhao J L, Yu L, et al. Collision detection algorithm based on AABB bounding box and space division[J]. Journal of Image and Graphics, 2018, 23(12):1925-1937. ] DOI:10.11834/jig.180050
 [19] 沈敬伟, 彭安琪, 周廷刚, 等. 基于并行元胞自动机的水体污染物扩散模拟[J]. 测绘科学技术学报, 2016, 33(1):105-110. [ Shen J W, Peng A Q, Zhou T G, et al. Water pollutant spreading simulation based on parallel cellular automata[J]. Journal of Geomatics Science and Technology, 2016, 33(1):105-110. ] DOI:10.3969/j.issn.1673-6338.2016.01.020
 [20] Karafyllidis I. A model for the prediction of oil slick movement and spreading using cellular automata[J]. Environment International, 1997, 23(6):839-850. DOI:10.1016/S0160-4120(97)00096-2
 [21] 李玉茹, 杨勤科, 王春梅, 等. 面向地形类型区分的地表粗糙度算法比较研究[J]. 西北农林科技大学学报(自然科学版), 2019, 47(8):134-143. [ Li Y R, Yang Q K, Wang C M, et al. Comparison of surface roughness algorithms for terrain type separation[J]. Journal of Northwest A & F University (Natural Science Edition), 2019, 47(8):134-143. ] DOI: 10.13207/j.cnki.jnwafu.2019.08.017
 [22] 张光辉. 坡面薄层流水动力学特性的实验研究[J]. 水科学进展, 2002. 13(2):159-165. [ Zhang G H. Study on hydraulic properties of shallow flow[J]. Advances in Water Science, 2002. 13(2):159-165. ] DOI:10.3321/j.issn:1001-6791.2002.02.005
 [23] 赵振国, 黄春花. 明渠均匀流研究[J]. 水利学报, 2013, 44(12):1482-1487. [ Zhao Z G, Huang C H. Study on the uniform flow in open channel[J]. Journal of Hydraulic Engineering, 2013, 44(12):1482-1487. ] DOI: 10.13243/j.cnki.slxb.2013.12.004
 [24] 张宽地, 王光谦, 王占礼, 等. 人工加糙床面薄层滚波流水力学特性试验[J]. 农业工程学报, 2011, 27(4):28-34. [ Zhang K D, Wang G Q, Wang Z L, et al. Experiments on hydraulic characteristics of roll wave for sheet flow with artificial rough bed[J]. Transactions of the Chinese Society of Agricultural Engineering, 2011, 27(4):28-34. ] DOI:10.3969/j.issn.1002-6819.2011.04.006
 [25] 吴望一. 流体力学[M]. 北京: 北京大学出版社,1982-1983. [ Wu W Y. Fluid mechanics[M]. Beijing: Peking University Press, 1982-1983. ]
 [26] 刘家琳, 张建林. 基于SWMM模型的山坡型公园子汇水区地表产流特征——以重庆地区为例[J]. 中国园林, 2018, 34(6):81-87. [ Liu J L, Zhang J L. Analysis on surface runoff characteristics of subcatchment in hillside park based on SWMM—A case study of Chongqing[J]. Chinese Landscape Architecture, 2018, 34(6):81-87. ] DOI:10.3969/j.issn.1000-6664.2018.06.015
 [27] 薛文宇. 城市暴雨积水及街道洪水模拟模型研究[D]. 天津: 天津大学, 2016. [ Xue W Y. Numerical simulation of stormwater and street flood in city[D]. Tianjin:Tianjin University, 2016. ]
 [28] Yang M Y, Zhang Y Y, Pan X Y. Improving the Horton infiltration equation by considering soil moisture variation[J]. Journal of Hydrology, 2020, 586:124864. DOI:10.1016/j.jhydrol.2020.124864
 [29] 史振宁, 戚双星, 付宏渊, 等. 降雨入渗条件下土质边坡含水率分布与浅层稳定性研究[J]. 岩土力学, 2020, 41(3):980-988. [ Shi Z N, Qi S X, Fu H Y, et al. A study of water content distribution and shallow stability of earth slopes subject to rainfall infiltration[J]. Rock and Soil Mechanics, 2020, 41(3):980-988.] DOI:10.16285/j.rsm.2019.0474
 [30] 范严伟, 赵文举, 毕贵权. Van Genuchten模型参数变化对土壤入渗特性的影响分析[J]. 中国农村水利水电, 2016(3):52-56. [ Fan Y W, Zhao W J, Bi G Q. The influence analysis of parameters variations in van genuchten model on the soil infiltration characteristics[J]. China Rural Water and Hydropower, 2016(3):52-56. ] DOI:10.3969/j.issn.1007-2284.2016.03.013
 [31] 杨坪坪, 王云琦, 张会兰, 等. 降雨强度和单宽流量与地表粗糙度交互作用下坡面流阻力特征[J]. 农业工程学报, 2018, 34(6):145-151. [ Yang P P, Wang Y Q, Zhang H L, et al. Characteristics of overland flow resistance under interaction of rainfall intensity and unit discharge and surface roughness[J]. Transactions of the Chinese Society of Agricultural Engineering, 2018, 34(6):145-151. ] DOI:10.11975/j.issn.1002-6819.2018.06.018
 [32] 张宽地, 王光谦, 孙晓敏, 等. 模拟植被覆盖条件下坡面流水动力学特性[J]. 水科学进展, 2014, 25(6):825-834. [ Zhang K D, Wang G Q, Sun X M, et al. Hydraulic characteristic of overland flow under different vegetation coverage[J]. Advances in Water Science, 2014, 25(6):825-834. ] DOI:10.14042/j.cnki.32.1309.2014.06.009
 [33] 程娅姗, 王中根, 李军, 等. 确定坡面径流过程曼宁糙率系数的实验方法研究[J]. 地理科学进展, 2020, 39(4):651-659. [ Cheng Y S, Wang Z G, Li J, et al. Experimental study on determining Manning roughness coefficient during slope runoff process[J]. Progress in Geography, 2020, 39(4):651-659. ] DOI:10.18306/dlkxjz.2020.04.012
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