A COUPLED 1-D AND 2-D CHANNEL NETWORK MATHEMATICAL MODEL USED FOR FLOW CALCULATIONS IN THE MIDDLE REACHES OF THE YANGTZE RIVER*

HAN Dong, FANG Hong-wei, BAI Jing, HE Guo-jian

Department of Hydraulic Engineering and State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China, E-mail: hand02@mails.tsinghua.edu.cn

A COUPLED 1-D AND 2-D CHANNEL NETWORK MATHEMATICAL MODEL USED FOR FLOW CALCULATIONS IN THE MIDDLE REACHES OF THE YANGTZE RIVER*

HAN Dong, FANG Hong-wei, BAI Jing, HE Guo-jian

Department of Hydraulic Engineering and State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China, E-mail: hand02@mails.tsinghua.edu.cn

A coupled one-dimensional (1-D) and two-dimensional (2-D) channel network mathematical model is proposed for flow calculations at nodes in a channel network system in this article. For the 1-D model, the finite difference method is used to discretize the Saint-Venant equations in all channels of a looped network. The Alternating Direction Implicit (ADI) method is adopted for the 2-D model at the nodes. In the coupled model, the 1-D model provides a good approximation with small computational effort, while the 2-D model is applied for complex topography to achieve a high accuracy. An Artificial Neural Network (ANN) method is used for the data exchange and the connectivity between the 1-D and 2-D models. The coupled model is applied to the Jingjiang-Dongting Lake region, to simulate the tremendous looped channel network system, and the results are compared with field data. The good agreement shows that the coupled hydraulic model is more effective than the conventional 1-D model.

Coupled 1-D and 2-D model, looped channel network system, Alternating Driection Implicit (ADI), Jingjiang-Dongting Lake region

Introduction

1-D mathematical models are commonly adopted to solve the looped channel network system in a large river basin[1,2]because 2-D mathematical models would take too much computational effort and the precise terrain data is not easy to obtain. In recent years, the parameter identification method sees a considerable development for modeling river networks[3,4]and the 1-D calculation accuracy is thus greatly improved. However, at some important nodes connecting more than 3 rivers, the 1-D model does not take into account of the impact of complex topography. This simplification reduces the calculation accuracy of the flow diversion, thereby would affect subsequent calculations of sediment transport. In order to solve this problem, Williams[5]provided an algorithm for fine-ssing 1-D hydraulic models into 2-D performance.

Horritt and Bates[6]evaluated the performance of HES-RAS, LISFLOOD-FP, and TELEMAC-2D to predict the river flood inundation. Lin et al.[7]proposed to integrate a 2-D model with ISIS to enhance its capability to simulate floodplain flows. Zhang et al.[8]used a 2-D hydrodynamic model to simulate a complex system composed of the Jingjiang River and Dongting Lake, and the model takes about 0.96 h to complete the computation of a 76 d flood.

Recently, in order to meet the needs of urban development, coupled 1-D and 2-D models see a great progress in urban flood simulations[9-11]. They were adopted in researches of the natural river, Lai and Wang[12]proposed a new method called overlapping-projecting method, which was applied to develop a coupled numerical model of unsteady flow, and Zhu et al.[13]carried out a coupled 1-D and 2-D hydrodynamic numerical simulation for a bay-channel network, in which a 1-D model is joined with a 2-D model at the river mouth. Xu and Yin[14]developed a coupled 1-D and 2-D hydrodynamic model for tidal rivers. Kuiry et al.[15]proposed a simplified numerical model for the simulation of the floodplain inundationresulting from natural floods in rivers, where the Saint-Venant equations were solved with a 1-D finite volume approach and an unstructured triangular grid to simulate the spread of excess flood water spilling overbank. Some models achieved good results in their calculations, but their implementation requires an upstream or downstream boundary condition at the node of the 1-D and 2-D models. In this article, the calculation domain with nodes that use 2-D models is Y-shaped, and will be embedded in the channel network structure. However, the 2-D model cannot be used to identify the actual boundary condition because of the interaction between the 1-D and 2-D models, therefore, a new coupling method is needed to solve this problem.

The proposed 1-D and 2-D models are used at the Songzi node, located in the connection of Jingjiang-Dongting Lake region, and an Artificial Neural Network (ANN) method is used for the data exchange and the connectivity between the 1-D and 2-D models to improve the accuracy of the calculation in the huge channel network system.

Fig.1 1-D and 2-D connection node

1. Coupled 1-D and 2-D mathematical model

1.11-D Channel network mathematical model and2-D mathematical model

Figure 1(a) shows a configuration of a simple channel network system composed of four channels in which the superscript number denotes different channels. Figure 1(b) shows the 2-D computational zone embedded in Fig.1(a).

For channels 1-4, the unsteady water flow dynamics in a channel can be described by the Saint-Venant equations,

wherexandtare spatial and temporal variables,Zis the water surface elevation,Qis flow discharge,qiis the lateral inflow per unit channel width,Ais the flow area,Bis the channel width,1αis the momentum correction factor,Cis the Chezy coefficient, defined asC=R1/6/n, withnbeing the Manning coefficient of roughness, andRis hydraulic radius.

In the region shown in Fig.1(b), the 2-D continuity equation is as follows

whereuis the water depth averaged velocity inxdirection,vis the water depth averaged velocity inydirection,his the water depth,νt= 0.0025His the viscosity coefficient of turbulence[18].

1.2Equations at nodes

The implicit four-point finite difference scheme proposed by Preissmann (1961) is used to discretize the governing equations in an individual channel and then an iteration procedure is adopted to obtain the relations between the flow discharge and the water level at the beginning and end sections. For example, the variablesQ1,Z1,Qn+!andZn+1in channel 1 are given as:

where R1-6are the recursive coefficients, which can be obtained by variables of the last time step.

1.2.1 Continuity equation

The water-balance equation is employed to link the discharge of the different channels meeting at a node, at the node M, the continuity equation can be written as

where i is the number of channels meeting at the node, and Ω is the storage capacity at the node.

1.2.2 Energy conservation equation

The energy conservation equation can be approximated by a kinematic balance condition neglecting the velocity and the loss of resistance. The flows in all channels meeting at a node are expressed as

The complemented interior boundary conditions at nodes involve Eqs.(8) and (9). After adding continuity and energy conservation equations, the total number of interior boundary conditions at nodes is equal to the variables of water level and discharge of channels meeting at the nodes. Other interior boundary conditions may be adopted in a single branch, for example, weirs or bridges. However, these interior conditions will not be discussed in this article.

The simplified energy conservation equation is totally in line with most accuracy requirements, however, to take into account of the complex topography and the water level variation at nodes, further analyses of the energy conservation equation are in order.

The 2-D mathematical model, Eqs.(3) to (5), is used to obtain the 2-D water level distribution and the results are used to correct the energy equation, then,

Equations (10) and (11), obtained according to the momentum conservation, are used to determine the water level relation of the nodes instead of the simplified energy conservation Eq.(9). Solving the 2-D mathematical model at the nodes, three boundary conditions are specified. For example, the discharge or the water level should be provided for channels 1, 2 and 3 in Fig.1(b). Generally speaking, the boundary can be obtained from the field data and the 1-D mathematical model.

The results of the 1-D calculation may directly provide a water level boundary condition for the 2-D model

However, the 2-D calculation needs the velocity to be specified as the boundary condition instead of the discharge obtained by the 1-D calculation, therefore, a transforming relationship is required as follows,

where Uiis the velocity at grid i, Q is the flow boundary, h is the depth at grid i.

Based on the 2-D calculation, a new water level relation can be obtained. However, the 2-D calculation would take a longer time. In order to improve the computation efficiency, the ANN method is used to estimate the water level. The ANN method[19]can provide functions in the following form,

2. Model applications

The middle reach of the Yangtze River with a complex river network is used to test the improved coupled 1-D and 2-D channel network mathematical model. Because the region's importance in flood control, many studies were carried out[20-24]. The calculation domain is from Yichang to Hankou, including Dongting Lake and Sankou distributary channels, respectively, Songzi, Taping and Ouchi, as shown in Fig.2. The Yangtze River main channel is about 659 km long, with six main channels. In the original map of the Yangtze River, there are 331 sections in total. We adopt 853 sections covering a length of about 1486 km in the Dongting Lake region and Sankou distributary channels. In the calculation, Δt= 3600s is used in the1-D model and 1 s in the 2-D model. The Yangtze River main channel terrain data were obtained in 1998 and the terrain data of the Dongting Lake region were obtained in 1995.

Fig.2 Configuration of channel network from Yichang to Hankou

In the 1-D calculation, the roughness coefficient of each section is determined by the measured water level and the discharge in the main hydrological stations in 1998 floods. The results show that the roughness coefficient of the main channel of the Yangtze River ranges from 0.013 to 0.030, and those of the Sankou distributary channel and the Dongting Lake range from 0.02 to 0.05. The results agree well with the Yangtze River Flood roughness coefficient analysis, and show that the adopted roughness coefficient of the channel is reasonable and reliable in the calculations.

In order to test the coupled 1-D and 2-D model, as well as the ANN simulation, the Songzi node is selected to connect the model and the ANN method. The Songzi node is located downstream of Zhicheng, its upper reach is defined as channel 1 and down reach is defined as channel 2, while channels 3 is the distributed reach. If there is a discharge flow in the Yangtze river, there would hardly be any flow distributed to channel 3.

2.1Simulation

Fig.3 Water level difference of the node

2.1.1 Analysis of the results of the 2-D mathematical model

2.1.2 Analysis of the results of ANN

Fig.6 Comparison of simulated and measure data at Songzi node

Fig.7 Deviation analysis at Songzi node

2.2Comparison of results based on 1-D model andcoupled model

In Fig.6, the horizontal axis is the measured data, while the vertical axis is the simulation of the 1-D model and the coupled model. As can be seen from the figure, the results are generally evenly distributed along the 45oline. This phenomenon shows that the simulation is generally consistent with the measurements at Songzi outlet from 1998. In Fig.7, the horizontal axis is the simulation of the 1-D model and the vertical axis is the simulation of the coupled model. In this figure, most of the deviation points are located in the lower right 45oline. This phenomenon shows that the deviation of the coupled model is smaller than that of the 1-D model. More details related to the comparison are shown in Table 1.

3. Conclusions

In this study, a coupled 1-D and 2-D mathematical model is proposed and an ANN method is used to link the 1-D and 2-D models. This is an effective concept for improving coupling models. In reference[8], with the 2-D model, 0.96 h is spent to complete the computation of a 76d flood, while in this study, for the same situation, with the coupled model, less than a minute is required. Compared with traditional coupled models, the new coupled model can deal with the zone which is embedded in the channel network structure and effectively increase the accuracy of the1-D model calculation with little additional computational effort. The model is efficient enough for engineering applications.

At important nodes, the new coupled model takes the complex topography and the momentum conservation into account in the calculation; as a consequence, more accurate results for the flow discharge and the water level can be obtained. The new coupled model can help the subsequent calculation of the sedi-ment. Further, the ANN method can be used to link 1-D and 3-D models or 2-D and 3-D models. Using the new coupled model, the advantages of each model will be retained with more accurate results.

Table 1 Results comparison of 1-D model and coupled model

[1] WU W. M., VIEIRA D. A. and WANG S. S. Y. Onedimensional numerical model for nonuniform sediment transport under unsteady flows in channel networks[J].Journal of Hydraulic Engineering,2004, 130(9): 914-923.

[2] FANG H. W., CHEN M. H. and CHEN Q. H. Onedimensional numerical simulation of non-uniform sediment transport under unsteady flows[J].International Journal of Sediment Research,2008, 23(4): 316-328.

[3] TANG Hong-wu, XIN Xiao-kang and DAI Wen-hong et al. Parameter identification for modeling river network using a genetic algorithm[J].Journal of Hydrodynamics,2010, 22(2): 246-253.

[4] BAO Wei-min, ZHANG Xiao-qin and QU Si-min. Dynamic correction of roughness in the hydrodynamic model[J].Journal of Hydrodynamics,2009, 21(2): 255-263.

[5] WILLIAMS D. T. Finessing 1-D hydraulic models into 2-D performance[C].Proceedings of the 2007 World Environmental and Water Resources Congress.Tampa, Florida, USA, 2007.

[6] HORRITT M. S., BATES P. D. Evaluation of 1D and 2D numerical models for predicting river flood inundation[J].Journal of Hydrology,2002, 268(1-4): 87-99.

[7] LIN B., WICKS J. M. and FALCONER R. A. et al. Integrating 1D and 2D hydrodynamic models for flood simulation[J].Proceedings of the Institution of Civil Engineers-Water Management,2006, 159(1): 19-25.

[8] ZHANG Xi-bing, HU De-chao and WANG Min. A 2-D hydrodynamic model for the river, lake and network system in the Jingjiang reach on the unstructured quadrangles[J].Journal of Hydrodynamics,2010, 22(3): 419-429.

[9] VOJINOVIC Z., TUTULIC D. On the use of 1D and coupled 1D-2D modelling approaches for assessment of flood damage in urban areas[J].Urban Water Journal,2009, 6(3): 183-199.

[10] LEANDRO J., CHEN A. S. and DJORDJEVIC S. et al. Comparison of 1D/1D and 1D/2D coupled (sewer/ surface) hydraulic models for urban flood simulation[J].Journal of Hydraulic Engineering,2009, 135(6): 495-504.

[11] LI W. F., CHEN Q. W. and MAO J. Q. Development of 1D and 2D coupled model to simulate urban inundation: An application to Beijing olympic village[J].Chinese Science Bulletin,2009, 54(9): 1613-1621.

[12] LAI Xi-jun, WANG De-guan. 1-D and 2-D coupling numerical model of unsteady flow[J].Hydro-Science and Engineering,2002, (2): 48-51(in Chinese).

[13] ZHU Yu-liang, YAN Yi-xin and LI Rui-jie et al. A coupled hydrodynamic numerical model for bay-river network[J].Advances in Water Science,2003, 14(2): 131-135(in Chinese).

[14] XU Zu-xin, Yin Hai-long. Development of two-dimensional hydrodynamic and water quality model for Huangpu River[J].Journal of Hydrodynamics, Ser. B,2003, 15(2): 1-11.

[15] KUIRY S. N., SEN D. and BATES P. D. Coupled 1DQuasi-2D flood inundation model with unstructured grids[J].Journal of Hydraulic Engineering,2010, 136(8): 493-506.

[16] LIU Bin, FANG Hong-wei and DUAN Jie-hui. Study cases of diagonal cartesian method for the two-dimensional numerical simulation of flow and suspended sediment transport over wet-dry boundaries[J].Journal of Sediment Research,2006, (2): 37-45(in Chinese).

[17] HUANG Bing-bin, FANG Hong-wei and LIU Bin. Diagonal cartesian method for numerical simulation of flow with complex boundary[J].Journal of Hydrodynamics Ser. A,2003, 18(6): 675-689(in Chinese).

[18] FANG H. W., LIU B. and HUANG B. B. Diagonal cartesian method for the numerical simulation of flow and suspended sediment transport over complex boundaries[J].Journal of Hydraulic Engineering,2006, 132(11): 1195-1205.

[19] LI Y. T., GU R. R. Modeling flow and sediment transport in a river system using an artificial neural network[J].Environmental Management,2003, 31(1): 122-134.

[20] MU Jin-bin, ZHANG Xiao-feng. Numerical simulation for sediment transport in the river network between Jingjiang River and Dongting Lake[J].Journal of Sediment Research,2009, (5): 52-57(in Chinese).

[21] CHANG J. A., LI J. B. and LU D. Q. et al. The hydrological effect between Jingjiang River and Dongting Lake during the initial period of Three Gorges project operation[J].Journal of Geographical Sciences,2010, 20(5): 771-786.

[22] YANG S. J., HU H. Y. and HAO Z. C. Trend forecast for the influence of the Three Gorges project on the water environmental capacity of Dongting Lake[C].Asia-Pacific Power and Energy Engineering Conference (APPEEC).Wuhan, 2009, 2454-2457.

[23] XUE L. Q., HAO Z. C. and LI D. et al. Dongting Lake floodwater diversion and storage modeling and control architecture based on the next generation network[C].Proceedings of the 7th International Conference on Computational Science, Part IV: ICCS 2007.2007, 841-848.

[24] LI Y. T., SUN Z. H. and LIU Y. et al. Channel degradation downstream from the Three Gorges project and its impacts on flood level[J].Journal of Hydraulic Engineering,2009, 135(9): 718-728.

June 27, 2010, Revised May 28, 2011)

* Project supported by the National Natural Science Foundation of China (Grant Nos. 10872110, 10902061).

Biography: HAN Dong (1985-), Male, Ph. D.

FANG Hong-wei,

E-mail: fanghw@tsinghua.edu.cn

2011,23(4):521-526

10.1016/S1001-6058(10)60145-X