Extraction of an analytical solution for flood routing in the river reaches (case study of Simineh River)

Introduction
Accurate flood routing through the river reaches is one of the essential issues in the river training activities and flood warning systems. Especially when the river passes near residential areas of cities, it is vital to have enough information about the maximum flow that can flow through the river without damaging its surroundings. Due to the complexity of the complete solution process of Saint-Venant equations, over the years, many researchers have tried to provide alternative models that, in addition to simplicity, have the necessary accuracy. Previous models usually have two significant drawbacks. First, the process of solving most of them is step-by-step, and to calculate the outflow discharge at each time step, the estimated flow in the previous step is required. Second, sometimes the model coefficients change during the resolution process. Therefore, in the present study, an attempt was made to provide a clear and direct relationship. Also, if the coefficients are known for determining the flow rate in each time step, there is no need for the values of the previous steps.
Methodology
In order to prove the prevailing analytical relationship, in this research, first, the two processes of flood transfer and flood dispersion in the river reaches were conceptually separated. For this purpose, the river reach was divided into three interconnected reservoirs. The first reservoir is an index of the flood convection, and the next two reservoirs were the index of flood propagation process. The runoff volume, obtained from the upper basin, was calculated using multiplying the runoff coefficient to the rainfall height. Then, it was suddenly applied to upstream of the river reach by using the Dirac delta function. By adding the spatial flow variation coefficient to the reservoirs of the propagation operation as well as applying the mass equilibrium and inclining the dimensions of the reservoirs to zero, the differential equations governing each reservoir were obtained. The outflow of each reservoir was used as the boundary condition of the next one, and the final equation, obtained from the interconnected reservoir system, was used as the output hydrograph relationship. In order to evaluate the performance of the introduced model, the data of four flood events that were recorded on (19 - 3 – 2017), (15 - 4 – 2017), (29 - 1 – 2019), and (31 - 3 - 2019) in Simineh River were used. Simineh River is located south of Lake Urmia and provides 11% of the lake's water. The flood data was recorded at three stations of BUCKAN Bridge, DASHBAND BUCKAN, and MIANDOAB Bridge with two-hours interval.

Results and discussion
The proposed model is a four-parameter model that works directly by operation of its parameters. Therefore, firstly the model parameters were estimated and then the output hydrograph was simulated at the end of the river reach. The simulated hydrographs by the proposed model were very consistent with the measured data at the end of the interval, indicating its efficiency. Statistical indicators of coefficient of determination (R2), root mean square error (RMSE), and Nash-Sutcliff (DC) were used to quantify the desirability of the model. The above-mentioned statistical parameters for all flood events were calculated as triple sets of (0.86, 0.07, 0.95), (0.82, 0.11, 0.8), (0.97, 0.07, 0.94), and (0.93, 0.1, 0.9), respectively which also proves its quantitative suitability. By creating linear relationships between the residence times of the flood in each of the interconnected reservoirs, the relevant volumes were calculated. It was also found that the length of each reservoir can be calculated separately by applying a mean cross sectional area in the river reach. The flood volume was calculated to be 30, 50, 63 and 37 million cubic meters for events of 1 to 4, respectively. This value is equal to the total volume of the assumed reservoirs in the river reach. Ratio (V/T) was calculated for all reach lengths and flood events, and it was found that its value decreases with the increasing of reach length, but its value for larger floods is higher than for smaller ones. Besides, it was found that the position of the dispersion reservoirs in the river reach can be exchanged with each other, and the total volume of them is the diffusion index.
Conclusion
Finally, it was observed that the proposed model has good compatibility with observational hydrographs, except in the initial points of raising limb. Optimization or numerical methods can also be used to obtain model parameters. Moreover, the explicitness and directness of the discharge calculation by this method is the most crucial advantage of this model. This model also has the capability of reconstructing hydrographs affected by the spatially varied flow.