Estimating stage-discharge curves in unsteady flows using the concept of isovel contours in natural rivers

Introduction
Investigating unsteady flows in compound channels and natural rivers is essential. Discharge measurements in medium and large rivers are often based on indirect methods of converting water level to discharge using stage-discharge curves of steady flow. However, these methods do not accurately estimate flow discharge in unsteady flow conditions. In the previous studies, many relationships have been proposed to modify the flow values in steady conditions to estimate the stage-discharge relations of unsteady flow. In most of the previous studies, the relationships are either oversimplified or have errors that make them not very generalizable. Considering the importance of estimating the rating curves in natural rivers and compound channels and the shortcomings of the studies in this field, this research aims to evaluate the stage-discharge curve and the output hydrograph in natural rivers with unsteady flow using a proposed novel method based on isovel contours.
Methodology
In order to analyze the flood flows, the combination of momentum and continuity equations was used, known as Saint-Venant's equations. Saint-Venant equations do not have an analytical solution, and numerical models must be used to solve them. The numerical model used in this paper was the four-point finite difference model, which is conventionally called the Preissmann implicit model.
Using the Bio-Savart law, the Maghregi’s 2006 method simulates the effect of the wall on the velocity distribution in the flow cross-section by considering the effects of the electromagnetic forces on a particle with a static charge placed in the electric field of a wire with an electric current. In the SPM method, using the Bio-Savart law, a relationship for determining the isovel contours was presented, similar to the magnetic field law. In this method, to determine the effects of the entire flow section wall on a point (uSPM), the value of uSPM was computed by integrating the impact of all boundary elements on each flow point. Then, using the power-law velocity, a relationship was obtained to calculate the average value of uSPM in the flow section known as USPM. In order to model the SPM method and estimate the parameters of this method, first, the values of the uSPM in a series of selected points of the flow section should be computed. It is worth mentioning that the pattern of arrangement of points is important in sections that do not have a regular geometric shape. One way to arrange the points was to cover the surfaces with triangular meshes. In this research, the Delaunay triangulation algorithm was used. The purpose of this was to maximize the angles of the triangles. After placing the triangular meshes on the flow section, it was enough to obtain the values of uSPM only in the centroid of each triangular element. In Wolfram Mathematica software, it is possible to use this grid type. The main effective parameters of the water discharge were listed as bed roughness (n), cross-sectional area (A), wetted perimeter (P), free water surface (T), bed slope (S0), and cross-sectional flow velocity (USPM). First, the A, P, T, and USPM should be calculated at each observation level. Having obtained the characteristics of the sections and the discharge of each section at different levels, the coefficient and exponents of the proposed discharge relation were computed using the genetic algorithm process based on error minimization.
Results and Discussion
The negligible difference between the observed data and estimated flow discharge based on the SPM method confirm the accuracy of this method. It is worth mentioning that this method can consider the effect of the shape of the section with any complexity on the water flow using the Bio-Savart law. This work was done by simulating the cross-sectional wall and water flows, respectively, with the wire flowing the electric current and the magnetic field around it. This method estimates the stage-discharge curve and flood routing with proper accuracy, even in the flow entering the floodplain where considering the shape of the cross-section is of particular importance.
The field data used in this research has been provided from the Tiber River in Italy. In order to solve Saint-Venant's equations and determine the hydrograph of flood output, the system of equations consisting of numerical modeling should be resolved. The Gauss elimination method was used to solve this system of equations. In this research, instead of using Manning's relation to solve Saint-Venant's equations, the proposed discharge equation obtained based on the theory of the Maghrebi method was used to determine the flood output hydrograph. The final results of flood routing, based on the aforementioned method, showed that the values of Root Mean Square Error (RMSE), Normalized Root Mean Square Error (NRMSE) and Mean Absolute Percentage Error (MAPE) for the outflow stage hydrograph were 0.1196 (m3/s), 0.037 and 4.10%, respectively, and for the outflow discharge hydrograph were 10.12(m3/s), 0.029 and 11.97%, respectively. In addition, the error of the proposed method in estimating the peak discharge and the peak stage was less than 4% and also, in the case of their occurrence time, was about 2%.
Conclusion
In the proposed approach of this research, the common discharge relationships in the Saint-Venant equations have been substituted by ones extracted from the Maghrebi method (equation 24). Based on this method, the error of the discharge estimation in natural rivers can be reduced compared to other methods, especially when the flow enters the floodplain. Finally, the estimated outflow hydrographs based on the proposed approach showed that the results were entirely consistent with the observation data at the beginning and end of the flood occurrence range. Also, the error of the considered method was negligible in the range of the peak stage and discharge and their occurrence time. Besides, the peak stage and discharge and the time of their occurrence, which are accounted as the essential indicators in hydrograph estimation, have been calculated using the proposed method with excellent accuracy.