# Influence of Erodible Beds on Shallow Water Hydrodynamics during Flood Events

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Two-Dimensional Depth-Integrated Shallow-Water Equations (SWE)

#### 2.2. Erosion Model

#### 2.3. Solution of the System of PDEs

## 3. Results and Discussion. Model Validation with Laboratory-Scale Channel Junctions

#### 3.1. Validation of the Hydraulic Model

#### 3.2. Validation of the Erosion Model

#### 3.3. Sensitivity Analysis

#### 3.3.1. Mesh Refinement Study

#### 3.3.2. Bed Properties: Friction Angle of Sediments

#### 3.3.3. Bed Properties: Effective Roughness

## 4. Results and Discussion. Model Application to Field-Scale Channel Junction

#### 4.1. Case Study

#### 4.2. Effect of Bed Erosion on Hydrodynamics During Flood

#### 4.3. Effect of Flooding on Erosion Patterns

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Validation of the numerical hydraulic model with laboratory experiments. (

**a**) Setup of the laboratory experiments. (

**b**) Contour plot of the water depth computed with the numerical model. Experiment # 10. (

**c**) Contour plot of the x-velocity gradient given by $\partial {U}_{x}/\partial y$ computed with the numerical model. Experiment # 10. (

**d**) Water depth for the model with $\theta ={30}^{\circ}$ along the main channel, and (

**e**) along the secondary channel. (

**f**) Water depth for the model with $\theta ={60}^{\circ}$ along the main channel, and (

**g**) along the secondary channel.

**Figure 3.**Sketch of bed erosion experiment setup and results. Panel (

**a**) depicts the setup. It consists of a perpendicular confluence of rectangular concordant-bed open-channels. The dimensions of the channels are included in the figure. Panel (

**b**) depicts a contour of the steady-state bed elevation provided by the laboratory experiment. The initial bed elevation was $5.5$ cm. The plot is reproduced from [45]. Panel (

**c**) includes the steady-state bed elevation computed with the numerical model. Units are in cm.

**Figure 4.**Evolution of bed elevation, water head, water velocity, and stream lines computed with the mathematical model. The first row plots the contour plot of the erosion at time 30 s—panel (

**a**), 3600 s—panel (

**b**), and 10,800 s—panel (

**c**). Erosion is positive and sedimentation negative. The second row shows the water depth at time 30 s—panel (

**d**), 3600 s—panel (

**e**), and 10,800 s—panel (

**f**). The third row includes the water velocity at time 30 s—panel (

**g**), 3600 s—panel (

**h**), and 10,800 s—panel (

**i**). The last row includes the stream lines at time 30 s—panel (

**j**), 3600 s—panel (

**k**), and 10,800 s—panel (

**l**).

**Figure 5.**Mesh refinement study. We plot the erosion patterns for four increasingly refined meshes: in panel (

**a**) the mesh size, $\Delta \delta $, is $0.5$ cm, in panel (

**b**) 1 cm, in panel (

**c**) 2 cm, and in panel (

**d**) 4 cm. We plot two erosion profiles of the main channel at x-coordinate 0 cm in panel (

**e**) and $x=20$ cm in panel (

**f**). The reference system is depicted in Figure 3a. Erosion is positive and sedimentation negative.

**Figure 6.**Effect of the friction angle of sediment on the erosion pattern. We plot the erosion contour plot for four friction angles, $\varphi $: in panel (

**a**) $\varphi $ is ${20}^{\circ}$, in panel (

**b**) $\varphi ={25}^{\circ}$, in panel (

**c**) $\varphi ={30}^{\circ}$, and in panel (

**d**) $\varphi ={35}^{\circ}$. We plot two erosion profiles of the main channel at x-coordinate 0 cm in panel (

**e**) and $x=20$ cm in panel (

**f**). The reference system is depicted in Figure 3a. Erosion is positive and sedimentation negative.

**Figure 7.**Effect of roughness on the erosion pattern. We plot the erosion contour plot for four Manning’s coefficients, n: in panel (

**a**) n is $0.015$ s·m${}^{-1/3}$, in panel (

**b**) $n=0.018$ s·m${}^{-1/3}$, in panel (

**c**) $n=0.020$ s·m${}^{-1/3}$, and in panel (

**d**) $n=0.025$ s·m${}^{-1/3}$. We plot two erosion profiles of the main channel at x-coordinate 0 cm in panel (

**e**) and $x=20$ cm in panel (

**f**). The reference system is depicted in Figure 3a. Erosion is positive and sedimentation negative.

**Figure 8.**Model geometry and input flow-rates. In panel (

**a**) we plot the location of the study area and the elevation map of our domain. Panel (

**b**) includes the flood hydrographs for three return periods: T = 2, 10, and 25 years. Panel (

**c**) depicts two flood hydrographs for the return period T = 10 years. The plot includes two discharge ratios, i.e., two flow distribution between the Nacimiento and Almarax rivers.

**Figure 9.**Effect of erosion on the flow patterns. In panel (

**a**) we depict the contour plot of erosion after the 25-year flood event. Erosion is positive and sedimentation negative. Panel (

**b**) depicts the water depth at time step $2.5$ h for the 25-year flood event computed with bed-mobile model, whereas panel (

**c**) plots the result provided by a model without bed erosion. Panel (

**d**) depicts the water velocity at time step $2.5$ h for the 25-year flood event computed with a bed mobile model, and panel (

**e**) plots the result provided by a model without bed erosion.

**Figure 10.**Effect of flood return period on the erosion pattern. We plot the erosion patterns for three flood events with return periods: (

**a**) T = 5 years, (

**b**) T = 10 years, (

**c**) T = 25 years. Erosion is positive and sedimentation negative.

**Figure 11.**Effect of the discharge ratio on the erosion patterns. Here we plot the erosion patterns under the 10-year flood event. In panel (

**a**) the total flow distributes 80% in the Andarax river and 20% in the Nacimiento river, in panel (

**b**) the distribution is 40% in the Andarax river and 60% in the Nacimiento river, and in panel (

**c**) it is 20% in the Andarax river and 80% in the Nacimiento river.

**Table 1.**Experimental results taken from [70].

# Experiment | $\mathit{\theta}$ | ${\mathit{Q}}_{1}$ (L/s) | ${\mathit{Q}}_{2}$ (L/s) | ${\mathit{y}}_{1}$ (cm) | ${\mathit{y}}_{2}$ (cm) | ${\mathit{y}}_{3}$ (cm) | ${\mathit{y}}_{4}$ (cm) | ${\mathit{y}}_{5}$ (cm) |
---|---|---|---|---|---|---|---|---|

1 | ${30}^{\circ}$ | $5.1$ | $5.4$ | $8.0$ | $8.2$ | $7.9$ | $8.0$ | $6.7$ |

2 | ${30}^{\circ}$ | $5.0$ | $10.1$ | $10.2$ | $10.3$ | $10.1$ | $10.1$ | $8.3$ |

3 | ${30}^{\circ}$ | $5.0$ | $14.9$ | $11.9$ | $12.1$ | $11.7$ | $11.8$ | $10.1$ |

4 | ${30}^{\circ}$ | $14.9$ | $9.8$ | $14.0$ | $14.1$ | $13.9$ | $14.0$ | $11.5$ |

5 | ${30}^{\circ}$ | $14.9$ | $14.9$ | $15.9$ | $16.1$ | $15.8$ | $15.9$ | $12.7$ |

6 | ${60}^{\circ}$ | $5.1$ | $4.9$ | $8.1$ | $8.1$ | $7.4$ | $7.8$ | $6.1$ |

7 | ${60}^{\circ}$ | $5.1$ | $9.9$ | $10.6$ | $10.6$ | $9.7$ | $10.4$ | $8.2$ |

8 | ${60}^{\circ}$ | $5.1$ | $14.8$ | $12.6$ | $12.6$ | $11.7$ | $12.2$ | $9.9$ |

9 | ${60}^{\circ}$ | $15.1$ | $9.9$ | $14.5$ | $14.4$ | $13.9$ | $14.3$ | $11.5$ |

10 | ${60}^{\circ}$ | $15.1$ | $14.8$ | $16.3$ | $16.2$ | $15.7$ | $16.0$ | $12.7$ |

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**MDPI and ACS Style**

Santillán, D.; Cueto-Felgueroso, L.; Sordo-Ward, A.; Garrote, L.
Influence of Erodible Beds on Shallow Water Hydrodynamics during Flood Events. *Water* **2020**, *12*, 3340.
https://doi.org/10.3390/w12123340

**AMA Style**

Santillán D, Cueto-Felgueroso L, Sordo-Ward A, Garrote L.
Influence of Erodible Beds on Shallow Water Hydrodynamics during Flood Events. *Water*. 2020; 12(12):3340.
https://doi.org/10.3390/w12123340

**Chicago/Turabian Style**

Santillán, David, Luis Cueto-Felgueroso, Alvaro Sordo-Ward, and Luis Garrote.
2020. "Influence of Erodible Beds on Shallow Water Hydrodynamics during Flood Events" *Water* 12, no. 12: 3340.
https://doi.org/10.3390/w12123340