# 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

- Smith, K. Environmental Hazards: Assessing Risk and Reducing Disaster; Routledge: London, UK; New York, NY, USA, 2013. [Google Scholar]
- Kundzewicz, Z.W.; Kanae, S.; Seneviratne, S.I.; Handmer, J.; Nicholls, N.; Peduzzi, P.; Mechler, R.; Bouwer, L.M.; Arnell, N.; Mach, K.; et al. Flood risk and climate change: Global and regional perspectives. Hydrol. Sci. J.-J. Sci. Hydrol.
**2014**, 59, 1–28. [Google Scholar] [CrossRef][Green Version] - Mediero, L.; Santillán, D.; Garrote, L.; Granados, A. Detection and attribution of trends in magnitude, frequency and timing of floods in Spain. J. Hydrol.
**2014**, 517, 1072–1088. [Google Scholar] [CrossRef] - Tayefi, V.; Lane, S.; Hardy, R.; Yu, D. A comparison of one-and two-dimensional approaches to modelling flood inundation over complex upland floodplains. Hydrol. Process.
**2007**, 21, 3190–3202. [Google Scholar] [CrossRef] - Sellin, R.; Ervine, D.; Willetts, B. Behaviour of meandering two-stage channels. Proc. Inst. Civ. Eng.-Water Marit. Energy
**1993**, 101, 99–111. [Google Scholar] [CrossRef] - Toro, E.F. Shock–Capturing Methods for Free–Surface Shallow Flows; John Wiley & Sons, Ltd.: Chichester, UK, 2001. [Google Scholar]
- Deltares Systems. Simulation of Multidimensional Hydrodynamic Flows and Transport Phenomena, Including Sediments; User Manual: Delft, The Netherlands, 2011. [Google Scholar]
- Bladé, E.; Cea, L.; Corestein, G.; Escolano, E.; Puertas, J.; Vázquez-Cendón, E.; Dolz, J.; Coll, A. Iber: Herramienta de simulación numérica del flujo en ríos. Rev. Int. Métodos Numér. Cálc. Diseño Ing.
**2014**, 30, 1–10. [Google Scholar] [CrossRef][Green Version] - Pérez-Sánchez, M.; Sánchez-Romero, F.; Redón-Santafé, M.; Torregrosa Soler, J.; Ferrer Gisbert, C.; Ferrán Gozálvez, J.; Ferrer Gisbert, A.; López-Jiménez, P. Numerical study for the development of flood maps considering the break hypothesis in irrigation reservoirs. Ing. Agua
**2019**, 23, 1–18. [Google Scholar] [CrossRef] - Sanz-Ramos, M.; Bladé, E.; Escolano, E. Optimization of the Floodplain Encroachment calculation with hydraulic criteria. Ing. Agua
**2020**, 24, 203–218. [Google Scholar] [CrossRef] - Zerihun, Y.T. A Numerical Study on Curvilinear Free Surface Flows in Venturi Flumes. Fluids
**2016**, 1, 21. [Google Scholar] [CrossRef] - Zerihun, Y.T.; Fenton, J.D. One-dimensional simulation model for steady transcritical free surface flows at short length transitions. Adv. Water Resour.
**2006**, 29, 1598–1607. [Google Scholar] [CrossRef] - Berger, R.C.; Carey, G.F. Free-surface flow over curved surfaces Part II: Computational model. Int. J. Numer. Methods Fluids
**1998**, 28, 201–213. [Google Scholar] [CrossRef] - Shimozono, T.; Sato, S. Coastal vulnerability analysis during tsunami-induced levee overflow and breaching by a high-resolution flood model. Coast. Eng.
**2016**, 107, 116–126. [Google Scholar] [CrossRef] - Sánchez-Cordero, E.; Gómez, M.; Bladé, E. 3D numerical analysis of flow characteristics in an open—Channel bend. Ing. Agua
**2020**, 24, 157–168. [Google Scholar] [CrossRef] - Duró, G.; Crosato, A.; Tassi, P. Numerical study on river bar response to spatial variations of channel width. Adv. Water Res.
**2016**, 93, 21–38. [Google Scholar] [CrossRef] - Martín-Vide, J.; Plana-Casado, A.; Sambola, A.; Capapé, S. Bedload transport in a river confluence. Geomorphology
**2015**, 250, 15–28. [Google Scholar] [CrossRef] - Herrero, H.S.; Lozada, J.M.D.; García, C.M.; Szupiany, R.N.; Best, J.; Pagot, M. The influence of tributary flow density differences on the hydrodynamic behavior of a confluent meander bend and implications for flow mixing. Geomorphology
**2018**, 304, 99–112. [Google Scholar] [CrossRef] - Baki, A.B.M.; Gan, T.Y. Riverbank migration and island dynamics of the braided Jamuna River of the Ganges—Brahmaputra basin using multi-temporal Landsat images. Quat. Int.
**2012**, 263, 148–161. [Google Scholar] [CrossRef] - Hicks, D.M.; Gomez, B.; Trustrum, N.A. Erosion thresholds and suspended sediment yields, Waipaoa River basin, New Zealand. Water Resour. Res.
**2000**, 36, 1129–1142. [Google Scholar] [CrossRef] - R. Grove, J.; Croke, J.; Thompson, C. Quantifying different riverbank erosion processes during an extreme flood event. Earth Surf. Process. Landf.
**2013**, 38, 1393–1406. [Google Scholar] - Croke, J.; Fryirs, K.; Thompson, C. Channel–floodplain connectivity during an extreme flood event: Implications for sediment erosion, deposition, and delivery. Earth Surf. Process. Landf.
**2013**, 38, 1444–1456. [Google Scholar] [CrossRef] - Lamb, M.P.; Fonstad, M.A. Rapid formation of a modern bedrock canyon by a single flood event. Nat. Geosci.
**2010**, 3, 477–481. [Google Scholar] [CrossRef][Green Version] - Bilal, A.; Xie, Q.; Zhai, Y. Flow, Sediment, and Morpho-Dynamics of River Confluence in Tidal and Non-Tidal Environments. J. Mar. Sci. Eng.
**2020**, 8, 591. [Google Scholar] [CrossRef] - Arega, F.; Lee, J.H.; Tang, H.W. Hydraulic jet control for river junction design of Yuen Long Bypass Floodway, Hong Kong. J. Hydraul. Eng.
**2008**, 134, 23–33. [Google Scholar] [CrossRef] - Ferrarin, C.; Madricardo, F.; Rizzetto, F.; Kiver, W.M.; Bellafiore, D.; Umgiesser, G.; Kruss, A.; Zaggia, L.; Foglini, F.; Ceregato, A.; et al. Geomorphology of scour holes at tidal channel confluences. J. Geophys. Res. Earth Surf.
**2018**, 123, 1386–1406. [Google Scholar] [CrossRef] - Xie, Q.; Yang, J.; Lundström, S.; Dai, W. Understanding morphodynamic changes of a tidal river confluence through field measurements and numerical modeling. Water
**2018**, 10, 1424. [Google Scholar] [CrossRef][Green Version] - Bennett, S.; Best, J. Mean flow and turbulence structure over fixed, two-dimensional dunes: Implications for sediment transport and bedform stability. Sedimentology
**1995**, 42, 491–513. [Google Scholar] [CrossRef] - Best, J.L.; Roy, A.G. Mixing-layer distortion at the confluence of channels of different depth. Nature
**1991**, 350, 411–413. [Google Scholar] [CrossRef] - Biron, P.; Best, J.L.; Roy, A.G. Effects of bed discordance on flow dynamics at open channel confluences. J. Hydraul. Eng.
**1996**, 122, 676–682. [Google Scholar] [CrossRef] - Biron, P.; Roy, A.; Best, J. Turbulent flow structure at concordant and discordant open-channel confluences. Exp. Fluids
**1996**, 21, 437–446. [Google Scholar] [CrossRef] - Boyer, C.; Roy, A.G.; Best, J.L. Dynamics of a river channel confluence with discordant beds: Flow turbulence, bed load sediment transport, and bed morphology. J. Geophys. Res. Solid Earth
**2006**, 111. [Google Scholar] [CrossRef] - Best, J.L.; Reid, I. Separation zone at open-channel junctions. J. Hydraul. Eng.
**1984**, 110, 1588–1594. [Google Scholar] [CrossRef] - Webber, N.B.; Greated, C.A. An investigation of flow behaviour at the junction of rectangular channels. Proc. Inst. Civil. Eng.
**1966**, 34, 321–334. [Google Scholar] [CrossRef] - Tang, H.; Zhang, H.; Yuan, S. Hydrodynamics and contaminant transport on a degraded bed at a 90-degree channel confluence. Environ. Fluid Mech.
**2018**, 18, 443–463. [Google Scholar] [CrossRef] - Yuan, S.; Tang, H.; Xiao, Y.; Xia, Y.; Melching, C.; Li, Z. Phosphorus contamination of the surface sediment at a river confluence. J. Hydrol.
**2019**, 573, 568–580. [Google Scholar] [CrossRef] - Cheng, Z.; Constantinescu, G. Stratification effects on hydrodynamics and mixing at a river confluence with discordant bed. Environ. Fluid Mech.
**2019**, 20, 843–872. [Google Scholar] [CrossRef] - Gurram, S.K.; Karki, K.S.; Hager, W.H. Subcritical junction flow. J. Hydraul. Eng.
**1997**, 123, 447–455. [Google Scholar] [CrossRef] - Hsu, C.C.; Lee, W.J.; Chang, C.H. Subcritical open-channel junction flow. J. Hydraul. Eng.
**1998**, 124, 847–855. [Google Scholar] [CrossRef] - Weber, L.J.; Schumate, E.D.; Mawer, N. Experiments on flow at a 90 open-channel junction. J. Hydraul. Eng.
**2001**, 127, 340–350. [Google Scholar] [CrossRef] - Guillén-Ludeña, S.; Franca, M.; Cardoso, A.; Schleiss, A. Evolution of the hydromorphodynamics of mountain river confluences for varying discharge ratios and junction angles. Geomorphology
**2016**, 255, 1–15. [Google Scholar] [CrossRef] - Leite Ribeiro, M.; Blanckaert, K.; Roy, A.; Schleiss, A.J. Flow and sediment dynamics in channel confluences. J. Geophys. Res. Earth Surf.
**2012**, 117. [Google Scholar] [CrossRef][Green Version] - Best, J.L. Sediment transport and bed morphology at river channel confluences. Sedimentology
**1988**, 35, 481–498. [Google Scholar] [CrossRef] - Nazari-Giglou, A.; Jabbari-Sahebari, A.; Shakibaeinia, A.; Borghei, S.M. An experimental study of sediment transport in channel confluences. Int. J. Sediment Res.
**2016**, 31, 87–96. [Google Scholar] [CrossRef] - Yuan, S.; Tang, H.; Xiao, Y.; Qiu, X.; Xia, Y. Water flow and sediment transport at open-channel confluences: An experimental study. J. Hydraul. Res.
**2018**, 56, 333–350. [Google Scholar] [CrossRef] - Rhoads, B.L.; Sukhodolov, A.N. Spatial and temporal structure of shear layer turbulence at a stream confluence. Water Resour. Res.
**2004**, 40. [Google Scholar] [CrossRef] - Rhoads, B.L.; Sukhodolov, A.N. Lateral momentum flux and the spatial evolution of flow within a confluence mixing interface. Water Resour. Res.
**2008**, 44. [Google Scholar] [CrossRef] - Rhoads, B.L.; Sukhodolov, A.N. Field investigation of three-dimensional flow structure at stream confluences: 1. Thermal mixing and time-averaged velocities. Water Resour. Res.
**2001**, 37, 2393–2410. [Google Scholar] [CrossRef] - Sukhodolov, A.N.; Rhoads, B.L. Field investigation of three-dimensional flow structure at stream confluences: 2. Turbulence. Water Resour. Res.
**2001**, 37, 2411–2424. [Google Scholar] [CrossRef] - Mosley, M.P. An experimental study of channel confluences. J. Geol.
**1976**, 84, 535–562. [Google Scholar] [CrossRef] - Wallis, E.; Nally, R.M.; Lake, P.S. A Bayesian analysis of physical habitat changes at tributary confluences in cobble-bed upland streams of the Acheron River basin, Australia. Water Resour. Res.
**2008**, 44. [Google Scholar] [CrossRef][Green Version] - Best, J.L.; Rhoads, B.L. River Confluences, Tributaries and the Fluvial Network; John Wiley & Sons, Ltd.: Chichester, UK, 2008; Chapter Sediment transport, bed morphology and the sedimentology of river channel confluences; pp. 45–72. [Google Scholar]
- Cea, L.; Puertas, J.; Vázquez-Cendón, M.E. Depth Averaged Modelling of Turbulent Shallow Water Flow with Wet-Dry Fronts. Arch. Comput. Methods Eng.
**2007**, 14, 303–341. [Google Scholar] [CrossRef] - Rastogi, A.K.; Rodi, W. Predictions of heat and mass transfer in open channels. J. Hydraul. Div.
**1978**, 104, 397–420. [Google Scholar] - Ashworth, P.; Ferguson, R. Interrelationships of Channel Processes, Changes and Sediments in a Proglacial Braided River. Geogr. Ann. Ser. A Phys. Geogr.
**1986**, 68, 361–371. [Google Scholar] [CrossRef] - Leeder, M.R. On the Interactions between Turbulent Flow, Sediment Transport and Bedform Mechanics in Channelized Flows. In Modern and Ancient Fluvial Systems; John Wiley & Sons, Ltd.: Hoboken, NJ, USA, 2009; pp. 3–18. [Google Scholar] [CrossRef]
- Anderson, R.S. Modeling the tor-dotted crests, bedrock edges, and parabolic profiles of high alpine surfaces of the Wind River Range, Wyoming. Geomorphology
**2002**, 46, 35–58. [Google Scholar] [CrossRef] - Mudd, S.M.; Furbish, D.J. Influence of chemical denudation on hillslope morphology. J. Geophys. Res. Earth Surf.
**2004**, 109. [Google Scholar] [CrossRef][Green Version] - Paola, C.; Voller, V.R. A generalized Exner equation for sediment mass balance. J. Geophys. Res. Earth Surf.
**2005**, 110. [Google Scholar] [CrossRef] - Van Rijn, L.C. Sediment Transport, Part I: Bed Load Transport. J. Hydraul. Eng.
**1984**, 110, 1431–1456. [Google Scholar] [CrossRef][Green Version] - Shields, A. Anwendung Der Aehnlichkeitsmechanik Und Der Turbulenzforschung Auf Die Geschiebebewegung. Ph.D. Thesis, Technical University Berlin, Berlin, Germany, 1936. [Google Scholar]
- Van Rijn, L.C. Sediment Transport, Part II: Suspended Load Transport. J. Hydraul. Eng.
**1984**, 110, 1613–1641. [Google Scholar] [CrossRef] - Van Leer, B. Towards the Ultimate Conservative Difference Scheme, V. A Second Order Sequel to Godunov’s Method. J. Comput. Phys.
**1979**, 32, 101–136. [Google Scholar] [CrossRef] - Alcrudo, A.; García-Navarro, P. A High Resolution Godunov–Type Scheme in Finite Volumes for the 2D Shallow Water Equations. Int. J. Numer. Methods Eng.
**1993**, 16, 489–505. [Google Scholar] [CrossRef] - García-Navarro, P.; Hubbard, M.E.; Priestley, A. Genuinely Multidimensional Upwinding for the 2D Shallow Water Equations. J. Comput. Phys.
**1995**, 121, 79–93. [Google Scholar] [CrossRef] - Toro, E.F.; García-Navarro, P. Godunov–Type Methods for Free–Surface Shallow Flows: A Review. J. Hydraul. Res.
**2007**, 45, 736–751. [Google Scholar] [CrossRef] - Cueto-Felgueroso, L.; Colominas, I. High-order finite volume methods and multiresolution reproducing kernels. Arch. Comput. Methods Eng.
**2008**, 15, 185–228. [Google Scholar] [CrossRef] - Cueto-Felgueroso, L.; Colominas, I.; Fe, J.; Navarrina, F.; Casteleiro, M. High-order finite volume schemes on unstructured grids using moving least-squares reconstruction. Application to shallow water dynamics. Int. J. Numer. Methods Eng.
**2006**, 65, 295–331. [Google Scholar] [CrossRef][Green Version] - Navas-Montilla, A.; Murillo, J.; García-Navarro, P. High order simulation models for the resolution of wave propagation phenomena in turbulent free surface flows. Ing. Agua
**2019**, 23, 275–287. [Google Scholar] [CrossRef] - Pinto Coelho, M.M.L. Experimental determination of free surface levels at open-channel junctions. J. Hydraul. Res.
**2015**, 53, 394–399. [Google Scholar] [CrossRef] - Cueto-Felgueroso, L.; Santillán, D.; García-Palacios, J.H.; Garrote, L. Comparison between 2D Shallow-Water Simulations and Energy-Momentum Computations for Transcritical Flow Past Channel Contractions. Water
**2019**, 11, 1476. [Google Scholar] [CrossRef][Green Version] - Muller, R. Theoretische Grundlagen der Flussund Wildbachverbauungen; Leemann: Zurich, Switzerland, 1943. [Google Scholar]
- Strickler, A. Beitrage zur Frage der Geschwindigkeitsformel und der Rauhigkeitszahlen fur Strome, Kanale und Geschlossene Leitungen; Amt. f. Wasserwirtschaft: Berna, Switzerland, 1923. [Google Scholar]
- Bellin, N.; Vanacker, V.; Van Wesemael, B.; Solé-Benet, A.; Bakker, M. Natural and anthropogenic controls on soil erosion in the Internal Betic Cordillera (southeast Spain). Catena
**2011**, 87, 190–200. [Google Scholar] [CrossRef][Green Version] - Martinez-Mena, M.; Castillo, V.; Albaladejo, J. Hydrological and erosional response to natural rainfall in a semi-arid area of south-east Spain. Hydrol. Process.
**2001**, 15, 557–571. [Google Scholar] [CrossRef] - Álvarez, A.J.; Montañés, C.G.; Orduña, L.M.; Caballer, L.I.; Revilla, J.G. El mapa de caudales máximos de las cuencas intercomunitarias. Rev. Obras Públicas
**2012**, 3533, 7–32. [Google Scholar] - Santillán, D.; Mediero, L.; Garrote, L. Modelling uncertainty of flood quantile estimations at ungauged sites by Bayesian networks. J. Hydroinform.
**2014**, 16, 822–838. [Google Scholar] [CrossRef][Green Version] - Odgaard, A.J.; Bergs, M.A. Flow processes in a curved alluvial channel. Water Resour. Res.
**1988**, 24, 45–56. [Google Scholar] [CrossRef] - Wu, W.; Rodi, W.; Wenka, T. 3D numerical modeling of flow and sediment transport in open channels. J. Hydraul. Eng.
**2000**, 126, 4–15. [Google Scholar] [CrossRef]

**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