# Non-Equilibrium Bedload Transport Model Applied to Erosive Overtopping Dambreach

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Modelling of 2D Generalized Bedload Transport

## 3. Numerical Scheme for the 2D Bedload Transport Model

^{˜}indicates edge-averaged quantities. The pseudo-Jacobian matrix ${\tilde{\mathbf{M}}}_{k}={(\tilde{\mathbf{J}}-\tilde{\mathbf{H}})}_{k}$ of the coupled system includes the approximated Jacobian matrix of the conservative convective flux ${\tilde{\mathbf{J}}}_{k}$ and the non-conservative matrix ${\tilde{\mathbf{H}}}_{k}$ of the bed-pressure momentum contribution at the edge. On the right hand side, the term ${\tilde{\mathbf{T}}}_{k}$ denotes the momentum dissipation due to flow-bed frictional stress, integrated through the kth edge. Details on the flux computation have been extensively reported in Martínez-Aranda et al. [34] but, for the sake of brevity, ${\mathcal{F}}_{k}^{\downarrow -}$ at the kth edge is computed as:

#### 3.1. Transport Layer Updating with Capacity and Non-Capacity Approaches

- Equilibrium hypothesis: The new transport layer thickness is directly computed as:$${\eta}_{i}^{n+1}=\frac{{k}_{e}}{{r}_{s}{k}_{d}}{(\Delta \theta )}_{i}^{n+1}{d}_{s}$$
- Non-capacity approach: This leads to the necessity of solving Equation (9) each time step. The updating formula for the transport layer thickness $\eta $ is expressed as:$${\eta}_{i}^{n+1}={\eta}_{i}^{n}-\frac{\Delta t}{{A}_{i}}\sum _{k=1}^{NE}{F}_{k}^{\eta \downarrow}{l}_{k}+\Delta t{({\dot{\eta}}_{e}-{\dot{\eta}}_{d})}_{i}^{n}$$$$\begin{array}{cc}\hfill & \frac{\partial \eta}{\partial t}+{\tilde{\lambda}}_{\eta ,k}\frac{\partial \eta}{\partial \widehat{n}}=0\hfill \end{array}$$$$\begin{array}{cc}\hfill & \mathrm{with}:\eta (\widehat{n},0)=\left\{\begin{array}{ccc}{\eta}_{i}^{n}& \mathrm{if}& \widehat{n}<0\\ {\eta}_{j}^{n}& \mathrm{if}& \widehat{n}>0\end{array}\right.\hfill \end{array}$$$${\tilde{\lambda}}_{\eta ,k}={\left(\frac{\delta {q}_{bn}}{\delta \eta}\right)}_{k}$$$$\begin{array}{cc}\hfill & {F}_{k}^{\eta \downarrow}=\left\{\begin{array}{ccc}{({q}_{bx}{n}_{x}+{q}_{by}{n}_{y})}_{i}^{n}& \mathrm{if}& {\tilde{\lambda}}_{\eta ,k}>0\\ {({q}_{bx}{n}_{x}+{q}_{by}{n}_{y})}_{j}^{n}& \mathrm{if}& {\tilde{\lambda}}_{\eta ,k}<0\end{array}\right.\hfill \end{array}$$The cell-centered exchange rates ${\dot{\eta}}_{e,i}^{n}$ and ${\dot{\eta}}_{d,i}^{n}$ between the underlying static stratum and the transport layer are computed as:$${\left({\dot{\eta}}_{e}\right)}_{i}^{n}={k}_{e}\frac{{(\Delta \theta )}_{i}^{n}}{{r}_{s}}\sqrt{({r}_{s}-1)g{d}_{s}}\text{}{\left({\dot{\eta}}_{d}\right)}_{i}^{n}={k}_{d}\frac{{\eta}_{i}^{n}}{{d}_{s}}\sqrt{({r}_{s}-1)g{d}_{s}}$$

#### 3.2. Morphological Collapse Mechanism

- Positive bed slope if $0<{({z}_{bR}-{z}_{bL})}^{n+1}<tan{\delta}_{b}{d}_{n}$$${q}_{b}^{\downarrow}\le \frac{tan{\delta}_{b}{d}_{n}-{({z}_{bR}-{z}_{bL})}^{n}}{l\Delta t}\frac{{A}_{L}{A}_{R}}{{A}_{L}+{A}_{R}}$$
- Negative bed slope if $-(tan{\delta}_{b}{d}_{n})<{({z}_{bR}-{z}_{bL})}^{n+1}<0$$${q}_{b}^{\downarrow}\ge \frac{-\left[tan{\delta}_{b}{d}_{n}+{({z}_{bR}-{z}_{bL})}^{n}\right]}{l\Delta t}\frac{{A}_{L}{A}_{R}}{{A}_{L}+{A}_{R}}$$

## 4. Numerical Results and Discussion

#### 4.1. Adaptation of the Non-Equilibrium Bedload Rate to Equilibrium States

#### 4.2. Dike Breaking by Overtopping Flow Erosion

#### 4.3. Breach Opening in Homogeneous Dam

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FV | Finite Volume |

RMSE | Root Mean Square Error |

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**Figure 3.**Bedload transport rate evolution for the generalised non-equilibrium model with (

**top**) ${k}_{e}=0.06$ and (

**bottom**) ${k}_{e}=0.015$.

**Figure 4.**Normalised bedload transport rate for the generalised non-equilibrium model with increasing Shields stresses $\theta $.

**Figure 5.**Dam profile evolution for (

**top row**) case C1 and (

**bottom row**) case C2 with equilibrium and non-equilibrium approaches: (

**left**) dam profile at different times and (

**right**) bed level evolution at P1, P2 and P3.

**Figure 6.**Temporal evolution of the transport layer thickness at P1, P2 and P3: (

**left**) case C1 and (

**right**) case C2.

**Figure 7.**Computed and measured temporal evolution of (

**left column**) the discharge at the dyke crest and (

**right column**) the upstream reservoir water surface level: (

**top row**) case C1 and (

**bottom row**) case C2.

**Figure 8.**3D sketch of the experimental setup (initial condition) at the Laboratório Nacional de Engenharia Civil (LNEC, Lisbon). Detail of the initial triangular notch at the dam crest.

**Figure 9.**Dambreach opening at $t=1500$ s with non-equilibrium model: (

**left**) 3D view of the dam level and water surface; (

**right top**) thickness of the bedload transport layer and (

**right bottom**) depth-averaged velocity field.

**Figure 10.**Dambreach opening at $t=2650$ s with non-equilibrium model: (

**left**) 3D view of the dam level and water surface; (

**right top**) thickness of the bedload transport layer and (

**right bottom**) depth-averaged velocity field.

**Figure 11.**Dambreach opening at $t=3566$ s with non-equilibrium model: (

**left**) 3D view of the dam level and water surface; (

**right top**) thickness of the bedload transport layer and (

**right bottom**) depth-averaged velocity field.

**Figure 12.**Dambreach opening at $t=4317$ s with non-equilibrium model: (

**left**) 3D view of the dam level and water surface; (

**right top**) thickness of the bedload transport layer and (

**right bottom**) depth-averaged velocity field.

**Figure 13.**Dambreach evolution with equilibrium and non-equilibrium approaches: (

**left**) dam discharge and (

**right**) upstream reservoir level.

**Figure 14.**Dambreach evolution with equilibrium and non-equilibrium approaches: (

**left**) cross-section wetted-area of the breach and (

**right**) averaged flow velocity at the dam crest section.

Formulation | ${\mathbf{\Gamma}}_{1}\left(\mathit{h}\right)$ | ${\mathbf{\Gamma}}_{2}\left(\mathit{\theta}\right)$ | ${\mathbf{\Gamma}}_{3}\left(\mathit{\eta}\right)$ | ${\mathit{\theta}}_{\mathit{c}}$ |
---|---|---|---|---|

MPM | $\frac{{n}^{3}\sqrt{g}}{({r}_{s}-1)\sqrt{h}}$ | $8\sqrt{\Delta \theta}{\theta}^{-3/2}$ | $\frac{{r}_{s}{k}_{d}}{{k}_{e}}\frac{\eta}{{d}_{s}}$ | 0.047 |

Nielsen | $\frac{{n}^{3}\sqrt{g}}{({r}_{s}-1)\sqrt{h}}$ | $12{\theta}^{-1}$ | $\frac{{r}_{s}{k}_{d}}{{k}_{e}}\frac{\eta}{{d}_{s}}$ | 0.047 |

Fernandez-Luque | $\frac{{n}^{3}\sqrt{g}}{({r}_{s}-1)\sqrt{h}}$ | $5.7\sqrt{\Delta \theta}{\theta}^{-3/2}$ | $\frac{{r}_{s}{k}_{d}}{{k}_{e}}\frac{\eta}{{d}_{s}}$ | 0.037 |

Wong | $\frac{{n}^{3}\sqrt{g}}{({r}_{s}-1)\sqrt{h}}$ | $3.97\sqrt{\Delta \theta}{\theta}^{-3/2}$ | $\frac{{\rho}_{s}/{\rho}_{f}{k}_{d}}{{k}_{e}}\frac{\eta}{{d}_{s}}$ | 0.0495 |

Smart | $\frac{{n}^{2}}{({r}_{s}-1){h}^{1/3}}$ | $4.2{S}_{b}^{0.6}{\theta}^{-1}$ | $\frac{{r}_{s}{k}_{d}}{{k}_{e}}\frac{\eta}{{d}_{s}}$ | 0.047 |

Wu | $\frac{{n}^{3}\sqrt{g}}{({r}_{s}-1)\sqrt{h}}$ | $0.0053{\theta}_{c}^{-2.2}\sqrt{\Delta \theta}{\theta}^{-3/2}$ | $\frac{{r}_{s}{k}_{d}}{{k}_{e}}\frac{\eta}{{d}_{s}}$ | 0.030 |

Case | ${\mathit{S}}_{\mathit{u}}$ | ${\mathit{S}}_{\mathit{d}}$ | ${\mathit{q}}_{\mathbf{inlet}}$ (L/s) | ${\mathit{k}}_{\mathit{e}}$ | ${\mathit{k}}_{\mathit{d}}$ |
---|---|---|---|---|---|

C1 | 1V:3H | 1V:5H | 1.05 | 0.24 | 0.012 |

C2 | 1V:3H | 1V:3H | 1.23 | 0.34 | 0.017 |

Data Series | Equil. | No-Equil. |
---|---|---|

C1—Bed level in P1 | 0.057 m | 0.023 m |

C1—Bed level in P2 | 0.053 m | 0.022 m |

C1—Bed level in P3 | 0.033 m | 0.020 m |

C2—Bed level profile at $t=30$ s | 0.112 m | 0.061 m |

C2—Bed level profile at $t=60$ s | 0.029 m | 0.039 m |

Data Series | Equil. | No-Equil. |
---|---|---|

C1—Dam discharge | 10.94 L/s | 3.10 L/s |

C1—Reservoir level | 0.055 m | 0.019 m |

C2—Dam discharge | 45.02 L/s | 11.76 L/s |

C2—Reservoir level | 0.116 m | 0.034 m |

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**MDPI and ACS Style**

Martínez-Aranda, S.; Fernández-Pato, J.; García-Navarro, P.
Non-Equilibrium Bedload Transport Model Applied to Erosive Overtopping Dambreach. *Water* **2023**, *15*, 3094.
https://doi.org/10.3390/w15173094

**AMA Style**

Martínez-Aranda S, Fernández-Pato J, García-Navarro P.
Non-Equilibrium Bedload Transport Model Applied to Erosive Overtopping Dambreach. *Water*. 2023; 15(17):3094.
https://doi.org/10.3390/w15173094

**Chicago/Turabian Style**

Martínez-Aranda, Sergio, Javier Fernández-Pato, and Pilar García-Navarro.
2023. "Non-Equilibrium Bedload Transport Model Applied to Erosive Overtopping Dambreach" *Water* 15, no. 17: 3094.
https://doi.org/10.3390/w15173094