# A 2D Hydraulic Simulation Model Including Dynamic Piping and Overtopping Dambreach

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations and Computational Model

#### 2.1. Governing Equations

#### 2.2. Dambreach by Piping-Erosion Model

#### 2.2.1. Flow Discharge through the Pipe

- $A={b}^{2}+\frac{1}{8}\pi {b}^{2}$ is the cross-sectional area of the pipe;
- b is the width of the base of the pipe;
- ${z}_{pb}={z}_{b}+\frac{1}{2}b$ is the elevation of the pipe centre line;
- ${z}_{b}$ is the elevation of the pipe bottom;
- ${z}_{s}=h+{z}_{bed}$ is the elevation of the water surface level;
- L is the pipe length;
- $R=A/P$ is the pipe hydraulic radius;
- $P=(3+0.5\pi )b$ is the pipe wetted perimeter;
- $f=8g{n}^{2}{R}^{-1/3}$ is the Darcy–Weisbach friction factor of the pipe surface;
- $n=\frac{1}{{A}_{n}}{d}_{50}^{1/6}$ is Manning’s roughness coefficient of the pipe surface;
- ${A}_{n}=12$ is a constant (Wu [17]).

- ${k}_{sm}$ is a dimensionless submergence correction for tailwater effects ([18]);
- $\beta $ is the breach side slope angle with respect to the horizontal;
- ${c}_{r}=1.7\phantom{\rule{0.166667em}{0ex}}\sqrt{m}/s$ is a discharge coefficient for the rectangular part of the breach section [19];
- ${c}_{t}=1.2\phantom{\rule{0.166667em}{0ex}}\sqrt{m}/s$ is a discharge coefficient for the triangular part of the breach section [19].

#### 2.2.2. Pipe Erosion

- ${k}_{d}$ is the measured erosion coefficient at the breach;
- ${\tau}_{c}$ is the critical stress required to initiate detachment for the material;
- ${\tau}_{e}=\frac{{\rho}_{w}g{n}^{2}{Q}_{b}^{2}}{{A}^{2}{R}^{1/3}}$ is the bed shear stress in the pipe surface;
- ${\rho}_{w}$ is the water density.

**Figure 1.**Cross-section of the expansion due to piping process before the dam collapse (

**left**) and trapezoidal breach evolution after the dam collapse (

**right**).

#### 2.2.3. Pipe Roof Collapse

- ${L}_{1}$ is the dam crest width;
- ${L}_{2}={L}_{1}+\frac{{z}_{crest}-{z}_{s}}{tan{\alpha}_{uw}}+\frac{{z}_{crest}-{z}_{s}}{tan{\alpha}_{dw}}$;
- ${L}_{3}={L}_{2}+\frac{{z}_{s}-({z}_{b}+b)}{tan{\alpha}_{uw}}+\frac{{z}_{s}-({z}_{b}+b)}{tan{\alpha}_{dw}}$;
- ${\alpha}_{uw}$ is the upward dam slope angle;
- ${\alpha}_{dw}$ is the downward dam slope angle.

#### 2.3. Numerical Scheme

**E**are evaluated at cell boundaries:

**E**, can be defined as:

#### 2.4. Dambreach Flow as Internal Boundary Condition

- (1)
- If the roof collapse does not occur, the pipe is considered filled with water and pressurized so that the enforced cell discharge in pipe during the next time level is computed using (7) as:$${Q}_{b}^{n+1}=A\sqrt{\frac{2g({z}_{s,L}^{n+1}-{z}_{b}^{n+1})}{1+fL/\left(4R\right)}}$$
- (2)
- If the roof collapse condition is satisfied, the dambreach is assumed open and the enforced cell discharge in pipe during the next time level is computed using (8) as:$${Q}_{b}^{n+1}={k}_{sm}\left({c}_{r}{b}^{n+1}{({z}_{s,L}^{n+1}-{z}_{b}^{n+1})}^{1.5}+\frac{{c}_{t}{({z}_{s,L}^{n+1}-{z}_{b}^{n+1})}^{2.5}}{tan\beta}\right)$$

## 3. Numerical Results and Discussion

#### 3.1. Test Case 1: Synthetic Dambreach through Piping Process

#### 3.2. Test Case 2: Synthetic Set of Dams

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

FV | Finite Volume |

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**Figure 5.**Bed level (

**a**) and detail of the dam geometry together with the triangular computational mesh (

**b**).

**Figure 6.**Temporal evolution of the surface water depth h after the dam breach at the initial state, $t=0.32$ h, $t=0.36$ h, $t=0.60$ h, $t=1$ h and $t=2$ h, respectively.

**Figure 7.**Pipe/breach discharge (Q) temporal evolution for a piping starting time ${t}_{initial}=0.3$ h. Full simulation time (

**a**) and detail at the collapse time (

**b**).

**Figure 8.**Temporal evolution of the top (B) and bottom (b) width of the breach for a piping process starting at time ${t}_{initial}=0.3$ h. Full simulation time (

**a**) and detail at the collapse time (

**b**).

**Figure 9.**Temporal evolution of the top (${z}_{top}$) and bottom level (${z}_{b}$) of the breach for a piping process starting at time time ${t}_{initial}=0.3$ h.

**Figure 12.**Peak breach discharge as a function of dam height for synthetic sets (continuous line) and comparison with the results provided by WinDam C (dashed line).

Dam Height ${\mathit{h}}_{\mathit{d}}$ (ft) | Dam Height ${\mathit{h}}_{\mathit{d}}$ (m) | Top Width (ft) | Top Width (m) |
---|---|---|---|

4 | 1.22 | 8 | 2.44 |

8 | 2.44 | 8 | 2.44 |

16 | 4.88 | 10 | 3.05 |

32 | 9.75 | 14 | 4.27 |

64 | 19.51 | 14 | 4.27 |

128 | 39.01 | 16 | 4.88 |

High | Medium | Low | |
---|---|---|---|

Erodibility ${k}_{d}$ m(${}^{3}$ N ${}^{-1}$ s $-1$) | $1.24\times {10}^{-4}$ | $2.30\times {10}^{-6}$ | $5.31\times {10}^{-7}$ |

Critical shear stress ${\tau}_{c}$ (Pa) | 0 | 9.576 | 9.576 |

${D}_{50}$ (mm) | 0.14 | 0.04 | 0.02 |

**Table 3.**Computational times for GPU and CPU models and the speed-up factor. GPU1 = NVIDIA GeForce GTX Titan Black (NVIDIA Corporation, Santa Clara, CA, USA), GPU2 = NVIDIA GeForce GTX 1080Ti (NVIDIA Corporation, Santa Clara, CA, USA), CPU = Intel Core i7-3770K (Intel Corporation, Santa Clara, CA, USA).

CPU (1 Core) | CPU (4 Cores) | GPU1 | GPU2 | |
---|---|---|---|---|

Computational time (s) | 3802 | 1056 | 90 | 49 |

Speed-up | - | 3.6 | 42.3 | 76.1 |

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**MDPI and ACS Style**

Fernández-Pato, J.; Martínez-Aranda, S.; García-Navarro, P.
A 2D Hydraulic Simulation Model Including Dynamic Piping and Overtopping Dambreach. *Water* **2023**, *15*, 3268.
https://doi.org/10.3390/w15183268

**AMA Style**

Fernández-Pato J, Martínez-Aranda S, García-Navarro P.
A 2D Hydraulic Simulation Model Including Dynamic Piping and Overtopping Dambreach. *Water*. 2023; 15(18):3268.
https://doi.org/10.3390/w15183268

**Chicago/Turabian Style**

Fernández-Pato, Javier, Sergio Martínez-Aranda, and Pilar García-Navarro.
2023. "A 2D Hydraulic Simulation Model Including Dynamic Piping and Overtopping Dambreach" *Water* 15, no. 18: 3268.
https://doi.org/10.3390/w15183268