# 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:$${Q}_{b}^{n+1}={k}_{sm}\left(\right)open="("\; close=")">{c}_{r}{b}^{n+1}{({z}_{s,L}^{n+1}-{z}_{b}^{n+1})}^{1.5}+\frac{{c}_{t}{({z}_{s,L}^{n+1}-{z}_{b}^{n+1})}^{2.5}}{tan\beta}$$

## 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 |

## References

- Wetmore, J.; Fread, D. The NWS Simplified Dam-Break Model; Technical Report; National Weather Service; NOAA: Washington, DC, USA, 1983. [Google Scholar]
- Fread, D. NWS FLDWAV Model: The replacement of DAMBRK for dam-break flood prediction. In Proceedings of the 10th Annual Conference of the ASDSO, Kansas, MO, USA, 26–29 September 1993; pp. 177–184. [Google Scholar]
- Albu, L.M.; Enea, A.; Iosub, M.; Breabăn, I.G. Dam Breach Size Comparison for Flood Simulations. A HEC-RAS Based, GIS Approach for Drăcșani Lake, Sitna River, Romania. Water
**2020**, 12, 1090. [Google Scholar] [CrossRef] - Jiang, H.; Zhao, B.; Dapeng, Z.; Zhu, K. Numerical Simulation of Two-Dimensional Dam Failure and Free-Side Deformation Flow Studies. Water
**2023**, 15, 1515. [Google Scholar] [CrossRef] - Mirauda, D.; Albano, R.; Sole, A.; Adamowski, J. Smoothed Particle Hydrodynamics Modeling with Advanced Boundary Conditions for Two-Dimensional Dam-Break Floods. Water
**2020**, 12, 1142. [Google Scholar] [CrossRef] - Kocaman, S.; Güzel, H.; Evangelista, S.; Ozmen-Cagatay, H.; Viccione, G. Experimental and Numerical Analysis of a Dam-Break Flow through Different Contraction Geometries of the Channel. Water
**2020**, 12, 1124. [Google Scholar] [CrossRef] - Dekay, M.; McClelland, G. Setting Decision Thresholds for Dam Failure Warnings: A Practical Theory-Based Approach; Technical Report No. 328; Center for Research on Judgment and Policy CRJP, University of Colorado: Boulder, CO, USA, 1991; 145p. [Google Scholar]
- Martínez-Aranda, S.; Fernández-Pato, J.; Echeverribar, I.; Navas-Montilla, A.; Morales-Hernández, M.; Brufau, P.; Murillo, J.; García-Navarro, P. Finite Volume models and Efficient Simulation Tools (EST) for Shallow Flows. In Advances in Fluid Mechanics: Modelling and Simulations; Springer Nature: Singapore, 2022; ISBN 978-9-81191-437-9. [Google Scholar]
- Lacasta, A.; Morales-Hernández, M.; Murillo, J.; García-Navarro, P. An optimized GPU implementation of a 2D free surface simulation model on unstructured meshes. Adv. Eng. Softw.
**2014**, 78, 1–15. [Google Scholar] [CrossRef] - Echeverribar, I.; Martínez-Aranda, S.; Fernández-Pato, J.; García-Navarro, P. A GPU-based 2D viscous flow model with variable density and heat exchange. Adv. Eng. Softw.
**2023**, 175, 103340. [Google Scholar] [CrossRef] - Martínez-Aranda, S.; Murillo, J.; García-Navarro, P. A GPU-accelerated Efficient Simulation Tool (EST) for 2D variable-density mud/debris flows over non-uniform erodible beds. Eng. Geol.
**2022**, 296, 106462. [Google Scholar] [CrossRef] - Fernández-Pato, J.; García-Navarro, P. An efficient GPU implementation of a coupled overland-sewer hydraulic model with pollutant transport. Hydrology
**2021**, 8, 146. [Google Scholar] [CrossRef] - Gordillo, G.; Morales-Hernández, M.; Echeverribar, I.; Fernández-Pato, J.; García-Navarro, P. A GPU-based 2D Shallow Water quality model. J. Hydroinformatics
**2020**, 22, 1182–1197. [Google Scholar] [CrossRef] - Cunge, J.A.; Holly, F.M.; Verwey, A. Practical Aspects of Computational River Hydraulics; Pitman Pub. Inc.: Wetherby, UK, 1989. [Google Scholar]
- Arcement, G.; Schneider, V. Guide for Selecting Manning’s Roughness Coefficients for Natural Channels and Flood Plains; Number 2339 in U.S. Geological Survey; Water-Supply Paper, U.S.G.P.O (Government Publishing Office); For sale by the Books and Open-File Reports Section; U.S. Geological Survey: Reston, VA, USA, 1984. [Google Scholar]
- Chen, S.S.; Zhong, Q.M.; Shen, G.Z. Numerical modeling of earthen dam breach due to piping failure. Water Sci. Eng.
**2019**, 12, 169–178. [Google Scholar] [CrossRef] - Wu, W. Simplified Physically Based Model of Earthen Embankment Breaching. J. Hydraul. Eng.
**2013**, 139, 837–851. [Google Scholar] [CrossRef] - Fread, D. Dambreak: The NWS Dam Break Flood Forecasting Model; National Weather Services: Silver Spring, MD, USA, 1984. [Google Scholar]
- Singh, V. Dam Breach Modeling Technology; Kluwer Academic Publishes: Dordrecht, The Netherlands, 1996. [Google Scholar]
- Roe, P. Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys.
**1981**, 43, 357–372. [Google Scholar] [CrossRef] - Morales-Hernández, M.; Murillo, J.; García-Navarro, P. The formulation of internal boundary conditions in unsteady 2-D shallow water flows: Application to flood regulation. Water Resour. Res.
**2013**, 49, 471–487. [Google Scholar] [CrossRef] - Echeverribar, I.; Morales-Hernández, M.; Brufau, P.; García-Navarro, P. Use of internal boundary conditions for levees representation: Application to river flood management. Environ. Fluid Mech.
**2019**, 19, 1253–1271. [Google Scholar] [CrossRef] - Tejral, R.; Hunt, S. Comparing process-based breach models for earthen embankments subjected to internal erosion. In Proceedings of the 5th Federal Interagency Hydrologic Modeling Conference and the 10th Federal Interagency Sedimentation, Reno, NV, USA, 19–23 April 2015. [Google Scholar]
- Hunt, S.; Temple, D.; Neilsen, M.; Ali, A.; Tejral, R. WinDAM C: Analysis Tool for Predicting Breach Erosion Processes of Embankment Dams Due to Overtopping or Internal Erosion. Appl. Eng. Agric.
**2021**, 37, 523–534. [Google Scholar] [CrossRef] - Bureau of Reclamation U.B. Design of Small Dams; Technical Report; U.S. Governement Printing Office: Washington, DC, USA, 1987. [Google Scholar]

**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