# SPH Simulation of Sediment Movement from Dam Breaks

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fundamental SPH Formulations

## 3. Standard SPH Fluid Model

^{3}is the initial density; the constant $B=\frac{{c}^{2}{\rho}_{o}}{\gamma}$, of which $c=10\times V$max; $V$max is the maximum flow velocity.

## 4. Two-Phase SPH Water–Sediment Model

#### 4.1. Lagrangian Form of Governing Equations

- (i)
- Equation for conservation of mass of water flow:$$\frac{D({\alpha}^{f}{\rho}^{f})}{Dt}=-{\alpha}^{f}{\rho}^{f}\nabla \cdot {\overrightarrow{u}}^{f}$$
- (ii)
- Conservation of momentum equation for water flow:$$\frac{D{\overrightarrow{u}}^{f}}{Dt}=-\frac{1}{{\rho}^{f}}\nabla {P}^{f}+\frac{1}{{\alpha}^{f}{\rho}^{f}}\nabla \cdot {\overrightarrow{\tau}}^{f}+\overrightarrow{g}-\frac{{\alpha}^{s}}{{\rho}^{f}}K({\overrightarrow{u}}^{f}-{\overrightarrow{u}}^{s})+\frac{1}{{\rho}^{f}}{S}_{US}K{\upsilon}_{t}^{f}\nabla {\alpha}^{s}$$
- (iii)
- Sediment mass conservation equation:$$\frac{D{\alpha}^{s}}{Dt}=({\overrightarrow{u}}^{f}-{\overrightarrow{u}}^{s})\cdot \nabla {\alpha}^{s}-{\alpha}^{s}\nabla \cdot {\overrightarrow{u}}^{s}$$
- (iv)
- Conservation of sediment momentum equation:

#### 4.2. SPH Discretizations in Two-Phase Equations

- (i)
- Conservation of water mass:$$\frac{{\left.{\alpha}_{a}^{f}\right|}^{(t+\Delta t)}{\left.{\rho}_{a}^{f}\right|}^{(t+\Delta t)}-{\left.{\alpha}_{a}^{f}\right|}^{(t)}{\left.{\rho}_{a}^{f}\right|}^{(t)}}{\Delta t}={\alpha}_{a}^{f}{\rho}_{a}^{f}{\displaystyle \sum _{b}\Delta {V}_{b}({\overrightarrow{u}}_{a}^{f}-{\overrightarrow{u}}_{b}^{f})}\cdot {\nabla}_{a}{W}_{ab}$$
- (ii)
- Conservation of water momentum:$$\begin{array}{l}\frac{{\left.{\overrightarrow{u}}_{a}^{f}\right|}^{(t+\Delta t)}-{\left.{\overrightarrow{u}}_{a}^{f}\right|}^{(t)}}{\Delta t}=-\frac{1}{{\rho}_{a}{}^{f}}{\displaystyle \sum _{b}\Delta {V}_{b}({P}_{a}^{f}+{P}_{b}^{f})}{\nabla}_{a}{W}_{ab}+\frac{1}{{\alpha}_{a}{}^{f}{\rho}_{a}{}^{f}}{\displaystyle \sum _{b}\Delta {V}_{b}({\overrightarrow{\tau}}_{a}^{f}+{\overrightarrow{\tau}}_{b}^{f})}\cdot {\nabla}_{a}{W}_{ab}\\ +\overrightarrow{g}-\frac{{\alpha}_{a}{}^{s}}{{\rho}_{a}{}^{f}}{K}_{a}({\overrightarrow{u}}_{a}^{f}-{\overrightarrow{u}}_{a}^{s})-\frac{1}{{\rho}_{a}{}^{f}}{S}_{US}{K}_{a}{\left.{\upsilon}_{t}^{f}\right|}_{a}{\displaystyle \sum _{b}\Delta {V}_{b}({\alpha}_{a}{}^{s}-{\alpha}_{b}{}^{s})}{\nabla}_{a}{W}_{ab}\end{array}$$
- (iii)
- Sediment mass conservation:$$\begin{array}{l}\frac{{\left.{\alpha}_{a}{}^{s}\right|}^{(t+\Delta t)}-{\left.{\alpha}_{a}{}^{s}\right|}^{(t)}}{\Delta t}=\\ {\alpha}_{a}^{s}{\displaystyle \sum _{b}\Delta {V}_{b}({\overrightarrow{u}}_{a}^{s}{{}_{}}^{}-{\overrightarrow{u}}_{b}^{s})}\cdot {\nabla}_{a}{W}_{ab}-{\displaystyle \sum _{b}\Delta {V}_{b}({\alpha}_{a}^{s}-{\alpha}_{b}^{s})({\overrightarrow{u}}_{a}^{f}-{\overrightarrow{u}}_{a}^{s})}\cdot {\nabla}_{a}{W}_{ab}\end{array}$$
- (iv)
- Sediment momentum conservation:$$\begin{array}{l}\frac{{\left.{\overrightarrow{u}}_{a}^{s}\right|}^{(t+\Delta t)}-{\left.{\overrightarrow{u}}_{a}^{s}\right|}^{(t)}}{\Delta t}=-\frac{1}{{\rho}_{a}{}^{s}}{\displaystyle \sum _{b}\Delta {V}_{b}({P}_{a}^{f}+{P}_{b}^{f})}{\nabla}_{a}{W}_{ab}-\frac{1}{{\alpha}_{a}{}^{s}{\rho}_{a}{}^{s}}{\displaystyle \sum _{b}\Delta {V}_{b}({\tilde{P}}_{a}^{s}+{\tilde{P}}_{b}^{s})}{\nabla}_{a}{W}_{ab}\\ +\frac{{\alpha}_{a}^{f}}{{\alpha}_{a}{}^{s}{\rho}_{a}{}^{s}}{S}_{US}{K}_{a}{\left.{\upsilon}_{t}^{f}\right|}_{a}{\displaystyle \sum _{b}\Delta {V}_{b}({\alpha}_{a}^{s}-{\alpha}_{b}^{s})}{\nabla}_{a}{W}_{ab}-{\displaystyle \sum _{b}\Delta {V}_{b}({\overrightarrow{u}}_{a}^{s}-{\overrightarrow{u}}_{b}^{s})({\overrightarrow{u}}_{a}^{f}-{\overrightarrow{u}}_{a}^{s})}\cdot {\nabla}_{a}{W}_{ab}\\ +\frac{1}{{\alpha}_{a}{}^{s}{\rho}_{a}{}^{s}}{\displaystyle \sum _{b}\Delta {V}_{b}({\overrightarrow{\tau}}_{a}^{s}+{\overrightarrow{\tau}}_{b}^{s})}\cdot {\nabla}_{a}{W}_{ab}+\overrightarrow{g}+\frac{{\alpha}_{a}^{f}}{{\rho}_{a}^{s}}{K}_{a}({\overrightarrow{u}}_{a}^{f}-{\overrightarrow{u}}_{a}^{s})\end{array}$$

## 5. Model Applications on Different Cases

#### 5.1. Model Application I

^{3}; the average particle size is ${d}_{s}=0.3$ mm; the angle of internal friction is 25° ± 0.4°; the roughness of the flume side walls is neglected. The experimental results of the initial sediment phase volume fraction α

^{s}= 0.53 ± 0.005 were selected for validation; the initial moment of the accumulation body is submerged in 0.1 m deep water; fluid density is ${\mathsf{\rho}}^{f}=$1000 kg/m

^{3}; kinematic viscosity coefficient is ${\upsilon}_{o}^{f}=1\times {10}^{-6}$ m

^{2}/s. After the test starts, the baffle that maintains the initial shape of the accumulation body is released instantaneously, and the effect of the baffle is neglected, and the accumulation body starts to collapse and reaches the final stable state within a few seconds.

#### 5.2. Model Application II

#### 5.3. Model Application III

## 6. Discussion

#### 6.1. Summary of the Study

- -
- The initial collapse;
- -
- The mixing of water and sediments released;
- -
- The flow velocity distribution of the upper flow layer, which is significantly larger than that of the lower sediment layer;
- -
- The flow velocities of the two layers when water and sediments are fully mixed.

#### 6.2. Limitations of the Approach

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${\alpha}^{f}$ | volume fraction of water phase | - |

${\alpha}^{s}$ | volume fraction of sand phase | - |

${c}_{0}$ | speed of sound | m/s |

$\rho $ | density | Kg/m^{3} |

${\rho}_{0}$ | SPH particle’s reference density | Kg/m^{3} |

$\Delta t$ | time step size | s |

$\overrightarrow{g}$ | acceleration of gravity | m/s^{2} |

$K$ | drag force coefficient | - |

${m}_{0}{}^{s}$ | sediment mass | Kg |

${m}_{0}{}^{f}$ | fluid mass | Kg |

${P}^{f}$ | pressure of water | N/m^{2} |

$\tilde{{P}^{s}}$ | positive stress of the sediment phase | N/m^{2} |

${S}_{US}$ | reciprocal of Schmidt’s number | - |

t | time | s |

$\overrightarrow{{\tau}^{f}}$ | shear stress of the water flow | N/m^{2} |

$\overrightarrow{{\tau}^{s}}$ | shear stress of the sediment phase | N/m^{2} |

$\overrightarrow{{u}^{f}}$ | velocity of water phase | m/s |

$\overrightarrow{{u}^{s}}$ | velocity of sand phase | m/s |

${\upsilon}_{t}^{f}$ | turbulent viscosity coefficient | m^{2}/s |

$r$ | position vector of SPH particle | m |

$h$ | smoothing length | m |

$W$ | kernel function | - |

${\upsilon}_{o}$ | coefficient of laminar motion viscosity | m^{2}/s |

$\Delta V$ | particle’s volume | m^{3} |

${H}_{o}$ | initial water depth | m |

CFL | Courant–Friedrichs–Lewy coefficient | - |

${d}_{s}$ | Average sediment grain size | m |

## References

- Ivarson, M.M.; Trivedi, C.; Vereide, K. Investigations of Rake and Rib Structures in Sand Traps to Prevent Sediment Transport in Hydropower Plants. Energies
**2021**, 14, 3882. [Google Scholar] [CrossRef] - Morche, D.; Schmidt, K.H. Sediment transport in an alpine river before and after a dambreak flood event. Earth Surf. Process. Landf.
**2011**, 37, 347–353. [Google Scholar] [CrossRef] - Wang, G.; Tian, S.; Hu, B.; Xu, Z.; Chen, J.; Kong, X. Evolution Pattern of Tailings Flow from Dam Failure and the Buffering Effect of Debris Blocking Dams. Water
**2019**, 11, 2388. [Google Scholar] [CrossRef] - Xiong, J.; Tang, C.; Gong, L.; Chen, M.; Li, N.; Shi, Q.; Zhang, X.; Chang, M.; Li, M. How landslide sediments are transferred out of an alpine basin: Evidence from the epicentre of the Wenchuan earthquake. Catena
**2022**, 208, 105781. [Google Scholar] [CrossRef] - Quecedo, M.; Pastor, M.; Herreros, M.I. Numerical modelling of impulse wave generated by fast landslides. Int. J. Numer. Methods Eng.
**2004**, 59, 1633–1656. [Google Scholar] [CrossRef] - Hajigholizadeh, M.; Melesse, A.M.; Fuentes, H.R. Erosion and Sediment Transport Modelling in Shallow Waters: A Review on Approaches, Models and Applications. Int. J. Environ. Res. Public Health
**2018**, 15, 518. [Google Scholar] [CrossRef] [PubMed] - Sun, X.; Zhang, G.; Wang, J.; Li, C.; Wu, S.; Li, Y. Spatiotemporal variation of flash floods in the Hengduan Mountains region affected by rainfall properties and land use. Nat. Hazards
**2022**, 111, 465–488. [Google Scholar] [CrossRef] - Palu, M.C.; Yulien, P.Y. Modeling the Sediment Load of the Doce River after the Fundão Tailings Dam Collapse, Brazil. J. Hydraul. Eng.
**2019**, 145. [Google Scholar] [CrossRef] - Bosa, S.; Petti, M. A numerical model of the wave that overtopped the Vajont Dam in 1963. Water Resour. Manag.
**2013**, 27, 1763–1779. [Google Scholar] [CrossRef] - Gabreil, E.; Wu, H.; Chen, C.; Li, J.; Rubinato, M.; Zheng, X.; Shao, S. Three-dimensional smoothed particle hydrodynamics modeling of near-shore current flows over rough topographic surface. Front. Mar. Sci.
**2022**, 9, 935098. [Google Scholar] - Li, J.; Liu, H.; Gong, K.; Tan, S.K.; Shao, S. SPH modeling of solitary wave fissions over uneven bottoms. Coast. Eng.
**2012**, 60, 261–275. [Google Scholar] [CrossRef] - Shu, A.; Wang, S.; Rubinato, M.; Wang, M.; Qin, J.; Zhu, F. Numerical Modeling of Debris Flows Induced by Dam-Break Using the Smoothed Particle Hydrodynamics (SPH) Method. Appl. Sci.
**2020**, 10, 2954. [Google Scholar] [CrossRef] - Wang, S.; Shu, A.; Rubinato, M.; Wang, M.; Qin, J. Numerical Simulation of Non-Homogeneous Viscous Debris-Flows based on the Smoothed Particle Hydrodynamics (SPH) Method. Water
**2019**, 11, 2314. [Google Scholar] [CrossRef] - Zhang, Y.; Rubinato, M.; Kazemi, E.; Pu, J.H.; Huang, H.; Lin, P. Numerical and experimental analysis of shallow turbulent flows over complex roughness beds. Int. J. Comput. Fluid Dyn.
**2019**, 33, 202–221. [Google Scholar] [CrossRef] - Li, Z.; Dufour, F.; Darve, F. Modelling rainfall-induced mudflows using FEMLIP and a unified hydro-elasto-plastic model with solid-fluid transition. Eur. J. Environ. Civ. Eng.
**2018**, 22, 491–521. [Google Scholar] [CrossRef] - Jakob, M.; McDougall, S.; Weatherly, H.; Ripley, N. Debris-flow simulations on Cheekye River, British Columbia. Landslides
**2013**, 10, 685–699. [Google Scholar] [CrossRef] - Hirt, C.W.; Nichols, B.D. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys.
**1981**, 39, 201–225. [Google Scholar] [CrossRef] - Sethian, J.A. Level Set Methods and Fast Marching Methods: Evolving Interfaces in Geometry Fluid Mechanics, Computer Vision and Materials Science; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Liu, G.R.; Liu, M.B.; Li, S. Smoothed particle hydrodynamics—A meshfree method. Comput. Mech.
**2004**, 33, 491. [Google Scholar] [CrossRef] - Vacondio, R.; Rogers, B.D.; Stansby, P.K.; Mignosa, P. SPH modeling of shallow flow with open boundaries for practical flood simulation. J. Hydraul. Eng.
**2012**, 138, 530–541. [Google Scholar] [CrossRef] - Shao, S.; Edmond, L.Y.M. Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Adv. Water Resour.
**2003**, 26, 787–800. [Google Scholar] [CrossRef] - Hosseini, S.M.; Manzari, M.T.; Hannani, S.K. A fully explicit three-step SPH algorithm for simulation of non-Newtonian fluid flow. Int. J. Numer. Methods Heat Fluid Flow
**2007**, 17, 715–735. [Google Scholar] [CrossRef] - Ran, Q.; Tong, J.; Shao, S.; Fu, X.; Xu, Y. Incompressible SPH scour model for movable bed dam break flows. Adv. Water Resour.
**2015**, 82, 39–50. [Google Scholar] [CrossRef] - Fourtakas, G.; Rogers, B.D. Modelling multi-phase liquid-sediment scour and resuspension induced by rapid flows using Smoothed Particle Hydrodynamics (SPH) accelerated with a Graphics Processing Unit (GPU). Adv. Water Resour.
**2016**, 92, 186–199. [Google Scholar] [CrossRef] - Bui, H.H.; Sako, K.; Fukagawa, R. Numerical simulation of soil–water interaction using smoothed particle hydrodynamics (SPH) method. J. Terramech.
**2007**, 44, 339–346. [Google Scholar] [CrossRef] - Pahar, G.; Dhar, A. Coupled incompressible Smoothed Particle Hydrodynamics model for continuum-based modelling sediment transport. Adv. Water Resour.
**2017**, 102, 84–98. [Google Scholar] [CrossRef] - Ulrich, C.; Leonardi, M.; Rung, T. Multi-physics SPH simulation of complex marine-engineering hydrodynamic problems. Ocean. Eng.
**2013**, 64, 109–121. [Google Scholar] [CrossRef] - Wang, D.; Li, S.; Arikawa, T.; Gen, H. ISPH simulation of scour behind seawall due to continuous tsunami overflow. Coast. Eng. J.
**2016**, 58, 1650014. [Google Scholar] [CrossRef] - Manenti, S.; Sibilla, S.; Gallati, M.; Agate, G. SPH simulation of sediment flushing induced by a rapid water flow. J. Hydraul. Eng.
**2012**, 138, 272–284. [Google Scholar] [CrossRef] - Shi, H.B. Two-phase flow models and their application to sediment transport. Ph.D. Thesis, Tsinghua University, Beijing, China, 2016. (In Chinese). [Google Scholar]
- Shi, H.; Si, P.; Dong, P.; Dong, P.; Yu, X. A two-phase SPH model for massive sediment motion in free surface flows. Adv. Water Resour.
**2019**, 129, 80–98. [Google Scholar] [CrossRef] - Chauchat, J.; Cheng, Z.; Nagel, T.; Bonamy, C.; Hsu, T.J. SedFoam-2.0: A 3-D two-phase flow numerical model for sediment transport. Geosci. Model Dev.
**2017**, 10, 4367–4392. [Google Scholar] [CrossRef] - Lucy, L.B. A numerical approach to the testing of the fission hypothesis. Astrophys. J.
**1977**, 8, 1013–1024. [Google Scholar] [CrossRef] - Gingold, R.A.; Monaghan, J.J. Smoothed particle hydrodynamics: Theory and application to non-spherical stars. Mon. Not. R. Astron. Soc.
**1977**, 181, 375–389. [Google Scholar] [CrossRef] - Monaghan, J.J. Simulating free surface flows with SPH. J. Comput. Phys.
**1994**, 110, 399–406. [Google Scholar] [CrossRef] - Monaghan, J.J.; Kos, A. Solitary waves on a Cretan Beach. J. Waterw. Port Coast. Ocean. Eng.
**1999**, 125, 145–155. [Google Scholar] [CrossRef] - Monaghan, J.J. Smoothed Particle Hydrodynamics. Annu. Rev. Astron. Astrophys.
**1992**, 30, 543–574. [Google Scholar] [CrossRef] - Kazemi, E.; Koll, K.; Tait, S.; Shao, S. SPH modelling of turbulent open channel flow over and within natural gravel beds with rough interfacial boundaries. Adv. Water Resour.
**2020**, 140, 103557. [Google Scholar] [CrossRef] - Vosoughi, F.; Rakhshandehroo, G.R.; Nikoo, M.R.; Sadegh, M. Experimental study and numerical verification of silted-up dam break. J. Hydrol.
**2020**, 590, 125267. [Google Scholar] [CrossRef] - Wang, C.; Wang, Y.; Peng, C.; Meng, X. Two-fluid smoothed particle hydrodynamics simulation of submerged granular column collapse. Mech. Res. Commun.
**2017**, 79, 15–23. [Google Scholar] [CrossRef] - Rondon, L.; Pouliquen, O.; Aussillous, P. Granular collapse in a fluid: Role of the initial volume fraction. Phys. Fluids
**2011**, 23, 73301. [Google Scholar] [CrossRef] - Wang, C.; Wang, Y.; Peng, C.; Meng, X. Dilatancy and compaction effects on the submerged granular column collapse. Phys. Fluids
**2017**, 29, 103307. [Google Scholar] [CrossRef] - Costa, J.E.; Schuster, R.L. The formation and failure of natural dams. Geol. Soc. Am. Bull.
**1988**, 100, 1054–1068. [Google Scholar] [CrossRef] - Rubinato, M.; Luo, M.; Zheng, X.; Pu, J.H.; Shao, S. Advances in Modelling and Prediction on the Impact of Human Activities and Extreme Events on Environments. Water
**2020**, 12, 1768. [Google Scholar] [CrossRef] - Rubinato, M.; Nichols, A.; Peng, Y.; Zhang, J.; Lashford, C.; Cai, Y.; Lin, P.; Tait, S. Urban and river flooding: Comparison of flood risk management approaches in the UK and China and an assessment of future knowledge needs. Water Sci. Eng.
**2019**, 12, 274–283. [Google Scholar] [CrossRef] - Pope, S.B. Turbulent Flows; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]

**Figure 4.**Validation of accumulation body collapse process [42].

**Figure 6.**Simulation results of a dam break of a reservoir silting (sediment phase volume fraction map).

**Figure 7.**Simulation results of a dam break of a reservoir silting (flow velocity distribution map).

**Figure 8.**Dam break validation results of a reservoir silting (experiment, VOF model, and Euler model results from literature Vosoughi et al. [39]). Adapted with permission from Elsevier, Journal of Hydrology, License ID 1383966-1.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zheng, X.; Rubinato, M.; Liu, X.; Ding, Y.; Chen, R.; Kazemi, E.
SPH Simulation of Sediment Movement from Dam Breaks. *Water* **2023**, *15*, 3033.
https://doi.org/10.3390/w15173033

**AMA Style**

Zheng X, Rubinato M, Liu X, Ding Y, Chen R, Kazemi E.
SPH Simulation of Sediment Movement from Dam Breaks. *Water*. 2023; 15(17):3033.
https://doi.org/10.3390/w15173033

**Chicago/Turabian Style**

Zheng, Xiaogang, Matteo Rubinato, Xingnian Liu, Yufei Ding, Ridong Chen, and Ehsan Kazemi.
2023. "SPH Simulation of Sediment Movement from Dam Breaks" *Water* 15, no. 17: 3033.
https://doi.org/10.3390/w15173033