# Two-Phase MPM Simulation of Surge Waves Generated by a Granular Landslide on an Erodible Slope

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Governing Equations

#### 2.1. Mass Balance Equations

#### 2.2. Momentum Balance Equations

**g**is the gravity acceleration vector, ${f}_{sw}$ is the force vector per unit volume due to soil–water interaction and can be defined by [39]

#### 2.3. Constitutive Models

**I**is identity matrix, $\mu $ is the dynamic viscosity of water, and ${\dot{\mathsf{\epsilon}}}_{w}$ is the strain rate tensor. The pressure is updated using Equation (8), the strain rate is calculated by

_{max}in this paper. When the porosity n is less than the maximum porosity n

_{max}, the soil–water mixture behaves like a saturated soil and the foresaid elastoplastic constitutive model used to model the solid–like response. When the porosity n is beyond the maximum porosity n

_{max}, the soil grains separate from each other and the effective stress vanished, the liquid–like behavior of soil–water mixture was modeled as weakly compressible fluid.

## 3. Numerical Examples

#### 3.1. Topsoil Erosion by Granular Landslides

^{4}kPa, Poisson ratio ν = 0.3, granular density ${\rho}_{s}=1500$ kg/m

^{3}, average granular diameter D

_{p}= 2 mm, internal friction angle ${\phi}^{\prime}$ = 25°, expansion angle $\psi $ = 0°, initial porosity n

_{0}= 0.3, cohesive force ${c}^{\prime}$ = 0 kPa. Similarly, the erodible slope was simulated using the Mohr–Coulomb intrinsic model with the following material properties: elasticity modulus E = 2.0 × 10

^{4}kPa, Poisson ratio ν = 0.3, the density of granules on the slope ${\rho}_{s}$ = 1800 kg/m

^{3}, average granular diameter D

_{p}= 2 mm, internal friction angle ${\phi}^{\prime}$ = 23°, expansion angle $\psi $ = 0°, initial porosity n

_{0}= 0.3, cohesive force ${c}^{\prime}$ = 3 kPa.

#### 3.2. Surge Waves by Dry Granules Sliding on a Rigid Slope

_{0}= 0.6. The bottom of the device is filled with water, and the initial water depth h

_{0}= 15 cm. When the gate is opened, the granular body will begin to slide along the slope and fall into the water. The height of the wave generated by the granular body into the water is measured by a wave height meter at two different locations, they are 0.45 m and 0.75 m from the horizontal distance of the gate, respectively.

^{3}, mean diameter D

_{p}= 1.5 mm, internal friction angle ${\phi}^{\prime}$= 15°, expansion angle $\psi $ = 0°, initial porosity n

_{0}= 0.4. Water is considered to be a weakly compressible fluid with a material characteristic density of ${\rho}_{w}$ = 1000 kg/m

^{3}, a dynamic viscosity ${\mu}_{w}$ = 1.00 × 10

^{−6}kPa·s and bulk modulus K

_{w}= 2.15 × 10

^{4}kPa.

^{−5}s and a total duration of 1.2 s.

_{s}= 1200, 1500, 1900, 2250, and 2500 kg/m

^{3}) are used in the simulation, and other conditions remain unchanged. Figure 11 displays the change in free water surface height over time at locations X1 and X2 for different densities and displays the correlation of sand density. The numerical results show that the impact of density on the water wave is significant. The wave height increases with the density. Moreover, the crest height linearly increases with the sand density.

#### 3.3. Surge Waves by Dry and Saturated Granules Sliding on Erodible Slope

^{−5}s for a total duration of 2.2 s.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Computational model of a granular landslide sliding along an erodible slope: (

**a**) Schematic of the initial model; (

**b**) MPM discretization.

**Figure 6.**Calculation model for wave generation from dry granules into water along a slope: (

**a**) Diagram of the initial form; (

**b**) MPM discretization.

**Figure 7.**Morphology of water waves generated by dry granular bodies entering the water at different moments.

**Figure 9.**Comparison of wave height test results with numerical results at two different locations: (

**a**) X1 = −0.3 m; (

**b**) X2 = −0.6 m.

**Figure 10.**Influence of internal friction angle of sliding sand: time history curve of the free water surface at two different locations: (

**a**) X1 = −0.3 m; (

**b**) X2 = −0.6 m.

**Figure 11.**Influence of density of sliding sand: time history curve of the free water surface at two different locations: (

**a**) X1 = −0.3 m; (

**b**) X2 = −0.6 m.

**Figure 12.**Influence of elasticity modulus of sliding sand: time history curve of the free water surface at two different locations: (

**a**) X1 = −0.3 m; (

**b**) X2 = −0.6 m.

**Figure 13.**Influence of Poisson ratio of sliding sand: time history curve of the free water surface at two different locations: (

**a**) X1 = −0.3 m; (

**b**) X2 = −0.6 m.

**Figure 14.**Influence of dilatancy angle of sliding sand: time history curve of the free water surface at two different locations: (

**a**) X1 = −0.3 m; (

**b**) X2 = −0.6 m.

**Figure 15.**Schematic diagram of a computational model for water waves generated by dry particles sliding along a rigid slope: (

**a**) Schematic of the initial model; (

**b**) MPM discretization.

**Figure 16.**Snapshots of the landslide process of dry particles at different moments on (

**a**) rigid slopes; and (

**b**) erodible slopes.

**Figure 17.**Snapshots of the landslide process of saturated particles at different moments on (

**a**) rigid slopes; and (

**b**) erodible slopes.

**Figure 18.**Temporal evolution of wave heights generated by granular landslides at the observation point (X = 0 m) under different conditions.

**Figure 19.**Temporal evolution of leading wave heights generated by granular landslides under different conditions.

Case | Slope Type | Elevation of Water Surface (m) | Tank Bottom Length (m) | The Dimensions of a Triangular Deposit (cm) | Slope Angle (°) |
---|---|---|---|---|---|

1 | Rigid | 0.15 | 2.9 | 14.4 × 38 (dry/saturated) | 22 |

2 | Erodible | 0.15 | 2.9 | 14.4 × 38 (dry/saturated) | 22 |

Material | Parameter | Numerical Values |
---|---|---|

Landslides | Density (kg/m^{3}) | 1900 |

Modulus of elasticity (kPa) | 1.0 × 10^{4} | |

Poisson ratio | 0.3 | |

Internal friction angle (°) | 15 | |

Expansion angle (°) | 0 | |

Cohesion (kPa) | 0 | |

Initial porosity | 0.4 | |

Maximum porosity | 0.5 | |

Mean diameter (mm) | 2 | |

Rigid slope | Density (kg/m^{3}) | 1900 |

Modulus of elasticity (kPa) | 1.0 × 10^{4} | |

Poisson ratio | 0.3 | |

Erodible slope | Density (kg/m^{3}) | 1900 |

Modulus of elasticity (kPa) | 1.0 × 10^{4} | |

Poisson ratio | 0.3 | |

Expansion angle (°) | 0 | |

Internal friction angle (°) | 15 | |

Cohesion (kPa) | 0.1 | |

Water | Density (kg/m^{3}) | 1000 |

Bulk modulus (kPa) | 2.15 × 10^{4} | |

Dynamic viscosity (kPa·s) | 1.00 × 10^{−6} |

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**MDPI and ACS Style**

Zhao, K.-L.; Qiu, L.-C.; Yuan, T.-J.; Wang, Y.; Liu, Y.
Two-Phase MPM Simulation of Surge Waves Generated by a Granular Landslide on an Erodible Slope. *Water* **2023**, *15*, 1307.
https://doi.org/10.3390/w15071307

**AMA Style**

Zhao K-L, Qiu L-C, Yuan T-J, Wang Y, Liu Y.
Two-Phase MPM Simulation of Surge Waves Generated by a Granular Landslide on an Erodible Slope. *Water*. 2023; 15(7):1307.
https://doi.org/10.3390/w15071307

**Chicago/Turabian Style**

Zhao, Kai-Li, Liu-Chao Qiu, Tang-Jin Yuan, Yang Wang, and Yi Liu.
2023. "Two-Phase MPM Simulation of Surge Waves Generated by a Granular Landslide on an Erodible Slope" *Water* 15, no. 7: 1307.
https://doi.org/10.3390/w15071307