# Simulating the Overtopping Failure of Homogeneous Embankment by a Double-Point Two-Phase MPM

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

**g**is the gravity vector, ${a}_{\mathrm{L}}$ is the liquid phase acceleration, ${a}_{\mathrm{S}}$ is the solid phase acceleration. ${\overline{\mathsf{\sigma}}}_{\mathrm{S}}={\mathsf{\sigma}}^{\prime}-{n}_{\mathrm{S}}{\mathsf{\sigma}}_{\mathrm{L}}$ and ${\overline{\mathsf{\sigma}}}_{\mathrm{L}}={n}_{\mathrm{L}}{\mathsf{\sigma}}_{\mathrm{L}}$ are the partial stresses of solid and liquid phases respectively, ${\mathsf{\sigma}}^{\prime}$ is the effective stress tensor, and ${\mathsf{\sigma}}_{\mathrm{L}}$ is the liquid phase stress tensor. ${f}_{\mathrm{L}}^{\mathrm{d}}$ is the drag force, which represents the water–soil interaction due to the velocity difference between the two phases, it can be calculated by [34]

**D**is the tangent stiffness matrix, ${\mathsf{\sigma}}^{\prime}$ is the effective stress vector, and $\mathsf{\epsilon}$ is the total strain. For the liquid phase, the volumetric stresses is updated by

_{max}is used to distinguish the two aforementioned states. When the soil porosity is less than the maximum porosity (${n}_{\mathrm{L}}=1-{n}_{\mathrm{S}}<{n}_{\mathrm{max}}$), the mean effective stress decreases as the porosity increases and the mean effective stress vanishes once the grains are not in contact. When the porosity is larger than n

_{max}, fluidization occurs.

## 3. Numerical Examples

#### 3.1. Flow through a Porous Block

#### 3.2. Overtopping Failure of Homogeneous Embankments

_{max}was used to distinguish the saturated soils as a solid-like or liquid-like state. When the porosity was larger than n

_{max}, fluidization occurred. The grains are not in contact and float together with the liquid phase. Here, three cases of different porosities (n

_{max}= 0.45, 0.50, 0.55) were conducted to study the effect of maximum porosity on the embankment erosion process, and the other conditions were the same as in the previous simulations. The calculated embankment profiles for different cases at t = 1.4 s and t = 3.0 s are shown in Figure 16. Results show that the maximum porosity plays an important role on the dam failure due to overtopping flow. The larger the maximum porosity is, the more difficult it is for soil particles to reach the fluidization state. Hence, it is very important to get the right maximum porosity in the simulations, and more accurate simulation results can only be obtained with the appropriate value.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The schematic model for the double-point two-phase material point method (MPM) [32].

**Figure 2.**Schematic diagram for soil behavior [34].

**Figure 5.**Comparison of free surface profile for flow through porous block: Simulation (▪) and experiment (⚬).

**Figure 6.**Experimental setup [12] and MPM model for the embankment overtopping problem.

**Figure 7.**Snapshots of the simulated water surface and the embankment profiles along the channel (H = 0.2 m).

**Figure 8.**Comparisons of the free water surface and embankment profiles between the simulation and experiment (H = 0.2 m).

**Figure 9.**Snapshots of the simulated water surface and the embankment profiles along the channel (H = 0.25 m).

**Figure 10.**Comparisons of the free water surface and embankment profiles between the simulation and experiment (H = 0.25 m).

**Figure 11.**Snapshots of the simulated water surface and the embankment profiles along the channel (H = 0.395 m).

**Figure 12.**Comparisons of the free water surface and embankment profiles between the simulation and experiment (H = 0.395 m).

Material | Parameter | Value |
---|---|---|

Porous block | Density (kg/m^{3}) | 2700 |

Young modulus (kPa) | 1000 | |

Poisson’s ratio | 0.3 | |

Initial porosity | 0.49 | |

Mean diameter (mm) | 15.9 | |

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

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

Dynamic viscosity (kPa·s) | 1 × 10^{−6} |

Material | Parameter | Value |
---|---|---|

Sand | Density (kg/m^{3}) | 2680 |

Young modulus (kPa) | 1000 | |

Poisson’s ratio | 0.3 | |

Internal friction angle (°) | 36 | |

Cohesion (kN) | 0 | |

Initial porosity Maximum porosity | 0.40 0.45 | |

Mean diameter (mm) | 0.2 | |

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

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

Dynamic viscosity (kPa·s) | 1 × 10^{−6} |

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**MDPI and ACS Style**

Yang, Y.-S.; Yang, T.-T.; Qiu, L.-C.; Han, Y.
Simulating the Overtopping Failure of Homogeneous Embankment by a Double-Point Two-Phase MPM. *Water* **2019**, *11*, 1636.
https://doi.org/10.3390/w11081636

**AMA Style**

Yang Y-S, Yang T-T, Qiu L-C, Han Y.
Simulating the Overtopping Failure of Homogeneous Embankment by a Double-Point Two-Phase MPM. *Water*. 2019; 11(8):1636.
https://doi.org/10.3390/w11081636

**Chicago/Turabian Style**

Yang, Yong-Sen, Ting-Ting Yang, Liu-Chao Qiu, and Yu Han.
2019. "Simulating the Overtopping Failure of Homogeneous Embankment by a Double-Point Two-Phase MPM" *Water* 11, no. 8: 1636.
https://doi.org/10.3390/w11081636