# 2D Numerical Study of the Stability of Trench under Wave Action in the Immersing Process of Tunnel Element

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

#### 2.1. Fluid Dynamic Model

#### 2.2. Seabed Model

#### 2.3. Tunnel Model

#### 2.4. Boundary Conditions

#### 2.4.1. Seabed Boundary Conditions

#### 2.4.2. Tunnel Boundary Conditions

#### 2.4.3. Free Water Surface Boundary Conditions

#### 2.5. Integration of Fluid Dynamic Model and Seabed Model

## 3. Model Validation and Numerical Results

#### 3.1. Model Validation

#### 3.2. Consolidation of the Seabed

#### 3.3. Dynamic Responses of the Seabed

#### 3.4. Wave-Induced Liquefaction

^{4}Pa, smaller than the 2.01 × 10

^{4}Pa in case (a). This may be due to a certain resistance to the wave propagation from the tunnel.

#### 3.5. Wave-Induced Shear Failure

#### 3.6. Influence of Wave Characters on Liquefaction

#### 3.7. Influence of Slope Rate on Liquefaction

## 4. Conclusions

- (1)
- Due to the existence of the trench, the pore pressure amplitude on the weather side slope is significantly smaller than that on the lee side slope.
- (2)
- The maximum depth of liquefaction in the case after the tunnel element is placed is smaller than that after the foundation groove is excavated.
- (3)
- Due to the existence of the tunnel structure, the distribution of the flow field and pressure field change dramatically; thus, the dynamic responses and the failure area in the seabed change accordingly.
- (4)
- In the case of the specific wave and seabed parameters, the liquefaction characteristics in the trench have an obvious fold point with the change of slope rate. That means that there is an optimal slope rate to minimize the failure possibility of the slope. Moreover, the specific failure mode deserves further research.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**The comparison of the water surface elevation in the present wave model and analytical solution.

**Figure 4.**The vertical distributions of the maximum pore pressure $\left({p}_{s}/{p}_{0}\right)$ versus the seabed depth $\left(z/h\right)$.

**Figure 7.**The distribution of (

**a**) ${\sigma}_{x}/{p}_{0}$, (

**b**) ${\sigma}_{y}/{p}_{0}$, (

**c**) ${\tau}_{xz}/{p}_{0}$, (

**d**) ${p}_{s}/{p}_{0}$ in two cases: after the foundation groove is excavated (t = 29 s) and after the tunnel element is placed (t = 29 s). (${p}_{0}$ is the water pressure at the seabed surface.)

**Figure 8.**The pore pressure at two asymmetric points on the two-side slopes: (

**a**) without immersed tunnel and (

**b**) with immersed tunnel.

**Figure 9.**The pore pressure at 1 m below the positions of two bottom corners of the tunnel: (

**a**) without immersed tunnel and (

**b**) with immersed tunnel.

**Figure 10.**The liquefied area in the trench in two cases: (

**a**) after the foundation groove is excavated (t = 36 s) and (

**b**) after the tunnel element is placed (t = 36 s).

**Figure 11.**The shear failure area in the trench in two cases: (

**a**) after the foundation groove is excavated (t = 36 s) and (

**b**) after the tunnel element is placed (t = 36 s).

**Figure 13.**The liquefied conditions around the trench with different slope angles: (

**a**) depth of liquefied seabed, (

**b**) area of liquefied seabed, (

**c**) width of liquefied seabed.

shear modulus ($G$) | 1.27 × 10^{7} N/m^{2} |

poison’s ratio ($\upsilon $) | 0.3 |

soil permeability (${k}_{f}$) | 1.8 × 10^{−4} m/s |

soil porosity ($n$) | 0.425 |

saturation degree (${S}_{r}$) | 0.995 |

seabed thickness ($h$) | 1.8 m |

Wave Parameters | Value | Unit |
---|---|---|

wave height ($H$) | 2 | m |

wave period ($T$) | 8 | s |

wave length ($L$) | 83.4 | m |

water depth ($d$) | 16 | m |

Soil Parameters | ||

seabed thickness ($h$) | 30.5 | m |

shear modulus ($G$) | 5 × 10^{6} | N/m^{2} |

soil porosity ($n$) | 0.45 | - |

poison’s ratio ($\upsilon $) | 0.27 | - |

elastic modulus ($E$) | 3 × 10^{7} | N/m^{2} |

soil permeability (${k}_{f}$) | 10^{−6} | m/s |

saturation degree (${S}_{r}$) | 0.975 | - |

density of soil grain (${\rho}_{s}$) | 2650 | kg/m^{3} |

internal cohesion ($c$) | 0 | kPa |

internal friction angle ($\varphi $) | 30 | deg |

Water Parameters | ||

shear modulus ($G$) | 2 × 10^{9} | N/m^{2} |

density of water (${\rho}_{w}$) | 986 | kg/m^{3} |

Tunnel Parameters | ||

elastic modulus (${E}_{t}$) | 3.5 × 10^{10} | N/m^{2} |

poison’s ratio (${\upsilon}_{t}$) | 0.18 | - |

density of tunnel (${\rho}_{t}$) | 2700 | kg/m^{3} |

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**MDPI and ACS Style**

Chen, W.-Y.; Liu, C.-L.; Duan, L.-L.; Qiu, H.-M.; Wang, Z.-H.
2D Numerical Study of the Stability of Trench under Wave Action in the Immersing Process of Tunnel Element. *J. Mar. Sci. Eng.* **2019**, *7*, 57.
https://doi.org/10.3390/jmse7030057

**AMA Style**

Chen W-Y, Liu C-L, Duan L-L, Qiu H-M, Wang Z-H.
2D Numerical Study of the Stability of Trench under Wave Action in the Immersing Process of Tunnel Element. *Journal of Marine Science and Engineering*. 2019; 7(3):57.
https://doi.org/10.3390/jmse7030057

**Chicago/Turabian Style**

Chen, Wei-Yun, Cheng-Lin Liu, Lun-Liang Duan, Hao-Miao Qiu, and Zhi-Hua Wang.
2019. "2D Numerical Study of the Stability of Trench under Wave Action in the Immersing Process of Tunnel Element" *Journal of Marine Science and Engineering* 7, no. 3: 57.
https://doi.org/10.3390/jmse7030057