# On the Slope Stability of the Submerged Trench of the Immersed Tunnel Subjected to Solitary Wave

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Formulations

_{1}is the thickness of soil layer 1, h

_{2}is the thickness of soil layer 2, D is the relative distance between the wave crest and the slope top of the left slope, H is the height of the solitary wave, W is the width of the trench, B is the height of the trench, d is the water depth, and U

_{0}is the initial current velocity.

#### 2.1. Wave–Current Sub-Model

#### 2.1.1. Continuity Equations and Momentum Equations

#### 2.1.2. Turbulence Models

#### 2.1.3. Boundary Conditions for Solitary Wave Generation

#### 2.2. Seabed Sub-Model

#### 2.2.1. Seepage Pressure

#### 2.2.2. Strength Reduction Method for the Seabed

_{1}is the first invariant stress tensor, J

_{2}is the second invariant deviatoric stress tensor. m, ${\alpha}_{0}$ and ${k}_{0}$ are the parameters related to soil material parameters:

#### 2.3. Boundary Conditions

#### 2.4. Integration of Sub-Models

#### 2.5. Convergence of the FEM Meshes

_{ec}| is the maximum pore pressure, and σ′

_{0}is the initial effective stress at point A. To achieve the computational accuracy, a mesh number with the smallest standard deviation was selected. The FEM mesh adopted in the computation for the seabed in the vicinity of the foundation trench is shown in Figure 3. The mesh refinement near the foundation trench was adopted to achieve satisfactory calculation.

## 3. Model Validation

## 4. Results and Discussion

#### 4.1. Consolidation of the Seabed

#### 4.2. Stability Index for the One-Stage Slope under Solitary Wave Loading

_{min}) was regarded as the stability index for the slope with the corresponding slope ratio. With the decrease of slope ratio, the FOS

_{min}increased as expected. It was observed that FOS

_{min}was bigger than 1 when the slope ratio was 1:2.5 in this case. Therefore, when the slope ratio was smaller than 1:2.5, the slope was stable under the combined actions of self-weight, wave pressure, and induced seepage force in the seabed in this study. With the decrease of slope ratio, the FOS

_{min}increased as expected.

#### 4.3. Influence of Soil Strength Parameters on the Slope Stability

_{min}, are also shown in Figure 7. As expected, the stability of slope increases with the increase of soil strength.

#### 4.4. Influence of the Slope Ratio on Slope Stability

#### 4.5. Influence of Current Direction on Slope Stability

_{min}in Figure 10a,d were smaller than those in Figure 10b,c. This phenomenon may be attributed to the fact that, when the currents propagated away from the slope surface, the current-induced pressure acting on the slope surface may have reduced the stability of slope.

#### 4.6. Influence of Slope Ratio on Two-Stage Slope

_{min}with the slope ratio of upper slope. It could be noted that FOS

_{min}increased slightly as the upper slope ratio increased. Thus, it was concluded that the slope ratio of the lower slope had more significant influence on the stability of the whole slope compared with the upper slope ratio.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Liu, C.; Cui, J.; Zhang, Z.; Liu, H.; Huang, X.; Zhang, C. The role of TBM asymmetric tail-grouting on surface settlement in coarse-grained soils of urban area: Field tests and FEA modelling. Tunn. Undergr. Space Technol.
**2021**, 111, 103857. [Google Scholar] [CrossRef] - Chen, W.Y.; Wang, Z.H.; Chen, G.X.; Jeng, D.S.; Wu, M.; Zhao, H.Y. Effect of vertical seismic motion on the dynamic response and instantaneous liquefaction in a two-layer porous seabed. Comput. Geotech.
**2018**, 99, 165–176. [Google Scholar] [CrossRef] [Green Version] - Chen, W.Y.; Huang, Y.; Wang, Z.H.; He, R.; Chen, G.X.; Li, X.J. Horizontal and vertical motion at surface of a gassy ocean sediment layer induced by obliquely incident SV waves. Eng. Geol.
**2017**, 227, 43–53. [Google Scholar] [CrossRef] - Chen, G.X.; Ruan, B.; Zhao, K.; Chen, W.Y.; Zhuang, H.Y.; Du, X.L.; Khoshnevisan, S.; Juang, C.H. Nonlinear Response Characteristics of Undersea Shield Tunnel Subjected to Strong Earthquake Motions. J. Earthq. Eng.
**2020**, 24, 351–380. [Google Scholar] [CrossRef] - Xu, L.Y.; Song, C.X.; Chen, W.Y.; Cai, F.; Li, Y.Y.; Chen, G.X. Liquefaction-induced settlement of the pile group under vertical and horizontal ground motions. Soil Dyn. Earthq. Eng.
**2021**, 144, 106709. [Google Scholar] [CrossRef] - Henkel, D. The Role of Waves in Causing Submarine Landslides. Geotechnique
**1970**, 20, 75–80. [Google Scholar] [CrossRef] - Bubel, J.; Grabe, J. Stability of Submarine Foundation Pits Under Wave Loads. In Proceedings of the Asme International Conference on Ocean, Rio de Janeiro, Brazil, 1–6 July 2012; p. 11. [Google Scholar]
- Jeng, D.S. Wave-induced liquefaction potential at the tip of a breakwater: An analytical solution. Appl. Ocean Res.
**1996**, 18, 229–241. [Google Scholar] [CrossRef] - Madsen, O.S. Wave-induced pore pressures and effective stresses in a porous bed. Géotechnique
**1978**, 28, 377–393. [Google Scholar] [CrossRef] - Yamamoto, T. Wave-induced pore pressures and effective stresses in inhomogeneous seabed foundations. Ocean Eng.
**1981**, 8, 1–16. [Google Scholar] [CrossRef] - Mei, C.C.; Foda, M.A. Wave-induced responses in a fluid-filled poro-elastic solid with a free surface—a boundary layer theory. Geophys. J. Int.
**1981**, 66, 597–631. [Google Scholar] [CrossRef] - Liu, B.; Jeng, D.S.; Ye, G.L.; Yang, B. Laboratory study for pore pressures in sandy deposit under wave loading. Ocean Eng.
**2015**, 106, 207–219. [Google Scholar] [CrossRef] - Zhang, C.; Titi, S.; Zheng, J.H.; Xie, M.X.; Nguyen, V.T. Modelling wave-induced 3D non-homogeneous seabed response. Appl. Ocean Res.
**2016**, 61, 101–114. [Google Scholar] [CrossRef] - Lin, Z.B.; Guo, Y.K.; Jeng, D.S.; Liao, C.C.; Rey, N. An integrated numerical model for wave–soil–pipeline interactions. Coast. Eng.
**2016**, 108, 25–35. [Google Scholar] [CrossRef] [Green Version] - Lin, Z.B.; Pokrajac, D.; Guo, Y.K.; Jeng, D.S.; Tang, T.; Rey, N.; Zheng, J.H.; Zhang, J.S. Investigation of nonlinear wave-induced seabed response around mono-pile foundation. Coast. Eng.
**2017**, 121, 197–211. [Google Scholar] [CrossRef] [Green Version] - Duan, L.L.; Liao, C.C.; Jeng, D.S.; Chen, L.Y. 2D numerical study of wave and current-induced oscillatory non-cohesive soil liquefaction around a partially buried pipeline in a trench. Ocean Eng.
**2017**, 135, 39–51. [Google Scholar] [CrossRef] [Green Version] - Zhai, Y.Y.; He, R.; Zhao, J.L.; Zhang, J.S.; Jeng, D.S.; Li, L. Physical Model of wave-induced seabed response around trenched pipeline in sandy seabed. Appl. Ocean Res.
**2018**, 75, 37–52. [Google Scholar] [CrossRef] [Green Version] - Zhang, Q.; Zhou, X.L.; Wang, J.H.; Guo, J.J. Wave-induced seabed response around an offshore pile foundation platform. Ocean Eng.
**2017**, 130, 567–582. [Google Scholar] [CrossRef] - Zhao, H.Y.; Zhu, J.F.; Zheng, J.H.; Zhang, J.S. Numerical modelling of the fluid–seabed-structure interactions considering the impact of principal stress axes rotations. Soil Dyn. Earthq. Eng.
**2020**, 136, 106242. [Google Scholar] [CrossRef] - Hu, Z.N.; Xie, Y.L.; Wang, J. Challenges and strategies involved in designing and constructing a 6 km immersed tunnel: A case study of the Hong Kong–Zhuhai–Macao Bridge. Tunn. Undergr. Space Technol.
**2015**, 50, 171–177. [Google Scholar] [CrossRef] - Lara, J.L.; Losada, I.J.; Maza, M.; Guanche, R. Breaking solitary wave evolution over a porous underwater step. Coast. Eng.
**2011**, 58, 837–850. [Google Scholar] [CrossRef] - Synolakis, C.E.; Bernard, E.N. Tsunami science before and beyond Boxing Day 2004. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci.
**2006**, 364, 2231–2265. [Google Scholar] [CrossRef] [PubMed] - Hsiao, S.C.; Lin, T.C. Tsunami-like solitary waves impinging and overtopping an impermeable seawall: Experiment and RANS modeling. Coast. Eng.
**2010**, 57, 1–18. [Google Scholar] [CrossRef] - Synolakis, C.E. The Runup of Long Waves. Ph.D. Thesis, California Institute of Technology, Pasadena, CA, USA, 1986. [Google Scholar]
- Sumer, B.M.; Sen, M.B.; Karagali, I.; Ceren, B.; Fredsøe, J.; Sottile, M.; Zilioli, L.; Fuhrman, D.R. Flow and sediment transport induced by a plunging solitary wave. J. Geophys. Res.
**2011**, 116, C1008. [Google Scholar] [CrossRef] [Green Version] - Young, Y.L.; White, J.A.; Xiao, H.; Borja, R.I. Liquefaction potential of coastal slopes induced by solitary waves. Acta Geotech.
**2009**, 4, 17–34. [Google Scholar] [CrossRef] [Green Version] - Xiao, H.; Young, Y.L.; Prévost, J.H. Parametric study of breaking solitary wave induced liquefaction of coastal sandyslopes. Ocean Eng.
**2010**, 37, 1546–1553. [Google Scholar] [CrossRef] - Nichols, B.D.; Hirt, C.W.; Hotchkiss, R.S. SOLA-VOF: A Solution Algorithm for Transient Fluid Flow with Multiple Free Boundaries; Technology Report LA-8355; Los Alamos Scientific Lab.: Los Alamos, NM, USA, 1980; Volume 12, p. 39.
- 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] - Hirt, C.; Sicilian, J. A Porosity Technique for the Definition of Obstacles in Rectangular Cell Meshes. In Proceedings of the 4th International Conference on Numerical Ship Hydrodynamics, Washington, DC, USA, 1 January 1985; p. 19. [Google Scholar]
- Launder, B.E.; Spalding, D.B. The numerical computation of turbulent flows. Comput. Method Appl. Mech. Eng.
**1974**, 3, 269–289. [Google Scholar] [CrossRef] - Harlow, F.H.; Nakayama, P.I. Turbulence Transport Equations. Phys. Fluids
**1967**, 10, 2323–2332. [Google Scholar] [CrossRef] - Franke, R.; Rodi, W. Calculation of Vortex Shedding Past a Square Cylinder with Various Turbulence Models, Berlin, Heidelberg, 1 January 1993; Durst, F., Friedrich, R., Launder, B.E., Schmidt, F.W., Schumann, U., Whitelaw, J.H., Eds.; Springer: Berlin/Heidelberg, Germany, 1993; pp. 189–204. [Google Scholar]
- McCowan, J. VII. On the solitary wave. Lond. Edinb. Dublin Philos. Mag. J. Sci.
**1891**, 32, 45–58. [Google Scholar] [CrossRef] [Green Version] - Boussinesq, J.V. Théorie de l’intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire. C. R. Hebd. Séances Acad. Sci.
**1871**, 72, 755–759. [Google Scholar] - Munk, W.H. The solitary wave theory and its application to surf problems. Ann. N. Y. Acad. Sci.
**1949**, 51, 376–424. [Google Scholar] [CrossRef] - Matsui, T.; San, K. Finite Element Slope Stability Analysis by Shear Strength Reduction Technique. Soils Found.
**1992**, 32, 59–70. [Google Scholar] [CrossRef] [Green Version] - Chen, W.Y.; Liu, C.L.; Li, Y.; Chen, G.X.; Yu, J. An integrated numerical model for the stability of artificial submarine slope under wave load. Coast. Eng.
**2020**, 158, 103698. [Google Scholar] [CrossRef] - Cheng, Y.M.; Länsivaara, T.; Wei, W.B. Two-dimensional slope stability analysis by limit equilibrium and strength reduction methods. Comput. Geotech.
**2007**, 34, 137–150. [Google Scholar] [CrossRef]

**Figure 2.**Variation of the wave induced maximum excess pore pressure (|u

_{ec}|/σ′

_{0}) (at point A in Figure 1) with the mesh number N.

**Figure 4.**Comparison of wave surface profile between the proposed model and the experimental data [23]: (

**a**) t = 10 s, (

**b**) t = 20 s, (

**c**) t = 25 s, (

**d**) t = 30 s.

**Figure 5.**Initial state of the trench after excavation: (

**a**) horizontal displacement, (

**b**) vertical displacement, (

**c**) stress state.

**Figure 7.**Destruction areas of the slope with different soil parameters: (

**a**) c′ = 15 kPa, 20 kPa, 25 kPa, 30 kPa; (

**b**) φ′ = 12°, 16°, 20°, 24°.

**Figure 8.**Plastic penetration zones in the slope with different slope ratios: (

**a**) slope ratio 1:2.5, (

**b**) slope ratio 1:3, (

**c**) slope ratio 1:3.5, (

**d**) slope ratio 1:4.

**Figure 9.**Sliding displacements in the slope with different slope ratios (scale factor is 20): (

**a**) slope ratio 1:2.5, (

**b**) slope ratio 1:3, (

**c**) slope ratio 1:3.5, (

**d**) slope ratio 1:4.

**Figure 10.**Deformation of the trench slopes under the currents in different directions: (

**a**) left slope, 1 m/s; (

**b**) right slope, 1 m/s; (

**c**) left slope, −1 m/s; (

**d**) right slope, −1 m/s.

**Figure 11.**Variation of FOS

_{min}with the slope ratio of upper slope for various lower slope ratios.

Parameters | Characteristics | Value | Unit |
---|---|---|---|

Wave Parameters | Wave height (H) | 3 | m |

Water depth (d) | 10 | m | |

Soil Parameters | Seabed thickness (h) | 40.5 | m |

Shear modulus (G) | 6.56 × 10^{6} | Pa | |

Soil porosity (n) | 0.41 | - | |

Poison’s Ratio (μ) | 0.35 | - | |

Elastic modulus (E) | 1.77 × 10^{7} | Pa | |

Soil permeability (k) | 8 × 10^{−6} | m/s | |

Density of soil grain (ρ_{s}) | 2.71 × 10^{3} | kg/m^{3} | |

Effective cohesion (c′) | 15 | kPa | |

Effective internal friction angle (φ′) | 20 | ° | |

Trench width (W) | 34 | m | |

Trench height (B) | 18 | m | |

Water parameters | Bulk modulus (K_{w}) | 2 × 10^{9} | Pa |

Density of water (ρ_{w}) | 1000 | kg/m^{3} |

Parameters | Characteristics | Value | Unit |
---|---|---|---|

Soil Parameters in Upper Slope | Seabed thickness (h_{1}) | 8 | m |

Shear modulus (G_{1}) | 4.33 × 10^{6} | Pa | |

Soil porosity (n_{1}) | 0.56 | - | |

Poison’s ratio (μ_{1}) | 0.35 | - | |

Elastic modulus (E_{1}) | 1.17 × 10^{7} | Pa | |

Soil permeability (k_{1}) | 1 × 10^{−9} | m/s | |

Density of soil grain (ρ_{s1}) | 2.75 × 10^{3} | kg/m^{3} | |

Effective cohesion (c_{1}′) | 12 | kPa | |

Effective internal friction angle (φ_{1}′) | 13 | ||

Soil Parameters in Lower Slope | Seabed thickness (h_{2}) | 32.5 | m |

Shear modulus (G_{2}) | 6.56 × 10^{6} | Pa | |

Soil porosity (n_{2}) | 0.41 | - | |

Poison’s ratio (μ_{2}) | 0.35 | - | |

Elastic modulus (E_{2}) | 1.77 × 10^{7} | Pa | |

Soil permeability (k_{2}) | 8 × 10^{−6} | m/s | |

Density of soil grain (ρ_{s2}) | 2.71 × 10^{3} | kg/m^{3} | |

Effective cohesion(c_{2}′) | 15 | kPa | |

Effective internal friction angle (φ_{2}′) | 20 |

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**MDPI and ACS Style**

Chen, W.; Wang, D.; Xu, L.; Lv, Z.; Wang, Z.; Gao, H.
On the Slope Stability of the Submerged Trench of the Immersed Tunnel Subjected to Solitary Wave. *J. Mar. Sci. Eng.* **2021**, *9*, 526.
https://doi.org/10.3390/jmse9050526

**AMA Style**

Chen W, Wang D, Xu L, Lv Z, Wang Z, Gao H.
On the Slope Stability of the Submerged Trench of the Immersed Tunnel Subjected to Solitary Wave. *Journal of Marine Science and Engineering*. 2021; 9(5):526.
https://doi.org/10.3390/jmse9050526

**Chicago/Turabian Style**

Chen, Weiyun, Dan Wang, Lingyu Xu, Zhenyu Lv, Zhihua Wang, and Hongmei Gao.
2021. "On the Slope Stability of the Submerged Trench of the Immersed Tunnel Subjected to Solitary Wave" *Journal of Marine Science and Engineering* 9, no. 5: 526.
https://doi.org/10.3390/jmse9050526