# Wave (Current)-Induced Pore Pressure in Offshore Deposits: A Coupled Finite Element Model

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## Abstract

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## 1. Introduction

## 2. Theory and Methods

#### 2.1. Wave Module

#### 2.2. Seabed Module

#### 2.2.1. Oscillatory Response of Soil

#### 2.2.2. Residual Response of Soil

#### 2.3. Coupling Method

#### 2.3.1. Time Scheme

#### 2.3.2. Mesh Scheme

#### 2.3.3. Boundary Conditions

#### 2.3.4. Coupled Process

## 3. Model Validation

#### 3.1. Wave Verification: Comparison with an Analytical Solution

#### 3.2. Seabed Verification: Comparison with Experimental Data

#### 3.2.1. Validation of the Oscillatory Pore Pressure

#### 3.2.2. Validation of the Residual Pore Pressure

#### 3.3. Wave-Seabed Interaction Verification

_{0}= −0.1, 0, and +0.1 m/s. The properties of the soil provided in their paper were: Shear modulus, $G=1\times {10}^{7}{\mathrm{N}/\mathrm{m}}^{2}$; Poisson’s ratio,$\mu =0.3$; permeability, $K=1.88\times {10}^{4}\mathrm{m}/\mathrm{s}$; the void ratio, $e=0.771$; and the soil was almost fully saturated.

## 4. Model Application

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbol | Description | Units |

u_{i} | Fluid velocity | [m/s] |

x_{i} | Coordinate | [m] |

t | Time | [s] |

$\rho $ | Fluid density | [kg/m^{3}] |

p | Fluid pressure | [N/m^{2}] |

g_{i} | Gravitational force | [m/s^{2}] |

${\tau}_{ij}$ | Viscous stress tensor | [N/m^{2}] |

$\mathsf{\Omega}$ | Source Region | [-] |

S_{i} | Momentum source function | [m/s^{2}] |

$\omega $ | Angular frequency | [/s] |

k | Wave number | [/m] |

$\theta $ | Wave obliquity | [1] |

${A}_{0}$ | Wave amplitude | [m] |

L | Wavelength | [m] |

C | Wave velocity | [m/s] |

$\eta \left(t\right)$ | Free surface elevation | [m] |

p_{e} | Wave-induce pore pressure | [Pa] |

${p}_{e}^{\left(1\right)}$ | Oscillatory pore pressure | [Pa] |

${p}_{e}^{\left(2\right)}$ | Residual pore pressure | [Pa] |

${\gamma}_{w}$ | Unite weight of water | [N/m^{3}] |

n_{s} | Soil Porosity | [1] |

${\epsilon}_{v}$ | Volume strain | [1] |

${\beta}_{s}$ | Compressibility of pore fluid | [/Pa] |

u_{s}, w_{s} | Soil displacements | [m] |

K_{w} | True elasticity modulus of pore water | [Pa] |

P_{w0} | Absolute water pressure | [Pa] |

S | Seabed degree of saturation | [1] |

${\sigma}_{ij}$ | Total stress | [Pa] |

${\sigma}_{ij}^{\prime}$ | Effective stress | [Pa] |

${\delta}_{ij}$ | Kronecker delta | [1] |

G | Shear modulus | [Pa] |

${\mu}_{s}$ | Poisson’s ratio | [1] |

K_{v} | Bulk modulus of soil | [Pa] |

${\epsilon}_{p}$ | Plastic volumetric strain | [1] |

R | Material parameters | [1] |

$\mathsf{\chi}$ | Cyclic stress ratio | [1] |

$\tau \left(x,z\right)$ | Maximum amplitude of shear stress | [Pa] |

${\sigma}_{v0}^{\prime}\left(z\right)$ | Initial effective stress in vertical direction | [Pa] |

${P}_{b}\left(x,t\right)$ | Wave pressure on seabed surface | [Pa] |

${\tau}_{b}\left(x,t\right)$ | Shear stress at the seabed surface | [Pa] |

h | Seabed thickness | [m] |

$\epsilon $ | Turbulent dissipation rate | [1] |

$\mathsf{\nu}$ | Kinetic viscosity | [kg/m/s] |

${\nu}_{t}$ | Eddy viscosity | [kg/m/s] |

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**Figure 2.**Comparison of (

**a**) free surface elevation and (

**b**) water pressure between the coupled model and the analytical solution. The small amplitude wave theory is applied for the analytical solution.

**Figure 3.**Schematic diagram of one-dimensional cylinder equipment (adapted from [25]).

**Figure 4.**Comparison of oscillatory pore pressure with one-dimensional experimental data. The experimental data include pore pressure records of ten gauges (P0 to P9).

**Figure 6.**Comparison of residual pore pressure with standing wave centrifugal test data [20]. Time history of pore pressure is from the soil element at elevation z/h = 0.25.

**Figure 7.**Schematic diagram of three-dimensional water flume system (adapted from [27]). (Units are m).

**Figure 8.**Comparison of (

**a**) flow velocity and (

**b**) pore pressure between the present coupled model and experimental data (U

_{0}= −0.1 m/s). The flow velocity represents the fluid velocity at 0.2 m above the seabed surface and pore pressure is from the soil element that was located at point C in Figure 7.

**Figure 9.**Comparison of (

**a**) flow velocity and (

**b**) pore pressure between the present coupled model and experimental data (U

_{0}= 0 m/s). The flow velocity represents the fluid velocity at 0.2 m above the seabed surface and pore pressure is from the soil element that was located at point C in Figure 7.

**Figure 10.**Comparison of (

**a**) flow velocity and (

**b**) pore pressure between the present coupled model and experimental data (U

_{0}= +0.1 m/s). The flow velocity represents the fluid velocity at 0.2 m above the seabed surface and pore pressure is from the soil element that was located at point C in Figure 7.

**Figure 11.**Comparison of (

**a**) pore pressure and (

**b**) liquefaction depth within the seabed between coupled and uncoupled models.

Wave Characteristics | Value | Soil characteristics | Value |
---|---|---|---|

Wave period (T) | 12.0 s | Permeability (K) | 1.0 × 10^{−4} m/s |

Porosity (n_{e}) | 0.30 | ||

Wave length (H) | 170.0 m | Shear modulus (G) | 1.0 × 10^{7} N/m^{2} |

Thickness (h) | ∞ m | ||

Water depth (d) | 30.0 m | Poisson’s ratio (μ) | 0.35 |

Degree of saturation (S) | 1 | ||

Wave amplitude (η) | 2.5 m |

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**MDPI and ACS Style**

Liao, C.; Jeng, D.; Lin, Z.; Guo, Y.; Zhang, Q.
Wave (Current)-Induced Pore Pressure in Offshore Deposits: A Coupled Finite Element Model. *J. Mar. Sci. Eng.* **2018**, *6*, 83.
https://doi.org/10.3390/jmse6030083

**AMA Style**

Liao C, Jeng D, Lin Z, Guo Y, Zhang Q.
Wave (Current)-Induced Pore Pressure in Offshore Deposits: A Coupled Finite Element Model. *Journal of Marine Science and Engineering*. 2018; 6(3):83.
https://doi.org/10.3390/jmse6030083

**Chicago/Turabian Style**

Liao, Chencong, Dongsheng Jeng, Zaibin Lin, Yakun Guo, and Qi Zhang.
2018. "Wave (Current)-Induced Pore Pressure in Offshore Deposits: A Coupled Finite Element Model" *Journal of Marine Science and Engineering* 6, no. 3: 83.
https://doi.org/10.3390/jmse6030083