# Two-Dimensional Numerical Study of Seabed Response around a Buried Pipeline under Wave and Current Loading

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Wave–Current Model

^{3}), $t$ is the time, $g$ is the acceleration (m/s

^{2}), $\mu $ is the dynamic viscosity (Pa·s), and $-{\rho}_{f}\langle {u}_{i}^{\prime}{u}_{j}^{\prime}\rangle $ is the Reynolds stress tensor. By applying the eddy-viscosity assumptions, Reynold stress term can be expressed as:

^{2}/s

^{2}), and ${\delta}_{ij}$ is the Kronecker delta. By substituting Equation (3) into Equation (2), the following equation can be obtained:

^{2}/s

^{2}), $\epsilon $ is the turbulence dissipation rate, ${C}_{\mu},{\sigma}_{k},{\sigma}_{\epsilon},{C}_{\epsilon 1},$ and ${C}_{\epsilon 2}$ are the empirical coefficients determined from experiments. The empirical coefficients used in this study are based on previous studies [29]:

#### 2.2. Seabed Model

#### 2.2.1. Oscillatory Soil Response

^{3}), ${n}_{s}$ is the soil porosity, ${k}_{s}$ is the soil permeability (m/s), $G$ denotes the shear modulus of the seabed soil (N/m

^{2}), $\nu $ is the Poisson’s ratio, and ${u}_{e}$ represents the wave–current induced oscillatory soil response. The compressibility of the pore fluid (${\beta}_{s}$) and the elastic volume strain of the soil matrix (${\epsilon}_{s}$) can be defined as:

^{9}N/m

^{2}), ${S}_{r}$ is the degree of saturation, and ${P}_{wo}$ is the absolute water pressure. ${u}_{s}$ and ${w}_{s}$ represent the soil displacement at x- and z-direction respectively.

#### 2.2.2. Residual Soil Response

#### 2.3. Boundary Condition

#### 2.4. Numerical Scheme

_{s}is the unit weight of seabed soil, γ

_{w}is the unit weight of water, z represents depth of a specific point in the seabed, P

_{s}is the wave–current-induced pore pressure, and P

_{b}is the wave pressure on the seabed surface.

## 3. Results and Discussion

#### 3.1. Wave Characteristics

#### Influence of Current Velocities

#### 3.2. Seabed Characteristics

#### 3.2.1. Effects of Soil Permeability

^{−12}to 1.0 × 10

^{−2}m/s. In this analysis, three different values of soil permeability (i.e., k = 1.0 × 10

^{−2}, 1.0 × 10

^{−3}and 1.0 × 10

^{−4}m/s) are evaluated. Figure 5a,b illustrate the distribution of wave and current induced oscillatory pore pressure and residual pore pressure at the point x = 50 m and z = −10 m respectively under different soil permeability conditions. The results expressed in Figure 5 are for the case in which shear modulus (G) and relative density (${D}_{r}$) set as 5.0 × 10

^{6}N/m

^{2}and 0.5, respectively.

_{s}= 1.0 × 10

^{−4}m/s) tends to generate a higher value of residual pore pressure. It is because water cannot dissipate efficiently from low permeable soils and eventually resulted in the build-up of excess pore pressure.

#### 3.2.2. Effects of Shear Modulus

^{6}and 1.5 × 10

^{7}N/m

^{2}). In this section, the soil permeability and relative density are fixed at 0.0001 m/s and 0.5 respectively. Shear modulus is one of the soil properties that used to describe the tendency of an object deforms in shape at constant volume when acted upon by the opposing forces. Consider that the soil to be elastic, shear modulus (G) has a relationship with Young’s modulus (E) and Poisson’s ratio (v), which can be calculated from:

^{6}N/m

^{2}and 1.0 × 10

^{7}N/m

^{2}respectively. The soil becomes denser when shear modulus and Young’s modulus increases. From the reference table below (Table 2), soil with Young’s modulus of 13.5 MPa is said to be loose sand while soil with Young’s modulus of 40.5 MPa is said to be medium dense sand. When loose, saturated soil is subjected to shear force, the soil particles tend to rearrange themselves into a denser manner, i.e., fewer void spaces as the water particles are being forced out of the voids. However, once the pore water drainage is blocked, the pore water pressure will increase progressively with shear force. The stress is transferred from the soil skeleton to pore pressure, which eventually leads to a reduction in effective stress and shear resistance. As for dense sand under monotonically shearing, the soil skeleton contracts and then dilates. The soil volume increases when the soil is saturated with poor drainage, which will result in a decrease in pore pressure. Hence, there is an increase in effective stress and shear strength. Therefore, it concludes that loose sand (contractive) tends to generate higher residual pore pressure than dense sand (dilative).

#### 3.2.3. Effects of Relative Density

^{−4}m/s and 5.0 × 10

^{6}N/m

^{2}respectively while other parameters remain unchanged as shown in Table 1. Figure 7a shows that oscillatory pore pressure is not affected by the change in relative density. However, the generation of residual pore pressure varies drastically as shown in Figure 7b. Higher accumulation of pore pressure is generated when the relative density is of a smaller value, i.e., looser soil. This concludes that loose sand tends to generate higher residual pore pressure (same explanation as Section 3.2.1).

#### 3.3. Around the Vicinity of the Pipeline

^{6}N/m

^{2}, 1.0 × 10

^{–4}m/s and 0.2 respectively.

## 4. Conclusions

- (1)
- In the analysis of wave-seabed-structure interaction, the current should be taken into consideration—an increase in the current flow results in an increase in wave pressure. When wave pressure increases, the oscillatory pore pressure tends to increase with increasing current velocity.
- (2)
- Soil permeability governs the seepage of fluid passing through or flowing out of the seabed. Low permeability, i.e., pore fluids cannot dissipate efficiently, resulted in higher residual pore pressure due to the increase in the buildup of excess pore pressure.
- (3)
- Shear modulus has a relationship with Young’s modulus and Poisson’s ratio, which describe the rigidity of the seabed. Keeping the Poisson’s ratio as a constant value, a higher value of shear modulus generates a higher value of Young’s modulus, which represents a denser soil. As presented above, loose sand tends to produce a higher value of residual pore pressure.
- (4)
- Relative density controls the empirical coefficients α
_{r}and β_{r}in source term, which affects the generation of residual pore pressure. It concludes that a smaller value of relative density results in a higher value of residual pore pressure. However, there is no visible difference in the oscillatory pore pressure.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The conceptual sketch of the pore pressure mechanisms (Adapted from Jeng [7]).

**Figure 3.**Comparison between different mesh sizes for COMSOL Multiphysics in terms of (

**a**) the number of elements and the computational time, and (

**b**) residual pore pressure, ${u}_{p}$ at a duration of 100 s.

**Figure 5.**Variations of (

**a**) oscillatory pore pressure, ${u}_{e}$ and (

**b**) residual pore pressure, ${u}_{p}$ at a specific point for different soil permeability.

**Figure 6.**Variations of (

**a**) oscillatory pore pressure,$\text{}{u}_{e}$ and (

**b**) residual pore pressure, ${u}_{p}$ at a specific point under the different shear modulus.

**Figure 7.**Variation of (

**a**) oscillatory pore pressure, ${u}_{e}$ and (

**b**) residual pore pressure, ${u}_{p}$_p at a specific point under the different relative density.

**Figure 8.**Variation in (

**a**) oscillatory pore pressure, ${u}_{e}$, (

**b**) residual pore pressure, ${u}_{p}$ and (

**c**) displacement, ${u}_{s}$ under various current velocity.

**Figure 9.**Distribution of seabed responses such as (

**a**) horizontal displacement ${u}_{s}$, (

**b**) vertical displacement ${w}_{s}$, (

**c**) oscillatory pore pressure,$\text{}{u}_{e}$, and residual pore pressure ${u}_{p}$ around the vicinity of the buried pipeline due to wave and current loading when current velocity is set at ${v}_{c}$ = 0.5 m/s.

Module | Parameter | Notation | Magnitude | Unit |
---|---|---|---|---|

Wave | Water Depth | d | 12 | m |

Wave Height | H | 4 | m | |

Wave Period | T | 10 | s | |

Current | Velocity | v_{c} | 0, 0.25, 0.5 | m/s |

Seabed | Permeability | k_{s} | 1.0 × 10^{−2}, 1.0 × 10^{−3}, 1.0 × 10^{−4} | m/s |

Degree of Saturation | S_{r} | 1 | - | |

Shear Modulus | G | 5.0 × 10^{6}, 1.5 × 10^{7} | N/m^{2} | |

Poisson’s Ratio | ν | 0.35 | - | |

Relative Density | ${D}_{r}$ | 0.2, 0.3, 0.5 | - | |

Porosity | n_{s} | 0.4 | - | |

Pipeline | Pipe Diameter | D | 2.0 | m |

Burial Depth | e | 3.0 | m | |

Young Modulus | E_{P} | 2.09 × 10^{11} | N/m^{2} | |

Shear Modulus | ${G}_{p}$ | 6.8 × 10^{10} | N/m^{2} |

**Table 2.**Selected elastic constants for soils (adapted from Das, 2007 [36]).

Type of Soil | Young’s Modulus, E (MPa) | Poisson’s Ratio, v |
---|---|---|

Loose sand | 10.5–24.0 | 0.20–0.40 |

Medium dense sand | 17.25–27.60 | 0.25–0.40 |

Dense sand | 34.50–55.20 | 0.30–0.45 |

Silty sand | 10.35–17.25 | 0.20–0.40 |

Sand and gravel | 69.00–172.50 | 0.15–0.35 |

Soft clay | 4.1–20.7 | - |

Medium clay | 20.7–41.4 | 0.20–0.50 |

Stiff clay | 41.4–96.6 | - |

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**MDPI and ACS Style**

Foo, C.S.X.; Liao, C.; Chen, J.
Two-Dimensional Numerical Study of Seabed Response around a Buried Pipeline under Wave and Current Loading. *J. Mar. Sci. Eng.* **2019**, *7*, 66.
https://doi.org/10.3390/jmse7030066

**AMA Style**

Foo CSX, Liao C, Chen J.
Two-Dimensional Numerical Study of Seabed Response around a Buried Pipeline under Wave and Current Loading. *Journal of Marine Science and Engineering*. 2019; 7(3):66.
https://doi.org/10.3390/jmse7030066

**Chicago/Turabian Style**

Foo, Cynthia Su Xin, Chencong Liao, and Jinjian Chen.
2019. "Two-Dimensional Numerical Study of Seabed Response around a Buried Pipeline under Wave and Current Loading" *Journal of Marine Science and Engineering* 7, no. 3: 66.
https://doi.org/10.3390/jmse7030066