# Numerical Simulations of Wave-Induced Soil Erosion in Silty Sand Seabeds

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Analytical Solution for Wave-Induced Pore-Pressure Accumulation

_{osc}is the oscillatory pore-pressure corresponding to the elastic deformation of the soil skeleton. P

_{osc}fluctuates in both temporal and spatial domains, and the fluctuation is accompanied by the attenuation of the amplitude and phase lag under wave actions [13,14,15]. P

_{res}is the residual pore-pressure that is period-averaged, and is the result of accumulated plastic deformation of the soil skeleton. It has been acknowledged recently that with the accumulation of pore-pressure, continuous seepage flow appears near the seabed surface and may lead to obvious particle migration [16,17,18].

_{v}is the consolidation coefficient, K

_{0}is the coefficient of lateral earth pressure, β and η are empirical constants, which can be confirmed based on the soil type and the relative density [24], k

_{s}is the wave number, T is the wave period, and P

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

## 3. Theoretical Model for Soil Erosion Process

#### 3.1. Definition of Three-Phase Soil Model

_{f}, dW

_{f s}, and dW

_{s}are the volumes of the soil element, soil skeleton, pore fluid, and fluidized soil particles, respectively. The masses of soil element, soil skeleton, pore fluid, and fluidized soil particles are represented by dM, dM

_{f}, dM

_{fs}, and dM

_{s}, respectively.

_{f}, ρ

_{s}are the densities of the pore fluid and the solid skeleton.

#### 3.2. Mass Conservation Equations

_{1,}ρ

_{2}are the densities of the soil phase and the mixture phase, v

_{1}, v

_{2}are the velocities of the soil phase and the mixture phase.

#### 3.3. Constitutive Laws of Mass Generation

#### 3.4. Darcy Flow Law

#### 3.5. Governing Equations for Soil Erosion

## 4. Numerical Implement of Seabed Erosion Model and Simulations

_{s}is equal to 30.00 m and the average mesh size is 0.1 m. More parameters can be listed as follows: water depth d

_{w}= 10.00m, wave height H = 2.00 m, wave period T = 5.00 s, wave length L = 36.59 m. According to the judgement criterion about the seabed depth [12], d

_{s}/L = 0.82>0.3, and thus the depth of seabed can be treated as infinite thickness. For the soil condition, the initial porosity φ

_{0}= 0.42, initial concentration of the fluid soil particles c

_{0}= 0.001. More details can be found in Table 1. For a typical wave condition, a series of numerical studies have been performed. It is known that the wave-induced erosion is not only associated with soil properties, but also closely related to wave characteristics. So, the influences of wave height H, wave period T, and critical concentration of the fluidized soil particles ${c}_{cr}$ on the process of wave-induced erosion were discussed. The simulation cases are listed in Table 2.

## 5. Time Characteristics of Wave-Induced Soil Erosion Process

_{b}decreases from 1.00 on the seabed surface to 0 at the −30.00 m depth. The liquefaction of the seabed can be divided into the oscillatory and residual liquefactions [5]. According to Jeng et al. [8] and Okusa [29], the criterions of oscillatory and residual liquefactions are $\frac{{P}_{osc}}{{\sigma}_{0}^{\prime}}\ge 1$ and $\frac{{P}_{res}}{{\sigma}_{0}^{\prime}}\ge 1$, respectively (${\sigma}_{0}^{\prime}$ is the effective vertical stress of soil). Figure 5b indicates that the oscillatory liquefaction will not occur under the typical wave condition. Figure 5c shows the evolution of the residual pore-pressure along depths. It is noted that the residual pore-pressure develops gradually with the extension of wave acting time and tends to be stable. The maximum value of P

_{res}occurs at about −5 m to −10 m (below the seabed surface) depth in the whole process of wave actions. In Figure 5d, it also reveals that there is no potential soil liquefaction in the seabed with the accumulation of P

_{res}. Under normal sea state, the soil erosion is the common behavior for the silty sand seabed.

_{0}increases with the extension of wave acting time and the maximum value reaches 4.10 at −0.50 m depth after 24 d. When t = 30 d, the depth with k/k

_{0}over 2.00 is around −2.30 m. These results indicate that the soil permeability increases significantly with the extension of wave acting time at the shallow seabed.

## 6. Results for Affecting Factors and Interpretations

#### 6.1. Effect of Wave Height

#### 6.2. Effect of Wave Period

#### 6.3. Effect of Critical Concentration of Fluidized Soil Particles

_{cr}were selected as 0.10, 0.20, 0.30, 0.40, and 0.50, respectively. Figure 16 gives the simulation results of soil porosity versus the depth for different c

_{cr}. It can be seen that the soil porosity increases mainly at shallow depths (within −4 m) with the growth of c

_{cr}. The soil at deep depths is not affected by c

_{cr}. The bigger the c

_{cr}, the more severe the soil erosion is. As shown in Figure 17, the erosion rate $\frac{\partial \phi}{\partial t}$ is obviously affected by c

_{cr}at shallow depths. Combined with Figure 17a–e, it can be seen that the peak values of $\frac{\partial \phi}{\partial t}$ for the selected depths increase obviously and $\frac{\partial \phi}{\partial t}$ reach the peak values later with the growth of c

_{cr}. Furthermore, the value of $\frac{\partial \phi}{\partial t}$ becomes negative in the later stage of the erosion process when ${c}_{cr}\ge 0.30$. It can be concluded that the bigger the c

_{cr}, the more remarkable the deposition effect.

## 7. Conclusions

- The wave-induced erosion mainly occurred at the shallow depth of the seabed. For the typical wave condition, the depth affected by the wave-induced erosion is within approximately −5.00 m. In the erosion process, the concentration of the fluidized particles increases to the critical value and then remains at a stable state within −2.00 m depth. The soil porosity and soil permeability increase significantly in the shallow seabed. The maximum values of soil porosity and soil permeability occurred at depths of about −0.50 m. It is also found that the deeper the soil, the slower the erosion rate, and the later the peak erosion rate can reach. The numerical model proposed in this paper can be used for the analysis of the seabed coarsening phenomenon.
- With the increase of wave height, the soil porosity, the affected depth, and the erosion rate increase obviously. When the wave height is over 2.00 m, the soil erosion on the seabed surface develops rapidly. In the later stage of the erosion process, the change rate of soil porosity can be negative, which illustrates that the deposition effect of fine particles plays an obvious role in the later stage of the erosion process.
- The wave period has an obvious effect on the soil porosity and the erosion rate, but the effect is not always promotional to the soil erosion. This is because the development of the residual pore-pressure is controlled by a competition mechanism between the accumulation and the dissipation. There exists a particular wave period to make the erosion induced by waves the fastest and most severe.
- The critical concentration of the fluidized soil particles has an obvious effect on the evolution of wave-induced erosion, including erosion rate and erosion degree. The bigger the critical concentration of the fluidized soil particles, the more severe the soil erosion. The erosion depth of seabeds is not affected by the critical concentration of the fluidized soil particles.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

c | concentration of the fluidized soil particles | P_{osc} | oscillatory pore-pressure |

c_{cr} | critical concentration of the fluidized soil particles | P_{res} | residual pore-pressure |

c_{v} | consolidation coefficient | P_{b} | amplitude of the dynamic wave pressure |

d_{s} | depth of the seabed | $\overline{q}$ | volume flow rate |

d_{w} | depth of the water | T | wave period |

dW | volume of the soil element | v_{fs} | velocity of the fluidized soil particles |

dW_{f} | volume of the soil skeleton | v_{f} | velocity of the pore fluid |

dW_{f s} | volume of the pore fluid | $\overline{v}$ | velocity of the mixture |

dW_{s} | volume of the fluidized soil particles | v_{s} | velocity of the soil skeleton |

dM | masse of the soil element | v_{1} | velocity of the soil phase |

dM_{f} | masse of the soil skeleton | v_{2} | velocity of the mixture phase |

dM_{fs} | masse of the pore fluid | ρ_{1} | density of the soil phase |

dM_{s} | masse of the fluidized soil particles | ρ_{2} | density of the mixture phase |

$d\overline{s}$ | pore part of ds | ρ_{f} | density of the pore fluid |

k | soil permeability | ρ_{s} | density of the solid skeleton |

K | reference permeability | $\overline{\rho}$ | density of the mixture |

K_{w} | bulk modulus of pore water | ${\overline{\rho}}_{fs}$ | apparent density of the fluidized soil particles |

K_{0} | coefficient of lateral earth pressure | φ | soil porosity |

k_{s} | wave number | γ’ | effective unit weight of soil |

L_{w} | wave length | β, η | empirical constants for soil type, relative density |

${\dot{m}}_{\alpha}$ | mass generation term | η_{k} | kinematic viscosity of the mixture |

${\dot{m}}_{er}$ | rate of eroded mass | μ | Poisson’s ratio of soil |

${\dot{m}}_{dep}$ | rate of deposited mass | ||

P | total excess pore-pressure | ||

$d\overline{W}$ | the volume of the mixture through the cross-sectional ds within dt time | ||

α | the fluidized particles phase or solid phase or fluid phase | ||

λ | the parameter used to describe the spatial frequency of the potential erosion starter points |

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**Figure 1.**Plumes of sediment and micro-holes in silty sediment seabed: (

**a**) The plumes of sediment on the silty sediment surface; (

**b**) The micro-holes due to erosion. [4].

**Figure 4.**Mass conservation of the three phases: (

**a**) Fluidized particles; (

**b**) soil skeleton; (

**c**) Pore fluid.

**Figure 5.**Distributions of the oscillatory pore-pressure and the residual pore-pressure: (

**a**) Vertical distribution of $\left|{P}_{osc}\right|/{P}_{b}$; (

**b**) vertical distribution of $({P}_{b}-\left|{P}_{osc}\right|)/{\sigma}_{0}^{\prime}$; (

**c**) vertical distribution of P

_{res}for different times; (

**d**) vertical distribution of ${P}_{res}/{\sigma}_{0}^{\prime}$ for different times.

**Figure 6.**Variations of the concentration of moving particles with the increase of wave acting time: (

**a**) Diagram of three-dimensions; (

**b**) diagram of two-dimensions.

**Figure 7.**Variations of the soil porosity with the increase of wave acting time: (

**a**) Diagram of three-dimensions; (

**b**) diagram of two-dimensions.

**Figure 8.**Variations of the soil permeability with the increase of wave acting time: (

**a**) Diagram of three-dimensions; (

**b**) diagram of two-dimensions.

**Figure 9.**Variations of $\frac{\partial c}{\partial t},\frac{\partial \phi}{\partial t}$ at different depths with the increase of wave acting time: (

**a**) $\frac{\partial c}{\partial t}-t$; (

**b**) $\frac{\partial \phi}{\partial t}-t$.

**Figure 10.**Distributions of the oscillatory pore-pressure and the residual pore-pressure for different wave heights: (

**a**) Vertical distribution of $\left|{P}_{osc}\right|/{P}_{b}$ for different H; (

**b**) vertical distribution of $({P}_{b}-\left|{P}_{osc}\right|)/{\sigma}_{0}^{\prime}$ for different H; (

**c**) vertical distribution of P

_{res}for different H; (

**d**) vertical distribution of ${P}_{res}/{\sigma}_{0}^{\prime}$ for different H.

**Figure 12.**Variations of $\frac{\partial \phi}{\partial t}$ for different wave heights: (

**a**) H = 1.50 m; (

**b**) H = 2.00 m; (

**c**) H = 2.25 m; (

**d**) H = 2.50 m.

**Figure 13.**Distributions of the oscillatory pore-pressure and the residual pore-pressure for different wave periods: (

**a**) Vertical distribution of $\left|{P}_{osc}\right|/{P}_{b}$ for different T; (

**b**) vertical distribution of $({P}_{b}-\left|{P}_{osc}\right|)/{\sigma}_{0}^{\prime}$ for different T; (

**c**) vertical distribution of P

_{res}for different T; (

**d**) vertical distribution of ${P}_{res}/{\sigma}_{0}^{\prime}$ for different T.

**Figure 15.**Variations of $\frac{\partial \phi}{\partial t}$ for different wave periods: (

**a**) T = 5 s; (

**b**) T = 10 s; (

**c**) T = 15 s; (

**d**) T = 20 s.

**Figure 16.**Variations of the soil porosity for different critical concentrations of the fluidized soil particles.

**Figure 17.**Variations of $\frac{\partial \phi}{\partial t}$ for different critical concentrations of the fluidized soil particles: (

**a**) c

_{cr}= 0.10; (

**b**) c

_{cr}= 0.20; (

**c**) c

_{cr}= 0.30; (

**d**) c

_{cr}= 0.40; (

**e**) c

_{cr}= 0.50.

Properties | Value |
---|---|

Wave height H_{w} (m) | 2.00 |

Wave period T (s) | 5.00 |

Wave length L_{w} (m) | 36.59 |

Water depth d_{w} (m) | 10.00 |

Depth of the seabed d_{s} (m) | 30.00 |

Density of the fluidized soil particles ρ_{fs} (kg/m^{3}) | 2650.00 |

Effective unit weight of soil γ’ (kN/m^{3}) | 10.20 |

Density of the fluid ρ_{f} (kg/m^{3}) | 980.00 |

Shear modulus of the soil skeleton G_{s} (MPa) | 50.00 |

Poisson’s ratio of soil μ | 0.33 |

Bulk modulus of pore water K_{w} (MPa) | 2.0e3 |

Coefficient of lateral earth pressure K_{0} | 0.40 |

Initial concentration of the fluid soil particles c_{0} | 0.001 |

Critical concentration of the fluid soil particles c_{cr} | 0.30 |

Initial porosity of soil in seabed φ_{0} | 0.42 |

Variables | Value |
---|---|

Wave height H_{w} (m) | 1.50, 1.75, 2.00, 2.25, 2.50 |

Wave period T (s) | 2.00, 5.00, 10.00, 15.00, 20.00 |

Critical concentration of fluid soil particles c_{cr} | 0.10, 0.20, 0.30, 0.40, 0.50 |

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## Share and Cite

**MDPI and ACS Style**

Guo, Z.; Zhou, W.; Zhu, C.; Yuan, F.; Rui, S.
Numerical Simulations of Wave-Induced Soil Erosion in Silty Sand Seabeds. *J. Mar. Sci. Eng.* **2019**, *7*, 52.
https://doi.org/10.3390/jmse7020052

**AMA Style**

Guo Z, Zhou W, Zhu C, Yuan F, Rui S.
Numerical Simulations of Wave-Induced Soil Erosion in Silty Sand Seabeds. *Journal of Marine Science and Engineering*. 2019; 7(2):52.
https://doi.org/10.3390/jmse7020052

**Chicago/Turabian Style**

Guo, Zhen, Wenjie Zhou, Congbo Zhu, Feng Yuan, and Shengjie Rui.
2019. "Numerical Simulations of Wave-Induced Soil Erosion in Silty Sand Seabeds" *Journal of Marine Science and Engineering* 7, no. 2: 52.
https://doi.org/10.3390/jmse7020052