Mathematical Problems in Engineering

Volume 2015, Article ID 184621, 13 pages

http://dx.doi.org/10.1155/2015/184621

## Numerical Study on Effects of the Embedded Monopile Foundation on Local Wave-Induced Porous Seabed Response

^{1}State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing, Jiangsu 210098, China^{2}Key Laboratory of Engineering Sediment of Ministry of Transport, Tianjin Research Institute for Water Transport Engineering, M.O.T., Tianjin, Tianjin 300456, China^{3}Fujian Provincial Investigation Design and Research Institute of Port and Waterway, Fuzhou, Fujian 350002, China^{4}Nanjing Hydraulic Research Institute, Nanjing, Jiangsu 210024, China

Received 15 April 2015; Revised 21 July 2015; Accepted 26 July 2015

Academic Editor: Carla Faraci

Copyright © 2015 Chi Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Effects of the embedded monopile foundation on the local distributions of pore water pressure, soil stresses, and liquefaction are investigated in this study using a three-dimensional integrated numerical model. The model is based on a Reynolds-Averaged Navier-Stokes wave module and a fully dynamic poroelastic seabed module and has been validated with the analytical solution and experimental data. Results show that, compared to the situation without an embedded foundation, the embedded monopile foundation increases and decreases the maximum pore water pressure in the seabed around and below the foundation, respectively. The embedded monopile foundation also significantly modifies the distributions of the maximum effective soil stress around the foundation and causes a local concentration of soil stress below the two lower corners of foundation. A parametric study reveals that the effects of embedded monopile foundation on pore water pressure increase as the degrees of saturation and soil permeability decrease. The embedded monopile foundation tends to decrease the liquefaction depth around the structure, and this effect is relatively more obvious for greater degrees of saturation, greater soil permeabilities, and smaller wave heights.

#### 1. Introduction

Monopile is a common structure used in ocean engineering, underneath which the porous seabed stability is a great concern for the structure safety [1, 2]. The existence of a monopile will affect the wave-induced seabed response in two ways. First, a monopile will modify the propagation pattern of nearby waves (e.g., wave reflection and diffraction), which in turn results in the variations of pore water pressure and soil stresses inside the seabed. Second, the embedded monopile foundation will disturb the transmission of pore water pressure and soil stresses, leading to the local redistributions of these two quantities. Wave-seabed-monopile interaction may cause the liquefaction of surface soil layer and eventually the structure destruction. Better understanding and modeling of these mechanisms are important for the monopile design and maintenance in ocean engineering.

The numerical modeling of wave-seabed-monopile interaction has only recently become available [3]. This is mainly because the simulation of these processes requires an advanced three-dimensional (3D) integrated model including both complex wave transformation and seabed response around the monopile. Li et al. [4] developed a 3D numerical model for simulating wave-induced pore water pressure response around a monopile foundation. It is found that the amplitude of transient pore pressure decreases with the decrease of soil permeability. The development of pore pressure is also affected by wave nonlinearity and pile diameter. However, the second-order Stokes progressive wave theory was used in their study and both wave reflection and diffraction were not considered. Based on Reynolds-Averaged Navier-Stokes (RANS) equations and poroelastic seabed equations, a 3D integrated model was developed by Zhao et al. [5] to investigate wave-induced seabed response around breakwater heads, in which the seabed module was developed within COMSOL environment. This model was also applied by Chang and Jeng [6] to simulate the wave-seabed-structure interaction around the high-rising offshore wind turbine foundation used in the Donghai offshore wind farm, China. They found that the existence of the structure has a significant effect on the wave transformation and the distribution of wave-induced pore pressure. They also investigated effects of various wave and soil parameters on dynamic soil behavior and found that replacing the original seabed with coarse sand is efficient to prevent the seabed around pile tips from liquefaction. More recently, Sui et al. [7] used a 3D fully dynamic seabed model to investigate the seabed response beneath wave transformation around a monopile. They confirmed that wave reflection and diffraction have significant effects on pore water pressure and soil displacements around the monopile, and these effects increase with an increasing wave number. However, they did not investigate the seabed liquefaction around the monopile. The aforementioned studies have shown that the state-of-the-art numerical models can reliably simulate the wave-seabed-monopile interaction and are very useful to investigate the underlying mechanisms. However, most of them focused on the seabed response to wave reflection and diffraction around a monopile, while the pure effects of the embedded monopile foundation on local seabed response are still not clear.

In this study, a 3D RANS wave model is integrated with a 3D fully dynamic poroelastic seabed model to investigate the effects of an embedded monopile foundation on the local distributions of pore water pressure, soil stresses, and liquefaction zone, under wave reflection and diffraction around a monopile. Both inertial terms of pore fluid and soil skeleton are included in the fully dynamic seabed model. In particular, we focus on the changes of maximum pore water pressure, maximum vertical effective normal soil stress, and maximum liquefaction depth due to the existence of an embedded monopile foundation. The significance of these effects with respect to various embedded depths and soil parameters is also discussed.

#### 2. Numerical Model

The 3D numerical model used in this study was developed by integrating a wave module based on RANS equations and a seabed module based on Biot’s poroelastic theory. The wave-induced dynamic water pressure at the seabed surface calculated by the wave module was employed as the external boundary condition of the seabed module.

##### 2.1. Wave Module

The RANS equations for describing water wave motion can be expressed aswhere is the water density, is the velocity vector, is time, is water pressure, is the viscous force, and , , and are the body force, surface force, and drag force, respectively, in which only the body force is included in this study. The - two-equation turbulence model is used to provide turbulence closure for wave module [8]. Waves are generated within the computational domain by the internal wave-maker of Lin and Liu [9]. The free surface of water waves is captured by the volume of fluid (VOF) method.

##### 2.2. Seabed Module

Based on Biot’s poroelastic theory [10–13], a fully dynamic mathematical description of the overall equilibrium of soil, the equilibrium of pore fluid flow, and the mass balance for porous seabed includes the accelerations of both soil and pore fluid. These can be expressed aswhere is the total stress, is the pore water pressure, is the total density of the porous medium, is the gravitational acceleration, is the displacement of soil matrix, is the average relative displacement of the fluid to the soil skeleton, is permeability of porous medium, is the porosity, is the strain of the soil skeleton defined as , and is the compressibility of pore fluid defined as , where is the bulk modulus of pore water, is the water depth, and is the degree of saturation. is typically taken as 1.95 × 10^{9} N/m^{2} following numerous studies [13, 14]. is usually less than unity because of the gas storage in marine soil skeletons [15], and its value depends on the content of gas in marine sediments. It was reported that the degree of saturation significantly affects the wave-induced seabed response [13]. It is noted that the definition of is only suitable for nearly saturated seabed [16], that is, when is close to 1.0. This definition is assumed to be applicable to the test values of (=0.975–0.995) in this study in order to investigate the relationship between and the embedded monopile foundation effects.

The total stresses are given in terms of the effective stresses () and pore pressure ():where is the Kronecker delta denotation, , is shear modulus, and is Poisson’s ratio.

In the present model, the linear elastic behavior of soil skeleton is considered. While the nonlinear or plastic soil behavior may be more obvious for large strains under long-time action of extreme waves, this study preliminarily focuses on the instantaneous seabed response on a relatively short time scale, for which the linear elastic concept is used as the first approximation due to its simplicity. This assumption was commonly made in the previous studies for wave-induced seabed response and gave satisfactory results [4, 6, 14, 17–20].

##### 2.3. Boundary Conditions

In the wave module, the sponge layers are applied to eliminate wave reflection at the side/outlet boundaries. The boundary conditions for the momentum equations are based on the bottom stress estimated from the log-law. At the wave-seabed and wave-structure boundaries, nonslip conditions for velocities are imposed, and the turbulent kinetic energy and its dissipation rate are specified from the “law of the wall” boundary condition following the traditional approach of Rodi [8]. At the air-water interface, zero surface tension is assumed, and both and are implemented with the zero-gradient boundary conditions.

In the seabed module, the bottom and the lateral boundaries of the seabed are considered impermeable and rigid, where the soil and pore fluid displacements and the normal gradient of pore water pressure are zero. At the seabed surface, the pore water pressure is equal to the wave-induced dynamic water pressure, and the vertical effective normal stress and shear stresses of soil are negligible as they are very small compared to the wave-induced dynamic pressure (less than 2% in this study). Since the excess pore water pressure is the dominant factor for liquefaction, the neglect of soil stresses at the seabed surface is considered to have little influence on liquefaction calculation. This treatment was adopted in most numerical studies [18–21].

For dealing with the effects of embedded monopile foundation on seabed response, appropriate seabed-structure boundary conditions are necessary. Unlike other model studies that solve the responses of seabed and structure as a whole system, the present model includes an internal seabed-structure boundary condition to ensure the normal gradient of pore water pressure equal to zero () at the impermeable and rigid structure surface. In addition, no relative displacement of soil with respect to structure () and the total stress equilibrium (, ) are also imposed at the seabed-structure boundaries.

##### 2.4. Numerical Scheme

In the wave module, the RANS governing equations are solved using a two-step projection method with a finite volume discretization [22]. A set of unstructured triangular grids are adopted to discretize the computational domain. In the vicinity of monopile foundation, the grid size is 0.05 m in the horizontal plane and 0.5 m in the vertical direction, respectively. The fluid variables, such as the pressure and the velocities, are defined at the cell centroids. To convert cell centroid data to the face centroid to evaluate the gradient of the quantity at the cell centroid, the least square linear reconstruction method developed by Barth [23] is applied. The forward time difference method is used for the discretization of the time derivative. To obtain computational stability, the time interval is automatically adjusted at each time step to satisfy the Courant-Friedrichs-Lewy condition and the diffusive limit condition [24], with a range between 0.005 s and 0.05 s.

In the seabed module, the second-order Crank-Nicolson type implicit Finite-Difference-Method is used to discretize governing equations, in combination with a staggered nonuniform rectangular grid. The Alternating-Direction-Implicit method and the Leap-Frog method are used to solve the multivariables in the differential equations with multidimensions. The underrelaxation technique is employed in iterative procedure to obtain convergent solutions. The grid size is 0.25 m in the horizontal plane and is 0.07 m in the vertical direction near the monopile, respectively. The time interval of seabed module is the same to wave module. The computational convergence of the numerical model is achieved when the maximum relative difference of solutions is less than 0.001 between any two successive iterations. This convergence is typically obtained within 100 iterations at each time step.

Considering that the movement of monopile is very small and has little effects on wave propagation, the integration of wave and seabed module is established as a so-called one-way coupling. The meshes of two modules are not required to match with each other at the wave-seabed interface. At each time step, the 3D field of dynamic wave pressure calculated by wave module is interpolated to the grid points of seabed module at the interface, driving the seabed module as the pressure boundary condition. This 3D interpolation of wave pressure is processed by an open source program of KT3D [25]. KT3D provides a fairly advanced 3D kriging program for points or blocks by simple kriging, ordinary kriging, or kriging with a polynomial trend model with up to nine monomial terms. More information of this program is available in reference [25].

#### 3. Model Validation

In this section, we provide model validation with both the analytical solution and the experimental data. Figure 1 shows comparisons of the maximum pore water pressure and effective stresses between numerical results and the analytical solution of Hsu and Jeng [26] for seabed response under obliquely incident linear waves with an angle of 45 degrees. Figure 2 presents comparisons of the maximum pore water pressure between numerical results and the experimental data of Maeno and Hasegawa [27] for normally incident linear wave-induced seabed response. As shown, the model agrees well with both the analytical solution and the experimental data, demonstrating the numerical accuracy of the present model. More validation cases for the seabed model with different datasets are available in Sui et al. [7].