PZIII is the Pastor–Zienkiewicz Mark-III.

**Figure 6.** Excess pore pressure at point A subject to progressive wave loading. Notation: red lines = the present model, blue lines = Sassa's model [35].

#### **4. Results and Discussion**

In this study, two new features were incorporated into an existing model for soil response: (1) PSR effects and (2) the combined wave and current loading. In this section, the seabed is considered to be an elastoplastic medium, and the discussed results are from simulations using the generalized plasticity model PZIII and modified PZIII model with PSR in the finite element analysis program DIANA-SWANDYNE II. Nevada dense sand was adopted for a seabed with elastoplastic behavior, and the soil parameters, given by Sassa and Sekiguchi [40], were determined experimentally and are specified in Table 3. In the computations, the seabed length *Ls* = 180 m, and seabed thickness *d* = 30 m. In order to ensure the numerical convergence suggested by Ye et al. [42], in the numerical model, the maximum horizontal mesh size was less than the wavelength *Lw*/40, where *Lw* is 88 m, and the

maximum vertical mesh size was half of the horizontal mesh size. Therefore, the horizontal mesh size and vertical mesh size were 1 m and 0.5 m, respectively. Furthermore, the time step Δt was T/40, where T is the wave period and equal to 8 s in this study.


**Table 3.** Parameters used in dynamic constitutive model for parametric study.

#### *4.1. Seabed Liquefaction*

Generally, the literature reports two different mechanisms of fluid-induced soil liquefaction [3]: momentary liquefaction and residual liquefaction. Momentary liquefaction normally occurs near the wave trough and in an unsaturated seabed when the upward seepage force is higher than the overlying pressure. However, the effect of momentary liquefaction is much smaller than that of residual liquefaction on the stability of offshore structures. As mentioned previously, the soil gradually loses stability as the wave spreads over the surface of the seabed. When the pore-water pressure increases, the effective stress between soil particles decreases. The condition for residual liquefaction occurs when the pore pressure reaches the maximum value and the effective stress approaches zero, at which point the soil loses its bearing capacity. The seabed's instability is a consequence of the horizontal or vertical movement of soil particles [1]. In such a situation, the soil acquires the behavior of a liquid. The liquefaction of the seabed has an essential impact on the safety of offshore structures.

In order to quantitatively study the liquefaction characteristics of the sandy seabed involving PSR effects, a parameter called liquefaction potential is introduced, as defined below.

$$L\_{potential} = \frac{\sigma\_{zd}^{\prime}}{|\sigma\_{z0}^{\prime}|} \tag{25}$$

where *σ zd* is the wave (current)-induced dynamic vertical effective stress and *σ <sup>z</sup>*<sup>0</sup> is the initial vertical effective stress.

With the liquefaction criterion proposed by Okusa [60], the sandy seabed liquefies at *Lp* = 1. In practice, however, the value of *Lp* may not reach 1. This is because sand is a non-viscous granular material and cannot withstand any tensile stress. Wu et al. [61] suggested that the adjustment coefficient *α<sup>r</sup>* should be 0.78–0.99 for liquefaction in sandy soils according to different soil characteristics. In this study, it is assumed that soil liquefaction occurs when the liquefaction potential reaches 0.9.

Figure 7 shows the liquefaction zone of soils at different times for the same wave condition. The numerical results for two cases—with and without PSR taken into account—are included. As shown in the figure, after 10 wave cycles, the seabed in the original PZIII model just begins to liquefy, and the liquefaction depth is less than 1 m. However, the seabed in the PSR model is markedly liquefied, and the depth is about 4 m. After 20 cycles of continuous wave action on the seabed surface, the depth of soil liquefaction changes slightly in the original PZIII model. However, in the present model, the liquefaction depth increases from 4 meters to 8 m. At the end of the simulation, the soil liquefaction depth in the original PZIII model is about 2 m. However, when considering the effect of PSR, the liquefaction depth increases to 14 m in the present model. It can be inferred that PSR plays a vital role in the stability of the seabed, and it also has an essential influence on the liquefaction depth. If the influence of PSR is neglected, the likelihood of the liquefaction of a sandy seabed is seriously underestimated, which poses a significant threat to coastal engineering.

(**a**) The original PZIII model

**Figure 7.** *Cont*.

(**b**) The Present model

**Figure 7.** Liquefaction process in a sandy seabed according to (**a**) the original PZIII (Pastor–Zienkiewicz Mark-III) model and (**b**) the present model.

Please note that the present model does not consider the process of progressive liquefaction (i.e., post-liquefaction). This is why the predicted liquefaction depth in Figure 7 is large (up to 14 m). As reported in the literature regarding wave-induced post-liquefaction [20,62], the maximum liquefaction depth approaches a constant value when the concept of progressive liquefaction is taken into account. However, this concept is not included in the present model. Therefore, the predicted liquefaction depth at *t*/*T* = 30 may be overestimated. This indicates that the existing model requires further improvement.

#### *4.2. Effect of Currents*

Waves and currents usually coexist in the natural marine environment, and the effects of currents on the seabed cannot be ignored. Currents not only change the length and direction of wave propagation but also affect the stability of the seabed. In this section, to demonstrate the effects of currents with PSR on the seabed response, a velocity of 1 m/s for both following and opposing currents was added to the present model to compare the soil response with the principal stress axis rotation (subject to combined wave and current loading). The term "following current" means that waves and currents propagate in the same direction, and the term "opposing current" indicates the opposite direction of waves and currents.

Figure 8 plots the variation in pore-water pressure and effective stress between soil particles for different current directions (following current *U*<sup>0</sup> = 1 m/s, no current *U*<sup>0</sup> = 0, and opposing current *U*<sup>0</sup> = −1 m/s). As seen from the figure, among the three cases, the soil for which the direction of current propagation is the same as that of wave propagation is liquefied first. When the current's direction is opposite of the wave propagation direction, the instability of soil liquefaction is effectively prevented. It is also noted that the pattern of the liquefaction zone is wave-like in appearance. A possible explanation is that the water particles move in the horizontal direction with combined wave and current loading, causing the wave pattern in the liquefaction zone. Unfortunately, there is no experimental evidence available to confirm these two findings. More detailed experimental works are required in the future. As shown in the figure, for the case with a following current, the seabed at *z* = −15 m is liquefied after 22 wave cycles, while liquefaction occurs after 28 wave cycles for the case without a current. However, the soil with reverse current experiences 30 wave cycles and does not show liquefaction.

**Figure 8.** *Cont*.

(**b**) Comparison of vertical effective stresses

**Figure 8.** Cyclic response of seabed under various current conditions at different locations: (**a**) pore pressures; (**b**) vertical effective stresses.

Figure 9 shows the horizontal and vertical displacements of soil particles for different directional currents at a depth of 5 m. It can be seen from the figure that the lateral displacement of soil remains almost unchanged before soil liquefaction, but the vertical displacement increases continuously. After 13 wave cycles, the liquefaction of soils occurs for both cases (with following current and no current), and the transverse displacement increases rapidly with time. This trend is more evident in the presence of the following current. In the vertical direction, the vibration amplitude of soil displacement is gentle, but when the soil is liquefied, the vibration of vertical displacement is significant. This phenomenon shows that before soil liquefaction, soil particles are continuously compressed, and pore-water pressure gradually rises but does not dissipate. These conditions eventually lead to soil liquefaction when pore pressure is higher than the vertical effective stress. When the soil loses its stability after liquefaction, the transverse and vertical displacements change significantly.

(**b**) Vertical displacement

**Figure 9.** Displacements under different current conditions: (**a**) horizontal displacement, (**b**) vertical displacement.

Figure 10 illustrates the distribution of liquefaction potential when considering the impact of PSR in the vertical direction of soil for different current conditions. It clearly shows that the soil near the surface is more prone to liquefaction. Moreover, the depth of soil liquefaction gradually increases. It also shows that when the soil depth is the same, the liquefaction potential in the following current case is the most significant, so soil liquefaction occurs more easily. On the other hand, when the liquefaction potential is the same, the maximum soil liquefaction depth in the following current situation is greater than the others. In other words, this result shows that the following current accelerates the process of soil instability, making soil liquefaction more likely to occur. Reverse currents have an opposite effect, so they are conducive to soil stability.

**Figure 10.** Effect of currents *U*<sup>0</sup> on the vertical distribution of liquefaction potential on *z* = −5 m at four typical time points.

#### *4.3. Effect of Principal Stress Rotation with Various Wave and Soil Parameters*

It is well known that when waves propagate on a porous seabed, wave parameters, including wave height and wave period, are closely related to the liquefaction of the seabed [4]. Generally, with a longer wavelength (*Lw*) and higher wave height (*H*), the seabed liquefaction depth is more obvious. Furthermore, different soil parameters, including soil permeability (*Ks*) and saturation rate (*Sr*), have a significant impact on seabed liquefaction. This section compares the effects of different wave periods, wave heights, soil permeabilities, and saturation levels on soil liquefaction with and without consideration of PSR. Four different time stages are used to illustrate the relationship between the vertical direction of soil and the liquefaction potential.

Figure 11 shows the effects of wave height on the vertical distribution of wave (current)-induced liquefaction potential with and without consideration of PSR conditions. As can be seen from the figure, in both situations, the liquefaction potential becomes more significant with increasing wave height. Also, the depth of soil liquefaction gradually increases over time. When *t*/*T* = 10 and *z*/*h* = 0.2, the value of *LP* increases from 0.25 to 0.82 with an increase in wave height from 1 m to 3 m when considering PSR. However, the liquefaction potential value increases from 0.15 m to 0.2 m when the effects of PSR are not considered.

Figure 12 illustrates the relationship between liquefaction potential and wave period with the change in soil depth. It can be seen in the figures that as the wave period increases, the value of the liquefaction potential increases and the soil is more liable to liquefy. This phenomenon is due to the increase in wavelength or wave height, both of which result in more energy. It can strengthen the interaction between the wave and seabed foundation. Also, at the same soil depth, the value of the liquefaction potential increases more significantly when PSR is taken into account. Therefore, it can be said that given different wave parameters, PSR increases the liquefaction depth and liquefaction potential of the soil.

Figure 13 shows the impact of soil permeability on the liquefaction potential along the vertical direction in the sandy seabed. Similarly, there is a steady increase in the liquefaction potential with the wave propagation for 35 wave periods. Also, the results of *Ks* = 10−<sup>5</sup> m/s and *Ks* = 10−<sup>7</sup> m/s are almost indistinguishable. Compared with *Ks* = 10−<sup>2</sup> m/s, when soil permeability is lower, the soil is more likely to liquefy. This is because the permeability of the soil is large, and the pore pressure between the soil particles dissipates rapidly during the wave propagation and does not increase cumulatively. Thus, the effective stress between soil particles is sufficient to maintain the stability of the soil, and this stability prevents the occurrence of soil liquefaction. When the permeability of the soil is relatively low, the pore-water pressure between soils does not dissipate efficiently along with wave propagation, resulting in the faster accumulation of pressure. Therefore, soil liquefaction easily occurs when soil permeability is small. When considering the existence of PSR, the liquefaction potential increases rapidly in the same situation. Therefore, it can be concluded that PSR has a considerable impact on soil liquefaction for different permeabilities.

(**a**) The original PZIII model without PSR

**Figure 11.** *Cont*.

*J. Mar. Sci. Eng.* **2019**, *7*, 123

(**b**) The present model with PSR

**Figure 11.** Effect of wave height *H* on the vertical distribution of liquefaction potential on *z* = −5 m at four typical time points. (**a**) The original PZIII model without PSR (principal stress rotation) and (**b**) the present model with PSR.

**Figure 12.** *Cont*.

**Figure 12.** Effect of wave period *T* considering PSR on the vertical distribution of liquefaction potential on *z* = −5 m at four typical time points. (**a**) The original PZIII model without PSR and (**b**) the present model with PSR.

**Figure 13.** *Cont*.

(**b**) The present model with PSR

**Figure 13.** Effect of soil permeability *Ks* considering PSR on the vertical distribution of liquefaction potential on *z* = −5 m at four typical time points. (**a**) The original PZIII model without PSR and (**b**) the present model with PSR.

Figure 14 presents the variation trend of the liquefaction potential with soil depth at different saturation levels at four typical time points. On the whole, the relationship between *LP* and different saturation levels shows the opposite trend, with a certain soil depth as the limit. Since the liquefaction potential is set near *Lp* = 0.9, the region with low *Lp* values effectively means that there is no liquefaction at all. Therefore, we only need to investigate the region near the seabed surface. In the surface layer of the soil, the liquefaction potential decreases with increasing saturation.

**Figure 14.** *Cont*.

*J. Mar. Sci. Eng.* **2019**, *7*, 123

**Figure 14.** Effect of the degree of *Sr* considering PSR on the vertical distribution of liquefaction potential on *z* = −5 m at four typical time points. (**a**) The original PZIII model without PSR and (**b**) the present model with PSR.

#### **5. Conclusions**

Principal stress rotation (PSR) is an important factor in the evaluation of wave (current)-induced seabed instability. In this study, a one-way coupled numerical model that incorporates a wave model and soil model was developed to investigate the effect of PSR on the wave (current)-induced dynamic response of an elastoplastic seabed foundation. The comparisons show that the proposed model agrees well with laboratory wave flume tests, geotechnical centrifuge tests, and previous numerical results. On the basis of the numerical examples, the effects of PSR and currents with various soil parameters were examined, and the following conclusions are drawn:


Please note that the above conclusions are based on the numerical examples presented in this manuscript, and comparable experimental data are not available in the literature. The above findings require further confirmation by experimental evidence in the future.

In this study, we adopted the model proposed by Zhu et al. [39] for the soil response to combined wave and current loading. Note that Liu et al. [63] proposed another model to modify the previous model [35], which can also be used for the present problem.

**Author Contributions:** Conceptualization, D.-S.J.; methodology, J.-F.Z., H.Z. & Z.L.; validation, J.-F.Z. & Z.L.; formal analysis, Z.L.; writing—original draft preparation, Z.L.; writing—review and editing, D.-S.J., J.-F.Z. & H.Z.; supervision, D.-S.J.

**Funding:** This research was funded by the National Nature Fund of China (Grant No. 51879133).

**Acknowledgments:** The first author is grateful for the support of the High Performance Computing Cluster "Gowanda" to complete this research and the support of a scholarship from Griffith University. The third author is grateful for the support from the National Nature Fund of China (Grant No. 51879133), who are acknowledged for their financial support. The authors express their gratitude to WG Qi and Professor FP Gao at the Institute of Mechanics, Chinese Academy of Sciences, for their experimental data.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Meshfree Model for Wave-Seabed Interactions Around Offshore Pipelines**

**Xiao Xiao Wang 1, Dong-Sheng Jeng 1,\*, Chia-Cheng Tsai 2,3**


Received: 26 February 2019; Accepted: 22 March 2019; Published: 28 March 2019

**Abstract:** The evaluation of the wave-induced seabed instability around a submarine pipeline is particularly important for coastal engineers involved in the design of pipelines protection. Unlike previous studies, a meshfree model is developed to investigate the wave-induced soil response in the vicinity of a submarine pipeline. In the present model, Reynolds-Averaged Navier-Stokes (RANS) equations are employed to simulate the wave loading, while Biot's consolidation equations are adopted to investigate the wave-induced soil response. Momentary liquefaction around an offshore pipeline in a trench is examined. Validation of the present seabed model was conducted by comparing with the analytical solution, experimental data, and numerical models available in the literature, which demonstrates the capacity of the present model. Based on the newly proposed model, a parametric study is carried out to investigate the influence of soil properties and wave characteristics for the soil response around the pipeline. The numerical results conclude that the liquefaction depth at the bottom of the pipeline increases with increasing water period (*T*) and wave height (*H*), but decreases as backfilled depth (*Hb*), degree of saturation (*Sr*) and soil permeability (*K*) increase.

**Keywords:** oscillatory liquefaction; wave-soil-pipeline interactions; meshfree model; local radial basis functions collocation method

#### **1. Introduction**

Offshore pipelines have been a commonly used facility for transportation of offshore oil and gas. In addition to construction causes, another key failure mode is the wave-induced seabed instability in the vicinity of pipelines [1,2]. Therefore, the evaluation of seabed stability around the pipeline is one of key factors that needs to be considered in an offshore pipeline project.

In general, ocean waves will exert fluctuations of dynamic pressures over the sea floor, which will further induce excess pore pressures and effective stresses within the seabed. The shear resistance in the vicinity of pipelines may be loss due to the liquefaction of surrounding soil, when the excess pore pressure increases. Thus, it is particularly important to understand the process of the wave-pipeline-soil interactions for the design of submarine pipelines [3]. The mechanisms of the wave-induced soil liquefaction can be classified into two categories, residual and oscillatory, in accordance with the way how the excess pore pressure is generated [4]. The residual liquefaction mechanism is resulted from the build-up of pore pressure induced by volumetric contraction under cyclic loading [5]. Momentary liquefaction usually appears in the seabed under wave troughs where the pore pressure is accompanied with some damping and phase lag [6]. This study focuses on the second mechanism.

Numerous investigations for the wave-induced soil response around submarine pipelines by adopting conventional numerical methods have been carried out since the 1980s, such as finite-element method (FEM), finite difference method (FDM) and boundary element method (BEM). Among these, Cheng and Liu [7] introduced the Boundary Integral Equation to obtain the distribution of pore-water pressure around a pipe fully buried in a sediment-filled trench. In their model, the inertia terms were considered, i.e., it is a *u* − *p* approximation. Thomas [8,9] established a one-dimensional finite-element model to analyze the wave-induced soil response in saturated and unsaturated soil. Later, the model was extended to investigate the influences of variable permeability and shear modulus of soil and non-linearity of wave loading [10,11]. Furthermore, by using finite-element formulation, Madga [12–14] further estimated the wave-induced pore pressure and uplift force acting on submarine pipelines. Based on the assumption of no slipping at the interface between pipeline and soil, Jeng and Lin [15] developed a finite-element model to examine the wave-pipeline-seabed interactions in an in-homogeneous seabed. Then, the inertial forces and soil-pipe contact effects were involved in Luan et al. [16]'s model. With the commercial software (ABQUS), the effects of combined non-linear wave and current loading were considered by Wen et al. [17]. Recently, by considering pre-consolidation due to self-weight of the pipeline, Zhao et al. [18] investigated the build-up pore pressures. Recently, Zhao and Jeng [19] extended the integrated numerical model to evaluate the influence of backfilled depth of trench layer. Later, Duan et al. [20] proposed a two-dimensional model to investigate the wave and current-induced soil response around a partially buried pipeline in a trench layer.

The aforementioned investigations have employed the conventional approaches with meshes, for example, the principle of FEM is to divide the computational domain into small elements, and these elements do not overlap each other. A field function was established within each element by adopting simple interpolation functions. If the element is severely distorted, the shape function of this element would be of poor quality, which may lead to an unacceptable numerical result. Unlike the finite-element technique, the interpolation functions are established directly on nodes instead of elements, when meshless methods are applied. This could avoid the drawback of the conventional mesh-based techniques such as FEM and FDM. In recent years, meshless methods have attracted increasing attention from numerical modelers due to the faster formulation process with less data storage and no extensive mesh.

The most commonly used meshless methods are method of fundamental solution (MFS), method of particular solutions (MPS), smooth particle hydrodynamics (SPH), global radial basis functions (RBF) collocation method (GRBFCM) and local RBF collocation method (LRBFCM). One of the earliest meshless methods is SPH [21,22]. Randles and Libersky [23] was the first group to apply SPH in solid mechanics, and later this method was improved and was adopted in more fields. The main drawbacks of this method are inaccurate results near boundaries and tension instability, which was first studied in 1995. Over the ensuring decades, more meshless methods have been proposed. Karim et al. [24] presented a two-dimensional model using the element-free Galerkin method to investigate transient response of saturated porous elastic soil under cyclic loading system. A radial point interpolation meshless method (radial PIM) was developed by Wang et al. [25] to avoid the occurrence of singularity associated with only polynomial basis. Later, the radial PIM was applied to solve Biot's consolidation problem [26] and wave-induced seabed response [27]. Existing meshless models have investigated water-soil interaction without any structure. Thus, the present study firstly establishes a meshless model to investigate the wave-seabed-pipeline interactions under various wave loading.

Among meshfree methods, the RBF are commonly used. The global RBF collocation method was first proposed for multivariate data interpolation and partial differential equations [28,29]. This method was further modified [30–32]. The GRBFCM can be used to deal with arbitrary and complex domains, but it usually leads to an ill-conditioned system matrix when high resolutions are required. Later, a localization procedure (LRBFCM) was proposed to overcome prescribed difficulty, to transform the dense system matrices into sparse ones [33]. Based on the multi-quadric RBF [34], Lee et al. [33] first

proposed the local RBF collocation method (LRBFCM), and then this method was applied to various problems [35–37].

In this study, LRBFCM is employed to investigate the wave-induced oscillatory liquefaction around a pipeline in a trench layer. The proposed seabed model is validated with the analytical solution [38], experimental data [39,40] and numerical models [7,41]. Then, a parametric study is conducted to evaluate the influence of pipeline configuration, wave characteristics, and seabed properties for the wave-induced pore pressure around a partially buried pipeline in a trench.

#### **2. Theoretical Models**

In this study, an impermeable pipeline is considered with a radius of *R* which is partially buried in a trench with a finite thickness (*h*), as shown in Figure 1. The propagation direction of waves is along the positive *x*-direction.

**Figure 1.** Sketch for wave-seabed-pipeline interactions.

The present model consists of two sub-models: wave and seabed sub-models. By using opensource CFD toolbox OpenFoam (Open Field Operation and Manipulation), the wave model proposed by Higuera et al. [42] is adopted for generating various wave loadings. A new seabed model is established based on LRBFCM.

By means of Volume-Averaged Reynolds-Averaged Navier-Stokes (VARANS) equations, the wave model was developed for coastal engineering applications by dealing with three-dimensional two-phase flow which was based on a solver in OpenFOAM, IHFOAM [42]. More detailed information about IHFOAM and its applications in coastal engineering, readers can refer to the publications of Higuera and his co-workers [42–45].

#### *2.1. Boundary Value Problem for the Seabed Model*

By neglecting the inertial effect, a quasi-static seabed model is established for the wave-induced seabed response based on the assumption of homogeneous seabed and compressible pore fluid. The effects of inertial terms on the wave-induced soil response has been reported in Jeng et al. [46]. For a two-dimensional problem, the governing equation for compressible homogeneous soil and compressible pore fluid can be represented as [47]:

$$K\nabla^2 p - \gamma\_{\rm wr} n' \beta \frac{\partial p}{\partial t} - \gamma\_{\rm w} \frac{\partial \varepsilon}{\partial t} = 0,\tag{1}$$

where *p* is pore-water pressure, *γ<sup>w</sup>* is the unit weight of water, *n* is soil porosity, *t* is the time. In (1), the volume strain () and compressibility of pore fluid (*β*) are defined as

*J. Mar. Sci. Eng.* **2019**, *7*, 87

$$
\epsilon = \frac{\partial u}{\partial x} + \frac{\partial w}{\partial z'}, \text{ and } \beta = \frac{1}{K\_{\text{uv}}} + \frac{1 - S\_r}{P\_{\text{uo}}}, \tag{2}
$$

where *u* and *w* are the soil displacements in the *x*- and *z*-direction, respectively; *Kw* is the true modulus of elasticity (*Kw* = <sup>2</sup> × <sup>10</sup><sup>9</sup> N/m2, [6]), *Sr* is the degree of saturation and *Pwo* is related to the absolute water pressure.

Based on Newton's second law, the force balance for the porous flow in *x*− and *z*− directions can be expressed respectively as

$$\frac{\partial \sigma\_x^{\prime}}{\partial x} + \frac{\partial \tau\_{xz}}{\partial z} = \frac{\partial p}{\partial x},\tag{3}$$

$$\frac{\partial \mathfrak{r}\_{xz}}{\partial x} + \frac{\partial \sigma'\_z}{\partial z} = \frac{\partial p}{\partial z'} \tag{4}$$

where *σ <sup>x</sup>* and *σ <sup>z</sup>* are effective normal stresses; *τxz* denotes shear stress component. In this study, tension is determined as positive.

Based on Hook's law, the effective normal stresses and shear stress can be expressed in term of soil displacements, i.e.,

$$
\sigma\_x' = 2G \left[ \frac{\partial u}{\partial x} + \frac{\mu}{1 - 2\mu} \epsilon \right],
\tag{5}
$$

$$
\sigma\_z' = 2G \left[ \frac{\partial w}{\partial z} + \frac{\mu}{1 - 2\mu} \epsilon \right],
\tag{6}
$$

$$\tau\_{\text{xz}} = G \left[ \frac{\partial u}{\partial z} + \frac{\partial w}{\partial x} \right] \,\text{\,\,\,}\tag{7}$$

where the shear modulus *G* is defined with Young's modulus (*E*) and the Poisson's ratio (*μ*) in the form of *E*/2(1 + *μ*).

Substituting (5)∼(7) into (3) and (4), the force equilibrium can be represented as

$$
\delta G \nabla^2 \mu + \frac{G}{1 - 2\mu} \frac{\partial \epsilon}{\partial x} - \frac{\partial p}{\partial x} = 0,\tag{8}
$$

$$G\nabla^2 w + \frac{G}{1 - 2\mu} \frac{\partial \epsilon}{\partial z} - \frac{\partial p}{\partial z} = 0. \tag{9}$$

To solve the pore pressures and soil displacements in (1), (8) and (9), the following boundary conditions are required.

• At seabed surface (*z* = 0) and trench surface, the vertical effective stress and shear stress vanish, and the pore pressure is equal to dynamic wave pressure.

$$
\sigma'\_z = \tau\_{\overline{x}\overline{z}} = 0,\text{ and }\ p = P\_{b\prime} \tag{10}
$$

where *Pb* is the dynamic wave pressure at the seabed surface, which is obtained from the wave model (IHFOAM).

• At the impermeable seabed bottom (*z* = −*h*), zero displacements and no vertical flow are specified, i.e.,

$$
\mu = w = 0, \quad \text{and} \quad \frac{\partial p}{\partial z} = 0,\tag{11}
$$

• The pipeline surface is assumed to be impermeable wall. Thus, there is no flow through the pipeline surface, i.e.,

$$\frac{\partial p}{\partial n} = 0,\tag{12}$$

where *n* denotes normal vector of the pipe surface.

#### *2.2. Meshfree Model for the Seabed Domain*

In this study, a rigid pipeline is considered to be partially buried in a trench. The computational domain is discretized into *N* nodes non-uniformly. Therefore, a linear equation of the following form is required to be established:

$$[A]\_{N \times N} \begin{bmatrix} \Phi \end{bmatrix}\_{N \times 1} = [B]\_{N \times 1} \tag{13}$$

where [Φ]*N*×<sup>1</sup> is the sought solution, [*B*]*N*×<sup>1</sup> is a column vector, and [*A*]*N*×*<sup>N</sup>* is a sparse system matrix. Similar structures of [*A*]*N*×*<sup>N</sup>* can be found in the FDM and the finite-element method.

For constructing a linear equation for each node **y***<sup>n</sup>* in the computational domain, Φ in (13) is assumed as Φ(**x**) by RBFs:

$$\Phi(\mathbf{x}) \approx \sum\_{m=1}^{K} \alpha\_m \chi(r\_m),\tag{14}$$

where Φ denotes either *p* or *ui* in the governing equations, *α<sup>m</sup>* refers to the corresponding undetermined coefficient and *rm* = **x** − **x***<sup>m</sup>* is the Euclidean distance from **x** to **x***m*. The group of **x***<sup>m</sup>* denote the locations of the *K*¯ nearest neighbor nodes surrounding the prescribed center **x**1. In this study, the kd-tree algorithm is applied to search the *K*¯ nearest neighbor nodes efficiently [48]. Furthermore, the multi-quadric RBF is expressed as

$$
\chi(r\_m) = \sqrt{r\_m^2 + c^2},
\tag{15}
$$

with the shape parameter (*c*) [34].

A localization process [33,35,49] is presented here for the sake of preventing unnecessary ill-conditioned system matrix. Firstly, the expression of *rm* is substituted into (14) as

$$\Phi(\mathbf{x}\_n) = \sum\_{m=1}^{\mathbb{R}} \mathfrak{a}\_m \chi(\left| \left| \mathbf{x}\_m - \mathbf{x}\_n \right| \right), \tag{16}$$

or in matrix-vector form as

$$\left[\Phi\right]\_{\mathbb{R}\times 1} = \left[\chi\right]\_{\mathbb{R}\times \mathbb{R}}\left[\alpha\right]\_{\mathbb{R}\times 1'}\tag{17}$$

where

$$[\Phi]\_{\vec{\mathbb{K}} \times 1} = \begin{bmatrix} \Phi(\mathbf{x}\_1) \\ \Phi(\mathbf{x}\_2) \\ \vdots \\ \Phi(\mathbf{x}\_{\vec{\mathbb{K}}}) \end{bmatrix} \tag{18}$$

$$\begin{aligned} [\chi]\_{\mathcal{R}\times\mathbb{R}} &= \begin{bmatrix} \chi(\|\left\|\mathbf{x}\_{1}-\mathbf{x}\_{1}\|) & \chi(\|\left\|\mathbf{x}\_{1}-\mathbf{x}\_{2}\|) & \cdots & \chi(\|\left\|\mathbf{x}\_{1}-\mathbf{x}\_{\mathcal{R}}\|) \\ \chi(\|\left\|\mathbf{x}\_{2}-\mathbf{x}\_{1}\|) & \chi(\|\left\|\mathbf{x}\_{2}-\mathbf{x}\_{2}\|) \right\| & \cdots & \chi(\|\left\|\mathbf{x}\_{2}-\mathbf{x}\_{\mathcal{R}}\|) \right\|) \\ \vdots & \vdots & \ddots & \vdots \\ \chi(\|\left\|\mathbf{x}\_{\mathcal{R}}-\mathbf{x}\_{1}\|) & \chi(\left\|\left\|\mathbf{x}\_{\mathcal{R}}-\mathbf{x}\_{2}\|\right\|) \right\| & \cdots & \chi(\left\|\left\|\mathbf{x}\_{\mathcal{R}}-\mathbf{x}\_{\mathcal{R}}\|\right\|) \end{bmatrix} \end{aligned} \tag{19}$$

and

$$[\boldsymbol{\alpha}]\_{\mathbb{K}\times1} = \begin{bmatrix} \boldsymbol{\alpha}\_1 \\ \boldsymbol{\alpha}\_2 \\ \vdots \\ \boldsymbol{\alpha}\_{\mathbb{K}} \end{bmatrix}.\tag{20}$$

*J. Mar. Sci. Eng.* **2019**, *7*, 87

Then, (17) can be inverted as

$$\left[\mathfrak{a}\right]\_{\mathbb{K}\times1} = \left[\chi\right]\_{\mathbb{K}\times\mathbb{K}}{}^{-1}\left[\Phi\right]\_{\mathbb{K}\times1} \,. \tag{21}$$

Now, *L*Φ(**x**) is considered to replace Φ(**x**) defined in (14), where *L* is a linear differential operator related to both the governing equation and the boundary conditions. The collocation of *L*Φ(**x**) on **x**<sup>1</sup> = **y***<sup>n</sup>* gives

$$L\Phi(\mathbf{y}\_n) = \sum\_{m=1}^{\mathcal{K}} \alpha\_m L\chi(r\_m) \mid\_{\mathbf{x} = \mathbf{x}\_{1'}} \tag{22}$$

or in matrix-vector form as

$$L\Phi(\mathbf{y}\_n) = [L\chi]\_{1\times\mathcal{K}} \, [a]\_{\mathcal{K}\times 1} \,. \tag{23}$$

In (23), the existence of *Lχ*(*rm*) is as a result of the influence of operator *L* on the RBF *χ*(*rm*). Thus, (17) and (23) can be combined as

$$L\Phi(\mathbf{y}\_n) = [\mathbb{C}]\_{1 \times \mathbb{R}} \, [\Phi]\_{\mathbb{R} \times 1 \, \prime} \, \tag{24}$$

with

$$\left[\mathbb{C}\right]\_{1\times\mathbb{R}} = \left[L\chi\right]\_{1\times\mathbb{R}}\left[\chi\right]\_{\mathbb{R}\times\mathbb{R}}{}^{-1}\text{.}\tag{25}$$

and

$$L[\boldsymbol{L}\boldsymbol{\chi}]\_{1\times\mathbb{K}} = \begin{bmatrix} L\boldsymbol{\chi}(r\_1) \ |\_{\mathbf{x}=\mathbf{x}\_1} & L\boldsymbol{\chi}(r\_2) \ |\_{\mathbf{x}=\mathbf{x}\_2} & \cdots & L\boldsymbol{\chi}(r\_{\mathbb{K}}) \ |\_{\mathbf{x}=\mathbf{x}\_{\mathbb{K}}} \end{bmatrix}.\tag{26}$$

From (24)–(26), it can be found that the row vector [*C*]1×*K*¯ can be obtained if all the values of *L*, *χ* and **x***<sup>j</sup>* are known. These equations can be assembled into the system matrix, and finally the resultant sparse system is solved by using the direct solver of SuperLU in this study, which finished the procedure of LRBFCM.

Please note that the radial PIM was adopted to solve Biot's consolidation problem [26] and wave-induced soil response [27], while the present model uses LRBFCM. The mixed bases of polynomial and radial bases are needed in the radial PIM for the accuracy of polynomials [25]. Compared with the radial PIM, the choice of basis functions in LRBFCM is easier. Furthermore, no submarine structure was included in their model [27]. Thus, the present seabed model is the first model by applying LRBFCM to investigate the wave-seabed interactions around a structure such as pipelines.

#### *2.3. Effects of Lateral Boundary Conditions*

This section presents two ways to handle the lateral boundary conditions: periodic and fixed. Generally speaking, the horizontal and vertical displacements and the pore pressure do not vanish at lateral boundaries. To deal with the problem of boundaries for wave-seabed interactions, Jeng et al. [50] applied the principle of repeatability [51]. However, the condition of employing periodic boundary condition is that the length of seabed must be an integer number of wavelength. Moreover, periodic boundary condition is not applicable for seabed with structures. Thus, Ye and Jeng [52] suggested another method by which employing a large computational domain and meanwhile fixing both the lateral boundaries in the horizontal direction, namely considering the boundary as impermeable. This method is under the assumption that the fixed lateral boundary only influences the region nearby. Similar with Ye and Jeng [52], in this section, both periodic and fixed lateral boundary conditions by LRBFCM are examined.

Theoretically, the larger the computational domain, the smaller effects of lateral boundaries. However, a large computational domain requires more computational resources. Thus, the length of computational domain is assumed as 3 times of the periodic wavelength in this study. The input data used is listed in Table 1. The maximum pore pressure and effective stresses of soil of these three sections are depicted in Figure 2.


**Table 1.** Input data for numerical examples demonstrating effects of lateral boundary conditions.

As depicted in Figure 2, sections *A* − *A* (*x* = 50 m), *B* − *B* (*x* = 125 m) and *C* − *C* (*x* = 200 m) are in the range of the first wave length, the second wavelength and the third wavelength, separately. Solid lines and dashed lines represent the soil response of the case with periodic and fixed boundaries, respectively. As shown in Figure 2a,c, the effect of fixed boundary condition is minor for vertical effective stress and pore pressure in sections *A* − *A* and *C* − *C* , but considerable difference can be observed from the horizontal effective stress, which leads to the conclusion that fixed lateral boundaries affect the soil response of the soil region near lateral boundaries significantly. Furthermore, the seabed response at section *B* − *B* under periodic boundary conditions is in a complete agreement with that under fixed boundary conditions, which means that the influence of fixed lateral boundaries vanish in the section far away from the boundary. Thus, fixed lateral boundary condition is employed in this study for investigating the wave-induced soil response.

**Figure 2.** Comparison of wave-induced seabed response between the cases with periodic and fixed lateral boundaries.

#### *2.4. Convergent Tests*

The present seabed model is a new model established by employing LRBFCM for the wave-induced soil response in the vicinity of submarine pipelines, it is necessary to check its convergence. Figure 3 presents three tests for model convergence which is with respect to node number of the whole computational domain, a model parameter *c* (where *c* is equal to "30× the maximum distance between each two nodes in the local region") and the value of *K*¯ (where *K*¯ indicates the node number of the local region). The pipeline is considered to be impermeable, and input data are as follows: *d* = 0.533 m, *L* = 1.25 m, *h* = 0.826 m, *μ* = 0.33, *n* = 0.42, *K* = 0.0011 m/s, *S* = 0.997, *R* = 0.084 m, *b* = 0.167 m, *Lx* = 4.57 m.

In principle, instability of the trend of soil response around pipeline occurs at the beginning stage of node number increase. The results should remain unchanged after the node number is increased to a certain extent. With fixed node numbers, the numerical results should not be changed when the value of *c* is in a reasonable range, which can prove that the model is convergent and reliable. As presented in Figure 3a, the wave-induced pore pressure keep changed when the node number varies from 16,558 to 45,000 approximately, but the values maintain a steady state in the process of the node number increases from 45,375 to 53,351 and even 65,231, which verifies the stability of the model. *c* is one of coefficients of the present model. In Figure 3b, the node number is determined as 45,375, then the trend of pore pressure can be observed through changing the value of *c*. There is almost no change for the pore pressure in the vicinity of the pipeline when *c* is equal to 0.3, 0.548 and 0.8, respectively, which can be evidence of the model convergence. *K*¯ refers to the number of the nearest neighbor nodes of unknown node **x**. Usually, the value of *K*¯ can be regarded as 5, 9, and 13. From Figure 3c, it can be found that 9 or 13 is applicable for the present model, and the result looks more smooth when *K*¯ is equal to 9. Thus, the convergence of present model is verified from these three cases. Combined Figure 3a–c, it can be concluded that the numerical result of this case scale is satisfactory when the node number, and the value of *c* and *K*¯ are determined as 45375, 0.548 and 9, respectively.

**Figure 3.** The wave-induced pore pressure in the vicinity of pipeline with various node number, *c* and *K*¯.

#### **3. Model Validation**

#### *3.1. Comparison with the Analytical Solution for Wave-Seabed Interactions*

For a homogeneous seabed, the previous analytical solution for the wave-induced soil response [38] will be compared with present numerical results. This comparison is to confirm the capacity of the present model.

In this comparison, the following input data are used: wave period *T* = 15.0 sec, water depth *d* = 70 m, wavelength *L* = 311.59 m, thickness of seabed *h* = 25 m, Poisson's ratio *μ* = 0.333, soil porosity *n* = 0.3, soil permeability *K* = 10−<sup>4</sup> m/s for fine sand and 10−<sup>2</sup> m/s for coarse sand, degree of saturation *S* = 0.932 for unsaturated soil, shear modulus *G* = 107 N/m2 . The numerical results of the comparison are presented in Figure 4. In the figure, the present results are presented by lines and the analytical solution [38] is denoted as circles. The vertical distributions of the maximum amplitude of the wave-induced pore pressure (|*p*|/*p*0) and effective stresses (|*σ <sup>x</sup>*|/*p*0, |*σ <sup>z</sup>*|/*p*0), and shear stress (|*τxz*/*p*0|) versus *z*/*h* are presented. In the figure, *p*<sup>0</sup> is the amplitude of linear wave pressure at the seabed surface, which is defined as *p*<sup>0</sup> = *γwH*//2 cosh *kd*. It is found that the present results are in complete accordance with the analytical solution of Hsu and Jeng [38]. The difference between the analytical solution and the present model is less than 10−<sup>3</sup> for both fine and coarse sands.

**Figure 4.** Comparison of the vertical distribution of pore pressure (|*p*|/*p*0) and stresses (|*σ <sup>x</sup>*|/*p*0, |*σ <sup>z</sup>*|/*p*<sup>0</sup> and |*τxz*|/*p*0) versus *z*/*h* between the present model (the solid lines) and analytical solutions [38] (symbols).

#### *3.2. Comparison with Experimental Data and FEM Results for Wave-Pipeline-Seabed Interaction*

The second validation is the comparison between present numerical results and experimental results [39] with respect to the linear wave-induced soil response around a fully buried pipeline. Considering an impermeable pipeline with a radius of *R* is fully buried within a porous elastic seabed

with a finite thickness (*h*). The propagation direction of waves is regarded along the positive *x*-direction. The input data employed in this validation is same as Section 2.4. Turcotte et al. [39] reported seven experiments with different wave period and wave height, and only three typical comparisons are presented here which are that with the longest wave length (*L* = 4.91 m, *T* = 2.3 s, *H* = 0.0302 m), the medium wave length (*L* = 3.536 m, *T* = 1.75 s, *H* = 0.143 m) and the shortest wave length (*L* = 1.25 m, *T* = 0.9 s, *H* = 0.0524 m). The water depth was a constant of 0.533 m for all tests. In the comparison, the results from previous boundary element model [7] and the COMSOL finite-element model are also included. The COMSOL model was based on the one proposed by Jeng and Zhao [41] and applied to the case with a buried pipeline.

From Figure 5a, slight difference can be found from the comparison between the numerical results and experimental data. For the intermediate wave period illustrated in Figure 5b, the present model is the closest to the experimental data than other two numerical models. From the third circumstance shown in Figure 5, it can be seen that the error of the finite-element model with the experimental data is relatively higher than the error of present results with the experimental data. Furthermore, the length of the computational domain was fixed as 4.57 m. Hence, the fixed lateral boundary condition is not applicable for the situation of *L* = 4.91 m. However, Figure 5 still presents that case for a complete comparison with the numerical solution of Cheng and Liu [7]. It can be observed that the amplitude of wave-induced pore pressure increases with the increase of the wavelength.

**Figure 5.** The wave-induced pore pressure in the vicinity of a fully buried pipeline. (red dashed line: numerical results of Cheng and Liu [7]; blue star: FEM results by COMSOL model [41]; circle line: experimental data [39]; solid line: present results).

#### *3.3. Comparison with Experimental Data for Wave-Induced Soil Response Around a Pipeline Buried in a Trench*

The third validation is to compare the model with experimental data [40] for the case of a homogeneous seabed, in which the pipeline is in a trench. Stoke II wave loading simulated with OpenFoam is employed in this case. Sun et al. [40] conducted a series of laboratory experiments to examine the wave-induced pore-water pressure along the surface of a pipeline partially backfilled in a trench, and for brevity of presentation, only two typical comparisons are shown in this section in which the wave and seabed conditions are listed in Tables 2 and 3. The corresponding comparisons of wave-induced pore pressure around the pipeline are illustrated in Figure 6. Red lines denote present numerical results and circle denote experimental data. It can be observed that the present numerical model agrees well with the laboratory experiments.


**Table 2.** Input data for the comparison with experimental data [40].

**Table 3.** Wave and seabed conditions for the comparison with experimental data [40].

**Figure 6.** Distribution of wave-induced pore pressure around a trenched pipeline (red solid line: the present result; circle: experimental data [40]).

#### **4. Results and Discussion**

The aim of this study is to investigate the wave-pipeline-seabed interactions around a trenched pipeline by employing the proposed time-dependent meshless seabed model. In this section, the influence of soil properties, wave characteristics, and pipe configuration on the wave-induced

oscillatory liquefaction are examined. Zen and Yamazaki [53] introduced and verified the concept of "oscillatory" excess pore pressure by conducting a series of experiments, and their criterion to determine the soil oscillatory liquefaction is used in this study, which can be expressed as

$$
\sigma\_0'(z,0) \le u\_\mathfrak{e}(z,t) = -[p(0,t) - p(z,t)]\tag{27}
$$

where *σ* <sup>0</sup>(*z*, 0) is the initial effective stress, and *ue*(*z*, *t*) = −[*p*(0, *t*) − *p*(*z*, *t*)] means the excess pore pressure. *p*(0, *t*) and *p*(*z*, *t*) denote the wave pressure at seabed surface and wave-induced pore pressure, respectively.

As shown in Figure 1, the pipeline is buried in the partially backfilled trench, and the lateral boundaries are considered as impermeable in this case. The direction of wave propagates along the positive *x*-direction. The Stokes II wave loading is simulated by OpenFOAM. Wave and soil parameters are listed in Table 4. When the effect of one parameter for the wave-induced pore pressure is examined, values of other parameters are kept fixed.


**Table 4.** Input data for the parametric study.

#### *4.1. Effects of Soil Characteristics*

Soil characteristics are significant factors to affect the wave-induced oscillatory soil response in the vicinity of a partially buried pipeline. In this section, two parameters are examined in detail. They are the degree of saturation *Sr* and soil permeability *K*. Figure 7 presents the distribution of liquefaction depth around pipeline in the trench under various soil conditions, and Figure 8 shows the distributions of the maximum excess pore pressures under wave trough in the vertical section through the center of the pipeline. For Figures 7a and 8a, permeability is 4.5 × <sup>10</sup>−<sup>3</sup> m/s. For Figures 7b and 8b, degree of saturation is 0.98. Soil properties of backfills are chosen as same as bottom soil.

Three typical values of degree of saturation are considered in this section, they are: 95%, 97% and 99%, respectively. Figure 7a demonstrates that degree of saturation (*Sr*) significantly affects the liquefaction depth in the trench. The depth is deeper with decreasing degree of saturation. Furthermore, from Figure 8a, it can be found that the soil on the bottom of the pipeline is much easier to be liquefied when degree of saturation is relatively small.

To investigate the influence of soil permeability on the wave-induced soil response, Figure 7b illustrates the distribution of the liquefaction depth in the trench with variable value of permeability, <sup>4</sup> × <sup>10</sup>−<sup>3</sup> m/s, 1 × <sup>10</sup>−<sup>3</sup> m/s and 7 × <sup>10</sup>−<sup>4</sup> m/s, respectively. Figure 8b shows the distribution of the maximum excess pore pressure under wave trough along the vertical section through the center of pipeline with variable permeability. It can be concluded that the liquefaction depth become large with decreasing permeability, and soil around the impermeable pipeline is much easier to be liquefied when permeability is relatively small.

**Figure 7.** Distribution of the liquefaction depth around the partially buried pipeline for variable degree of saturation and permeability.

**Figure 8.** Distribution of the excess pore pressure (|*ue*|/*σ* <sup>0</sup>) under wave trough along the vertical section through the center of the pipeline for different soil properties.

#### *4.2. Effects of Wave Characteristics*

In addition to soil characteristics, wave parameters have been found to significantly influence the wave-pipeline-seabed interactions. The influences of two wave parameters, wave height (*H*) and period (*T*), are examined in this section. The wave height *H* can directly affect the magnitude of wave loading exerting on the seabed surface. Wave period *T* affects the wave-induced oscillatory excess pore pressure by affecting wavelength. Figure 9 illustrates the distribution of oscillatory liquefaction depth in the trench under various wave height (*H*) and period (*T*), and the wave period is 10 s for Figure 9a, the wave height (*H*) is 3 m for Figure 9b. Figure 10 shows the distribution of the maximum excess pore pressure |*ue*|/*σ* <sup>0</sup> along the vertical section under the center of the pipeline.

Three values of wave height are examined in this section: 2 m, 2.5 m and 3.1 m. From Figure 9a, it can be found that the liquefaction depth increases with increasing wave height. Meanwhile, from Figure 10a, it can be seen that soil under the pipeline is much easier to be liquefied when wave height is relatively large.

Figures 9b and 10b present the effect of wave period on the wave-induced oscillatory excess pore pressure around the trenched pipeline. Liquefaction depth from the seabed surface increases when wave period increases, and the soil on the bottom of pipeline is much easier to be liquefied when wave period is large.

**Figure 9.** Distribution of the liquefaction depth around the partially buried pipeline for variable wave height and period.

**Figure 10.** Distribution of the excess pore pressure (|*ue*|/*σ* <sup>0</sup>) under wave trough along the vertical section through the center of the pipeline for various wave characteristics.

#### *4.3. Effects of Backfill*

In this section, the effect of backfill depth (*Hb*) on the wave-induced oscillatory liquefaction around a pipeline buried in a trench under the Stokes II wave loading is investigated. Three variable backfill depth are examined: 0.3 m, 0.5 m and 0.9 m. Figure 11 depicts the distribution of oscillatory liquefaction depth around the partially buried pipeline in a trench for various backfill depths, and Figure 12 illustrates the distribution of oscillatory excess pore pressure of the vertical section on the bottom of the pipeline under wave trough for the same four backfill depths. Figure 11 demonstrates that the liquefaction depth is greater with decreasing backfill depth. Similarly, from Figure 12, it can be seen that the maximum excess pore pressure increases as the backfill depth decreases.

**Figure 11.** *Cont*.

**Figure 11.** Distribution of the liquefaction depth around the partially buried pipeline for variable backfill depth.

**Figure 12.** Distribution of the excess pore pressure (|*ue*|/*σ* <sup>0</sup>) under wave trough along the vertical section through the center of the pipeline for various backfill depth.

#### **5. Conclusions**

This study proposes a two-dimensional seabed model by LRBFCM to investigate the wave-induced oscillatory liquefaction around a partially buried pipeline in a trench under non-linear wave loading. The model is validated by comparison with analytical solution, experimental data, and previous numerical results. The effects of wave characteristics, soil properties, and backfill depth in the trench are examined. The following conclusions can be drawn:


(4) Pipe configuration is significantly important for the analysis of wave-pipeline-seabed interaction. In the process of increasing buried depth of pipe, the magnitude of oscillatory excess pore pressure at the bottom of the trenched pipeline decreases, which means that relatively large value of backfill depth can reduce the risk of liquefaction.

When the conventional methods with meshes are applied to analyze the computational domain with irregular boundaries, the elements or meshes may be distorted. Interpolation and re-meshing can be used to solve this problem. However, it requires more intensive work for complicated engineering problems. The meshless model presented in this study is designed for avoiding the poor mesh quality existing in conventional models. However, it needs to be further developed for different engineering problems if a huge number of nodes are required.

This study focuses on the wave-induced soil response under wave loading in two-dimensional. However, in real ocean environments, waves may approach the pipeline from any direction. Therefore, the effect of wave oblique on the soil response in the vicinity of pipelines will be examined in the future.

**Author Contributions:** conceptualization, D.-S.J.; methodology, C.-C.T. & X.X.W., validation, X.X.W.; formal analysis, X.X.W.; writing—original draft preparation, X.X.W.; writing—review and editing, D.-S.J. & C.-C.T.; supervision, D.-S.J. & C.-C.T.

**Funding:** This research received funding from the Ministry of Science and Technology of Taiwan under the grant no. MOST 106-2918-I-022-001.

**Acknowledgments:** The authors are grateful for Miss K Sun at Hohai University (China) kindly providing her experimental data for the validation of the present model. The first author is grateful for the support of High-Performance Computing Cluster "Gowanda" to complete this research, and the support of scholarship from Griffith University. The third author is grateful for the support of the Ministry of Science and Technology of Taiwan under the grant no. MOST 106-2918-I-022-001.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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

#### **Cynthia Su Xin Foo 1, Chencong Liao 1,\* and Jinjian Chen**

State Key Laboratory of Ocean Engineering, Department of Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China; Cynfsx-wkdtlstl.234@sjtu.edu.cn (C.S.X.F.); chenjj29@sjtu.edu.cn (J.C.)

**\*** Correspondence: billaday@sjtu.edu.cn; Tel.: +86-135-6479-6195

Received: 4 March 2019; Accepted: 8 March 2019; Published: 13 March 2019

**Abstract:** The evaluation of the wave-induced seabed response around a buried pipeline has been widely studied. However, the analysis of seabed response around marine structures under the wave and current loadings are still limited. In this paper, an integrated numerical model is proposed to examine the wave and current-induced pore pressure generation, for instance, oscillatory and residual pore pressure, around a buried pipeline. The present wave–current model is based on the Reynolds-Averaged Navier–Stokes (RANS) equation with *k*-*ε* turbulence while Biot's equation is adopted to govern the seabed model. Based on this numerical model, it is found that wave characteristics (i.e., wave period), current velocity and seabed characteristics such as soil permeability, relative density, and shear modulus have a significant effect on the generation of pore pressure around the buried pipeline.

**Keywords:** wave–current–seabed interaction; current velocity; Reynolds-Averaged Navier-Stokesequations; buried pipeline; *k*-*ε* turbulence model

#### **1. Introduction**

Submarine pipelines are frequently used in the transportation of hydrocarbons such as oil and natural gas from offshore platforms to onshore terminals and in the disposal of industrial and municipal waste in offshore engineering. Since these submarine pipelines play such an important role, it has become one of the significant concern in marine and geotechnical engineering. Generally, submarine pipelines are buried into the seabed or within a trench. In the ocean environment, as waves propagate over the ocean floor, they generate dynamic pressure fluctuations, which will further induce excess pore pressure and reduce effective stress within the seabed soil. As the excess pore pressure increases, the soil particles loses its strength and part of the seabed becomes unstable. In most of the cases, seabed instability is the primary cause that leads to marine structure failure. Therefore, it is essential for offshore engineers to evaluate the wave-induced seabed response.

To date, numerous studies on wave-induced seabed response with marine structures such as breakwaters [1], monopiles [2] and pipelines [3] or without marine structures [4,5] had already been carried out. Based on laboratory experiments and field studies [6], the wave-induced pore pressure classifies into two mechanisms, which are oscillatory and residual mechanisms, as shown in Figure 1. Oscillatory pore pressure generation is caused by the amplitude damping and phase lag in pore pressure whereas residual pore pressure generation is the build-up excess pore pressure, which caused by the contraction of soil resulted from cyclic loading [7].

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

In the real ocean environment, water waves and current flows exist simultaneously. Hence, in addition to wave loadings, current loadings should also be taken into consideration. Recently, numerous researchers [8–12] have conducted studies to understand the interaction between wave, current, and seabed. In general, to understand the phenomenon of wave–current–seabed interaction (WCSI) and wave–current–seabed-structure interaction (WCSSI), three common approaches are widely adopted, which are the analytical solution, laboratory experiments, and numerical methods. For the study of WCSI, Zhang et al. [13] and Liao et al. [14] developed an analytical approximation to calculate the soil response of a porous seabed under the combination of wave and current loading. In many cases, to simplify the problems, seabed condition is assumed to be isotropic. In Zhou's [15] research, the soil response in anisotropic seabed with a buried marine pipeline was calculated using the analytical method. In the analytical approach, most researchers utilized the third approximation wave–current interactions as the governing equation to determine the dynamic pressure from the wave model, which eventually used as the boundary condition for the porous seabed analysis in the seabed model.

Besides analytical method, laboratory experiments have also been reported in the literature. Liu et al. [16] conducted a series of experimental studies using a one-dimensional cylinder. The purpose of a one-dimensional cylinder experiment is to determine the wave-induced oscillatory soil response and to obtain the vertical profile of the pore pressure distribution. Two or three-dimensional wave flume tank experiments [17,18] and centrifuge modeling [19,20] were conducted in the laboratory to study WCSSI phenomenon and the stability of a pipeline. There are advantages and limitations to one-dimensional cylinder experiments and two-dimensional wave flume tank experiments or centrifuge experiments. For a one-dimensional cylinder experiment, many points can be measured due to their thick soil layers, but only oscillatory pore pressure and the vertical pore pressure distribution is obtained, whereas for a two-dimensional experiment, the accumulation of build-up excess pore pressure can be analyzed, but fewer surface points are measured due to the shallow soil layers. Recently, Yang et al. [21] conducted a flume experiment using a new method of the grey value of water's image to study the initial movement of mud particles due to current loading.

As the problem becomes more complicated to be solved by analytical solution and laboratory experiments, numerical methods are usually performed. Several researchers [11] had conducted numerical simulations to study the wave–current induced seabed response without any presence of marine structures. With the inclusion of marine structures such as pipelines in the seabed, the soil responses vary under wave and current loading. Zhao et al. conducted a two-dimensional model to study the influences of pore pressure accumulations around the vicinity of a fully buried pipeline in the seabed [22] and protected in trench layer with partial backfills [23]. Zhou et al. [24] simulated the pore pressures, effective stresses and liquefaction potential around a buried in a poroelastic seabed subjected to cnoidal wave loading. In both Zhao's and Zhou's studies, only wave loading takes into consideration. Recently, Duan et al. performed a two-dimensional [25] and three-dimensional [26] numerical simulation to study the wave–current induced soil liquefaction around the buried pipeline. However, in their research, only the oscillatory soil response under a wave and current loading have

been determined. The wave–current induced oscillatory soil response is influenced by the current characteristics (following or opposing currents), wave characteristics (i.e., wave height, wave period and water depth) and soil properties (i.e., backfill thickness, permeability, and degree of saturation). Relevant research on wave–current–seabed-structure interaction is still limited. Therefore, more studies still needed to be conducted to provide a better understanding of this phenomenon.

In the present study, the objective is to develop a numerical model to analyze the pore pressure accumulation around the vicinity of a fully buried pipeline when the seabed is subjected to wave and current loadings. This numerical analysis consists of two submodels, which are the wave–current model and the seabed model. In the wave–current model, RANS equation with *k*-*ε* turbulence governs the wave motion and current flow, whereas in the seabed model, Biot's quasi-static equations are employed to calculate soil response such as soil displacements, oscillatory pore pressure, residual pore pressure and the effective stress within the seabed. The influence of wave, current, and seabed characteristics such as current velocities, wave period, shear modulus, permeability, and relative density on the soil responses are presented and discussed in the later sections.

#### **2. Methods**

This numerical model is made up of two submodels: wave–current model and seabed model with a buried pipeline. These two submodels are integrated with one-way coupling method. For a one-way coupling method of fluid-seabed interactions [27], the wave pressure computed in the wave solver is introduced into the seabed solver as a boundary condition to solve for the seabed response. Figure 2 illustrates a wave–current–seabed-structure interaction (WCSSI), where *x* and *z* represents the Cartesian coordinates system, *h* is the seabed thickness, *d* is the water depth, *l* is the length of the computational domain, *e* represents the embedment depth of pipeline (i.e., the distance from the surface of the seabed to the center of the pipeline) and *u* is the initial current velocity. In this study, the length of the computational domain (*l*) is set to be three times the wavelength to ignore the influence of lateral boundaries.

**Figure 2.** Sketch of the wave–current–seabed–pipeline interactions.

#### *2.1. Wave–Current Model*

The wave and current interaction model is based on the Reynolds-Averaged Navier–Stokes equation to simulate the flow motion. For the incompressible fluids, the mass conservation equation and the momentum conservation equation can be expressed as below:

$$\frac{\partial u\_i}{\partial \mathbf{x}\_i} = \mathbf{0},\tag{1}$$

$$\frac{\partial \langle u\_{i} \rangle}{\partial t} + \frac{\partial \langle u\_{i} \rangle \langle u\_{j} \rangle}{\partial x\_{j}} = -\frac{1}{\rho\_{f}} \frac{\partial \langle p \rangle}{\partial x\_{i}} + g\_{i} + \frac{1}{\rho\_{f}} \frac{\partial}{\partial x\_{j}} \left[ \mu \left( \frac{\partial \langle u\_{i} \rangle}{\partial x\_{j}} + \frac{\partial \langle u\_{j} \rangle}{\partial x\_{i}} \right) \right] - \frac{\partial \langle u\_{i}^{\prime} u\_{j}^{\prime} \rangle}{\partial x\_{j}},\tag{2}$$

where *xi* represents the Cartesian coordinate, ()*<sup>i</sup>* and ()*<sup>j</sup>* represent the index tensor notion, *ui* is the ensemble mean velocity (m/s), *p* is the fluid pressure (Pa), *<sup>ρ</sup><sup>f</sup>* is the fluid density (kg/m3), *<sup>t</sup>* is the time, *<sup>g</sup>* is the acceleration (m/s2), *<sup>μ</sup>* is the dynamic viscosity (Pa·s), and −*ρfu i u j* is the Reynolds stress tensor. By applying the eddy-viscosity assumptions, Reynold stress term can be expressed as:

$$-\rho\_f \langle u\_i' u\_j' \rangle = \mu \left[ \frac{\partial \langle u\_i \rangle}{\partial x\_j} + \frac{\partial \langle u\_j \rangle}{\partial x\_i} \right] - \frac{2}{3} \rho\_f \delta\_{ij} k\_\prime \tag{3}$$

where *<sup>μ</sup><sup>t</sup>* is the turbulence viscosity (Pa·s), *<sup>k</sup>* is the turbulence kinetic energy (TKE, m2/s2), and *<sup>δ</sup>ij* is the Kronecker delta. By substituting Equation (3) into Equation (2), the following equation can be obtained:

$$\frac{\partial \rho\_f \langle u\_i \rangle}{\partial t} + \frac{\partial \rho\_f \langle u\_i \rangle \langle u\_j \rangle}{\partial \mathbf{x}\_j} = -\frac{\partial}{\partial \mathbf{x}\_i} \left[ \langle p \rangle + \frac{2}{3} \rho\_f k \right] + \rho\_f \mathbf{g}\_i + \frac{\partial}{\partial \mathbf{x}\_j} \left[ \mu\_{eff} \left( \frac{\partial \langle u\_i \rangle}{\partial \mathbf{x}\_j} + \frac{\partial \langle u\_j \rangle}{\partial \mathbf{x}\_i} \right) \right], \tag{4}$$

where *μeff* = *μ* + *μt*, which is the total effective viscosity (Pa·s).

For the prediction of a fully turbulent flow, the standard *k*-*ε* turbulence model [28] based on model transport equations for the turbulence kinetic energy and its rate of dissipation can be expressed as follows:

$$\frac{\partial \rho\_f k}{\partial t} + \frac{\partial \rho\_f \langle \mu\_j \rangle k}{\partial x\_j} = -\frac{\partial}{\partial x\_j} \left[ \left( \mu + \frac{\mu\_t}{\sigma\_k} \right) \frac{\partial k}{\partial x\_j} \right] + \rho\_f \,' \mathbf{P}\_k - \rho\_f \varepsilon\_t \tag{5}$$

$$\frac{\partial \rho\_f \varepsilon}{\partial t} + \frac{\partial \rho\_f \langle u\_{\rangle} \rangle \varepsilon}{\partial \mathbf{x}\_{\circ}} = -\frac{\partial}{\partial \mathbf{x}\_{\circ}} \left[ \left( \mu + \frac{\mu\_t}{\sigma\_t} \right) \frac{\partial \varepsilon}{\partial \mathbf{x}\_{\circ}} \right] + \frac{\varepsilon}{k} \left( \mathbb{C}\_{\varepsilon 1} \rho\_f \langle \mathbf{P}\_k - \mathbb{C}\_{\varepsilon 2} \rho\_f \varepsilon \right), \tag{6}$$

where *μ<sup>t</sup>* = *ρ<sup>f</sup> C<sup>μ</sup> k*2 *<sup>ε</sup>* , *<sup>k</sup>* is the turbulence kinetic energy (m2/s2), *<sup>ε</sup>* is the turbulence dissipation rate, *Cμ*, *σk*, *σε*, *Cε*1, and *Cε*<sup>2</sup> are the empirical coefficients determined from experiments. The empirical coefficients used in this study are based on previous studies [29]:

$$\mathbb{C}\_{\emptyset} = 0.09 \quad \mathbb{C}\_{\mathfrak{t}1} = 1.44 \quad \mathbb{C}\_{\mathfrak{t}2} = 1.92 \quad \sigma\_{\mathfrak{k}} = 1.00 \quad \sigma\_{\mathfrak{t}} = 1.30$$

#### *2.2. Seabed Model*

As mentioned earlier, pore pressure generation classifies into two mechanisms: oscillatory pore pressure and residual pore pressure. The pore pressure generation can be expressed as follows:

$$
\mu = \mu\_c + \mu\_p \tag{7}
$$

where *u* represents the wave and current-induced excess pore pressure at a specific point, *ue* is the oscillatory component whereas *up* is the residual component that is further expressed as *up* = <sup>1</sup> *T* % *T* <sup>0</sup> *udt* where *T* denotes the wave period.

#### 2.2.1. Oscillatory Soil Response

In this study, the seabed model is considered to be porous and hydraulically permeable. The soil skeleton and pore fluid are assumed to be compressible and obey the Hooke's Law. Therefore, Biot's theory [30] is utilized to govern the soil response. The mass conservation and force equilibrium equation can be written as follows:

$$
\nabla^2 u\_{\mathfrak{e}} - \frac{\gamma\_w n\_s \beta\_s}{k\_{\mathfrak{e}}} \frac{\partial u\_{\mathfrak{e}}}{\partial t} = \frac{\gamma\_w}{k\_{\mathfrak{e}}} \frac{\partial \varepsilon\_s}{\partial t} \tag{8}
$$

$$
\delta G \nabla^2 u\_\delta + \frac{G}{(1 - 2\nu)} \frac{\partial \varepsilon\_s}{\partial x} = -\frac{\partial u\_\delta}{\partial x} \,, \tag{9}
$$

$$-G\nabla^2 w\_s + \frac{G}{(1-2\nu)}\frac{\partial \varepsilon\_s}{\partial z} = -\frac{\partial u\_c}{\partial z},\tag{10}$$

where ∇<sup>2</sup> = *∂*<sup>2</sup> *∂x*<sup>2</sup> *∂*2 *∂z*<sup>2</sup> represents the Laplace operator, *γ<sup>w</sup>* is the unit weight of water (N/m3), *ns* is the soil porosity, *ks* is the soil permeability (m/s), *G* denotes the shear modulus of the seabed soil (N/m2), *ν* is the Poisson's ratio, and *ue* represents the wave–current induced oscillatory soil response. The compressibility of the pore fluid (*βs*) and the elastic volume strain of the soil matrix (*εs*) can be defined as:

$$\beta\_s = \frac{1}{K\_{\text{av}}} + \frac{1 - S\_r}{P\_{\text{voo}}},\tag{11}$$

$$
\varepsilon\_s = \frac{\partial u\_s}{\partial x} + \frac{\partial w\_s}{\partial z},
\tag{12}
$$

where *Kw* is the true modulus of water (which is taken as 2 × 109 N/m2), *Sr* is the degree of saturation, and *Pwo* is the absolute water pressure. *us* and *ws* represent the soil displacement at *x*- and *z*-direction respectively.

#### 2.2.2. Residual Soil Response

For homogeneous and isotropic soil, the residual pore pressure can be derived from the onedimensional Biot's consolidation equation, which can be expressed as follows:

$$\frac{\partial \mathfrak{u}\_p}{\partial t} = \mathbb{C}\_v \frac{\partial^2 \mathfrak{u}\_p}{\partial z^2} + f \tag{13}$$

where *up* is the wave–current induced residual pore pressure, *Cv* is the coefficient of consolidation and *f* represents the source term of the pore pressure generation.

With the presence of a buried pipeline in the seabed, the present study considers a two-dimensional plane strain problem. Hence, a slight modification is made to Equation (13), the governing equation for residual pore pressure [7,31] can be written as follows:

$$\frac{\partial u\_p}{\partial t} = \mathbb{C}\_v \left( \frac{\partial^2 u\_p}{\partial x^2} + \frac{\partial^2 u\_p}{\partial z^2} \right) + f(x, z, t) \tag{14}$$

where *Cv* is the coefficient of consolidation and *f*(*x*, *z*, *t*) is the source term, a function of space and time. These two parameters can be defined as follows respectively:

$$\mathcal{C}\_{v} = \frac{Gk\_{s}}{\gamma\_{w}(1 - 2v)}\tag{15}$$

$$f(\mathbf{x}, z, t) = \frac{\partial u\_{\mathcal{S}}}{\partial t} = \frac{\sigma\_0'}{T} \left[ \frac{|\mathbf{r}(\mathbf{x}, z, t)|}{a\_r \sigma\_0'} \right]^{-\frac{1}{\mathcal{V}r}} \tag{16}$$

where *ug* is the generation of pore-water pressure, *<sup>τ</sup>*(*x*, *<sup>z</sup>*, *<sup>t</sup>*) is the shear-stress term, <sup>|</sup>*τ*(*x*,*z*,*t*)<sup>|</sup> *αrσ* 0 represents the induced cyclic shear-stress ratio, which defines the pore pressure accumulation, *α<sup>r</sup>* and *β<sup>r</sup>* are empirical coefficients that obtain from large-scale simple shear tests. Both the empirical coefficients have a correlational relationship with the relative density of soil (*Dr*), which can be expressed as follows [32]:

$$
\alpha\_{\!\!\!\!=} = 0.34D\_{\!\!\!=} + 0.084 \qquad \beta\_{\!\!\!=} = 0.37D\_{\!\!\!=} - 0.46 \tag{17}
$$

#### *2.3. Boundary Condition*

Equation (8), (9), and (10) represents the governing equations to determine the oscillatory pore pressure and soil displacements within the seabed and should be solved with the appropriate boundary conditions. Hence, boundary conditions need to be specified at the appropriate locations; at the surface of the seabed where wave interacts with the seabed, at the bottom of the seabed, at the lateral boundaries and the seabed–pipeline interface.

*J. Mar. Sci. Eng.* **2019**, *7*, 66

At the seabed surface, it is common that the vertical effective normal stress and shear stress vanishes, therefore the pore pressure at the seabed surface is assumed to be equal to wave pressure.

$$
\sigma\_{sz}^{\prime} = \tau\_{sxz} = 0 \text{ and } P\_s = \,^P\_b \qquad \text{at } \mathbf{z} \, = \, 0 \tag{18}
$$

where *Pb* denotes the wave pressure at the seabed surface, which can be obtained from the wave–current model.

At the lateral boundaries of the seabed, it is said to be impermeable (i.e., zero flux) and zero horizontal displacements occur, i.e.,

$$
\sigma'\_{s\mathbf{x}} = 0, \frac{\partial p}{\partial \mathbf{x}} = 0 \qquad \text{at } \mathbf{x} = 0 \text{ and } \mathbf{x} = L\_{\mathcal{S}} \tag{19}
$$

Since the seabed is resting on a rigid and impermeable base, it is assumed that there are zero displacements and no vertical flow occurring at the bottom of the seabed. For infinite seabed thickness,

$$u\_s = w\_s = 0, \frac{\partial p}{\partial z} = 0 \qquad \text{at } \mathbf{x} = -h\_w \tag{20}$$

The buried pipeline is assumed to be elastic and impermeable; hence, along the surface of the pipeline, the pore pressure to the normal gradient is assumed to be zero, which can be written as:

$$\frac{\partial p}{\partial n} = 0 \quad \text{at } r = \sqrt{(\mathbf{x} - \mathbf{x}\_0)^2 + (z - z\_0)^2} = R \tag{21}$$

It is also assumed that there is no relative displacement occurs between soil particles and pipeline at the interface,

$$u = u\_{pipe} \qquad w = w\_{pipe} \tag{22}$$

#### *2.4. Numerical Scheme*

In this study, the integrated model consists of a wave–current model and a seabed model. The linear wave along with a steady current is simulated using computational fluid dynamics (CFD) software, FLOW-3D v11.2 (Flow Science, Inc., Santa Fe, New Mexico, USA). From the wave–current simulation, wave pressure acts upon on the seabed surface can be obtained. Besides wave pressure, in FLOW-3D, the Volume of Fluid (VOF) method [33] is employed to get the free surface wave elevation. Then, the wave pressure on the seabed surface is introduced into the seabed model as a boundary condition.

As mentioned in Paulsen et al.'s report [34], an increase in spatial resolution leads to a reduction in errors. Therefore, the spatial resolution of an incident wave is recommended to be 15 points per wave height (p.p.w.h.). In this research, a total of 250,000 real cells with an element size of 0.2 m in both *x*- and *z*-direction (i.e., 20 points per wave height) are constructed for the simulation of the wave–current model.

The seabed model is the simulation of the seabed and buried pipeline using a finite element analysis software, COMSOL Multiphysics v5.3a (COMSOL Inc., Burlington, MA, USA). COMSOL Multiphysics is a finite-element-method (FEM) software utilized to model and solve various types of scientific and engineering problems [35]. Figure 3 shows a comparison between three different levels of predefined mesh range from COMSOL Multiphysics; i.e., fine (0.09 m–15.9 m), finer (0.0375 m–11.1 m) and extra fine (0.0225 m–6 m) mesh size. It can be observed in Figure 3a that smaller mesh sizes lead to an increase in the number of elements and longer computational time in solving the simulation. However, there is no visible difference in the result as shown in Figure 3b. The residual pore pressure over a specified period as shown in Figure 3b is taken from a point at *x* = 50 m and *z* = −5 m of the seabed. Hence, in this research, the seabed submodel consists of 2367 triangular elements with

predefined finer mesh, which has mesh sizes range from 0.0375 m (domain near the pipeline) to 11.1 m (domain further away from the pipeline).

Both softwares, i.e. FLOW-3D and COMSOL Multiphysics, are running on Intel®Core (TN) 17-6700 CPU @ 3.40GHz with available memory of 8 GB.

**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, *up* at a duration of 100 s.

After building and simulating the model in COMSOL Multiphysics, MATLAB code is used to perform the loop simulation for the *N* –th number of computational time we intend to study. The MATLAB version used in this research is R2014a (The MathWorks, Inc., Natick, MA, USA). The MATLAB script is created to perform a series of iterative calculation with transient studies. The solution from the previous transient study is then set as the initial condition for the next transient study. For instance, at the first step during the time interval Δ*t* = 1/*nT*, the wave load is introduced into the seabed submodel to calculate the soil response and study 1 is created. Then the solution from study 1 is set as the initial condition for study 2. The following steps repeat until the desired computational time is reached. The number of loop simulations corresponds to the number of wave load extracted from the wave–current submodel.

In this study, a total of 200 loop simulations were performed for a duration of 200 s, i.e., time interval of 1 s, to generate the dynamic soil responses for a specific model. The computational time is chosen as 200 s because at *t* = 200 s, the soil around the buried pipeline has begun to liquefy even though the oscillatory pore pressure and residual pore pressure from the simulations have not reached its steady state. The liquefaction criterion proposed Zen and Yamazaki [6] adopted to evaluate the liquefaction potential is expressed as:

$$-\left(\gamma\_{\mathfrak{s}} - \gamma\_{w}\right)z \le P\_{\mathfrak{s}} - P\_{\mathfrak{b}}\tag{23}$$

where *γ<sup>s</sup>* is the unit weight of seabed soil, *γ<sup>w</sup>* is the unit weight of water, *z* represents depth of a specific point in the seabed, *Ps* is the wave–current-induced pore pressure, and *Pb* is the wave pressure on the seabed surface.

The parameters utilized in the simulation of the numerical model are listed in Table 1.


**Table 1.** Input data for the numerical simulation.


**Table 1.** *Cont*.

#### **3. Results and Discussion**

#### *3.1. Wave Characteristics*

In the wave–current model simulation, the wave pressure and free surface elevation are simulated based on the wave characteristics that are input into the solver. The wave characteristics such as wave height (*H*), wave period (*T*), water depth (*d*) and the presence of current (*u*) influence the wave outcomes. The influence of the current velocities will be further discussed in the following section.

#### Influence of Current Velocities

Figure 4 shows the influence of different current flow on wave at a specific location over time. The information is extracted from the midpoint of the seabed surface within the computational domain (i.e., *x* = 150 m). Seabed surface midpoint (*x* = 150 m) is chosen as the specific point to be analyzed because wave trains are assumed to be more stable as there might be a minor disturbance at the start and end point of the computational domain (i.e., *x* = 0 m and *x* = 300 m respectively). At the starting point of the domain (i.e., *x* = 0 m), wave and current flow are first introduced. A wave absorber is placed after the computational domain (i.e., x = 300 m), which is for the dissipation of wave energy. From Figure 4, it is observed that as the current flow increases, wave pressure also increases.

**Figure 4.** Variation of wave pressure, *p* at the seabed bottom for different current velocities.

#### *3.2. Seabed Characteristics*

In COMSOL Multiphysics, several simulations were conducted to simulate the wave–current induced soil response, i.e., oscillatory pore pressure and residual pore pressure, and the parameters used are as listed in Table 1, and the current velocity is taken as 0.5m/s in all the following simulations in Section 3.2. The generation of pore pressure in the seabed are sensitive to the seabed characteristics, for instance, soil permeability (*ks*), shear modulus of the seabed (*G*), relative density (*Dr*) and degree of saturation (*Sr*). The degree of sensitivity of these parameters on the pore pressure generation is discussed in the following section.

#### 3.2.1. Effects of Soil Permeability

Soil permeability or also known as the hydraulic conductivity is a soil property, which allows the seepage of fluids to pass through its interconnected void spaces. Depending on the soil types, soil permeability can vary on a range from 1.0 × <sup>10</sup>−<sup>12</sup> to 1.0 × <sup>10</sup>−<sup>2</sup> m/s. In this analysis, three

different values of soil permeability (i.e., k = 1.0 × <sup>10</sup><sup>−</sup>2, 1.0 × <sup>10</sup>−<sup>3</sup> and 1.0 × <sup>10</sup>−<sup>4</sup> 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 (*Dr*) set as 5.0 × 106 N/m2 and 0.5, respectively.

From Figure 5a, it can be observed that changes in the soil permeability have minimal influence on the value for oscillatory pore pressure. However, as noted in Figure 5b, seabed with low permeability (i.e., *ks* = 1.0 × <sup>10</sup>−<sup>4</sup> 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.

**Figure 5.** Variations of (**a**) oscillatory pore pressure, *ue* and (**b**) residual pore pressure, *up* at a specific point for different soil permeability.

#### 3.2.2. Effects of Shear Modulus

Figure 6 illustrates the variations of wave–current induced pore pressure at *x* = 50 m and *z* = −10 m under two different values of shear modulus (i.e., *<sup>G</sup>* = 5.0 × 106 and 1.5 × <sup>10</sup><sup>7</sup> N/m2). 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:

$$G = \frac{E}{2(1+v)}\tag{24}$$

Therefore, with a fixed value of Poisson's ratio (*v*) at 0.35, Young's modulus can be obtained from Equation (24) as 13.5 MPa and 40.5 MPa for shear modulus (*G*) of 5.0 × 106 N/m<sup>2</sup> and 1.0 × <sup>10</sup><sup>7</sup> N/m2 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).


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

**Figure 6.** Variations of (**a**) oscillatory pore pressure, *ue* and (**b**) residual pore pressure, *up* at a specific point under the different shear modulus.

#### 3.2.3. Effects of Relative Density

Relative density (*Dr*) is a soil parameter that commonly used to indicate the in-situ denseness or looseness of granular soil [37]. It is defined as the ratio of the difference between the void ratios of a cohesionless soil in its loosest state and existing natural state to the difference between its void ratio in the loosest and densest states, which can be formulated as follows:

$$D\_I = \frac{\varepsilon\_{\max} - \varepsilon}{\varepsilon\_{\max} - \varepsilon\_{\min}} \tag{25}$$

As seen in Equation (17), when there is a change in the relative density of the soil, both the empirical coefficient *α<sup>r</sup>* and *β<sup>r</sup>* in the source term *f*(*x*, *z*, *t*) varies, which eventually leads to different values of residual pore pressure. Figure 6 shows the generation of oscillatory and residual pore pressure due to the influence of various relative density (*Dr* = 0.2, 0.3 and 0.5) over a certain period at point x = 50 m and z = –10 m. In this section, the soil permeability (k) and shear modulus (G) are set as 1.0 × <sup>10</sup>−<sup>4</sup> m/s and 5.0 × 106 N/m2 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).

**Figure 7.** Variation of (**a**) oscillatory pore pressure, *ue* and (**b**) residual pore pressure, *up*\_p at a specific point under the different relative density.

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

In this section, we will study the dynamic soil response around the vicinity of the buried pipeline when the seabed is subjected to wave and current loading. The parameters utilized in the comparison of soil responses such as displacement and pore pressure around the buried pipeline under different current velocity are stated in Table 1, however, the shear modulus, permeability and relative density of the seabed in the simulation are set at a fixed value of 5.0 × 106 N/m2, 1.0 × 10–4 m/s and 0.2 respectively.

As shown in the sketch of wave–current–seabed–pipeline interaction (Figure 2), the pipeline is located at *x* = 150 m and is embedded 3 m into the ground (Embedment depth is measured from seabed surface to the center of the pipeline). Hence, the analysis is taken at the point *x* = 150 m at *t* = 200 s. The graphs in Figure 8 show a comparison between various current velocity ranges from 0 m/s to 0.5 m/s in a gradient of 0.25, which are illustrated in terms of different soil response such as oscillatory pore pressure, residual pore pressure and displacement. The negative values of oscillatory pore pressure in Figure 8a is due to the wave trough phase at that specific time. It can be observed that with lower current values, the oscillatory pore pressure is higher. Figure 8b illustrates the residual pore pressure near the buried pipeline. It can be seen that at the surface of the seabed, residual pore

pressure equals to zero. As it goes into the seabed closer to the top of the pipeline, the values increase gradually. The residual pore pressure tends to be higher at the bottom of the pipeline than the top of the pipeline. Figure 8c presents the displacement of seabed when wave and current loading apply on the seabed. It is observed that displacement is higher at the mid-depth of the seabed and gradually decreases to zero as it gets deeper into the seabed. It indicates that at a greater depth, the wave and current loading does not affect the movement of the soil particles.

**Figure 8.** Variation in (**a**) oscillatory pore pressure, *ue*, (**b**) residual pore pressure, *up* and (**c**) displacement, *us* under various current velocity.

Figure 9 displays the distribution of wave and current induced seabed responses; oscillatory pore pressure, residual pore pressure, and soil displacements, *us* and *ws* around the periphery of the buried pipeline at a duration of 200 s (*t* = 200 s). Figure 9a,b show the horizontal and vertical soil displacement around the embedded pipeline respectively. There is no significant difference in the soil displacement at the top and bottom of the pipeline. At *t* = 200 s, a wave trough reaches the proximity of the buried pipeline; therefore, a negative value of oscillatory pore pressure is observed as shown in Figure 9c. As shown in Figure 9d, the residual pore pressure increases gradually with depth because pore water pressure can dissipate efficiently at the seabed surface. Meanwhile, as it goes deeper into the seabed and pipeline, excess pore pressure accumulates which results in an increase in residual pore pressure at the seabed bottom.

**Figure 9.** Distribution of seabed responses such as (**a**) horizontal displacement *us*, (**b**) vertical displacement *ws*, (**c**) oscillatory pore pressure, *ue*, and residual pore pressure *up* around the vicinity of the buried pipeline due to wave and current loading when current velocity is set at *vc* = 0.5 m/s.

#### **4. Conclusions**

This research proposes a two-dimensional model with a turbulence closure scheme (*k*-*ε* turbulence) to investigate the dynamic seabed responses around the vicinity of a buried pipeline under combined wave and current loading. However, according to Alberello et al. [38], there has no visible effect on the two-dimensional model with or without turbulence closure scheme whereas a substantial difference can be seen in a three-dimensional with turbulence model. Therefore, the current two-dimensional model is expected to expand into a three-dimensional model with a turbulence closure scheme in the future.

The effects of current, wave and soil characteristics on the wave–current-induced soil responses are examined. According to the numerical results presented above, the following conclusions can be drawn:


**Author Contributions:** Methodology, C.L.; Data curation, C.S.X.F.; Writing-Original Draft Preparation, C.S.X.F.; Writing-Review and Editing, J.C.; Supervision, C.L.

**Acknowledgments:** The authors are grateful for the financial support from the National Science Foundation of China (Grant No. 41727802, No. 51678360, No. 41602282).

**Conflicts of Interest:** The authors declare no conflicts of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Seismic Dynamics of Pipeline Buried in Dense Seabed Foundation**

**Yan Zhang 1, Jianhong Ye 2,\*, Kunpeng He 2,3 and Songgui Chen <sup>4</sup>**


Received: 1 February 2019; Accepted: 12 June 2019; Published: 20 June 2019

**Abstract:** Submarine pipeline is a type of important infrastructure in petroleum industry used for transporting crude oil or natural gas. However, submarine pipelines constructed in high seismic intensity zones are vulnerable of attacks from seismic waves. It is important and meaningful in engineering design to comprehensively understand the seismic wave-induced dynamics characteristics of submarine pipelines. In this study, taking the coupled numerical model FSSI-CAS 2D as the tool, the seismic dynamics of a submarine steel pipeline buried in dense soil is investigated. Computational results indicate that submarine pipeline buried in dense seabed soil strongly responds to seismic wave. The peak acceleration could be double of that of input seismic wave. There is no residual pore pressure in the dense seabed. Significant resonance of the pipeline is observed in horizontal direction. Comparative study shows that the lateral boundary condition which can avoid wave reflection on it, such as laminar boundary and absorbing boundary should be used for seabed foundation domain in computation. Finally, it is proven that the coupled numerical model FSSI-CAS 2D is applicable to evaluate the seismic dynamics of submarine pipeline.

**Keywords:** submarine pipeline; dense seabed foundation; seismic dynamics; resonance of submarine pipeline; FSSI-CAS 2D

#### **1. Introduction**

Submarine pipeline is a type of important infrastructure in petroleum industry widely used in offshore area for transporting crude oil or natural gas. Nowadays, several hundred thousands of kilometers of submarine pipelines have been constructed worldwide. The stability of submarine pipelines is important and crucial for guaranteeing their normal service performance in the designed service period. However, submarine pipelines are vulnerable of attacks from extreme ocean waves or strong seismic waves. Therefore, it is necessary and meaningful to understand the responding dynamics characteristics of submarine pipeline under the dynamic loading applied by ocean wave or seismic wave.

Generally, the instability of submarine pipelines would be attributed to scouring, ocean wave applications, or seismic wave attacks. Some valuable works have been conducted on the scouring of seabed floor near the submarine pipeline to understand the process and mechanism of seabed scouring around pipeline under ocean waves and currents [1–3]. On the ocean wave-induced dynamics of submarine pipeline, a series of research works have also been conducted, and a great number of literature is available. The research method mainly includes analytical solutions, numerical computations, and laboratory wave flume tests. Previous studies mainly focused attention on the wave and current-induced pore pressure and effective stress in seabed soil, seepage force [4,5] and buoyancy [6–8] of pipeline. In the field of marine geotechnical engineering, the investigation on the response of pore pressure and effective stress in seabed foundation to ocean wave around marine pipeline was the most popular topic. On this topic, the team led by Jeng D.S. conducted a number of works. For example, Jeng and Cheng [9] proposed an analytical solution to understand the wave-induced pore pressure around a pipeline buried in poro-elastic seabed. Then, Wang et.al [10] and Jeng [11] further investigated the wave-induced pore pressure around a pipeline buried in anisotropic or nonhomogeneous seabed. Later, the effect of nonlinear wave as well as soil-pipeline contact effects on pipeline dynamics were studied [12–14]. Recently, the dynamics of a pipeline buried in a single-layer or multi-layer seabed applied by conodal wave or linear wave were studied by Zhou et al. [15] and Zhou et al. [16]. Previous studies were basically limited to two dimensions. Zhang et al. [17] studied the wave-induced dynamics of a pipeline, adopting a three-dimensional model. In above-mentioned works, the seabed soils were all described as poro-elastic medium. However, there is another type of seabed soil widely distributes in offshore area. It is loosely deposited seabed soil, in which pore pressure could build up under ocean wave loading, resulting in seabed soil liquefaction. Recently, the wave-induced dynamics of a pipeline buried in loose seabed soil was tentatively investigated [18,19] by adopting some empirical-based soil models, such as the soil model proposed by seed [20,21]. There were also few investigations [17,22] which adopted a advanced soil model, such as PZIII model proposed by Zienkiewicz et al. [23], to do such work.

On the seismic dynamics of submarine pipeline buried in seabed floor, only a few investigations have been previously conducted. Actually, researchers mainly focused on the seismic dynamics and the stability of submarine pipeline from about 1980s. At the early stage, the seismic performance of free-spanning pipelines supported by a number of upholders was the focus of engineers and scientists [24,25]. To the authors' best knowledge, Wang and Cheng [26] first investigated the axial seismic dynamics of a buried pipeline adopting a simplified quasi-static method, in which the soil-pipeline interaction was modelled by some virtual springs. After that, Datta et al. [27] further investigated the seismic dynamics of a buried pipeline, adopting a three-dimensional numerical model where the seabed foundation was described by linear elastic model, and the steel pipeline was modelled by shell elements. It was found by Datta et al. [27] that the seismic dynamics of submarine pipeline was significantly controlled by the stiffness ratio between pipeline and its surrounding soil. Later, Datta and Mashaly [28] further analyzed the seismic dynamics of a buried pipelines by performing spectral analysis, where the earthquake was considered as a partially correlated stationary random process characterized by a power spectral density function (PSDF). After the 2000s, there were also a few works performed to study the seismic dynamics of submarine pipeline adopting numerical modelling, such as Ling et al. [29], Luan et al. [30], Zhang and Han [31], and Saeedzadeh and Hataf [32]. However, these works mainly focused their attention on the pore pressure and acceleration in soil foundation. The dynamics characteristics of effective stresses in soil foundation, as well as the dynamics of pipeline itself, were basically not demonstrated, resulting in the lack of comprehensive understanding on the seismic dynamics characteristics and the instability mechanism of submarine pipelines. Over the past 10 years, several numerical modelling works were conducted to study the deformation of steel pipelines buried in seabed after faults were moved in strong earthquake events [33,34]. Their works were beneficial to improve the seismic design ability of engineers, avoiding instability of submarine pipeline in earthquake events. In addition to numerical modelling, laboratory shaking table tests in centrifuge device were also performed to study the seismic dynamics of buried pipelines [35,36]. Their test results provided engineers with insights to further understand the seismic instability mechanism of submarine pipeline.

As we know, a seismic wave is a kind of significant and nonignorable environmental loading for marine structures. It brings a great threat to the stability of offshore structures constructed in high seismic intensity zones. The seismic stability of submarine pipelines has attracted much attention in offshore petroleum industry. Some national or industry association codes, such EU code EN 1594, Canadian code CSA Z662, and ASME codes B31.4 and B31.8 suggest to design engineers that the adverse impact of seismic wave should be considered in pipeline design, and some mitigation measures should be taken. However, there is basically no further detail information on how to quantitatively perform the anti-seismic design due to the fact that the seismic dynamics characteristics of submarine pipeline is not yet comprehensively understood. In this study, taking the coupled numerical model FSSI-CAS 2D as a tool, the seismic dynamics of a submarine steel pipeline buried in dense soil is investigated. The analysis results could further improve the understanding of ocean engineers on the seismic dynamics of submarine pipeline buried in seabed foundation.

#### **2. Coupled Numerical Model: FSSI-CAS 2D**

In the offshore environment, pipeline, seabed foundation, and overlying seawater are an integrated system. There is a strong interaction between them when subjected to environmental loading-ocean waves or seismic waves. To understand the complicated interaction between fluid, offshore structures, and their seabed foundations, an integrated numerical model FSSI-CAS 2D, as well as its three-dimensional version FSSI-CAS 3D, were successfully developed by Jeng et al. [37], Ye et al. [38], and Ye et al. [39] for the problem of fluid-structures-seabed foundation interaction. In FSSI-CAS 2D, the Volume Averaged Reynolds Averaged Navier–Stokes (VARANS) equation [40] governs the wave motion and porous flow in the porous seabed, solved using the finite difference method (FDM). The dynamic Biot's equation, known as 'u-p' approximation [41], is adopted to describe the dynamic behavior of offshore structure and its seabed foundation, which is solved in the finite element framework [23]. A coupled algorithm was developed to integrate these two governing equations together, forming a coupled/integrated numerical model for the problem of fluidstructures–seabed interaction (FSSI). More detailed information on solving the VARANS equation and the dynamic Biot's equation can be found in Ye et al. [38,42] and Zienkiewicz et al. [23]. FSSI-CAS 2D has the innate advantage for the problem of fluid-structures-seabed interaction. However, the main limitation of FSSI-CAS 2D is that the displacement discontinuity cannot be guaranteed on the interface between fluid and structures/seabed.

The developed coupled model FSSI-CAS 2D has been validated by analytical solutions, a series of wave flume tests, and a centrifuge test [39]. It has also been successfully applied to investigate the dynamics of breakwater and its seabed foundation to several types of ocean waves, such as regular waves, breaking waves [43], and tsunami waves, as well as seismic waves [44]. It is indicated that the coupled numerical model FSSI-CAS 2D is applicable for the seismic dynamics of pipeline.

#### **3. Computational Domain, Boundary Condition, Seismic Wave and Parameters**

As demonstrated in Figure 1, a pipeline transporting crude oil is buried in dense seabed foundation in offshore area with a water depth d = 10 m. The diameter of pipeline is 800 mm. The buried depth is 1.0 m (distance of pipeline center to the surface of seabed). The computational domain of seabed foundation is 200 m in length and 20 m in thickness. The pipeline is placed on the symmetrical line x = 100 m.

The bottom of the seabed foundation is fixed in x and z direction. The lateral sides of the seabed foundation are set as laminar boundary in x direction and set free in z direction. It means that there is no reflection of seismic wave on the lateral sides of the seabed foundation. On the surface of the seabed foundation, only the hydrostatic water pressure is applied (ocean wave loading is not considered in this study). Meanwhile, the effective stresses remain at zero at all times on the surface of seabed floor due to the fact that the seabed foundation is porous (have no relationship with water depth). In order to simulate the working status of the pipeline, a pressure with a value of 200 kPa driving the crude oil flowing in the pipeline is applied to the crude oil.

**Figure 1.** Sketch map of the pipeline-seabed system adopted in computation. A submarine pipeline is buried in the dense seabed foundation. Only the hydrostatic water pressure is applied on the surface of seabed, and the laminar boundary condition is applied on the two lateral sides.

The FE mesh system of the pipeline-seabed used in computation is illustrated in Figure 2. In total, 23,316 four-node elements are used. In the zone around the pipeline, the size of elements (0.02–0.2 m) are much smaller than that in the other zone (0.5–2.0 m). In the mesh system, the pipeline treated as impermeable and rigid steel circle (thickness = 2 cm), and the crude oil in it is also meshed. Two typical point A and B are labelled in Figure 2 to demonstrate the characteristics of seismic dynamics of seabed soil near to the pipeline thereafter.

**Figure 2.** Mesh system of the pipeline-seabed in computation (Noted: The crude oil in the pipeline is also considered, and only the mesh around the pipeline is shown).

In seismic analysis, the seismic wave truly recorded in offshore area would be the best choice to be the input seismic excitation (definitely better than a synthetic seismic wave based on an acceleration response spectrum). Here, the recorded seismic wave at the observation station MYGH03 (141.6412E, 38.9178N, buried depth = 120 m, at Karakuwa, Japan), which is near to the Pacific coastal line in Japan, 311 off-Pacific coast of Tohoku earthquake (ML = 9.0), is adopted as the input seismic wave (Figure 3). The input horizontal (E-W) and vertical (U-D) seismic acceleration wave are applied on the bottom of the seabed foundation simultaneously.

**Figure 3.** Input seismic wave after wave filtering adopting the recorded seismic wave at the station MYGH03 (141.6412E, 38.9178N, buried depth = 120 m) at Karakuwa, Japan during 311 off-Pacific earthquake event. Noted: Noncausal butterworth filter is used; filtering range: f ≤ 0.03 Hz and f ≥ 30 Hz.

In offshore area, there is not only loosely deposited seabed soil, but also relative dense seabed soil. For example, the hard layer sand widely distributes in the seabed floor at the Yellow River Estuary, Bohai, China (Zhang et al. (2009) [45]). Dense seabed soil generally is formed due to the multi-process of sand liquefaction-post densification under ocean wave or seismic wave. Previous investigations indicated that poro-elastic model was applicable to describe the behavior of dense seabed soil, so long as the magnitude of external loading is not too great. In this study, poro-elastic model is used for the dense seabed foundation (property parameters are listed in Table 1). Generally, marine pipeline is made of steel (density = 7.85 g/cm3). Therefore, it can be modelled by elastic model. Here, the pipeline is considered as a kind of impermeable medium without porosity. The crude oil transported by the pipeline is considered as a kind of incompressible and fluidized elastic medium with a small value of Young's elastic modulus. It means *v* = 0.5 and porosity *n* = 1.0. The density of crude oil is set as 0.85 g/cm3, which is significantly less than that of water. In computation, a great value of permeability 1.0 <sup>×</sup> 10−<sup>1</sup> m/s is given to the crude oil due to the fact that there is no a solid medium to block the flowing of crude oil in pipeline. In this study, the flowing process of crude oil in pipeline cannot be modelled in 2D condition. Consideration of the crude oil helps determine the effect of the crude oil mass on the seismic dynamics of pipeline-seabed system. In previous literature, such as Ling et al. [29], Luan et al. [30], and Zhang and Han [31], the pipeline is set as empty without any mass, resulting in that the effect of the mass of crude oil on the seismic dynamics of pipeline-seabed system is ignored. In this study, the consideration of crude oil in pipeline actually is an innovative point relative to previous studies.

**Table 1.** Model parameters of seabed foundation, pipeline and crude oil.


In computation, the density of pure pore water in seabed soil is 1.0 g/cm3, and the bulk modulus is 2.24 <sup>×</sup> 109 Pa. The saturation S*<sup>r</sup>* is set as 98% due to the fact that there are more or less NH3/CH4 or air bubbles in real seabed soil. It has been widely recognized and accepted that Biot's equation can accurately describe the mechanical behavior of seabed soil when its saturation is greater than 95% by introducing a parameter, bulk of compressibility β = <sup>1</sup> K*f* + <sup>1</sup>−S*<sup>r</sup> pw*<sup>0</sup> , where K*<sup>f</sup>* <sup>=</sup> 2.24 <sup>×</sup> 109 Pa is the bulk modulus of pure water, S*<sup>r</sup>* is the saturation of soil, and *p*w0 is the absolute water pressure. Furthermore, the effect of temperature on properties of soil and pore water is not considered. Elastic modulus, permeability, and saturation of seabed soil are constant in computation, not depending on the confining pressure.

For loosely deposited seabed foundation, the poro-elastic model is not applicable to describe its complicated behavior. In this circumstance, an elasto-plastic model must be used. The seismic dynamics of marine pipeline buried in loosely deposited seabed soil is an interesting topic. It would be further studied by FSSI-CAS 2D in the future adopting advanced elasto-plastic models.

#### **4. Results**

#### *4.1. Initial Status*

Before arrival of the seismic wave, there is an initial status for the pipeline-seabed foundation system. This initial status should be taken as the initial condition for the seismic dynamics analysis thereafter. The distributions of displacement and effective stresses of the pipeline-seabed foundation in the initial status are shown in the Figures 4 and 5. It is clearly observed that the existence of pipeline has significant effect on the distributions of displacement and effective stresses in the seabed foundation around the pipeline. In Figure 4, it can be seen that the vertical displacement of pipeline and crude oil is basically the same, and slightly greater than that of surrounding seabed. It is shown that the pipeline-crude oil system slightly subsides relative to its surrounding seabed soil.

**Figure 4.** Displacement distribution of the pipeline-seabed in consolidated status. The effect of the pipeline on horizontal displacement is obvious, and the pipeline slightly subsides relative to its surrounding seabed soil.

In Figure 5, it is found that the pore pressure is layered in the seabed foundation due to the fact that the pipeline is made of impermeable steel. The driven pressure of crude oil (200 kPa) in the pipeline is isolated with the pore pressure in the seabed outside of the pipeline. There is no excess pore pressure in the initial status before seismic wave arriving. Due to the effect of the pipeline, the distribution of vertical effective stress σz' is not layered. However, the zone where the effective stress is affected by the pipeline is limited in the range x = 98 m to 102 m, and z = 16 m to 20 m. In the other zone, the distribution of effective stress is basically layered. Additionally, it is interesting to find that there is a small zone (labelled by red color) in the seabed beneath the pipeline where the effective stress is very small, comparing with that in the zone near to it. The physical mechanism is that some volume of pore water is expelled by the pipeline, resulting in an upward buoyancy applied on the pipeline. As a result, the effective stress in the seabed soil beneath the pipeline of course decreases. In the surrounding seabed soil of pipeline, the magnitude of shear stress is significant (greater than 5 kPa), and the distribution has symmetrical characteristics. Furthermore, there is also shear stress in the pipeline itself. However, there is no shear stress in the crude oil due to the fact that fluid cannot resist shear stress.

**Figure 5.** Effective stress and pore pressure distribution of the pipeline-seabed in consolidated status. The distribution of the vertical effective stress σz' indicates that an upward buoyancy is applied on the pipeline.

#### *4.2. Seismic Dynamics of Pipeline*

Taking the initial status as the initial condition, the seismic dynamics of the pipeline is modelled adopting the coupled numerical model FSSI-CAS 2D. The time history of seismic acceleration of the pipeline is illustrated in Figure 6. It is observed that the peak acceleration of the pipeline is 0.242 g and 0.345 g, respectively, in E-W and U-D direction. Compared with the input seismic wave on the bottom of seabed foundation, the amplification factor of peak acceleration reaches up to 1.78 and 2.79, respectively, in E-W and U-D direction. It is indicated that the acceleration amplification of pipeline buried in dense seabed foundation in vertical direction is stronger than that in horizontal direction. Another interesting phenomenon observed in Figure 6 is that there is significant resonance in the horizontal acceleration response of the pipeline. However, there is no resonance in vertical direction due to the suppression effect of gravity. In horizontal direction, this resonance is very significant after t = 170 s. Even at the end of computation, the vibration of horizontal acceleration of the pipeline does not vanish.

**Figure 6.** Time history of acceleration of the pipeline responding to input seismic wave. It is shown that there is a significant resonance in horizontal direction.

The time history of displacement of the pipeline responding to the input seismic wave is shown in Figure 7. It is found that the maximum amplitude of horizontal displacement of the pipeline responding to the input seismic wave reaches up to 139.6 mm. Meanwhile, the maximum amplitude of vertical displacement of the pipeline is only 24.6 mm. It is indicated that the displacement response of the pipeline buried in dense seabed foundation is much stronger in horizontal direction than that in vertical direction. Furthermore, the resonance of the horizontal dynamics of the pipeline can also be observed in Figure 7. As demonstrated in Figure 3, the input seismic wave on bottom of the seabed foundation basically vanishes after t = 170 s. However, the horizontal displacement of the pipeline continuously vibrates in a regular way in time domain.

**Figure 7.** Time history of displacement of the pipeline responding to input seismic wave. It is shown that there is a significant resonance in horizontal direction.

In this study, the computation is actually a 2D case, without the ability to evaluate the risk of pipeline rupture due to excessive stress. The strength and elastic modulus of the steel pipeline is at least greater than 235 MPa, 210 GPa. Comparing with the surrounding seabed soil, the steel pipeline can be treated as a rigid body in computation. In the geometrical model, the thickness of the pipeline is only 2 cm, only two layers mesh are used to discretize the steel wall of the pipeline, as shown in Figure 2. As a result, the stress state in the steel wall of the pipeline would not have enough computational accuracy. If the risk of pipeline rupture is the focus in the future, then the computation must be three-dimensional, and more meshes are necessary to discretize the thin wall of pipeline.

It is necessary to explore the seismic dynamics characteristics of the dense seabed soil near to the impermeable and rigid steel pipeline. In Figure 8, the time history of pore pressure and mean effective stress I1 on the two typical positions, A and B, labelled in Figure 2, are demonstrated. It is found that the wave form of the time histories on the two typical positions are basically the same, regardless of the pore pressure or the mean effective stress. They are all similar to the wave form of the input seismic wave on the bottom of seabed foundation. Due to the fact that the seabed soil is dense, poro-elastic model is used to describe the behavior of dense seabed soil in computation. There is only oscillatory pore pressure in seabed soil without the build-up of residual pore pressure. These characteristics are completely different compared to that in loosely deposited seabed soil [46,47].

**Figure 8.** Time history of pore pressure and effective stress I1 at the two typical position **A** and **B** in the seabed foundation labelled in Figure 2. There is no residual pore pressure built up in dense seabed soil.

Except for the time history of dynamics of the pipeline in time domain, the spectrum characteristics in frequency domain is also necessary to be analyzed. The acceleration spectrum of the pipeline responding to the input seismic wave is illustrated in Figure 9. It is observed that there are two peak values in the spectrum of horizontal acceleration of the pipeline. The corresponding periods for the peak values are 0.6 s and 1.85 s, respectively. Meanwhile, there is only one peak value in the spectrum of vertical acceleration of the pipeline. The corresponding period is also 0.6 s. As observed in Figure 9, it is known that there are two resonance periods for the pipeline-crude oil-seabed foundation system.

**Figure 9.** Acceleration spectrum of the pipeline responding to input seismic wave. It is observed that there are two resonance periods (0.6 s and 1.85 s) for the pipeline-crude oil-seabed foundation system.

#### *4.3. E*ff*ect of Lateral Boundary Condition*

In this study, the laminar boundary condition is applied on the two lateral sides of the seabed foundation. This kind of boundary condition can guarantee that there is no seismic wave reflection on the lateral sides. Laminar boundary without wave reflection on lateral sides is much more approaching the real situation because the seabed is infinite in horizontal in offshore environment. However, it is also interesting to investigate the effect of fixed lateral boundary condition on the seismic dynamics of the pipeline.

As demonstrated in Figure 10, the effect of fixed lateral sides on the horizontal seismic dynamics of the pipeline is significant. However, this effect on the vertical seismic dynamics of the pipeline is negligible. If the fixed lateral boundary condition is applied, the acceleration and displacement of the pipeline in horizontal direction responding to the input seismic wave are both significantly greater than that in which the laminar lateral boundary condition is applied before t = 150 s. Furthermore, the resonance of the pipeline in horizontal is very significant after t = 170 s in the case laminar lateral boundary condition is applied, as illustrated in Figures 6 and 7. It is found in Figure 10 that there is also resonance phenomenon if the lateral sides of seabed foundation are fixed. However, the amplitude of acceleration and displacement of the pipeline are generally less than that if the laminar lateral boundary condition is applied. Therefore, it is concluded that the peak horizontal acceleration and displacement of marine pipeline will be overestimated. Meanwhile, the seismic wave-induced resonance of marine pipeline will be underestimated if fixed lateral boundary condition is applied to seabed foundation. The lateral boundary condition without seismic wave reflection, such as laminar boundary condition or absorbing boundary condition, should be used in computation.

**Figure 10.** Effect of the fixed lateral side boundary on the dynamics of pipeline. It is shown that there is a significant adverse effect of the fixed lateral boundary condition on the horizontal dynamics.

#### *4.4. Comparison with Pipeline-Gas System*

In the practice of engineering, marine pipeline is not only used to transport crude oil, but also natural gas (density is 0.7174 kg/m3). In this study, the seismic dynamics of pipeline-gas system buried in dense seabed foundation is also investigated under the same excitation of the input seismic wave. The time history of acceleration of the pipeline-gas system is demonstrated in Figure 11. Compared with the result of the pipeline-oil system shown in Figure 6, it is found that the difference of acceleration response between the two cases is not significant. The peak horizontal acceleration (0.242 g) of the pipeline-gas system is only slightly greater than that (0.225 g) of the pipeline-oil system. The peak vertical acceleration of the two systems are basically the same.

**Figure 11.** Time history of acceleration of pipeline-gas system responding to input seismic wave. There is also a significant resonance if natural gas is transported by the pipeline.

The comparison of the seismic dynamics between the pipeline-gas system and the pipeline-oil system in frequency domain is further illustrated in Figure 12. Figure 12 further proves that the difference of seismic dynamics is minor between the pipeline-gas system and the pipeline-oil system excited by the same seismic wave if buried in dense seabed.

**Figure 12.** Acceleration spectrum of pipeline-gas system responding to input seismic wave. It is shown that the difference of seismic dynamics of the pipeline-oil system and the pipeline-gas system is minor.

Previous studies have indicated that seabed soil could significantly amplify the peak acceleration from its bottom to its surface. Figure 13 also confirms this amplification effect of the seabed foundation. It is observed that the peak acceleration in horizontal and vertical direction generally increases with the distance to the bottom of seabed foundation. It is also found that the amplification effect of the seabed foundation basically is the same, regardless of pipeline-oil system or pipeline-gas system. Adopting a perspective considering time history, spectrum of acceleration of the pipeline, as well as the amplification effect of seabed foundation, it is found that the difference of seismic dynamics of pipeline-oil system and pipeline-gas system is minor.

**Figure 13.** Amplification effect of the seabed soil along depth. It is confirmed that the seabed foundation has significant amplification effect to the input seismic wave in both horizontal and vertical direction.

#### **5. Conclusions**

In this study, taking the coupled numerical model FSSI-CAS 2D as a tool, the seismic dynamics of a marine steel pipeline transporting crude oil or natural gas buried in dense seabed soil is investigated. The computational results indicate that the marine steel pipeline buried in dense seabed soil strongly responds to seismic waves. The response peak acceleration of the pipeline could be twice of the peak acceleration of the input seismic wave. There is only oscillatory pore pressure in the dense seabed soil surrounding the pipeline without the build-up of residual pore pressure under the excitation of seismic wave. The resonance phenomenon is very significant in the horizontal dynamics of the pipeline. However, there is no resonance for the vertical dynamics of the pipeline. Fixed lateral boundary condition on seabed foundation has ill-natured effect on the computational results. Any type of lateral boundary condition which could avoid the wave reflection, such as laminar boundary and absorbing boundary, should be used in computation. It is also found from the computation results that the difference on the seismic dynamics of marine pipeline between pipeline-oil system and pipeline-gas system is minor. Finally, it is proven that the coupled numerical model is applicable to study the seismic dynamics of marine pipeline.

**Author Contributions:** J.Y. was contributed to all numerical cases running, computational results analysis. Y.Z. was contributed to all figures drawing and manuscript writing. K.H. was contributed to the mesh generation and preparation of the input files for all numerical cases. S.C. was contributed to the results analysis and manuscript writing.

**Funding:** Jianhong Ye are grateful to the funding support from National Natural Science Foundation of China under project NO. 51879257.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Dynamic Response of O**ff**shore Open-Ended Pile under Lateral Cyclic Loadings**

**Junwei Liu 1,2, Zhen Guo 3,\*, Na Zhu 1, Hui Zhao 1, Ankit Garg 4, Longfei Xu 1, Tao Liu <sup>5</sup> and Changchun Fu <sup>6</sup>**


Received: 28 March 2019; Accepted: 30 April 2019; Published: 3 May 2019

**Abstract:** Foundations for offshore wind turbines (OWTs) are mainly open-ended piles that are subjected to cyclic loadings caused by winds, waves and currents. This study aims to investigate the dynamic responses of open-ended pipe pile under lateral cyclic loadings, as well as the characteristics of the soil plug and surrounding soil. Both large-scale indoor model test and discrete element simulation were adopted in this study. The test results show that the resistance of each part of the pipe pile increases linearly with depth during the process of pile driving. The pile side resistance degradation effect was also observed along with the friction fatigue. The soil plug formation rate decreases gradually with an increase in the pile depth. The influence range in the surrounding soil is about 5~6 times of the pile diameter. The cumulative displacement of the pile head increases with the number of cycles. Lateral tangential stiffness and lateral ultimate bearing capacity decreases with an increase in number of cycles. The severe disturbance range of soil around the pile is 2~3 times of the pile diameter. The center of rotation of the pile body is about 0.8 times of the pile body depth. The side frictional resistance and lateral pressure of the pile body is found to fluctuate along the pile body. Additionally, the lateral pressure and side friction resistance decreases gradually with decreasing tendency of the former more than the latter.

**Keywords:** open-ended pile; soil plug; offshore wind turbines; lateral cyclic loading; model test; discrete element simulation

#### **1. Introduction**

The open-ended pile is a common option of foundations for offshore wind turbines (OWTs). In the offshore environment, the piles are subjected to not only the vertical static load (e.g., its own weight), but also the lateral cyclic loadings caused by winds, waves and currents. The deformation characteristics of pile foundations under cyclic loadings are important for the safety of OWTs. There has been a number of theoretical and experimental researches about the offshore structures [1–11]. Matlock [1] proposed the p-y curve calculation method, which was adopted by the US American

Petroleum Institute (API). The p-y curve is the embodiment of the laterally loaded pile-soil interaction, where the soil resistance of the pile is gradually exerted with the application of lateral load. Reese et al. [2,3] developed and made some corrections on the calculation method of the p-y curve, which still is the most widely used calculation method for pile foundations under lateral loading. Rosquoet et al. [12] carried out a series of cyclic lateral loading tests of the pile foundation in sand, and proposed the calculation formula of the cyclic deformation of the pile body related to the cyclic load size. Leblanc et al. [13] carried out the centrifugal model test in sand to study the cumulative deformation of pile foundations under lateral cyclic loading. A series of indoor model test of pile foundations was performed to study the mechanical principle and deformation characteristics of rigid piles under different cyclic loads [14–17]. Li et al. [18] investigated weakening characteristics of offshore platform pile foundations under long-term cyclic loading, as well as used the pore-pressure development model to establish the residual soil shear strength model.

In addition, large amounts of research concerning the large-scale pile group have been conducted. These studies include not only the extensive experimental tests subjected to cyclic lateral loading [19–22], which reveals the highly nonlinear nature of the pile-soil-pile interaction, but also the analysis method for laterally loaded pile groups, that takes into account the non-linear behavior of the soil and the non-linear response of reinforced concrete pile sections simultaneously by the newly proposed Boundary Element Method (BEM) approach [23]. Recently, considering the limitations of test facilities and high costs for prototype (large-scale) tests, a series of cyclic lateral load tests was conducted on a stainless steel mono-pile in the centrifuge [24,25].

The aforementioned studies have significantly advanced the understanding of the response of pile under lateral cyclic loads. But there are few studies regarding the distinctive behaviors of open-ended piles due to the installation effect compared to that of the equivalent close-ended pile [26]. Thus, the dynamic response of the open-ended pile, particularly the soil plug inside the pile, still has considerable uncertainty. Moreover, understanding the micro-mechanisms is essential to interpret the macro-behavior in complex geotechnical issues [27]. In this paper, both the numerical simulation and model test were used to the reveal the comprehensive responses of the soil-pile system during the long-term lateral loading, including the micro-mechanisms and macro-behaviors both inside and outside the pile. A double-walled pile system was applied in both the tests and numerical simulation to separate the internal and external frictions.

#### **2. Model Test Design**

#### *2.1. Model Box and Soil Sample Preparations*

Figure 1 shows the model box utilized for testing of the open-ended pile. The inner dimensions of the model box are 3 m × 3 m × 2 m (length × width × height). The model box is provided with an unloading port system as well as visual windows. Figure 2 shows the mold particle gradation curve. The sand sample median grain size, non-uniformity coefficient and coefficient of curvature are 0.72 mm, 4.25 and 1.47, respectively. The soil sample is controlled by the sub-lamination compaction.

**Figure 1.** Model test arrangement.

**Figure 2.** Soil particle gradation curve.

#### *2.2. Model Pile and Sensor Layout*

The double-layer pipe wall pile model consists of two concentric pipes of 6063 aluminum alloy material. The outer diameter, inner diameter, wall thickness and the pile length are 140 mm, 120 mm, 10 mm and 1000 mm, respectively. Both the inner and outer tubes are instrumented with a fiber optic sensor. The outer tube in addition is also instrumented with a soil pressure sensor (Figure 3). Six installed soil pressure sensors are arranged in order from the pile bottom to the pile-top to measure the soil pressure in the pile-soil interface.

**Figure 3.** Instrumentation of model pile sensor layout for measuring strain and pressure.

#### *2.3. Test Programme*

Figure 4 shows that pile driving is proceeded using the step loading method by the lateral servo loading equipped and cyclic loading (refer to Figure 4 for illustration). According to the proposed damage standard of laterally loaded pile [28], the ultimate bearing capacity (*PR*) of a single pile foundation under the lateral static load conditions is the corresponding load when the pile top displacement reaches 0.1 times the pile diameter. Based on static the loading test, the lateral ultimate bearing capacity of the pipe pile is 1587 N. Leblanc et al. [13] defines two coefficients ζ<sup>b</sup> and ζ<sup>c</sup> to represent the characteristics of cyclic loading (Refer to Equations (1) and (2). ζ<sup>b</sup> is cyclic load ratio, ζ<sup>c</sup> is the ratio of minimum load *Pmin* to maximum load *Pmax*.

$$
\zeta\_b = \frac{P\_{\text{max}}}{P\_R} \tag{1}
$$

$$\zeta\_c = \frac{P\_{\text{min}}}{P\_{\text{max}}} \tag{2}$$

**Figure 4.** Cyclic loading control (Note: 1–6 is the displacement meter number; and the displacement from the pile is 0.1 m, 0.25 m and 0.6 m).

Loads in ocean environment are usually complicated and irregular. For simplifications, the cyclic loading form in model tests is shown in Figure 5. In this study, two types of cyclic loadings, i.e., Rc = 0.0 and Rb = 0.5 are simulated (according to Figure 5). Table 1 summarizes the testing program for this study. Among them, three tests on the open-ended pipe pile were subjected to biaxial loading while the other one was subjected to uniaxial loading. The load amplitudes adopted in this study are 200 N, 500 N, and 800 N, respectively, and the cyclic load ratios (ζb, refer to Equation (1) and *PR* = 1587 N) are 0.126, 0.315, and 0.504, respectively. The uniaxial cyclic load ratio is 0.113.

**Figure 5.** Schematic diagram of cyclic loading.


**Table 1.** Test programme.

\* Buried depth = 0.74 m; Frequency = 4 Hz; Cycles = 1000.

#### **3. Discrete Element Simulations**

The experimental program was established to quantify deformation characteristics, that will help in evaluating the performance of open-ended pipe pile under various loading conditions (refer to Table 1). In addition to the experimental program, the numerical simulation plan was also designed so as to understand the mechanism, which will help in the interpretation of measured deformation characteristics from the experimental program. Below is the description of the adopted numerical program and procedures in this study.

#### *3.1. Soil Sample Preparation*

The size of the model is 2.4 m × 2.4 m (width × height) (refer to Figure 6). The a modified particle generation method, referred to as the Grid-Method (GM), was used to generate soil samples [29]. The model was divided into 24 small squares. The soil layers were defined by simulating one-side particles from left to right and from bottom to top. This method effectively avoids the pressure realization in the process of soil formation. The maximum particle size, minimum particle size, median diameter (*D*50) and uneven coefficient (*C*u) are 3.52 mm, 2.25 mm, 2.92 mm and 1.26, respectively.

**Figure 6.** Preparation of soil samples by the GM method.

#### *3.2. Numerical Simulation Model*

The model pile has a diameter of 45 mm, a length of 0.5 m and a wall thickness of 2.475 mm. The model pile consists of particles with a radius of 1.125 mm. The particles overlap with each other. The distance between the centers of two adjacent particles is *d*pp (0.2 *R*) [30], as shown in Figure 7. In this simulation, the diameter of the particles forming the pile is much smaller than the diameter of the pile. Further, the distance between the particles is short. The roughness is close to the initial set value. The direction of the contact force between the particles and the pile is the same as the axial direction of the pile. With this, the axis resistance calculations are easier and more accurate. Since the proposed GM uses the explosive method for particle generation. Particles are created at their final radii in specify numbers to achieve the desired porosity and the number of every type size are calculated in advance. The following Equations (3)–(6) will be used for the calculation of the initial porosity *e*initial and particle number in every grid. *A*<sup>m</sup> is the area of the model, *A*pi and *N*(i) is the total particles area of the same specific diameter *r*(i) and the quantity of the corresponding diameter particles. The final selection of soil samples is shown in Table 2.

$$N\_{(i)} = NP\_{(i)} \tag{3}$$

$$\left|A\_{P}\right|^{2} = N\_{(i)}r\_{(i)}{}^{2}\pi\tag{4}$$

$$A\_P = A\_P^{-1} + A\_P^{-2} + \dots + A\_P^{-i} \tag{5}$$

$$x\_{initial} = \frac{A\_P}{A\_m} \tag{6}$$

**Figure 7.** Pile composition.

**Table 2.** Numerical simulation of the physical parameters of soil samples.


#### *3.3. Numerical Simulation Programme*

An open-ended rigid pipe pile has an outer diameter of 45 mm and a length of 0.5 m and a wall thickness of 2.475 mm. In order to better observe the soil sample deformation, the soil samples are set to different colors. The depth of the pile model test is 0.4 m, and the model pile is placed on the centerline of the model box. The minimum distance between the model pile and the model box wall is more than 7*D* (*D* is the pile diameter), and the distance between the pile end and the bottom of the box is more than 4*D*. This configuration avoids any boundary effects [31]. The horizontal load test is carried out and results show that the horizontal ultimate bearing capacity of the pipe pile is 8118 N. The specific parameters adopted in the numerical simulation program are summarized in Table 3.


**Table 3.** Numerical simulation program emphasizing input parameters.

#### **4. Test Results and Discussions**

#### *4.1. Measured Pile Top Cumulative Displacement under Lateral Cyclic Loadings*

Figure 8 shows the variation in the horizontal displacement with time for all different loading conditions (refer to open-ended pipe pile cases, M1-M4 in Table 1). It can be observed that trends of the displacement curves of the pile top are consistent under different cyclic loading modes. With the application of the sinusoidal load, the displacement of the pile also changes sinusoidally over time.

**Figure 8.** Pile top cumulative displacement.

The total displacement for tests M1, M2, M3 and M4 are 0.65 mm, 0.9 mm and 1.35 mm, 0.8 mm, respectively. As observed from Figure 8, the maximum cumulative displacement of the pile top gradually increases with an increase in the number of cycles and then gradually stabilizes beyond 100 cycles. The increase in the horizontal displacement is much faster in the first 100 cycles as compared to beyond. The displacement in the first 100 cycles for M1, M2, M3 and M4 are about 76.9%, 74.4%, 81.4%, 78.9% of the total displacement, respectively. The cumulative displacement for M2 and M3 increases higher than M1 by 27.7% and 51.8%, respectively. This is obviously due to an increase in load. It can be also observed that the cumulative displacement in the case of uniaxial loading is 18.75% higher than biaxial loads.

In this study, the cumulative displacement under the lateral cyclic loading is predicted mainly by establishing the relationship (refer to Equation (7)) between the displacement of the pile and the number of cycles. Hettler [32] carried out the cyclic triaxial test and model pile test in dry sand. It is considered that the relationship between the ratio of the lateral displacement (*y*N) of the pile under the cyclic load and the displacement *y*<sup>1</sup> of the pile after the first cycle and the number of cycles *N* are as follows:

$$y\_N = y\_1(1 + \mathbb{C}\_N + \ln N) \tag{7}$$

where *yN* is the horizontal displacement after *N* cycles, *CN* is the weakening coefficient. For cohesion-less soil, *CN* is usually 0.2. The weakening coefficients for M1, M2, M3 and M4 are found to be 0.159, 0.173, 0.181, and 0.186, respectively. The weakening coefficient is similar to that obtained by Zhu et al. [33].

#### *4.2. Measured Load-displacement Curve under Lateral Cycling Load*

Figure 9a–d shows the pile top load-displacement curves for cases M1-M4 under the lateral cyclic loading. The ratio of the maximum load and the change in the lateral displacement of the pile top is the lateral secant stiffness of the pile foundation. The stiffness of the soil around the pile changes under the cyclic load.

**Figure 9.** Load-displacement curve of pile top of Pile M1-M4.

As shown in the figures, there exists a hysteresis loop in load displacement curves of the pile top with each of its cycle overlapping partially. Generally for all the cases, the hysteresis loop is relatively small during the first ten cycles of loading. The hysteresis loop gradually tilts toward the displacement axis. The area within the hysteresis loop curve is gradually reduced with an increase in cycles. Ultimately, the load displacement curve seems stabilized. The lateral stiffness for M1, M2, M3 and M4 decreases by 11.6%, 14.0%, 17.2%, and 12.8% at the end of 1000 cycles. However, it should be noted that the major decrease in lateral stiffness for M1, M2, M3 and M4 are 6.97%, 9.9%, 13.5% and 8.01%, respectively in the first 100 cycles. These account for more than 60% of the overall decrease. It shows that cyclic loading can reduce the lateral secant stiffness of the pile foundation. To a certain extent, cyclic loading can reduce the lateral deformation modulus of the soil. This is similar to the law obtained by Zhang et al. [34]. As per their law, the soil around the pile will "plastically" deform and

gradually deflect with an increase in loading cycles. The gradual deformation of the soil will cause the weakening of the pile-soil system.

#### *4.3. Measured Surface Displacement under Lateral Cyclic Loading*

Figure 10 shows the variation in surface displacements (at points 1–6; refer to the experimental set up in Figure 4) for case M1 (refer to Table 1). It can be observed that there was a rapid increase in the displacement in the initial stage of application of the cyclic load. However, the rate of increase reduces after around 5 s. The displacement for point 2 (gauge no. 2; refer to Figure 4) on the right side of the pile is the largest, while on the left side, the displacement for gauge no. 5 is the largest. It can be observed that under the cyclic load, the soil will settle down in the range of 0.1 m near the pile. Between the range of 0.1 m~0.25 m, the soil uplift will occur on both sides of the pile.

**Figure 10.** Variation of surface displacement with time under lateral cyclic loading for case M1.

The displacement on the left side of the pile is larger than that on the right side, where the active pressure area is larger than the passive pressure area. The comparison shows that the maximum surface displacement of the pile under different loading conditions such as M1, M2, M3 and M4 are 0.25 mm, 0.35 mm, 0.4 mm, 0.28 mm, respectively. With an increase in the cyclic load, the surface displacement increases gradually. Also, the displacement under the uniaxial cyclic load is larger than the biaxial cyclic load. This law is similar to the cumulative displacement of the pile top (refer to Figure 9). The disturbance range of the soil around the pile is 2~3 times the diameter of the pile.

#### *4.4. Measured Pile Friction under Lateral Cyclic Loading*

Inclination occurs to the pile under the application of the lateral cyclic load to the soil around the pile. The lateral cyclic load makes changes to the direction of the pile, where the friction and lateral pressure of the pile differs from the traditional vertical loaded pile. The friction law of the pile body is similar under different loading conditions. This study provides mainly the curve of the frictional force of the pile body (for case M1 only) with the cycle period (as shown in Figure 11). As observed from Figure 11, the unit friction force near the pile bottom decreases with an increase in the number of cycles. The frictional force near the pile top tends to increase during the first 100 cycles. However, the change in the frictional force is minimal in the middle of the pile body during the first 100 cycles. The friction at the pile bottom is generally weakened by about 3.8%, and the friction at the pile top is increased by about 3.4%. The friction of the pile body is generally found to reduce with the application of the lateral cyclic load; nevertheless, the decay rate is about 3.8%, where the degradation degree mainly accounts for more than 70% of the total degradation degree in the first 100 cycles.

**Figure 11.** Unit side friction for case M1 (refer to Table 1).

The comparative analysis shows that the friction of the pile body is generally reduced with an application of the lateral cyclic load. The reduction tendency of the friction is found to slow down with an increase in the number of cycles. The overall reduction ranges of frictional forces for M1, M2, M3 and M4 are 3.6%, 3.8%, 4.2%, and 3.7%, respectively. As the magnitude of the cyclic load increases, the frictional decay amplitude increases, and the frictional force in the case of the axial cycle becomes weaker than the biaxial cycle. The friction inside the pile is mainly concentrated in the range of two times the pile diameter above the pile end, which can be called the "developing height" of the soil plug. In the range of the "developing height" of the soil plug, the frictional force in the pile changes more obviously. The disturbance of the soil plug at the end of the pile is directly proportional to the load amplitude. Moreover, the attenuation of the friction on the inner wall surface increases with the load amplitude.

#### *4.5. Measured Lateral Pressure of Pile under Lateral Cycling Load*

Figure 12a–d variation of the soil pressure with depth for different loading conditions (M1-M4, refer to Table 1). From Figure 12a, it is clear that the change in the lateral pressure at a depth of 0.58 m is zero. In general for all cases, the lateral soil pressure above the depth of 0.58 m increases under the cyclic load, while it decreases below the depth of 0.58 m. Hence, the depth of around 0.58 m is the center of the pile rotation. The center of the pile rotation is located approximately at about 0.8 times of the pile depth. During the cyclic loading process, the soil pressure sensor of the pile body is positioned in the "active zone", and a part of the sensor is located in the "passive zone". The definition of "active zone" and "passive zone" are established during the first cycle loading process. More specifically, when applying the horizontal loading in the first half cycle, the piles incline to the right around a rotation center, which brings about an "active zone" on the left of the pile while a "passive zone" on the right side. Herein, three soil press sensors (4#, 5# and 6#), above the rotation center, are located in the "passive zone"; on the contrary, 1# and 2# sensors are placed in the "active zone".

The lateral pressure of the active zone increases with the cyclic loading, while the lateral pressure of the passive zone decreases. The pressure in the passive zone increases with an increase in the number of cycles. The major increase occurs during the first 100 cycles, accounting for more than 70% of the total. The lateral pressure of the active zone shows a decreasing trend. The analysis shows that the overall lateral pressure for M1, M2, M3 and M4 are attenuated by 6.9%, 7.5%, 8.8% and 7.3%, respectively.

**Figure 12.** Variation of lateral soil pressures with depth for various loading conditions of Pile M1-M4.

#### *4.6. Measured Static p-y Curve under Lateral Cyclic Loadings*

In this study, the American Petroleum Institute API [35] and Reese [3] sand p-y curve models are used to calculate the horizontal load and displacement of single piles. Figure 13 shows the variation of the calculated load-displacement in the horizontal direction. As observed from Figure 13, the results obtained by the two above-mentioned methods are relatively more close to those of the static calculation (i.e., before the test cycle loading). With the application of the cyclic load, the soil around the pile is disturbed and the ultimate bearing capacity of the soil after the cyclic loading is reduced. The ultimate bearing capacity of the soil for cases M1, M2, M3 and M4 are reduced by about 11%, 14%, 17% and 13%, respectively. After 100 cycles, the results calculated by the two methods are quite different from the test results, indicating that the two static calculation methods cannot reflect accurately the influence of cyclic loading on the displacement of piles.

**Figure 13.** Static p-y curve based on p-y models proposed by API [26] and Reese [3] compared with physical model test results.

#### **5. Numerical Simulation Results**

#### *5.1. Computed Cumulative Displacement of Pile Top*

Figure 14 illustrates the maximum computed cumulative displacement curve of the pile top. Clearly, the cumulative displacement of the pile top under different loading conditions are similar. In all the cases (P2–P6), the displacement first increases rapidly during the initial 10 cycles and then gradually stabilizes. After 10 cycles, the displacements of P2, P4, P5 and P6 (Refer to Table 3 for numerical plan) were 0.35 mm, 0.6 mm, 0.98 mm and 0.45 mm, respectively. After the end of the cycle, the maximum cumulative displacement of the pile top for P2, P4, P5 and P6 are 0.41 mm, 0.71 mm, 1.13 mm and 0.55 mm, respectively.

**Figure 14.** Displacement of the pile top According to Equation (7), the weakening coefficients of P2, P4, P5, and P6 after fitting are 0.22, 0.24, 0.26, and 0.27, respectively. It can be seen that as the cyclic load ratio increases, the weakening coefficient also increases gradually. The axial cyclic load-weakening coefficient reaches its peak value at a much faster rate with an increase in the cyclic load ratio.

Obviously, the maximum cumulative displacement of the pile top gradually increases with the increase in the cycle period. The rate of change in the first 10 cycles is faster, and the velocity gradually becomes slower after 20 cycles. The cumulative displacement mainly occurs in the first 10 cycles. The displacement is about 74.5%, 84.5%, 88.2%, and 81.8% of the total displacement. The cumulative displacement of the pile top increases with the increase of the cyclic load ratio, which increases by 33.8% and 57.7%, respectively. After the axial cyclic loading, the cumulative displacement around the pile is increased by 14.6% over the biaxial.

The computed and measured cumulative displacement of the pile top seems to be similar. The cumulative displacement gradually increases with the increase of the period, and the increasing rate of increase gradually becomes slower. The computed weakening coefficient is slightly larger than the experimental weakening coefficient.

#### *5.2. Computed Load-displacement Curves*

Under the lateral cyclic loading, the interaction between the pile and soil is weakening. Carrying the normalization analysis through the test and simulation results, the corresponding load amplitude divides the lateral load, and the lateral displacement is divided by the maximum cumulative displacement. Figure 15 illustrates the load-displacement curves under normalized cyclic loading conditions. The curves basically resemble the hysteresis loop. As the cycle period increases, the area of the loop gradually increases, indicating the increase in the displacement of the pile top as well as the reduction in the lateral stiffness.

**Figure 15.** Load vs. displacement curves for various loading conditions of Pile P2, P4, P5 and P6.

The lateral secant stiffness k1 of the first cycle for P2, P4, P5 and P6 are 6.67 kN/mm, 8.57 kN/mm, 9.09 kN/mm, 5.88 kN/mm, respectively. The lateral secant stiffness k10 in the 10th cycle for P2, P4, P5 and P6 are 6.18 kN/mm, 7.65 kN/mm, 7.32 kN/mm, 5.21 kN/mm, respectively. The lateral tangential stiffness k100 in the 100th cycle for P2, P4, P5 and P6 are 5.88 kN/mm, 7.32 kN/mm, 6.94 kN/mm, 4.84 kN/mm, respectively. The overall reduction is 11.8%. 14.6%, 23.6%, and 17.7%, mainly occurred in the first 10 cycles, accounting for 62.0%, 73.6%, 82.3%, and 64.4% of the total.

#### *5.3. Computed Displacement Around Soil*

Figure 16 illustrates a computed displacement vector diagram and a displacement cloud diagram of soil around the P2 pile. As observed from the figure, with the application of the lateral load, the particles around the pile are disturbed, and the pile body is tilted in the direction of the loading force.

**Figure 16.** Displacement vector and Displacement cloud of surrounding soilaround P2 pile.

It can be seen from Figure 16a that the particle motion range of the soil in front of the pile and the soil behind the pile appears to be opposite. The direction of the movement of the soil plug is generally upward. It can be seen from Figure 16b that the influence range of the soil around the pile is the "butterfly" type. The soils around the pile can be approximately divided into different disturbance zones. The soil displacement in the pile top range is larger, whereas, the soil particle displacement in the pile bottom range is smaller. The displacement of the active zone is greater than the passive zone. Based on the figure, the displacement of soil particles is the largest in the range of four to five times of the pile diameter. The farther the distance from the pile, the smaller the particle displacement. The comparison test results show that the critical influence zone in the test lies at around 2~3 times of the pile diameter. The computed zone is slightly smaller than that measured in experiments.

#### *5.4. Computed Pile Side Friction*

Figure 17a,b shows the variation of the pile frictional resistance with depth for outer and inner tubes, respectively for case P2. The axial side frictional resistance appears to increase along the depth with some fluctuations for both the outer and inner tubes of the pile. It can be seen from Figure 17a that the side frictional resistance on the left side of the pile decreases at the vicinity of the pile top. The decreasing rate gradually becomes slower with the number of cycles. The side frictional resistance of the pile bottom tends to increase under the application of the load. The pile body rotates to the right around the center of rotation, and the friction between the soil and the pile on the left side of the pile top is reduced. The analysis of Figure 17a shows that the outer side frictional resistance above the center of rotation decreases. However, there is an increasing trend observed below the center of rotation. Variations in both these trends occur mainly during the first 10 cycles. The outer side frictional resistance on the right side of the pile is opposite to the left side. As compared to Figure 17a, the side frictional resistance (Figure 17b) on the left and right sides of the pile changes minimally during the period. Compared with the inner and outer frictional resistance, the side frictional resistance of the right side is greater than the side frictional resistance of the left side. The comparative analysis shows that the variation in the lateral frictional resistance is the highest in pile P5, while it is lowest for pile P2. For all the piles (P2, P4 and P5), the variations (i.e., gradually increase) of the lateral frictional resistance occur mainly during the first 10 cycles of loading. Additionally, it can be stated that the lateral frictional resistance of the pile (based on P2 and P6) is more under the axial cyclic load than the biaxial cyclic load.

**Figure 17.** Pile (P2) side frictional resistance at (**a**) outer tube and (**b**) inner tube.

Figure 18 shows the variation of the side frictional resistance of P2, P4, P5 and P6 with the number of cycles. As the number of cycles increase, the total side frictional resistance of the pile body decreases. The overall decrease for P2, P4, P5 and P6 are 3.4%, 3.8%, 5.1%, and 3.5%, respectively. Generally, for

all the cases (P2, P4, P5 and P6), the main decline in the side frictional resistance in the first 10 cycles accounted for more than 77% of the total.

**Figure 18.** Unit side frictional resistance of pile body.

#### *5.5. Computed Lateral Pressure of the Pile Body*

Figure 19a,b shows the variation of the lateral pressure distribution along with the depth for the outer and inner tube, respectively of the pile (P2). It can be observed that despite fluctuations, there is an increase in the lateral pressure along with the depth. Further, it can be observed that the outer lateral pressure on the left side of the pile is a positive value, while, the outer lateral pressure on the right side of the pile is a negative value. The difference among them tends to increase along with the depth.

**Figure 19.** Distribution of lateral outside and inside pressure of Pile P2 ((**a**) outer pressure; (**b**) inside pressure).

The pressure on the left side of the pile near the pile top decreases due to cyclic loadings, while the pressure on the left side near the pile end increases. The lateral pressure on the right side of the pile changes inversely to the left side. The lateral pressure near the center of rotation varies minimally during the loading period. It can be seen from Figure 19b that the sign of the lateral pressure on the inside of the pile is opposite to the outside. Overall, there is an increase in the lateral pressure along with the depth. The fluctuations are relatively smaller. The difference in pressures between the inside and outside of the pile increases with the depth. The maximum pressures on surfaces of the inner tube are approximately twice that of the outer tube. It indicates that the inner tube experiences larger lateral pressures than the outer tube. Among P4, P5, P6 and P2, the lateral pressure change mainly occurs in the first 10 cycles and then gradually stabilizes. However, the change extents are different for the different piles due to different load ratios, loading methods and loading amplitude.

Figure 20 compares the distribution of the total lateral pressure between piles P2, P4, P5, and P6 piles. It can be observed that the pressure of the pile-soil interface under axial and biaxial cyclic loading shows a reducing trend. An overall decrease of 8.9%, 9.3%, 10.1%, and 9.7% was observed for piles P2, P4, P5 and P6, respectively. The major change in soil pressure occurs mainly in the first 10 cycles, accounting for more than 74% of the total. Similar to the experimental results, the lateral pressure decreases with the number of cycles. The major change occurs in the first 100 cycles accounting for more than 70% of the total. The lateral pressure of the active zone shows a decreasing trend. The comparative analysis showed that the overall lateral pressure for piles P2, P4, P5 and P6 was attenuated by 6.9%, 7.5%, 8.8%, and 7.3%. It can be seen that the magnitude of the decline increases with an increase of the cyclic load ratio (i.e., lateral pressure higher in the case of axial loading than biaxial).

**Figure 20.** Comparison of computed unit lateral pressure distribution with cycles between P2, P4, P5 and P6.

Figure 21 shows the comparison of static calculation curves before and after cyclic loading between piles P2, P4, P5 and P6. It can be seen from Figure 21 that under the application of different cyclic loading modes, the lateral ultimate bearing capacity of P2, P4, P5 and P6 is reduced by 11.7%, 14.5%, 23.5%, and 17.7%, respectively. The comparative analysis shows that the cyclic load can reduce the lateral bearing capacity of the pile foundation. The magnitude of the reduction under different loading conditions is different. The rate of decrease of the amplitude is enhanced with the cyclic load ratio. The rate of decrease of the amplitude is higher in the case of uni-directional loading than biaxial loading.

**Figure 21.** Comparison of simulated static load curves between P2, P4, P5 and P6 after cyclic loading.

Both measured and computed static load curves for all types of loading conditions were normalized by dividing the actual value (load or displacement) with the corresponding maximum value. Figure 22 shows the comparison of measured as well as computed normalized static load curves. It can be seen from the test results that with the application of the cyclic load, the soil around the pile is disturbed, and the ultimate bearing capacity of the soil after circulation is reduced. The ultimate bearing capacity for M1, M2, M3 and M4 cycle is reduced by about 11%, 14%, 17%, 13%, respectively. In the simulation results, the lateral ultimate bearing capacity of the piles decreased by 11.7%, 14.5%, 23.5%, and 17.7% after 100 cycles. The ultimate bearing capacity increases with the cyclic load ratio. The decreasing amplitude is gradually increased, and the reduction of the bearing capacity in the case of axial cyclic loading is more than the reduction of biaxial cyclic loading.

**Figure 22.** Comparison of measured and computed normalized static loads between various loading conditions.

#### **6. Conclusions**

In this study, the dynamic response of open-ended pipe piles under lateral cyclic loadings is studied by large-scale indoor model tests and discrete element simulations. The main findings can be summarized as follows:

(1) Both the increases of cumulative displacement on the pile top and the decrease of the lateral secant stiffness occur mainly in the first 100 cycles, which is in the range of 10~25% but varies greatly with the change of the loading mode. Uni-axial cyclic loading causes more lateral displacement than biaxial loading. The ultimate bearing capacity of the pile decrease logarithmically with the increase of the period, and the weakening coefficients are different for loading modes but all in the range of 0.15~0.2.

(2) The cumulative displacement on the pile top increases with the increasing cyclic load ratio, but its increasing extent is less than that of the cyclic load ratio. The cumulative displacement reaches to around 1% of the pile diameter when the cyclic load ratio increases to about 0.5.

(3) The influence range of the soil around the pile under lateral loadings is the "butterfly" type. The surrounding clear disturbance range of the soil is 2~3 times of the pile diameter, and the rotation center position of the pile body is about 0.8 times of the buried depth of the pile body.

(4) Both the soil plug and outer friction contributed significantly to the pile lateral resistance, the "developing height" of the soil plug under lateral loading is in the range of two times the pile diameter above the pile end. The lateral pressure and frictional resistance of the active zone increases with the cyclic loading, while the lateral pressure and frictional resistance of the passive zone decreases with lateral loadings.

**Author Contributions:** Conceptualization and methodology, J.L. and Z.G.; software, N.Z.; validation, J.L., H.Z. and A.G.; formal analysis, J.L., Z.G. and A.G.; writing—original draft preparation, J.L. and Z.G.; writing—review and editing and visualization, L.X., T.L. and C.F.

**Funding:** This research was funded by National Natural Science Foundation of China (41772318), Shandong Key Research and Development Plan (2017GSF20107), Cooperative Innovation Center of Engineering Construction and Safety in Shandong Blue Economic Zone, Open Fund of State key Laboratory of Coastal and Offshore Engineering (LP1712), and National Key Foundation for Exploring Scientific Instrument Program (41627801).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Scour E**ff**ects on the Lateral Behavior of a Large-Diameter Monopile in Soft Clay: Role of Stress History**

#### **Ben He 1, Yongqing Lai 2, Lizhong Wang 2, Yi Hong 2,\* and Ronghua Zhu <sup>3</sup>**


Received: 23 March 2019; Accepted: 27 May 2019; Published: 1 June 2019

**Abstract:** Scouring of soil around large-diameter monopile will alter the stress history, and therefore the stiffness and strength of the soil at shallow depth, with important consequence to the lateral behavior of piles. The existing study is mainly focused on small-diameter piles under scouring, where the soil around a pile is analyzed with two simplified approaches: (I) simply removing the scour layers without changing the strength and stiffness of the remaining soils, or (II) solely considering the effects of stress history on the soil strength. This study aims to investigate and quantify the scour effect on the lateral behavior of monopile, based on an advanced hypoplastic model considering the influence of stress history on both soil stiffness and strength. It is revealed that ignorance about the stress history effect (due to scouring) underestimates the extent of the soil failure wedge around the monopile, while overestimates soil stiffness and strength. As a result, a large-diameter pile (diameter *D* = 5 m) in soft clay subjected to a souring depth of 0.5 *D* has experienced reductions in ultimate soil resistance and initial stiffness of the *p-y* curves by 40% and 26%, and thus an increase of pile head deflection by 49%. Due to the inadequacy to consider the stress history effects revealed above, the existing approach (I) has led to non-conservative estimation, while the approach (II) has resulted in an over-conservative prediction.

**Keywords:** scour; soft clay; monopile; stress history; hypoplastic model

#### **1. Introduction**

Scour is a process of soil erosion and can often occur around the foundations of offshore structures [1–5]. Currently, monopiles are the most widely employed foundation for offshore wind turbines, and its slenderness ratio of embedded pile length to pile diameter (*L*/*D*) are relatively small (typically in the range of 4–8). Scour reduces the pile embedded length and changes the stress history of the remaining soils, which significantly influences pile responses and the natural frequency of wind turbines [5]. Thus, scour should be well considered during the design of wind turbines.

Although extensive research efforts have been paid on sour effect on pile lateral responses [6–9], most of these studies have been largely limited to small-diameter piles, with ignorance of the stress history effect under scour conditions. The response of the laterally loaded pile under scouring is usually analyzed by two simplified approaches (I): simply removing the scour layers without changing the strength and stiffness of the remaining soils [7,8], or (II) solely considering the effects of stress history on the soil strength [10–12]. The approach (I) ignores the stress history effect due to scouring, which overestimates the undrained shear strength of the remaining soils (as shown in Figure 1), and

thus leads to a non-conservative estimation of monopile response. The influence of altering the stress history by scouring on the stiffness of soil, which governs the natural frequency and fatigue of monopile supported wind turbine, has not been taken into account in both approaches (I) and (II).

**Figure 1.** Schematic diagram of the stress-strain curve of the remaining soil after scour.

At present, the *p-y* method is widely used for the analysis of laterally loaded piles. In this method, the pile is considered as an elastic beam and the soil is represented by a series of discrete *p-y* springs. The most widely used *p-y* curves were proposed by Matlock [13], which had been adopted in design codes. The Matlock *p-y* curves are formulated as follows:

$$p = \begin{cases} \frac{p\_u}{\mathcal{Z}} \left(\frac{y}{y\_c}\right)^{1/3} & \text{for } y \le 8y\_c\\ p\_u & \text{for } y > 8y\_c \end{cases} \tag{1}$$

where *p* is lateral soil resistance per unit length of a pile; *y* is lateral pile deflection; *y*<sup>c</sup> is the lateral displacement at half the maximum soil stress, which can be determined by:

$$y\_c = 2.5 \varepsilon\_{50} D \tag{2}$$

where ε<sup>50</sup> is the strain at one-half the maximum stress.

*p*<sup>u</sup> is the ultimate soil resistance per length, which is equal to the smaller value of *p*u1 and *p*u2 calculated by:

$$p\_{\rm u1} = (3 + \frac{\gamma'}{s\_{\rm u}}z + \frac{f}{D}z)s\_{\rm u}D \tag{3}$$

$$p\_{\rm u2} = 9s\_{\rm u}D\tag{4}$$

where γ is the effective unit weight; *s*<sup>u</sup> is the soil undrained shear strength; *J* is a constant value; *z* is the depth below the post-scour mudline.

Lin et al. [10] modified Matlock *p-y* curves to consider the stress history effect. This was achieved by modifying the ultimate soil resistance, *p*u. The modification of *p*<sup>u</sup> depends on the change of the undrained shear strength and the effective unit weight of the remaining soils after scour, as follows [14]:

$$\frac{\left(\mathrm{s}\_{\mathrm{u}}^{\mathrm{sc}}/\sigma\_{\mathrm{sc}}^{\prime}\right)}{\left(\mathrm{s}\_{\mathrm{u}}^{\mathrm{int}}/\sigma\_{\mathrm{int}}^{\prime}\right)} = \frac{\left(\mathrm{s}\_{\mathrm{u}}/\sigma^{\prime}\right)\_{\mathrm{OC}}}{\left(\mathrm{s}\_{\mathrm{u}}/\sigma^{\prime}\right)\_{\mathrm{NC}}} = \mathrm{OCR}^{\Lambda} \tag{5}$$

$$\gamma\_{\rm sc}^{\prime} = \frac{1 + \varepsilon\_{\rm int}}{1 + \varepsilon\_{\rm int} + \mathbb{C}\_{\rm sur} \log[\frac{(\gamma\_{\rm int}^{\prime})(z + S\_{\rm d})}{(\gamma\_{\rm sc}^{\prime})z}]} \gamma\_{\rm int}^{\prime} \tag{6}$$

where *s*int <sup>u</sup> and *s*sc <sup>u</sup> are the soil undrained shear strength before and after scour, respectively; σ int and σ sc are the vertical effective stress before and after scour, respectively; *OCR* is the overconsolidated ratio of soil; Λ is a parameter (approximately 0.8); γ int and γ int are the effective unit weight before and after scour, respectively; *e*int is soil void ratio before scour; *S*<sup>d</sup> is scour depth; *C*<sup>c</sup> and *C*ur denote the compression and swelling indexes obtained from the oedometer tests.

By substituting Equations (5) and (6) into Equations (3) and (4), the equations for *p*u considering the stress history effect can be rewritten as follows:

$$p\_{\rm ul1} = (OCR)^{\Lambda} \gamma\_{\rm sc}' z (3D + fz) \left[ \frac{s\_{\rm u}^{\rm int}}{(z + S\_{\rm d}) \gamma\_{\rm int}'} \right] + \gamma\_{\rm sc}' zD \tag{7}$$

$$p\_{\rm u2} = 9D(\rm OCR)^{\Lambda} \gamma\_{\rm sc}' z \left[ \frac{s\_{\rm u}^{\rm int}}{(z + S\_{\rm d}) \gamma\_{\rm int}'} \right] \tag{8}$$

Lin's method [10] offered novel insights into the stress history effects due to scouring. Subsequently, Zhang et al. [12] and Liang et al. [13] further developed Lin's [10] *p-y* method to consider not only the stress history effect, but also scour-hole dimension and vertical load effect, respectively. However, the above-mentioned methods for considering the stress history effect are all based on Matlock *p-y* curves. It is well known that Matlock *p-y* curves were developed from a full-scale field lateral loading test on long, and flexible pile with a diameter of 0.324 m and the ratio of pile embedded length to diameter of 39.5. Its validity for large diameter (typically 4–6 m) monopiles of offshore wind turbines has been questioned by many researchers [15–18]. Thus, DNVGL [19] recommends that any proposed design method for large diameter monopiles should be validated by other means, such as by finite element calculations [18]. In addition, due to the stress–dependent behavior of soil, scour-induced stress loss decreases the soil stiffness of the remaining soils, which directly affects the initial stiffness of *p-y* curves. However, those methods for considering the stress history effect are based on the Matlock *p-y* curves, which make use of a parabolic curve shape, and thus the initial tangent modulus is infinite. In other words, those methods for considering the stress history effect are unable to reflect the influence on the initial stiffness of *p-y* curves.

The objective of this paper is to present a three-dimensional finite element method to investigate the stress history effect on the response of a monopile supported offshore wind turbine under scour conditions in soft clay. In this study, a full-scale offshore wind turbine was chosen as the reference structure. An advanced hypoplastic clay model that considers dependency of soil stiffness and strength on stress history was adopted. The three-dimensional model was validated with a published centrifuge test, and then analyses were performed to examine the stress history effect on monopile lateral responses at two different scour depths under the ultimate limit state (ULS) condition. Suitability of the *p-y* method, proposed by Lin et al. [10] for considering stress history was also assessed.

#### **2. Three-Dimensional Finite Element Analyses**

#### *2.1. Three-Dimensional Finite Element Model*

A three-dimensional finite element model of a full-scale monopile-supported wind turbine in clay was developed using the software ABAQUS. The properties of the wind turbine and monopile were taken from Ma et al. [20] and Shirzadeh et al. [21]. A schematic diagram of the overall structure is shown in Figure 2a. The monopile has a diameter of 5 m, a wall thickness of 0.07 m and an embedded length of 30 m. The diameter of the tower at the top and bottom are 3.4 and 4.4 m, respectively.

Figure 2b shows an isometric view of the finite element mesh and the boundary conditions adopted in this study. The lateral boundary of the finite element mesh was constrained by roller supports, while the bottom boundary was fixed against translation in all directions. The lateral boundary was 10 *D* (*D* is the pile diameter) from the center of the pile, which is sufficient to eliminate the boundary effect [22].

The soil and the whole wind turbine structure were modeled using Eight-node brick with pore pressure (C3D8P) elements and Eight-node brick (C3D8) elements, respectively. The monopile foundation was assumed to be linear elastic with typical properties of Young's modulus of *E*<sup>p</sup> = 210 GPa and Poisson's ratio of ν<sup>p</sup> = 0.3. An advanced hypoplastic clay model (to be presented in the following subsection) was used to represent the soil behavior. The interaction between the pile and the soil was simulated based on the Coulomb friction law. The frictional coefficient μ = 0.31 was adopted in this study based on the equation proposed by Randolph and Wroth [23]. The detachment between the pile and the clay was allowed [24].

The tower stiffness, which could have a potential impact on the simulation, was properly considered by adopting the geometry and material of the tower of a 3 MW offshore wind turbine [20,21]. The referred wind turbine was founded on a monopile with a diameter of 5 m, being identical to the pile diameter adopted in this numerical investigation.

**Figure 2.** Diagram of: (**a**) the offshore wind turbine supported on a monopile foundation modelled in this study; (**b**) three-dimensional finite element mesh and boundary conditions.

#### *2.2. An Advanced Hypoplastic Model for Clay Considering Stress History E*ff*ect*

An advanced hypoplastic clay model proposed by Masin [25] with considering stress history effect and small-strain stiffness was selected in this study to represent the soil constitutive model. A general formulation of the hypoplastic model can be written as [26–28]:

$$\stackrel{\circ}{T} = f\_s(L:D + f\_d \mathbf{N} || D ||)\tag{9}$$

where ◦ *T* and *D* represent the objective stress rate and the Euler stretching tensor, respectively.

The hypoelastic tensor *L* is

$$L = \left(c\_1 \mathbf{I} + c\_2 a^2 \mathbf{\hat{I}} \otimes \mathbf{\hat{I}}\right) \tag{10}$$

where **I** is a fourth-order identity tensor; *a*, *c*1, and *c*<sup>2</sup> are the three parameters, and can be calculated by

$$a = \frac{\sqrt{3}(3 - \sin \varphi\_{\circledast}')}{2\sqrt{2}\sin \varphi\_{\circledast}'} \tag{11}$$

$$c\_1 = \frac{2(3 + a^2 - 2^a a \sqrt{3})}{9v} \tag{12}$$

$$c\_2 = 1 + (1 - c\_1)\frac{3}{a^2} \tag{13}$$

where ϕ*'*<sup>c</sup> is a model parameter that denotes critical friction angle; ν is a model parameter that controls the soil shear stiffness at large strain; Parameter α can be calculated by

$$\alpha = \frac{1}{\ln 2} \ln[\frac{\lambda^\* - \kappa^\*}{\lambda^\* + \kappa^\*}(\frac{3 + a^2}{a\sqrt{3}})] \tag{14}$$

where λ\* and κ\* are model parameters defining the slope of the isotropic virgin compression and unloading line in the ln(1 + *e*) versus ln (*p*' ) plane, respectively. (*e* and *p*' denote void ratio and mean effective stress, respectively).

The *N* is a second-order constitutive tensor and can be expressed as

$$\mathbf{N} = L: (-\mathbf{Y}\frac{\mathbf{m}}{||\mathbf{m}||})\tag{15}$$

where *Y* and **m** denote the degree of non-linearity and tensorial quantity, respectively, and have the following equations

$$Y = (\frac{\sqrt{3}a}{3+a^2} - 1) \frac{\left[\frac{1}{2}\text{tr}T(T : T - \left(\text{tr}T\right)^2) + 9\text{det}T\right](1 - \sin^2\varphi\_c')}{8\text{det}T\sin^2\varphi\_c'} + \frac{\sqrt{3}a}{3+a^2} \tag{16}$$

$$\mathbf{m} = -\frac{a}{F} [\mathbf{\hat{T}} + \mathbf{\hat{T}}' - \frac{\mathbf{\hat{T}}}{3} (\frac{6\mathbf{\hat{T}} : \mathbf{\hat{T}} - 1}{\left(F/a\right)^2 + \mathbf{\hat{T}} : \mathbf{\hat{T}}})] \tag{17}$$

with factor *F* given by

$$F = \sqrt{\frac{1}{8}\tan^2\psi + \frac{2-\tan^2\psi}{2+\sqrt{2}\tan\psi\cos 3\theta}} - \frac{1}{2\sqrt{2}}\tan\psi\tag{18}$$

where

$$
\tan \psi = \sqrt{3} \| \mathcal{T}' \|\tag{19}
$$

$$\cos 3\theta = -\sqrt{6} \frac{\text{tr}(\mathcal{T}' \cdot \mathcal{T}' \cdot \mathcal{T}')}{\left(\mathcal{T}' : \mathcal{T}'\right)^{3/2}} \tag{20}$$

where *p*<sup>r</sup> = 1 kPa is a reference stress; *N* is a model parameter defining the position of the isotropic virgin compression line in the ln(1 + *e*) versus ln (*p*' ) plane.

The model is formulated based on incremental equations, which is distinctively different from the conventional elastoplastic framework, i.e., decomposing strains into elastic and plastic parts. The stress rate ◦ *T* varies nonlinearly with the strain rate *D* due to the nonlinear form given by the Euclidian norm *D*, and thus there is no need to define yield surface when predicting nonlinear behavior.

In summary, the basic hypoplastic clay model requires five parameters, i.e., ϕ*'*c, *N*, λ\*, κ\*, and ν. The parameters are equivalent to those defined in the modified Cam clay model.

The basic hypoplastic clay model can predict the monotonic behavior of clay in the medium to large strain range. In order to consider the small-strain stiffness and stress history effect of clay, the so-called intergranular strain concept [29] is combined in the enhancement of the basic model. The intergranular strain δ is used as a new tensorial state variable and the normalized magnitude of δ is:

$$
\rho = \frac{||\mathfrak{S}||}{R} \tag{21}
$$

where *R* is a model parameter that denotes the size of the elastic range.

The direction of the intergranular strain δ is:

$$
\hat{\mathfrak{S}} = \begin{cases}
\mathfrak{S} / \|\mathfrak{S}\| & \mathfrak{S} \neq \mathfrak{0} \\
& \mathfrak{0} = \mathfrak{0}
\end{cases}
\tag{22}
$$

The general stress-strain relation can be re-written as:

$$
\stackrel{\circ}{T} = \mathfrak{u} : \mathcal{D} \tag{23}
$$

where *u* is a fourth-order tensor that represents stiffness, and can be calculated using the following equation:

$$\mathfrak{u} = [\rho^{\chi} m\_{\text{rat}} m\_{\text{R}} + (1 - \rho^{\chi}) m\_{\text{R}}] f\_{\text{s}} L + \begin{cases} \rho^{\chi} (1 - m\_{\text{rat}} m\_{\text{R}}) f\_{\text{s}} L : \boldsymbol{\hat{\mathcal{S}}} \otimes \boldsymbol{\hat{\mathcal{S}}} + \rho^{\chi} f\_{\text{s}} f\_{\text{d}} N \boldsymbol{\hat{\mathcal{S}}} & \boldsymbol{\hat{\mathcal{S}}} : \boldsymbol{\mathcal{D}} > 0 \\\ \rho^{\chi} (m\_{\text{R}} - m\_{\text{rat}} m\_{\text{R}}) f\_{\text{s}} L : \boldsymbol{\hat{\mathcal{S}}} \otimes \boldsymbol{\hat{\mathcal{S}}} & \boldsymbol{\hat{\mathcal{S}}} : \boldsymbol{\mathcal{D}} \le 0 \end{cases} \tag{24}$$

where χ is a model parameter that controls the rate of stiffness degradation; *m*rat is a model parameter that can be quantified by the ratio between initial small-strain stiffness upon a 90◦ strain path reversal and the initial stiffness upon a 180◦ strain reversal; *m*<sup>R</sup> represent the initial small-strain stiffness upon a 180◦ strain path reversal. *m*<sup>R</sup> can be calibrated to fit the initial stiffness *G*0, which is formulated by Wroth and Houlsby [30], as follows:

$$G\_0 = p\_\mathbf{r} A\_\mathbf{g} (\frac{p'}{p\_\mathbf{r}})^{n\_\mathbf{g}} \tag{25}$$

where *A*<sup>g</sup> and *n*<sup>g</sup> are model parameters that reflect the stress dependency of small-strain stiffness.

The evolution equation for the δ is:

$$
\hat{\boldsymbol{\delta}} = \begin{cases}
(\boldsymbol{\Psi} - \hat{\boldsymbol{\delta}} \otimes \hat{\boldsymbol{\delta}} / \rho^{\beta\_r}) : \boldsymbol{D} & \hat{\boldsymbol{\delta}} : \boldsymbol{D} \succeq \mathbf{0} \\
\boldsymbol{D} & \boldsymbol{\delta} : \boldsymbol{D} \le 0
\end{cases} \tag{26}
$$

where β<sup>r</sup> is a model parameter that controls the rate of stiffness degradation.

In summary, the advanced hypoplastic clay model adopted in this study consists of 11 model parameters in total. Five out of the 11 parameters, i.e., ϕ*'*c, *N*, λ\*, κ\* and ν are for the basic model. The six other parameters, i.e., *R*, *m*rat, βr, χ, *A*g, and *n*<sup>g</sup> are for the intergranular concept.

#### *2.3. Parameter Calibration and Model Validation*

The basic model parameters of kaolin clay, i.e., ϕ' c, *N,* λ\*, and κ\* were obtained from Powrie [31] and Al-Tabbaa [32]. The parameters *R*, *m*rat, βr, and χ were calibrated against data reported by Benz [33] on small-strain stiffness of kaolin clay, as shown in Figure 3a. In order to calibrate the remaining parameters ν, *A*g, and *n*g, an undrained cyclic triaxial test was carried out. The kaolin clay sample was consolidated under an isotropic confining stress of 200 kPa, followed by 100 cycles of undrained cyclic compression. More details about the triaxial test can be found in He [34]. The confining stresses in the afore-mentioned elemental tests (for calibrating model parameters) generally do not exceed 200 kPa, except one case of 300 kPa in Benz [33]'s tests. This range of effective confining stress (i.e., *p*' ≤ 200 kPa) is relevant to that considered in the numerical investigation reported herein, i.e., *p*' value of the soil (γ*'* = 8 kN/m3, *K*<sup>0</sup> = 0.625, where *K*<sup>0</sup> is the coefficient of lateral earth pressure) along the 30 m deep

monopile fall within 180 kPa. It is worth noting that the Kaolin clay can differ a lot depending on the manufacture. The aforementioned databases for the calibration of the model parameters and the trixial test were all based on the same type of Kaolin, i.e., Speswhite Kaolin clay.

Results of the cyclic triaxial test and calibration are shown in Figure 3b for comparison. It can be found that the hypoplastic model can reasonably capture the soil behavior. All of the 11 model parameters used in this study are summarized in Table 1.

**Figure 3.** Comparison between measured and computed of: (**a**) small-strain stiffness; (**b**) stress-strain relationship of soil element subjected to cyclic triaxial shearing.


**Table 1.** Hypoplastic clay model parameters of kaolin clay.

Given the scarcity of published experimental results on large diameter rigid piles (e.g., *D* = 5 m, as simulated in this study), the hypoplastic clay model was verified against centrifuge test results on a semi-rigid pile in soft clay (Hong et al. [35]). The model pile has a diameter of *D* = 0.8 m in prototype. Its embedded length (*L*) and load eccentricity (*h*) are 13.2 and 2 m in prototype, respectively. The slenderness ratio (*L*/*D*) of the model pile is therefore 16.5. A three-dimensional finite element model was established to simulate the centrifuge model. Figure 4 shows the computed and measured monotonic lateral load-deflection relationship at the pile head. A good agreement can be found

between the computed and measured results. Hence, it can be concluded that the three dimensional finite element model adopted in this study is capable of modelling the pile-soil interaction. Although the hypoplastic model has reasonably predicted the lateral behavior of the semi-rigid pile, it is desirable to perform tests on large diameter rigid piles in the future for further verifying the predictive capability of the model against such piles.

**Figure 4.** Validation of the numerical model against the centrifuge test reported by Hong et al. [35].

#### *2.4. Load Case*

An ultimate design load case, i.e., the Extreme Turbulence Model (ETM) wind load at rated wind speed combined with the 50-year Extreme Wave Height (EWH) [36], was considered in this study. The environmental site conditions adopted in this study are summarized in Table 2 [20].


**Table 2.** Environmental site conditions [20].

The wind load acting on the hub *F*hub was estimated as [37,38]:

$$F\_{\rm hub} = 0.5 \rho\_{\rm a} A\_{\rm R} C\_{\rm T} \mathcal{U}^2 \tag{27}$$

where ρ<sup>a</sup> is the density of air; *A*<sup>R</sup> is the rotor swept area; *C*<sup>T</sup> is the thrust coefficient, and *U* is the wind speed.

The wind load acting on the tower *F*<sup>z</sup> tower of height *z* was calculated as [36]:

$$F\_{\text{tower}}^{\text{x}} = 0.5 \rho\_{\text{a}} A\_{\text{tower}}^{\text{x}} C\_{\text{S}} V\_{\text{x}}^{2} \tag{28}$$

where *A*<sup>z</sup> tower is the wind pressure area on the tower of height *z*; *C*<sup>s</sup> is shape coefficient which equals to 0.5 for the tubular tower; *V*<sup>z</sup> denotes the average wind speed as a function of height *z*. The normal wind speed profile is given by the power law [36]:

$$V\_{\mathbf{z}} = V\_{\text{hub}} \left(\frac{z}{z\_{\text{hub}}}\right)^{\alpha} \tag{29}$$

where *V*hub is the wind speed at the height of the hub *z*hub; α is the power law exponent, which is assumed to be 0.2.

Wave forces on the structure were calculated using the Morison's equation based on the linear Airy wave theory [36]:

$$dF\_{\rm wave} = \mathcal{C}\_{\rm M} \rho \pi \frac{D\_{\rm t}^2}{4} \ddot{\mathbf{x}} dz + \mathcal{C}\_{\rm D} \rho \frac{D\_{\rm t}}{2} \dot{\mathbf{x}} |\dot{\mathbf{x}}| dz \tag{30}$$

where d*F*wave is the horizontal wave load on a vertical element *dz* of the monopile at level *z*; *C*<sup>M</sup> is the inertia coefficient; *C*<sup>D</sup> is the drag coefficient; ρ is the mass density of the sea water; *D*<sup>t</sup> is the diameter of each section; . *<sup>x</sup>* and .. *x* are the wave-induced velocity and acceleration in the horizontal direction. Since current force is relatively small compared to wind and wave force, thus loads due to current are not considered for analysis.

#### *2.5. Numerical Modelling Procedure*

The validated 3D model was then developed to investigate the stress history effect on the lateral response of the monopile under scour conditions. The clay was assumed to be normally consolidated before scour. For the purpose of the case studies considered in this paper, two scour depths similar to Lin et al. [10], i.e., *Sd* = 0.2 *D* (1 m) and *S*<sup>d</sup> = 0.5 *D* (2.5 m) were examined. The detailed procedures are as follows:

Procedures of modelling scour ignoring stress history:


Procedures of modelling scour considering the stress history effect:


**Figure 5.** Schematic diagram of numerical modelling procedures of: (**a**) no scour; (**b**) scour ignoring the stress history effect; (**c**) scour considering stress history effect. Note: *S*<sup>d</sup> denotes scour depth; γ' denotes soil effective weight.

#### **3. Numerical Results**

#### *3.1. Undrained Shear Strength after Scour*

The distributions of the overconsolidation ratio (OCR) of the clay at two different scour depths are shown in Figure 6a. After scour, the normally consolidated clay becomes overconsolidated. The OCR increases with increasing scour depth and decreases with soil depth and gradually approaches a normally consolidated condition at a greater depth.

The properties and stresses of the soil elements at different depths after scour were extracted from the model to calculate the undrained shear strength. Figure 6b shows the undrained shear strength (*s*u) of the remaining clay after scour. When the stress history effect is ignored, the soil vertical stress and the OCR of the remaining soil keep unchanged. Therefore, the undrained shear strength of the remaining soil is almost the same as that in the condition of no scour, and could be fitted by a linear line, i.e., *s*u = 1.66 *z*. However, the undrained shear strength of the remaining soil which considers the stress history effect is found to be decreased when compared with that of ignoring the stress history. In this study, Λ = 0.78 in Equation (5) provides the best agreement with the computed results, as shown in Figure 6b.

**Figure 6.** Distribution of: (**a**) overconsolidation ratio; (**b**) undrained shear strength with depth.

#### *3.2. Lateral Load-Deflection Response*

Figure 7a,b show the computed load-deflection response at pile-head at the scour depth of *S*<sup>d</sup> = 0.2 *D* and *S*<sup>d</sup> = 0.5 *D*, respectively. The results of the case of no scour are also presented in the figures. All the detailed values are summarized in Table 3. It should be noted that, at any given scour depth, the percentage increases in pile head deflection presented in Table 3 are relative to the values of that with ignoring the stress history effects.

**Figure 7.** *Cont.*

**Figure 7.** Comparison between the computed and calculated lateral load-deflection relationships at the scour depth of: (**a**) *S*<sup>d</sup> = 0.2 *D*; (**b**) *S*<sup>d</sup> = 0.5 *D*.


**Table 3.** Lateral deflection at pile head (m).

Under the scour conditions, considering the stress history effect results in 13% (*S*<sup>d</sup> = 0.2 *D*) and 49% (*S*<sup>d</sup> = 0.5 *D*) higher pile-head deflection compared with the case in which the stress history effect is ignored. The percentage increase in pile-head deflection between considering and ignoring the stress history effect is found to increase with increasing scour depth. Therefore, ignoring the stress history of the remaining soil is likely to cause an unconservative analysis of the laterally loaded pile under scour conditions. A similar conclusion is also made by Lin et al. [10] and Zhang et al. [12]. In addition, compared to the result of no scour, considering scour and the resulted stress history effect leads to a 16% (*S*<sup>d</sup> = 0.2 *D*) and 64% (*S*<sup>d</sup> = 0.5 *D*) increase in lateral pile-head deflection. It is recommended that scour and the accompanying stress history effect should be well treated when designing the monopile-supported wind turbines in clay.

For comparison, the calculated lateral load-deflection relationships at pile head by using the modified *p-y* curves proposed by Lin et al. [10] (see Equations (1), (2) and (5)–(8)) at scour depth of *S*<sup>d</sup> = 0.2 *D* and *S*<sup>d</sup> = 0.5 *D* are also included in the Figures 7a and 7b, respectively. Since scour has an insignificant effect on the change of the effective unit weight of the remaining soil [10,12], thus it was ignored in this study.

As shown in the figures, at any given lateral pile-head displacement, the calculated force at the pile head based on Lin's [10] *p-y* method is lower than those computed. The difference is likely attributed to the factor that Lin's [10] *p-y* curves is developed based on Matlock's *p-y* method [11]. It has been well recognized that Matlock's *p-y* method [11] is mainly applicable to small-diameter flexible piles, but could significantly underestimate the lateral resistance of a large-diameter rigid pile, due to the

ignorance of resistances from base shear force, base moment and skin friction [15–18]. On the other hand, all these factors have been implicitly considered in the 3D finite element analyses reported herein. Nevertheless, the percentage difference between ignoring and considering the stress history effect are comparable. As can be seen in Table 3, by using Lin's [10] p-y method, the calculated pile-head deflection when considering stress history effect increases by 25% (*S*<sup>d</sup> = 0.2 *D*) and 75% (*S*<sup>d</sup> = 0.5 *D*) compared with that of ignoring stress history effect. The differences are higher than those computed by 3D FE analysis, i.e., 13% for *S*<sup>d</sup> = 0.2 *D* and 49% for *S*<sup>d</sup> = 0.5 *D*. The comparison shows that Lin's [10] *p-y* method overestimates the percentage difference in pile-head deflection between ignoring and considering stress history effect.

It would be not possible to model lateral behavior of piles under unsymmetrical and irregularly shaped scour with *p-y* curves. Instead, finite element method, as adopted in this study, has an advantage to account for these effects.

#### *3.3. Profiles of the Bending Moment*

Figure 8 shows the bending moment profiles at different scour depths. Generally speaking, the computed maximum bending moments in the pile are slightly lower than that calculated by Lin's [10] *p-y* method. This difference may be due to the inherent difference between the 3D FE analysis and the *p-y* method. When considering the stress history effect, the location of maximum bending moment shifts toward to a greater depth and results in 1% (*S*<sup>d</sup> = 0.2 *D*) and 2% (*S*<sup>d</sup> = 0.5 *D*) higher maximum bending moment compared with the case in which stress history is neglected. It can also be found that the percentage difference in the maximum bending moment, between considering and ignoring stress history effect increases insignificantly with increasing scour depth. The results indicate that the stress history effect may have a minor influence on the maximum bending moment in the pile. Besides, when the scour depth increases to 0.2 *D* and 0.5 *D*, the maximum bending moment increases by approximately 2% and 5%, respectively, compared with that under no scour condition. To some extent, the scour and the stress history effect on the maximum bending moment in the pile can also be ignored.

**Figure 8.** Profiles of the bending moment at the scour depth of: (**a**) *S*<sup>d</sup> = 0.2 *D*; (**b**) *S*<sup>d</sup> = 0.5 *D*.

#### *3.4. Soil Displacement Field*

The computed soil displacement fields, as well as displacement vectors are shown in Figures 9 and 10 at the scour depth of *S*<sup>d</sup> = 0.2 *D* and *S*<sup>d</sup> = 0.5 *D*, respectively. Two distinct soil flow mechanisms can be clearly identified for the large diameter monopile, namely a wedge mechanism near the ground surface and rotational soil flow near the pile toe. Similar failure mechanisms were also observed by Hong et al. [35] and Schroeder et al. [39]. When considering the stress history effect, the width of the wedge failure zone on the ground surface extends from 1.6 *D* to 1.8 *D* (*S*<sup>d</sup> = 0.2 *D*) and 1.7 *D* to 2.5 *D* (*S*<sup>d</sup> = 0.5 *D*). Meanwhile, the wedge failure zone is observed to extend to a greater depth, i.e., from 0.53 *L* (*L* = pile embedded length before scour) to 0.57 *L* (*S*<sup>d</sup> = 0.2 *D*) and 5.7 *L* to 6.7 *L* (*S*<sup>d</sup> = 0.5 *D*). As expected, the differences in the width and depth of the wedge failure zone between considering and ignoring stress history effect increase with increasing scour depth. As for the rotation center of the plane rotation zone, when considering stress history, it moves downward from 0.76 *L* to 0.78 *L* (*S*<sup>d</sup> = 0.2 *D*) and 0.78 *L* to 0.82 *L* (*S*<sup>d</sup> = 0.5 *D*). As a conclusion, soil failure mechanism of the large diameter monopile consists of two parts, namely wedge failure at shallow and rational soil flow at depth. Ignoring the stress history effect underestimates the width and depth of the wedge failure zone, while overestimates the location of the rotational soil flow zone.

**Figure 9.** Computed soil displacement field and displacement vectors at the scour depth of *S*<sup>d</sup> = 0.2 *D*: (**a**) ignoring the stress history effect; (**b**) considering the stress history effect.

**Figure 10.** Computed soil displacement field and displacement vectors at the scour depth of *S*<sup>d</sup> = 0.5 *D*: (**a**) ignoring the stress history effect; (**b**) considering the stress history effect.

#### *3.5. p-y Curves Derived from Finite Element Simulation Results*

To investigate the stress history effect on *p-y* curves and to assess the suitability of the modified *p-y* curves proposed by Lin et al. [10] for considering the stress history effect. The pile-soil contact force and the corresponding lateral pile displacement were extracted from the numerical results to deduce the *p-y* curves.

Figure 11a,b show the extracted typical *p-y* curves of the remaining soil after scour at the depth of 0.5 and 1.5 m (measured from the post-scour mudline), respectively. The *p-y* curves proposed by Lin et al. [10] are also presented in the figure for comparison. It can be seen that both computed and calculated *p-y* curves reflect a similar trend that is at any given lateral displacement, the *p-y* curves of considering the stress history effect have much lower lateral soil resistance than that of ignoring the stress history effect. The reduction in lateral soil resistance increases the pile-head deflection and the maximum bending moment in the pile.

**Figure 11.** Comparison of the *p-y* curves at the scour depth of: (**a**) *Sd* = 0.2 *D*; (**b**) *S*<sup>d</sup> = 0.5 *D*. Note: The depth of the *p-y* curve, *z*, was measured from the post-scour mudline.

It should be noted that at any given depth, the *p-y* curves proposed by Lin et al. [10] show much lower soil resistance than the computed *p-y* curves. This is because the *p-y* curves proposed by Lin et al. [10] are based on Matlock *p-y* curves which underestimate the ultimate lateral soil resistance. The underestimation has been identified by many researchers based on pile tests and numerical modelling [17,18,40,41]. It should also be noted that the soil resistance of the *p-y* curves proposed by Lin et al., [10] presented in the figure is far from reaching the ultimate soil resistance. In other words, the *p-y* curves proposed by Lin et al. [10] significantly overestimate the required lateral displacement to reach the ultimate soil resistance. Due to the much softer *p-y* curves, the calculated pile-head deflections are much larger than that computed by 3D FE analysis (see Table 3).

The limited number of three-dimensional analyses reported herein (see Figure 11 for example), which aim to illustrate the typical influences of changing stress history (by sour) on soil-pile interaction, are still not sufficient for rigorously formulating a modification of the *p-y* method. It will be the authors' future pursuit to propose modified *p-y* method considering different pile diameters, sour depths and geometries, by performing several series of comprehensive numerical parametric study.

Figure 12a,b present the distributions of the ultimate soil resistance along pile length at scour depth of 0.2 *D* and 0.5 *D*, respectively. The ultimate soil resistance calculated based on Lin's [10] *p-y* curves is also included in the figure. It should be pointed out that for 3D FE analysis, the soil resistance at deep depth may not be fully mobilized as the lateral displacement was quite small. Thus, those *p-y* curves were fitted by a hyperbolic function to obtain the ultimate soil resistance [42,43]. As expected, consideration of the stress history of the remaining soil results in a decrease in the ultimate soil resistance when compared with the results obtained when ignoring the stress history effect. A comparison between Figure 12a,b shows that, the difference in the ultimate soil resistance between considering and ignoring the stress history effect increases with increasing the scour depth.

**Figure 12.** Distributions of ultimate soil resistance at the scour depth of: (**a**) *S*<sup>d</sup> = 0.2 *D*; (**b**) *S*<sup>d</sup> = 0.5 *D*.

To quantitatively evaluate the effect of stress history on ultimate soil resistance, the percentage reduction in ultimate soil resistance between considering stress history and ignoring stress history is plotted in Figure 13. At any given scour depth, the percentage reductions in ultimate soil resistance in Figure 13 are relative to the values of that with ignoring the stress history effects. As presented in the figure, the percentage reduction decreases when the soil depth increases. Based on the computed results, the percentage reduction in ultimate soil resistance near ground surface between considering stress history and ignoring stress history can be up to 30.1% (*S*<sup>d</sup> = 0.2 *D*) and 39.8% (*S*<sup>d</sup> = 0.5 *D*), which is lower than that when using the *p-y* curves proposed by Lin et al. [10], i.e., the percentage reduction is 36.2% (*S*<sup>d</sup> = 0.2 *D*) and 52.3% (*S*<sup>d</sup> = 0.2 *D*). The comparison demonstrates that Lin's [10] *p-y* method overestimates the percentage reduction in ultimate soil resistance between considering and ignoring the stress history effect. The overestimation leads to a larger percentage difference in pile-head deflection when compared with that obtained from 3D FE analysis (see Table 3).

**Figure 13.** Percentage reduction in ultimate soil resistance.

Due to the stress dependency of soil behavior, the loss of the overburden stress caused by scour decreases the stiffness of the remaining soil, and finally results in a lower initial stiffness of the *p-y* curves. The *p-y* curves proposed by Lin et al. [10] are based on Matlock *p-y* curves, and thus adopt the same parabolic curve shape, making the initial stiffness of the *p-y* curves infinite. Therefore, the *p-y* curves proposed by Lin et al. [10] cannot reflect the stress history effect on the initial stiffness of the *p-y* curves which has a significant influence on natural frequency and fatigue life of wind turbine structure [44].

Since the initial stiffness of the *p-y* curves proposed by Lin et al. [10] are infinite, only the distributions of the initial stiffness of the computed *p-y* curves at scour depth of *S*<sup>d</sup> = 0.2 *D* and *S*<sup>d</sup> = 0.5 *D* are presented in Figure 14a,b, respectively. In this study, the secant modulus at a small displacement, i.e., *y* = *D*/1000, of the *p-y* curves were regarded as initial stiffness [44]. It can be clearly seen in the figure that considering stress history effect leads to a decrease in the initial stiffness of the *p-y* curves when compared with the results that ignoring the stress history. The percentage reduction in initial stiffness between considering and ignoring stress history effect is further examined in Figure 15. At any given scour depth, the percentage reductions in the initial stiffness of *p-y* curves in Figure 15 are relative to the values of that with ignoring the stress history effects. The figure reveals that the percentage reduction gradually decreases when the soil depth increases. On the other hand, the percentage reduction in the initial stiffness of *p-y* curves is found to increase with increasing scour depth. When the scour depth is 0.2 *D* and 0.5 *D*, the percentage reduction in the initial stiffness of the *p-y* curves can be up to 20.7% and 25.8%, respectively. The initial stiffness of *p-y* curves can affect the natural frequency of an offshore wind turbine directly. It was founded by Wang et al. [45] that 10–50% decrease in the initial stiffness of *p-y* curves could lead to a 4.6–6.6% drop in the natural frequency of the wind turbine, which has a dramatic effect on the wind turbine fatigue life [46]. Thus, it is recommended that the initial stiffness of *p-y* curves should be well evaluated.

It is worth noting that the soil properties and thus sour geometries generally vary spatially even within a homogeneous layer [47,48]. It will be the authors' future pursuit to integrate numerical modelling and spatial variability into the analysis of lateral behavior of piles under irregularly shaped scour.

**Figure 14.** Distributions of the initial stiffness of the *p-y* curves at scour depth of: (**a**) *S*<sup>d</sup> = 0.2 *D*; (**b**) *S*<sup>d</sup> = 0.5 *D*.

**Figure 15.** Percentage reduction in the initial stiffness of the *p-y* curves.

#### **4. Conclusions**

This paper presents a numerical investigation for studying the scour effect on the lateral behavior of monopile, based on an advanced hypoplastic model considering the influence of stress history on both soil stiffness and strength. Suitability of the *p-y* method proposed by Lin et al. [10] that considers the stress history effect was also assessed. Based on this study, the following conclusions can be drawn:


**Author Contributions:** B.H. and Y.L. guided this research and performed the numerical studies. L.W. and Y.H. drawn the main conclusions. R.Z. processed the results and plot some figures. All of the authors have participated in the writing of the paper. The final manuscript has been approved by all the authors.

**Funding:** This research was funded by National Key Research and Development Program, grant number 2016YFC0800200; National Natural Science Foundation of China, grant number 51779221; Key Research and Development Program of Zhejiang Province, grant number 51779221; Natural Science Foundation of Zhejiang Province, grant number Q19E090001 and Qianjiang Talent Plan Research Fund of Zhejiang Province, grant number QJD1602028.

**Acknowledgments:** The authors would like to acknowledge the supports from National Key Research and Development Program (2016YFC0800200), National Natural Science Foundation of China (51779221), Key Research and Development Program of Zhejiang Province (51779221), Natural Science Foundation of Zhejiang Province (Q19E090001) and Qianjiang Talent Plan Research Fund of Zhejiang Province (QJD1602028).

**Conflicts of Interest:** The authors declare no conflicts of interest.

#### **Nomenclature**



#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Dynamic Impedances of O**ff**shore Rock-Socketed Monopiles**

#### **Rui He 1,2,\*, Ji Ji 1,2, Jisheng Zhang 1,2, Wei Peng 1,2, Zufeng Sun <sup>3</sup> and Zhen Guo <sup>4</sup>**


Received: 1 March 2019; Accepted: 2 May 2019; Published: 9 May 2019

**Abstract:** With the development of offshore wind energy in China, more and more offshore wind turbines are being constructed in rock-based sea areas. However, the large diameter and thin-walled steel rock-socketed monopiles are very scarce at present, and both the construction and design are very difficult. For the design, the dynamic safety during the whole lifetime of the wind turbine is difficult to guarantee. Dynamic safety of a turbine is mostly controlled by the dynamic impedances of the rock-socketed monopile, which are still not well understood. How to choose the appropriate impedances of the socketed monopiles so that the wind turbines will neither resonant nor be too conservative is the main problem. Based on a numerical model in this study, the accurate impedances are obtained for different frequencies of excitation, different soil and rock parameters, and different rock-socketed lengths. The dynamic stiffness of monopile increases, while the radiative damping decreases as rock-socketed depth increases. When the weathering degree of rock increases, the dynamic stiffness of the monopile decreases, while the radiative damping increases.

**Keywords:** rock-socketed piles; monopiles; impedances; dynamic responses; offshore wind turbines

#### **1. Introduction**

As a source of clean energy, wind power generation has been increasingly supported and encouraged in China in recent years. However, as the constraints on onshore wind turbines increase consistently, the development prospects of offshore wind turbines are very broad. The offshore wind farms in Jiangsu, Zhejiang, Fujian, Guangdong, and other provinces in China are currently being constructed (Figure 1). Influenced by the "narrow tube effect" of the Taiwan Strait, the annual average wind speed in the coastal areas from mid-southern Fuzhou to the south of Quanzhou in Fujian Province exceeds 7.5 m/s at places 70 m above the ground [1], together with the stable wind direction and abundant wind power resources, making them suitable for large-scale development of offshore wind power. However, the geological conditions in Fujian province are more complicated than those in Jiangsu province, and monopiles for offshore wind turbines in Fujian province are mostly rock-socketed.

**Figure 1.** Part of the coast of China (from OpenStreetMap).

Investment in offshore wind turbine foundation accounts for a large proportion of total construction investment. Thus, it is of great significance to choose the appropriate foundation types to improve the economic benefits of the project. The commonly used foundations in China at present are: monopile foundation, jacket foundation, and multi-pile foundation. Compared with the soft soils in the Jiangsu sea area, the seabed in Fujian is mostly rock-based, with the depth of the rock-bed and weathering degree varying greatly. As soft soils at the surface of the seabed under complicated wave loads are easily damaged by liquefaction or scour [2–8], most of the piles need to be embedded in rocks. The large-diameter rock-socketed monopiles, similar to monopiles in pure soils [9–11], have the advantages of simpler form, lower cost, and a shorter construction and installing period than other foundation types, which are more conducive for promoting the development of offshore wind farms in rock-based sea areas, such as those in Guangdong and Fujian provinces in China.

The rock-socketed monopiles can be roughly divided into different categories according to the thickness of overlying soil layer and the weathering degree of rock layer. In 2017, the success in installing the large diameter rock-socketed monopile in the sea area of Nanri Island, Fujian province [12], indicating that the difficulties of its construction technology have been preliminarily solved. The question of how to ensure safe operation of offshore wind turbines in the following 25 years has become a major concern for the completion of construction. The overall dynamic safety of wind turbines corresponds to the fatigue limit state in the design [13], and the natural vibration frequency is one of the control conditions in the design process [14]. Considering the safe operation of the structure, the first natural frequency of the wind turbine system should lie between 1P and 3P [15–18]. Resonance will occur if the wind turbine runs in the non-safe frequency band, making the turbines prone to fatigue damage and directly reducing the operation life [19]. The main factor affecting the natural vibration frequency and displacements at mudline is the dynamic impedances

of the foundation, the understanding of which will effectively improve, solving the vibration-caused problems of offshore wind turbines.

The theoretical analyses, experimental research, and practical experiences of monopile are mainly concentrated in the shallow sea areas composed of silt, clay, and sand [20–24], and the application of monopile in the rock-based seabed is still in its infancy. For the socketed piles, the existing research mostly focuses on the vertical bearing capacities [25], while the study on the dynamic impedances of large diameter rock-socketed monopiles is very scarce. The dynamic impedances of monopiles are mainly affected by the pile length to diameter ratio, the pile to soil modulus ratio, the pile thickness to diameter ratio and load frequency, according to the study on monopiles in different soils [26–31]. However, the differences between the mechanical properties of rock and soil are obvious, which make the existing conclusions for monopiles in pure soil unable to be directly applied to rock-socketed monopiles. For example, the elastic modulus of the soil is very small compared to monopiles, and monopiles usually behave like a rigid or semi-rigid body; while the elastic modulus of rock is not very different from monopile, and the deformation mechanism of rock-socketed monopile is still uncertain.

To discover the dynamic impedances and responses of large-diameter monopile under different soil and rock conditions, ABAQUS is used in this study to establish the interaction between rock-socketed monopile and layered soil–rock seabed:


#### **2. The Finite Element Model Created by ABAQUS**

#### *2.1. Introduction to the Model*

The design and study of large diameter rock-socketed monopiles are not yet standardized. The main method for research now is numerical analysis, such as finite element. To analyze the 3*D* dynamic contact problem, it is assumed that: (1) both the soil and rock are homogeneous elastic medium; (2) monopile, soil, and rock always keep in good contact and no separation is allowed.

Due to the symmetry of the pile–soil interaction problem, half model is used. The dimensions of the monopile selected in the model are as follows: the radius (*r*) is 3.5 m, as the average radius of large-diameter monopiles is 4 m to 8 m; the wall thickness is 0.07 m, referring to the built offshore monopiles; depth of penetration (*L*) is 35 m (Figure 2), consistent with the ratio of embedded depth to radius in the literature. The calculation domain of rock and soil is 50 times the diameter of monopile, and an infinite element boundary layer is set to eliminate the reflection of stress waves from the boundaries, as illustrated in Figure 3. The geotechnical model is divided into two layers, of which the upper part is soil, while the lower part is rock.

**Figure 2.** Illustration of the rock-socketed monopile.

**Figure 3.** Combined finite element and infinite element model for the dynamic interaction problem.

The dynamic impedance of pile foundation, whose value is a complex number, refers to the ratio of the external load acting on the top of pile foundation to the corresponding displacement. If a simple harmonic horizontal force (*F*(*t*)) or bending moment (*M*(*t*)) is applied on the surface of a pile foundation, the stress wave is generated due to the vibration of pile–soil interface, of which the energy is partly dissipated during the radiation, resulting in the phase lag of displacement or rotation (ϕ) on the top of

the pile. In the model, the harmonic frequency (ω) is non-dimensionalized, and the corresponding dimensionless frequency (*a*) is represented as:

$$a = \omega r \sqrt{\frac{2(1+\nu)\rho\_s}{E\_s}}\tag{1}$$

where ν, ρs, and *Es* mean Poisson's ratio, density, and elastic modulus of the soil, respectively. Data related to the displacement (*u*(*t*)) and the rotation angle (ϕ(*t*)) of the monopile are extracted from the post-processing module, whose amplitude and phase lag can be obtained by data fitting. The horizontal dynamic impedance (*KH*), the coupled dynamic impedance (*KMH*), and the rotational dynamic impedance (*KM*) of monopile can then be calculated correspondingly. Taking *KH* as an example:

$$K\_{H} = k\_{H} + \mathrm{i}\omega\epsilon\_{H} = \frac{F(t)}{\mu(t)} = \frac{F\_{0}e^{\mathrm{i}\omega t}}{\mu\_{0}e^{\mathrm{i}(\omega t - \varphi)}} = \frac{F\_{0}}{\mu\_{0}}\cos(\varphi) + \mathrm{i}\frac{F\_{0}}{\mu\_{0}}\sin(\varphi)\tag{2}$$

In Equation (2), *kH* means the horizontal stiffness, while *cH* means the horizontal radiative damping. The horizontal dimensionless dynamic impedance (*Kh*) can be obtained by non-dimensionalization:

$$\operatorname{Re}(K\_{\rm h}) = k\_{\rm H} / (\mu r) = F\_0 \cos(\varphi) / (\mu r u\_0) \tag{3}$$

$$\operatorname{Im}(\mathbb{K}\_{\hbar}) = \omega \mathbb{c}\_{\hbar} / (\mu r) = F\_0 \sin(\varphi) / (\mu r u\_0) \tag{4}$$

$$
\mu = E\_s / 2(1+\nu) \tag{5}
$$

Similarly, the dimensionless coupled dynamic impedance (*Kmh*) and the dimensionless rotational dynamic impedance (*Km*) can be written as:

$$\operatorname{Re}(K\_{\text{mbl}}) = k\_{\text{MH}} / (\mu r^2) = F\_0 \cos(q) / (\mu r^2 q\_0) \tag{6}$$

$$\operatorname{Im}(K\_{\text{mbl}}) = \omega \mathfrak{c}\_{\text{Mfl}} / (\mu r^2) = F\_0 \sin(\varphi) / (\mu r^2 \varphi\_0) \tag{7}$$

$$\operatorname{Re}(K\_{\mathfrak{M}}) = k\_{\mathfrak{M}} / (\mu r^3) = M\_0 \cos(q) / (\mu r^3 q\_0) \tag{8}$$

$$\operatorname{Im}(K\_{\mathfrak{m}}) = a \ltimes\_{\mathcal{M}} / (\mu r^3) = M\_0 \sin(\varphi) / (\mu r^3 q \eta) \tag{9}$$

#### *2.2. Comparision with the Existing Solutions*

To verify the rationality of the model, results obtained in this model are compared with the results of pile in homogeneous soil condition in the literature [32]. It can be found in Figure 4 that the two models are in good agreement. The average errors of the real part and imaginary part are 1.39% and 4.80%, respectively. As the soil and rock are modeled as elastic materials in this study, just with different parameters, if this model works for pile in homogeneous soil conditions, it is reasonable to believe that the model also works for pile in layered soil–rock conditions.

**Figure 4.** Model Verification: (**a**) comparison of horizontal dynamic impedance (*Kh*); (**b**) comparison of coupled dynamic impedance (*Kmh*); (**c**) comparison of rotational dynamic impedance (*Km*).

In order to know the dynamic impedances of rock-socketed monopile under different soil layer depths (refer to case 1 in Table 1), the elastic modulus of monopile (*Ep*), soil (*Es*), and rock (*Er*) are set to be 210 GPa, 30 MPa, and 40 GPa, respectively, with the embedded length of the monopile in rock (*h*0) ranging from 1*D* to 3*D* (*D* means the diameter of the pile). Besides, in order to study the influences of rocks with different weathering conditions (refer to case 2 in Table 1), the elastic modulus of the upper soil is set to be 10 MPa, with the elastic modulus of rock being 0.5 GPa, 5 GPa, and 60 GPa, i.e., the ratios of *Es* to *Er* are 1:50, 1:500, and 1:6000, respectively.


**Table 1.** Parameters for soil, rock, and monopile.

Note: In case 1, *Es* = 30 MPa, *Er* = 40 GPa, *Ep* = 210 GPa, variable: *h*0; In case 2, *Es* = 10 MPa, *Ep* = 210 GPa, *h*<sup>0</sup> = 7 m, variable: *Er*.

#### **3. Numerical Results**

#### *3.1. Dynamic Impedance*

3.1.1. Effects of Rock-Socketed Depth on Dynamic Impedance of Monopile

For case 1, three different rock-socketed depths, with elastic modulus of overlying soil *Es* = 30 MPa, are used to calculate *Kh*, *Kmh*, and *Km*, as shown in Figure 5.

**Figure 5.** *Cont*.

**Figure 5.** Effects of rock-socketed depth on dynamic impedance of monopile: (**a**) variation of Re(*Kh*) for various *h*0; (**b**) variation of Im(*Kh*) for various *h*0; (**c**) variation of Re(*Kmh*) for various *h*0; (**d**) variation of Im(*Kmh*) for various *h*0; (**e**) variation of Re(*Km*) for various *h*0; (**f**) variation of Im(*Km*) for various *h*0.

The total depth of the soil layer plus *h*<sup>0</sup> always remains unchanged (equal to 5*D*, refer to Figure 2), which means the thickness of the upper soil layer increases while the rock-socketed depth decreases.

It can be seen from Figure 5 that the stiffness (real part of impedance) of monopile increases with the increase of rock-socketed depth, and the stiffness is frequency dependent, among which the horizontal stiffness has the lowest sensitivity to frequency. When the rock-socketed depth increases from 1*D* to 2*D*, Re(*Kh*), Re(*K*m*h*), and Re(*Km*) increase by about 35.5%, 19.6%, and 5.6%, respectively; when the rock-socketed depth increases from 1*D* to 3*D*, the above physical quantities increase by about 148.6%, 97.3%, and 31.1%, respectively. It can be seen that the deeper the monopile is embedded in the rock, the more obvious the effect of the rock on increasing the stiffness will be.

The radiative damping of monopile decreases with the increase of rock-socketed depth, that is, the deeper the monopile is embedded in the rock, the smaller the radiative damping will be. There is a small fluctuation at low dimensionless frequencies. When the rock-socketed depth increases from 1*D* to 2*D*, *cH*, *cMH*, and *cM* decrease by about 31.2%, 54.1%, and 19.7%, respectively; when the rock-socketed depth increases from 1*D* to 3*D*, the above physical quantities decrease by about 74.4%, 58.0%, and 43.5%, respectively, and the horizontal damping decreases greatly. This may be due to the fact that the deeper the monopile is embedded in the rock, the smaller the interface between soil and monopile will be, and less energies will be radiated from the interface.

#### 3.1.2. Influence of Elastic Modulus Ratio of Rock to Soil

Under the condition of retaining rock-socketed depth being 1*D* and elastic modulus of overlying soil being 10 MPa, the elastic modulus ratio of rock to soil is changed by changing the elastic modulus of rock under three different weathering conditions. The results are shown in Figure 6.

**Figure 6.** Effects of elastic modulus ratio of rock to soil on dynamic impedance of monopile: (**a**) variation of Re(*Kh*) for various *Es* to *Er* ratio; (**b**) variation of Im(*Kh*) for various *Es* to *Er* ratio; (**c**) variation of Re(*Kmh*) for various *Es* to *Er* ratio; (**d**) variation of Im(*Kmh*) for various *Es* to *Er* ratio; (**e**) variation of Re(*Km*) for various *Es* to *Er* ratio; (**f**) variation of Im(*Km*) for various *Es* to *Er* ratio.

As can be seen from the figures, with the change of the elastic modulus ratio of soil and rock from 1:6000 to 1:50, the weathering degree of rock increases, and the elastic modulus of rock is closer to that of soil, so the dynamic stiffness of monopile decreases accordingly. When the elastic modulus of rock decreases to 1/12 of the original one, Re(*Kh*), Re(*K*m*h*), and Re(*Km*) decrease by about 11.9%, 7.7%, and 3.2%, respectively; when it decreases to 1/120, Re(*Kh*), Re(*K*m*h*), and Re(*Km*) decrease by about 31.3%, 19.6%, and 5.6%, respectively, among which the horizontal dynamic stiffness decreases the most.

The variation of radiative damping of monopile is opposite to that of dynamic stiffness. With the decrease of rock modulus, the ability of rock to reduce the dissipated energies becomes weaker, leading to an increase of radiative damping. When the elastic modulus of rock decreases to 1/12 of the original one, *cH*, *cMH*, and *cM* increase by about 39.5%, 27.8%, and 40.9%, respectively; and when it decreases to 1/120 of the original one, *cH*, *cMH*, and *cM* increase by about 44.0%, 55.2%, and 62.9%, respectively, with *cM* increasing the most.

In conclusion, the influence of rock-socketed depth on dynamic impedances of monopile foundation is greater than that of the elastic modulus ratio of rock to soil. A reasonable rock-socketed depth can not only effectively increase the dynamic impedance of pile foundation, but also save piling costs.

#### *3.2. Analysis of Pile Deformation Under Simple Harmonic Horizontal Forces*

When the monopile is subjected to horizontal load, the main deformation is the horizontal deflection. The displacements along the monopile also change periodically under the action of horizontal harmonic load. The dynamic response of the pile in the last period of the total calculation time (2*T*~3*T*, *T* = 2π/ω) is basically stable. Five measuring points (with the polar coordinate θ = π) were arranged 0 m, 5 m, 10 m, 15 m, and 20 m away from the top of the pile, whose dimensionless deflections under horizontal load and bending moment (*u*<sup>1</sup> = *u*1*Esr*/*F* + *u*1*Esr*2/*M*) in the last calculation period were extracted and plotted in Figure 7. The figure shows that the time when the displacement amplitude of each point reaches its maximum is almost the same. Therefore, it will be fast and effective to draw the deformation curve of monopile if we extract the displacement of points along the monopile from the analysis step, in which the displacement of pile top reaches its maximum.

**Figure 7.** The sinusoidal curves of monopile deflection.

#### 3.2.1. Effect of Dimensionless Frequency on Monopile Deformation

The results of pile deflection under horizontal load and bending moment when rock-socketed depth is 1*D* are shown in Figure 8. *Z* means the distance from the calculated point to ground. When the dimensionless frequency changes from 0 to 0.5, the deformation of points along the pile in the upper soil layer firstly increases and then decreases, being the smallest under the action of static force, and the largest (about 1.28 times that of the smallest) when *a* = 0.2. The deflection at the bottom of the monopile is very small, as shown in Figure 9, due to the fastening effect of rock on monopile.

**Figure 8.** Effect of dimensionless frequency on monopile deflection.

**Figure 9.** The magnified view of the rock-embedded part of monopile.

3.2.2. Effect of Rock-Socketed Depth on Pile Deformation

The deformation curves of monopile under different socketed depth conditions are shown in Figure 10. With the increase of rock-socketed depth, the deflection of monopile decreases within the soil layer. When *h*<sup>0</sup> increases from 1*D* to 2*D*, the deflection of pile top decreases about 9.6%, and 46.6% when it increases from 1*D* to 3*D*. The pile deflection gradually decreases to almost 0 near the interface of rock and soil. While the slight reverse deformation exists, when *h*<sup>0</sup> is 1*D* and 2*D* in the rock layer, it disappears when the pile is embedded deeper in the rock.

**Figure 10.** Effect of rock-socketed depth on monopile deformation: (**a**) deflection of monopile under pure horizontal load; (**b**) deflection of monopile under pure bending moment.

#### 3.2.3. Effect of Elastic Modulus Ratio Between Soil and Rock on Monopile Deformation

Figure 11 shows the deformation of monopile gradually increases as the elastic modulus of rock decreases. When the monopile is embedded in slightly weathered rock, the point where deflection of the pile becomes 0 is still near the interface of soil and rock, with a small deformation of pile part embedded in rock. However, the point moves downward with an obvious deformation in the lower part of the pile as the rock is completely weathered, which indicates that the effect the rock has on fastening the pile is weakened. Viewed from the range of curve changing, the influence of rock-socketed depth on pile deformation is slightly greater than that of the elastic modulus ratio between soil and rock.

**Figure 11.** Effect of elastic modulus ratio between soil and rock on pile deformation: (**a**) deflection of monopile under pure horizontal load; (**b**) deflection of monopile under pure bending moment.

#### *3.3. Analysis of the Internal Force of Pile under Simple Harmonic Forces*

In this part, von-Mises stress along the monopile is extracted from the analysis step in which the displacement of pile reaches the maximum, to analyze the influences of different rock-socket depth and elastic modulus ratio between soil and rock. From the five measuring points in the previous section, it can be seen that when the displacement of the pile reaches its maximum, the corresponding dimensionless stress (σ*<sup>M</sup>* = σ*Mr*2/*F* + σ*Mr*3/*M*) are also approximately at the maximum (Figure 12).

**Figure 12.** The periodic change in von-Mises stress.

#### 3.3.1. Effect of Dimensionless Frequency on von-Mises Stress of Pile

Dimensionless von-Mises stresses at points along monopile are taken when *h*<sup>0</sup> = 1*D*. The results are shown in Figure 13. Stress of the pile first increases and then decreases with the increase of dimensionless frequency, reaching the maximum, when *a* = 0.2, in this case. At the interface of rock and soil, the stress of monopile changes very sharply: the stress of the pile under the interface is far less than that in the upper soil layer. Stress of the socketed part decreases rapidly, close to zero at the bottom of the pile.

**Figure 13.** Effect of dimensionless frequency on von-Mises stress of monopile.

#### 3.3.2. Effect of Rock-Socketed Depth on Von-Mises Stress

To study the effect of rock-socketed depth on von-Mises stress, the results are extracted when *a* = 0.5, as shown in Figure 14. The stress of pile in the soil layer increases with the increase of rock-socketed depth, and von-Mises stresses for monopile in the soil layer part are much larger than monopile in the rock layer part. The stress has a sudden drop near the interface between soil and rock, and decreases sharply in the rock, tending to zero at the bottom of pile.

**Figure 14.** Effect of rock-socketed depth on von-Mises stress: (**a**) von-Mises stress of monopile under pure horizontal load; (**b**) von-Mises stress of monopile under pure bending moment.

3.3.3. Effect of Elastic Modulus Ratio between Soil and Rock on Von-Mises Stress

The von-Mises stresses in Figure 15 are also extracted under the condition that *a* = 0.5, *h*<sup>0</sup> = 1*D*. With the increase of rock weathering degree, the stress of monopile in the upper soil layer part decreases while it increases in the lower rock layer part. The drop of stress near the interface under strong weathering rock condition is gentler than the weak weathering rock condition.

**Figure 15.** Effect of elastic modulus ratio between soil and rock on von-Mises stress: (**a**) von-Mises stress of monopile under pure horizontal load; (**b**) von-Mises stress of monopile under pure bending moment.

#### **4. Conclusions and Outlook**

#### *4.1. Basic Conclusions*

Based on a combined finite–infinite element model, the dynamic impedances and dynamic responses of large diameter rock-socketed monopiles under harmonic load are analyzed in this paper. The conclusions are as follows:

	- a. the dynamic stiffness of pile increases, while the sensitivity to dimensionless frequency decreases, indicating that the ability of pile to resist deformation increases under dynamic load, which is consistent with the results obtained from monopile deformation analysis;
	- b. the radiative damping of pile decreases, and the horizontal radiative damping decreases the most. When the contact surface between the pile and the soil becomes smaller, less stress wave energies will be generated and radiated;
	- c. the deformation of monopile reduces and the deformation of the rock-embedded part of the monopile is very small;
	- d. von-Mises stress of the monopile in the soil layer increases, and there is a sudden drop at the soil–rock interface.
	- a. the dynamic stiffness of the monopile reduces, and the closer the elastic modulus of rock is to that of soil, the faster its reduction rate is. When the elastic modulus of the rock is reduced, resulting in the weakened ability of the pile to resist deformation under external force;
	- b. the radiative damping increases, with the rotational radiative damping increasing the most. Compared with rock, it seems that the capability of the soil to radiate stress waves is stronger;
	- c. the deflection of the monopile increases and the point at which the displacement is 0 shifts downward, considering that the effect of rock on fastening the pile is reduced;
	- d. von-Mises stress of monopile in the soil layer decreases while increasing in the rock layer. The phenomenon of stress drop at the soil–rock interface is no longer obvious.

Besides, with the dynamic impedances obtained in this study, the resonant frequencies and dynamic responses of the offshore wind turbines can be calculated by the so-called substructure technique [33,34].

#### *4.2. Outlook on Further Study*

Due to the assumptions made for the model, there are still limitations in the research. Some differences may exist between the homogeneous constitutive model of rock and soil and the geological conditions in practical engineering. In addition, the finite element method can not be used to discover the dynamic impedance evolution of monopile under long periodic vibration loads. More work remains to solve:


**Author Contributions:** Conceptualization, R.H.; Software, J.J. and Z.G.; Supervision, J.Z.; Validation, W.P.; Visualization, Z.S.; Writing—original draft, J.J. and R.H.; Writing—review & editing, R.H., J.J., J.Z. and Z.G.

**Funding:** The first author would like to acknowledge the support of the National Natural Science Foundation of China, Grant No. 51879097 and the Fundamental Research Funds for the Central Universities, Grant No. 2018B12714.

**Conflicts of Interest:** The authors declare no conflict of interest. Rui He, Ji Ji, Jisheng Zhang, Wei Peng, Zufeng Sun and Zhen Guo.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Bridge Scour Identification and Field Application Based on Ambient Vibration Measurements of Superstructures**

#### **Wen Xiong 1,\*, C.S. Cai 2, Bo Kong 2, Xuefeng Zhang <sup>3</sup> and Pingbo Tang <sup>4</sup>**


Received: 16 March 2019; Accepted: 18 April 2019; Published: 26 April 2019

**Abstract:** A scour identification method was developed based on the ambient vibration measurements of superstructures. The Hangzhou Bay Bridge, a cable-stayed bridge with high scour potential, was selected to illustrate the application of this method. Firstly, two ambient vibration measurements were conducted in 2013 and 2016 by installing the acceleration sensors on the girders and pylon. By modal analysis, the natural frequencies of the superstructures were calculated with respect to different mode shapes. Then, by tracing the change of dynamic features between two measurements in 2013 and 2016, the discrepancies of the support boundary conditions, i.e., at the foundation of the Hangzhou Bay Bridge, were detected, which, in turn, qualitatively identified the existence of bridge foundation scour. Secondly, an FE model of the bridge considering soil-pile interaction was established to further quantify the scour depth in two steps. (1) The stiffness of the soil springs representing the support boundary of the bridge was initially identified by the model updating method. In this step, the principle for a successful identification is to make the simulation results best fit the measured natural frequencies of those modes insensitive to the scour. (2) Then, using the updated FE model, the scour depth was identified by updating the depth of supporting soils. In this step, the principle of model updating is to make the simulation results best fit the measured natural frequency changes of those modes sensitive to the scour. Finally, a comparison to the underwater terrain map of the Hangzhou Bay Bridge was carried out to verify the accuracy of the predicted scour depth. Based on the study in this paper, it shows that the proposed method for identifying bridge scour based on the ambient vibration measurements of superstructures is effective and convenient. It is feasible to quickly assess scour conditions for a large number of bridges without underwater devices and operations.

**Keywords:** bridge scour; identification; ambient vibration; field application; natural frequency; mode shape; superstructure; cable-stayed bridge

#### **1. Introduction**

Bridge scour is a significant concern around the world. In the past 40 years, more than 1500 bridges collapsed, and approximately 60% of these failures are related to the scour of foundations in the United States [1,2]. For example, the catastrophic collapse of the Schoharie Creek Bridge in New York in 1987 was caused by the cumulative effects of pier scour [3]. The Los Gatos Creek Bridge over I-5 in the state of California collapsed because of local pier scour during a flood event, and the underlying reason was due to the channel degradation during the 28 years of service [4]. However, the vulnerability of bridges subject to scouring cannot be predicted by routine hydraulic and geotechnical analyses, because more than 100,000 bridges over water in the United States are identified as "unknown foundations" [4,5].

In China, many cable-supported bridges with super-long spans have been recently constructed over rivers and water channels such as the Yangtze River, Yellow River, Qiantang River, etc. These bridges are enormously vulnerable to the hydrological environments such as the annually typhoon-induced floods. Thus, more serious bridge scour, especially at pylons, occurred very often and developed rapidly. Based on the investigation reports, the scour depths of three typical cable-stayed bridges crossing the Yangtze River in China, i.e., the 2nd Nanjing Yangtze River Bridge, Hangzhou Bay Bridge, and Runyang Yangtze River Bridge, were 20 m, 16 m, and 18 m, respectively [6]. Taking the devastating Typhoon Morakot in the summer of 2009 for a detailed example, as many as 3000 mm of rainfall poured in four days and led to the failures or severe damages of more than 110 bridges [7]. Field investigation indicates that the major cause is foundation scour that removes the bed materials surrounding the piers and abutments. Therefore, it is urgent to develop an effective and economical method to identify bridge scour.

An analytical solution or numerical simulation is the common way to predict the bridge scour depth due to its practical conveniences [8–14]. Along with continuous scour experiments, these formulas for calculation are similar to the real situations [15,16]. However, some assumptions may still be applied in the formulas in order to reduce their complexity since the number of the selected parameters is limited. Therefore, these formulas may not be able to reflect the real scouring situations. In the past few decades, many on-site monitoring methods or techniques were proposed to identify or directly measure bridge scour, including the visual inspection by divers [17], and the adoption of fiber Bragg grating (FBG) sensors [18–20], sliding magnetic collars [21], steel rod [21], multi-lens pier scour monitoring [22], microelectro-mechanical system (MEMS) sensors [23], and ultrasonic or radar [24–27]. All of these methods appear quite promising; however, the underwater operability, economic sustainability, or scour refilling process prevent them from further successful and wide application in field monitoring.

Bridge scour identification by tracing the changes in dynamic characteristics has attracted a lot of attention in recent years due to its simplicity, efficiency, and low cost. Although the temperature may also induce the change of dynamic characteristics, it has almost no contribution to such difference if compared to the scour influence [28–33]. The temperature effect can be removed by many statistical methods such as NLPCA (Nonlinear Principal Component Analysis), ANN (Artificial Neural Network), SVM (Support Vector Machine), etc. [34–37]. It can also be simply eliminated by selecting the same seasonal period for each dynamic measurement. The frequency change by the internal cracking or corrosion by changing local stiffness can also be simply ignored if compared to the scour effect by changing entire stiffness. Samizo et al. [38] stated that the natural frequencies of bridge piers would be reduced with the increase of scour depths. Foti and Sabia [39] monitored a bridge, which is affected by scouring and subjected to retrofitting, by measuring the traffic-induced vibrations and indicated a great potential for the use of dynamic tests in the scour assessment of bridges. Zarafshan et al. [40] proposed a scour depth detection concept based on measuring the fundamental vibration frequency of a rod embedded in the riverbed using a laboratory test, simulation, and measurements at the bridge site. Prendergast et al. [41] examined the effect that scour has on the frequency response of a driven pile foundation system and proposed a method to predict the scour depth based on a given pile frequency. Elsaid and Seracino [42] experimentally illustrated the use of the horizontally-displaced mode shapes and dynamic flexibility features to identify the scour from the response of the bridge superstructure. Kong and Cai [43] investigated the scour effect on the response of the entire bridge. The results demonstrated that the response changes of the bridge deck and vehicle are significant and can be used for scour damage detection. Prendergast et al. [44] also developed a similar approach to determine the bridge scour condition by using the vehicle-induced vibrations of superstructures and employed a vehicle-bridge-soil interaction (VBSI) model to trace the frequency change induced by the scour.

The above studies show the feasibility to establish an effective relationship between the scour effect and the dynamic features obtained from the vibration signals of bridge superstructures. However, most of the previous studies focus on the theoretical significance in the vibration-based method by either analytical solutions or Finite Element (FE) models. More research is still needed in numerous aspects to further improve the reliability and accuracy when applied to the field bridges. For instance, it is usually enormously difficult to accurately quantify the vibration signals of bridges in field environments. The actual condition of the soil surrounding the foundation system is very hard to be obtained, which directly determines the support boundary for the vibration of superstructures. Moreover, the limited number or irrational arrangement of sensors installed on the superstructures may highly reduce the sensitivity of the scour identification when tracing vibrations. It is generally believed that the last mile to the success of such vibration-based methods to identify the scour is the application of studies on real bridges under actual scour and in a field environment. Although Chen et al. [7] has already applied this method to the Kao-Ping-His cable-stayed bridge, more improvements are still needed in many aspects. For example, the information of the bridge condition before the scour is required to be pre-known in Chen et al.'s study [7] and there is no attention paid to the contributions from different vibration modes and structural components.

The present study aims to develop a scour identification method for existing bridges based on ambient vibration measurements of superstructures. The Hangzhou Bay Bridge, a 908m cable-stayed bridge, was selected to illustrate the application due to its high scour potential. Firstly, through the acceleration sensors installed on the girder and pylon, two ambient vibration measurements were conducted in 2013 and 2016. By applying the modal analysis on the measurements, the natural frequencies of different orders corresponding to different mode shapes of the superstructure were obtained. Then, by tracing the change of two dynamic features between two measurements, the discrepancies of the support boundary at the foundation of the Hangzhou Bay Bridge were detected, which can qualitatively identified the existence of foundation scour. Secondly, an FE model of the bridge considering soil-pile interactions was established to further quantitatively identify the scour depth in two steps. (1) The stiffness of the soil springs representing the support boundary of the bridge was identified by the model updating method. The principle for a successful identification at this step is to make the simulation results best fit the measured natural frequencies of those modes insensitive to the scour. (2) Then, based on the updated FE model, the scour depth was identified by updating the depth of supporting soils. The principle of model updating at this step is to make the simulation results best fit the measured natural frequency changes of those modes sensitive to the scour. Finally, a comparison with the underwater terrain map of the Hangzhou Bay Bridge was carried out to verify the accuracy of the predicted scour depth. This practical application shows that the proposed method for identifying bridge scour based on the ambient vibration measurements of superstructures is effective and convenient. It is feasible to quickly assess scour conditions for a large number of bridges without underwater devices and operations.

#### **2. The Hangzhou Bay Cable-Stayed Bridge**

#### *2.1. Bridge Information*

The Hangzhou Bay Bridge is a large-scale bridge across the Hangzhou Bay in the eastern coastal region of China with a total length of 36 km. It is among the ten longest trans-oceanic bridges in the world. The construction of this bridge was completed on 14 June 2007 and opened to traffic on 1 May 2008. It consists of a cable-stayed bridge with the main ship-channel and a large quantity of beam bridges (Figure 1). This cable-stayed bridge is selected for the ambient vibration measurement. In the following study, the Hangzhou Bay Bridge only refers to this cable-stayed bridge if not otherwise specified.

**Figure 1.** Overview of the cable-stayed bridge and beam bridges.

The Hangzhou Bay Bridge has one main span along with four side spans, each measuring 70 m, 160 m, 448 m, 160 m, and 70 m, as shown in Figure 2a. The connection between the girder and pylon is designed as a semi-floating system, which is beneficial to increasing the earthquake resistance at the limit stage. The side spans are additionally supported by the auxiliary piers in order to improve the dynamic behavior for daily service.

**Figure 2.** Layout of the Hangzhou Bay Bridge (Unit: cm). (**a**) Span arrangement of girder, (**b**) Pylon.

The pylon of the Hangzhou Bay Bridge is designed as a diamond-shaped structure, of which the height is 178.8 m from the base level to the top (Figure 2b) and 138.575 m from the girder. The cable system consists of 28 pairs of stay cables on each side of the pylon and is arranged in double-plane as a semi-fan shape. The minimum and maximum number of steel wires within a cable is 109 and 199, respectively. The main girder is primarily supported by the cable system and also sits on a transverse beam of the pylon. The flat steel box is applied as the girder's cross-section 3.5 m high and 37.1 m wide (Figure 3). Inside the box girder, the steel crossbeams are added at a 3.75 m interval to improve the girder's resistance to the torsion.

**Figure 3.** Cross-section of girder (Unit: mm).

#### *2.2. Soil Properties*

Based on the boring information at the bridge site, 17 layers of soils numbered from the top to the bottom are categorized as eight types, including the cohesionless sand 1 (Layer 1, elevation: −9.9~−18.4 m), cohesive clay 1 (Layers 2~5 elevation: −18.4~−39.9 m), cohesionless sand 2 (Layers 6~8 elevation: −39.9~−71.1 m), cohesive clay 2 (Layers 9~10 elevation: −71.1~−84.0 m), silty fine sand 1 (Layer 11 elevation: −84.0~−96.5 m), cohesive clay 3 (Layers 12~15 elevation: −96.5~−120.8 m), silty fine sand 2 (Layer 16 elevation: −120.8~−139.8 m), and rounded gravel (Layer 17 elevation: deeper than −139.8 m). The bottom of the piles is located in the 7th category (silty fine sand 2), which is regarded as a good supporting layer for the foundation [45]. The soil parameters of each layer are tested by the bridge design company and the details are not provided here due to the page limit.

#### *2.3. Potential Scour Development*

The Hangzhou Bay Bridge is located in the Hangzhou Bay where the sea flow is rapid and turbulent and the tide is intensive. There are many suspended sediments floating with the current, and erosion and deposition occur very often along the seabed. Especially at the foundation of the bridge, the angle between the current direction and piers or pylons is large, resulting in a rapid removal of the soils. In addition, considering the bridge has served for almost ten years, the foundations of the bridge, especially at the pylons, may already have been scoured. In this sense, the Hangzhou Bay Bridge is appropriate as a case study subject to demonstrate the feasibility and convenience of the vibration-based scour identification in engineering practice.

The final balanced scour depths of the pylon are predicted by the solutions of different design specifications and water-tank experiments as listed in Table 1. It can be observed that the scouring of the Hangzhou Bay Bridge could be very critical during its service time.


**Table 1.** Predicted results of scour depths (Pylon number: B10/B11).

#### **3. Ambient Vibration Measurements**

Two ambient vibration measurements of the Hangzhou Bay Bridge were conducted in 2013 and 2016 to obtain its dynamic features. The field environmental forces, i.e., earth pulsation, hydrodynamic forces, or random traffic flow, excite the ambient vibration. The measurements covered the interior of the steel box girders at the main and side spans and the pylon structures above the pile cap. The sensors employed in the measurements were the broad-band acceleration-meters INV9828 from COINV (China Orient Institute of Noise & Vibration). The sensitivity of the sensors is 0.17–100 Hz with four gear selections and the resolution response is 0.0004 m/s2.

Based on the theory of modal analysis and structural characteristics of the bridge, the following locations were selected to install the sensors and ensure sufficient information measured from the vibration.


The final arrangements of the sensors are illustrated in Figures 4 and 5 for the girder and pylon, respectively. The location of each sensor is denoted with a capital letter to categorize its position at different structural regions. L (L1-L15) and R (R1-R17) in Figure 4 denote the locations on the girder on the left and right sides of the pylon, respectively; U (U1-U10) and D (D1-D7) in Figure 5 denote the locations on the pylon above and below the girder, respectively. M (M0) in both Figures 4 and 5 denote the intersecting location of the girder and pylon. All the letters are followed by different numbers to indicate their relative distances to the pylon or girder.

**Figure 4.** Arrangement of vertical acceleration sensors along the girder.

**Figure 5.** Arrangement of transverse acceleration sensors along the pylon.

All the vibration measurements for the Hangzhou Bay Bridge were conducted under the ambient excitation without interrupting the traffic. A duration of 20 min was adopted for each time of recording with a sampling rate at 100 Hz. The measurements inside the steel box girder were targeted to the identification of the bending modes of the girder in the vertical direction. Horizontal direction was the target in the pylon case for its transverse bending modes. It is also noteworthy that the measurements need to be conducted by several subsequent steps with the common or shared reference points, i.e., M0, L1, and R1 for the girder and U6 and U7 for the pylon, due to the limited number of available sensors. Taking the reference point of M0 as an example, the acceleration signals recorded in the time domain from the sensor are provided in Figure 6.

**Figure 6.** Acceleration signals at the reference point of M0.

#### **4. Qualitative Scour Identification by Tracing Dynamic Features**

#### *4.1. Identification by the Change of Natural Frequencies*

The very high frequency components of the vibration signals measured in 2013 and 2016 were first removed by the wave filtering (low-pass wave filter) method to eliminate the vehicle-induced local vibration of the deck. Then, the modal analysis using the subspace iteration method was applied to the filtered vibration signals. In this way, the natural frequencies of the first eleven orders of the Hangzhou Bay Bridge were extracted from the two measurements and are provided in Table 2.


**Table 2.** Modal analysis results from the measurements in 2013 and 2016.

Note: L: lateral bending; V: vertical bending; LM: longitudinal moving; sym-: symmetric; anti-: antisymmetric. There were no sensors installed along the longitudinal direction of the girder; therefore, no result is given in Table 2 for the 1st order.

Comparing the extracted natural frequencies in the measurements of 2013 and 2016, a significant disparity can be clearly observed in Figure 7. In addition, the natural frequencies of almost all the orders decrease in 2016. The global frequency change should be mainly induced by the change of whole structural stiffness. For bridges, the whole structural stiffness can only be significantly influenced by the variation of the support boundary which in most cases is caused by the scour. In this sense, the

natural frequency change of the superstructure has a strong relationship with the development of the foundation scour. The decrease of the natural frequencies from 2013 to 2016 shown in Figure 7 clearly indicates and qualitatively identifies the scour occurrence at the Hangzhou Bay Bridge.

**Figure 7.** Disparity of natural frequencies between two measurements.

It is also observed from Figure 7 that the frequency range of low orders from the 2nd to 5th, especially for the modes of the pylon, shows more distinguishable disparity between two measurements than other orders. This observation reveals that the natural frequencies of different orders or components have different sensitivities to the scour development (the variation of the support boundary). In order to describe the sensitivity, a new parameter of the frequency change ratio, abbreviated to "*FCR*" hereafter, is proposed as follows:

$$FCR = \frac{\Delta\omega}{\omega} \tag{1}$$

ω = the natural frequency; and Δω = the change of the natural frequency (absolute values).

Figure 8 shows the values of *FCR* corresponding to different orders of vibration modes based on the two measurements. The values of *FCR* are generally above the level of 15% for the orders from the 2nd to 5th while under the level of 5% for the high orders. This is because the low-order modes of vibration contribute to almost all the measured ambient vibration. Once the vibration is affected by the scour, the low-order modes should be influenced much more significantly than others. Especially for the low-order modes with the pylon (the 3rd and 5th orders), they have higher values of *FCR* (20–25%) than those with the girder (15–20%). Although the high-order modes yield the low values of *FCR*, a locally higher value of *FCR* (5%) still can be observed for the 8th mode with the pylon (1st anti-L (girder) + 2nd anti-L (pylon)) than the nearby modes (mostly lower than 1%). This is because the scour, which directly determines the support boundary of the pylon, has more significant effects on the pylon stiffness than that on the girder.

**Figure 8.** Values of *FCR* corresponding to different orders of the vibration mode.

#### *4.2. Identification by the Change of Mode Shapes*

The mode shapes of the first eleven orders of the Hangzhou Bay Bridge were obtained using the subspace iteration method based on the ambient vibration measurements. Figure 9 compares four pairs of mode shapes of the girder which have the same order when extracted from the measurements of 2013 and 2016 (the shape scale was determined by the same normalization processing).

**Figure 9.** The change of the mode shapes of the girder. (**a**) The 1st vertical bending mode (symmetric), (**b**) The 1st vertical bending mode (antisymmetric), (**c**) The 2nd vertical bending mode (symmetric) (**d**) The 3rd vertical bending mode (symmetric).

The significant changes of these mode shapes after three years are highlighted in Figure 9 in the rectangle. Most changes of the mode shapes occur locally at the vibration crests and troughs of the girder especially in the main span close to the span center and pylon. Such shape changes become more distinguishable if the order of the mode decreases. For example, the 1st mode in Figure 9a presents a much more distinguishable shape change than the 3rd mode in Figure 9d. It is also noticed that besides the significant local changes, the mode shapes also vary along the whole bridge length (908 m). This global difference in the mode shapes can be mainly attributed to different supporting boundaries of the bridge. It has been widely believed that the bridge scour is the most probable and usually physical explanation for the variation of the supporting boundary. In this sense, the change of the mode shapes of the girder, especially at their local crests and troughs, becomes another convincing indicator for the scouring of the Hangzhou Bay Bridge.

Similar shape changes can also be found by comparing mode shapes of the pylon which have the same order when extracted from the measurements of 2013 and 2016. The comparative results of the lateral bending modes of the pylon are provided in Figure 10 (the shape scale was determined by the same normalization processing), where the left figure is based on the mode that two pylons vibrate antisymmetrically and the right one is based on the symmetric mode. The unsmooth mode shapes in the figure are due to the limited number of the sensors installed in this case. It can be seen from both modes that the upper section of the pylon presents a much more significant shape change than the bottom part. The lower the order of the mode is, the more distinguishable the shape change between two measurements is. If comparing the results of Figure 10 to Figure 9, the pylon presents a more visible change of mode shapes between two measurements than the girder. This is because the pylon vibrates as a rigid rotation around the foundation and the scour depth directly determines the unsupported height of the pylon. Therefore, by tracing the change of mode shapes of the pylon, especially on the top sections, the change validates again the previous identification results for the scour of Hangzhou Bay Bridge.

**Figure 10.** The change of the mode shapes of the pylon (only one pylon is shown). (**a**) The 1st lateral bending mode (antisymmetric) (**b**) The 1st lateral bending mode (symmetric).

Based on the analysis above, it can be concluded that: (1) the scour development of the Hangzhou Bay Bridge can be qualitatively identified by tracing the change of either natural frequencies or mode shapes between the measurements of 2013 and 2016; (2) the dynamic features of low orders are more sensitive to the scour than those of high orders; (3) the pylon presents more sensitive change of the dynamic features to the scour than the girder does, even for the high orders.

#### **5. Quantitative Scour Identification by FE Model Updating**

The change of pile-soil stiffness affects vibration modes by changing the restriction capacity provided by soils (such as fixed or joint connections). The change of scour depth affects vibration modes by changing the component length of foundations (pile, pier, or pylon). Considering the above different influential sources, all the vibration modes of bridges can be classified into two groups: scour-sensitive modes and scour-insensitive modes. Then, the quantitative scour identification was conducted in the following three steps: (1) First, an FE model was established as the object for the model updating; (2) Second, the soil stiffness (stiffness of soil-pile springs) was identified by model updating until the simulation results best fit the measurements of the scour-insensitive vibration modes. (3) Using the soil-updated model, the scour development was finally identified by model updating until the simulation results best fit the variation between two measurements of the scour-sensitive vibration modes. In the second step, the scour-insensitive vibration modes were used to update the values of soil stiffness. This is because all the global vibration modes of bridges should be influenced by different soil stiffness no matter if they are scour-sensitive or scour-insensitive. A similar phenomenon can also be found in the Kao-Ping-His cable-stayed bridge [7] and Jintang cable-stayed bridge whose frequencies of the scour-insensitive modes have 5% to 10% changes when the pile-soil stiffness increases by 75%.

#### *5.1. FE Model Establishment*

Based on the engineering drawings of the Hangzhou Bay Bridge, a 3D Finite Element (FE) model was created using the ANSYS program. A fish-bone structural system was selected to model the superstructure which can accurately simulate the stiffness of cable-stayed bridges by using beam elements (Figure 11) [47]. In the model, the main girder was simulated by a longitudinal single beam, i.e., Beam188 element, with all the material properties referring to the detailed design parameters. The Beam188 element has three translational and three rotational degrees-of-freedom (DOFs) for each node. The crossbeams intersected with the main girder were modeled by transverse beams using the same Beam188 element. The torsional moments of inertia of crossbeams were considered by the Mass21 element. All the stayed cables connected to the ends of crossbeams were modeled by the Link8 element with the consideration of the existing cable forces. The Link8 element has two translational DOFs for each node. The pylons were also modeled using the Beam188 element to reduce the computation effort. The cross sectional configurations and material properties for each component of the FE model are listed in Tables 3 and 4, respectively. The densities of the girder and cross beams were re-calculated after accounting for all the stiffening and non-structural components.

**Figure 11.** FE model using a fish-bone structural system. (**a**) Axonometric view, (**b**) Vertical view.


**Table 3.** Configurations for cross-sections of girder.


**Table 4.** Material properties for different components.

The key issue to establish the FE model of a scoured bridge is the pile-soil interaction and the corresponding effective area. Soil behavior is highly nonlinear and in particular its stiffness changes nonlinearly with the strain. The response of the pile-soil system is heavily dependent on the magnitude and type of external loads. However, in the service stage of bridges, the external loads over the bridge will only lead to a very small lateral strain being imparted into the soil surround the piles. Therefore, for the case of the Hangzhou Bay Bridge, it is assumed that the soil strains remain within the "small-strain" linear-elastic region of the soil response curve.

In the present study, the pile-soil interaction was represented by the stiffness of soils. All the soil layers in the FE model were simulated by the discrete and closed spaced spring elements (Combin14) at intervals of 0.5 m along the piles (Figure 12). The spring element of the Combin14 with a null mass matrix has two translational DOFs (degrees of freedom) and allows one-dimensional uniaxial movement along the axis of the spring. Each node of the pile connects two spring elements along the longitudinal and lateral directions of the bridge. For the purpose of dynamic interaction modeling, the springs are assumed to provide dynamic impedance only and inertial effects are ignored. Then, an appropriate determination of the element stiffness of the springs becomes the next pursuit, which plays a very important role in the soil behavior simulation.

**Figure 12.** Simulation of the pile-soil interaction.

According to the one-parameter Winkler soil model for piles, the lateral resistance of soils on the pile at a certain depth is linearly varied with the increase of the lateral displacement of the pile. This linear elastic expression is provided by the following mathematical equation.

$$
\sigma\_{\mathbf{z}} = \mathbb{C}u\_{\mathbf{z}} \tag{2}
$$

where σ<sup>z</sup> = lateral resistance of soils on the pile at the depth of z (kN/m2); *uz* = lateral displacement of the pile at the depth of z (m); and *C* = foundation coefficient (KN/m3), which is usually a function of the depth (Equation (3)).

$$C = kz^n\tag{3}$$

where *k* = coefficient related to the soil types, mechanical parameters, material properties, depth, etc.; *z* = soil depth; and *n* = exponent of *z*. Further assigning *n* a value of 1.0 in the present study yields Equation (4), a simplified Winkler-based relationship between *C* and *z*.

$$\mathbb{C} = m \times z \tag{4}$$

where *m* = coefficient (kN/m4),; its value can be found in the foundation design specifications of different countries. Considering that the Hangzhou Bay Bridge is located in China, the Chinese code for the design of the ground base and foundation of highway bridges and culverts (JTG D63-2007) was selected in the present study to specify the value of *m*.

Since the spring elements were modeled discretely at intervals of 0.5 m, Equation (4) as a continuous relationship between *C* and *z* cannot be directly applied to determine the stiffness of each spring. Therefore, the lateral resistance of soils between two close-by springs needs to be equivalently converted into two nodal forces at the nodes of the springs. Such equivalence between a continuous soil force and two discrete nodal spring forces was achieved by keeping the lateral restrain and bending moment of the pile the same. The specific conversion process of the lateral forces on the pile involves the following steps and the result is applicable to both the longitudinal and lateral springs.

Step (1): The pile embedded in the soil is divided into n elements from the top to bottom. The length of each pile element is *hj* (*j* = 1, 2, ... , *n*) and two nodes of the *j*th element are Nj and Nj+1, as shown in Figure 13.

**Figure 13.** Schematic presentation of the equivalence.

Step (2): Based on Equation (4), the stiffness distribution of soils along the pile forms a triangle shape starting from the top of the pile embedded in soils. Then, according to the equivalence principle specified earlier, the stiffness of discrete springs at the nodes N1 and N2 can be derived as:

$$K\_{1,1}^{\varepsilon} = \frac{K(h\_1) \times h\_1}{2} \times \frac{1}{3} = \frac{K(h\_1) \times h\_1}{6} \tag{5}$$

$$K\_{2,1}^{\varepsilon} = \frac{K(h\_1) \times h\_1}{2} \times \frac{2}{3} = \frac{K(h\_1) \times h\_1}{3} \tag{6}$$

where *K<sup>e</sup>* 1,1 and *<sup>K</sup><sup>e</sup>* 2,1 = stiffness of springs at the nodes N1 and N2 of the 1st element, respectively; and *K*(*h*1) = stiffness of soils at the depth h1 based on Equation (4), which is calculated as:

$$K(h\_1) = b\_0 \times m \times h\_1 \tag{7}$$

where *b*<sup>0</sup> = width of the pile.

Similarly, the stiffness of springs at the nodes Nj and Nj<sup>+</sup><sup>1</sup> of the *j*th element (*j* > 1) can be derived as:

$$K\_{j,j}^c = \frac{K(\sum\_{i=1}^j h\_i) \times h\_j}{2} + \frac{[K(\sum\_{i=1}^{j+1} h\_i) - K(\sum\_{i=1}^j h\_i)] \times h\_j}{6} \tag{8}$$

$$K\_{j+1,j}^{\epsilon} = \frac{K(\sum\_{i=1}^{j} h\_i) \times h\_j}{2} + \frac{[K(\sum\_{i=1}^{j+1} h\_i) - K(\sum\_{i=1}^{j} h\_i)] \times h\_j}{3} \tag{9}$$

where *K<sup>e</sup> <sup>j</sup>*,*<sup>j</sup>* and *<sup>K</sup><sup>e</sup> <sup>j</sup>*,*j*+<sup>1</sup> = stiffness of springs at the nodes Nj and Nj<sup>+</sup><sup>1</sup> of the *j*th element, respectively; and *K*( *j <sup>i</sup>*=<sup>1</sup> *hi*) and *K*( *j*+<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> *hi*) <sup>=</sup> stiffness of soils at the depths *<sup>j</sup> <sup>i</sup>*=<sup>1</sup> *hi* and *j*+<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> *hi*, respectively, which based on Equation (4) are calculated as:

$$\mathcal{K}(\sum\_{i=1}^{j} h\_i) = b\_0 \times m \times \sum\_{i=1}^{j} h\_i \tag{10}$$

$$K(\sum\_{i=1}^{j+1} h\_i) = b\_0 \times m \times \sum\_{i=1}^{j+1} h\_i \tag{11}$$

Step (3): Since one node connects two elements, the stiffness of the spring at each node (Dexcept node N1) should add up to the contributions from both connecting elements (Figure 13). Therefore, the stiffness of all the springs in the FE model can be assigned by the values based on the following equation.

$$\begin{aligned} K\_1 &= K\_{1,1}^\varepsilon \\ K\_2 &= K\_{2,1}^\varepsilon + K\_{2,2}^\varepsilon \\ \vdots \\ K\_n &= K\_{n,n-1}^\varepsilon + K\_{n,n}^\varepsilon \\ K\_{n+1} &= K\_{n+1,n}^\varepsilon \end{aligned} \tag{12}$$

#### *5.2. Identification of Soil Sti*ff*ness*

The above theoretical derivation provides the preliminary values for the stiffness of soils/springs (*Ki*). These *Ki* values need a further identification based on field measurements to best fit the actual response of the Hangzhou Bay Bridge. Based on Figure 8 and the corresponding discussion, the 6th, 7th, 9th, 10th, and 11th vibration modes have the indicator of *FCR* less than 5%, which shows that the natural frequencies of these modes are negligibly affected by the scouring between two measurements. In other words, the 6th, 7th, 9th, 10th, and 11th vibration modes are insensitive to the scour. The stiffness of soils/springs (*Ki*) becomes the last main reason to affect the natural frequencies of these five modes. Therefore, the real values of *Ki* for all the springs in the FE model can be identified by model updating until the simulated natural frequencies of the scour-insensitive vibration modes match the measurements. The adoption of scour-insensitive vibration modes significantly lowers the scour interference during the model updating of soil stiffness. This mode sensitivity can also be determined by the FE simulation of the bridge if there is not enough information from field measurements. By the parametric study based on the FE model, a scour-sensitive vibration mode still remains sensitive when the scour keeps developing. Considering *Ki* is a function of the coefficient of m based on Equations (4)–(12); the algorithm for identifying soil/spring stiffness by amending m to update the values of *Ki* is provided in Figure 14.

**Figure 14.** Algorithm for identifying soil stiffness.

Following the algorithm, the foundation soils of the Hangzhou Bay Bridge were first divided into eleven different layers according to the geological survey results. Since the bridge is located in China, all the soil layers were assigned with initial values of m based on the Chinese code for design of ground base and foundation of highway bridges and culverts (JTG D63-2007), as listed in Table 5. The stiffness of soil springs in the FE model with these initial values of m was accordingly calculated by Equations (4)–(12). By using this FE model, the simulated natural frequencies of different orders of the bridge were numerically obtained.

**Table 5.** Soil layers and initial values of m.


In order to quantitatively describe the difference between the simulated and measured natural frequencies, a new parameter is proposed in Equation (13).

$$D\_1 = \sum\_q \left( f\_{\rm sim}^q - f\_{\rm max}^q \right)^2 \tag{13}$$

where *D*<sup>1</sup> = difference between the simulated and measured natural frequencies; *f q sim* and *f q mea* = simulated and measured natural frequencies of the *q*th vibration mode, respectively; and *q* = concerned orders, which refer to the orders of the scour-insensitive vibration modes. It is noted that other forms of *D*1, i.e., Root-Mean-Squared-Error (RMSE) and the Nash-Sutcliffe Efficiency, can also be used in Equation (13), which should provide the same iteration results.

If *D*<sup>1</sup> is greater than a preset threshold, *m* for each soil layer needs to be further amended. Meanwhile, the natural frequencies need to be re-simulated based on the FE model with the newly amended m and accordingly a new *D*<sup>1</sup> can also be calculated. This iteration process for updating the value of *m* (stiffness of springs *Ki*) is repeated until the value of *D*<sup>1</sup> reaches the threshold.

For the Hangzhou Bay Bridge, the orders of the scour-insensitive vibration modes are the 6th, 7th, 9th, 10th, and 11th. Considering the soil stiffness barely changes, either the measurements in 2013 or 2016 can be selected as the data of modal analysis to calculate *D*1. The threshold of terminating the iteration is set as reaching the lowest value of *D*<sup>1</sup> during the updating process. The stiffness of each soil layer keeps being updated by revising the value of *m* at intervals of 100 kN/m4 per sub-step of the iteration. The values of *D*<sup>1</sup> calculated by Equation (13) at all the sub-steps of the iteration are provided in Figure 15.

**Figure 15.** Values of *D*<sup>1</sup> versus sub-steps of iteration. (**a**) Iteration results at all the 35 sub-step, (**b**) Iteration results in a selected range (10th–30th).

Based on Figure 15, and especially the selected results in Figure 15b, the value of *D*<sup>1</sup> keeps decreasing until the 24th sub-step of the iteration when it increases again. At this sub-step the difference between the simulated and measured natural frequencies of the 6th, 7th, 9th, 10th, and 11th vibration modes reaches its minimum during the entire iteration process. In other words, the pile-soil simulation in the FE model gradually approaches the actual situation of the bridge before this sub-step. Table 6 lists the values of m based on the results at the 24th sub-step of the iteration. Substituting the newly updated m into Equations (4)–(12), the new stiffness of soil springs *Ki* was subsequently obtained and then correspondingly updated in the FE model of the bridge. The updated values of *Ki* are listed in Table 7.


**Table 6.** Updated values of *m* after the iteration.

**Table 7.** Updated stiffness of soil springs after the iteration (Partially).


So far, the soil stiffness (stiffness of soil-pile springs) of the Hangzhou Bay Bridge was identified by best fitting the scour-insensitive vibration modes and accordingly updating the FE model. This updated FE model will be used in the following study as a reference model to further quantitatively identify the scour depth. It should also be noted that even if the soil stiffness is not identified accurately enough, the scour depth increasing is still the most significant reason for the change of vibration modes since the soil property hardly changes during the bridge service.

#### *5.3. Identification of Scour Depth (Soil Level)*

Considering there is no significant damage reported by the routine inspections on the superstructures of the Hangzhou Bay Bridge, the scour development should be the only reason for the variation of the dynamic features of the bridge. Therefore, in this case the scour depth was identified by updating the previous soil-updated FE model to best fit the measured natural frequency change from 2013 to 2016. Different from the soil updating, the scour-sensitive vibration modes were selected as the best fitting objects to update the scour depth. Using the scour-sensitive vibration modes can include the most scour effects on the variation of natural frequencies. In this way, the interference of other factors can be lowered to the maximum extent during the model updating of the scour depth. The algorithm for identifying the scour depth by model updating is provided in Figure 16.

**Figure 16.** Algorithm for identifying scour depth.

Another parameter *D*<sup>2</sup> is proposed in Equation (14) to quantitatively describe the difference between the simulated and measured natural frequency changes of scour-sensitive vibration modes.

$$D2 = \sum\_{p} \left[ (f\_{\sin, (h\_1 + \Delta h)}^p - f\_{\sin, h\_1}^p) - (f\_{\text{max,2016}}^p - f\_{\text{max,2013}}^p) \right]^2 \tag{14}$$

where *D*<sup>2</sup> = difference between the simulated and measured natural frequency changes; *f p sim*,(*h*1+Δ*h*) <sup>=</sup> simulated natural frequency of the *<sup>p</sup>*th vibration mode by the FE model with the scour depth of *h*<sup>1</sup> + Δ*h*, i.e., *h*2; *f p sim*,*h*<sup>1</sup> <sup>=</sup> simulated natural frequency of the *<sup>p</sup>*th vibration mode by the FE model with the scour depth of h1; *f p sim*,(*h*1+Δ*h*) <sup>−</sup> *<sup>f</sup> p sim*,*h*<sup>1</sup> <sup>=</sup> simulated natural frequency change induced by the increment of scour depth Δ*h*; *f p mea*,2016 = measured natural frequency of the *p*th vibration mode in 2016; *f p mea*,2013 = measured natural frequency of the *p*th vibration mode in 2013; *f p mea*,2016 − *f p mea*,2013 = measured natural frequency change induced by the increment of scour depth from 2013 to 2016; and *p* = the orders of the scour-sensitive vibration modes.

For the Hangzhou Bay Bridge, the scour-sensitive vibration modes are the 2nd, 3rd, 4th, and 5th modes with more than 15% *FCR* based on Figure 8. The initial value of scour depth *h*<sup>1</sup> set in the FE model before updating Δ*h* was determined based on the underwater inspection report in 2013. The Δ*h* was amended at intervals of 0.5 m per sub-step of the iteration until the *D*<sup>2</sup> was less than the pre-set threshold. Subsequently, new natural frequencies were simulated based on the FE model with the newly amended Δ*h* and a new *D*<sup>2</sup> was also obtained. The threshold for terminating the iteration is set as reaching the lowest value of *D*2. The values of *D*<sup>2</sup> calculated by Equation (14) at all the sub-steps of the iteration are provided in Table 8. Figure 17 plots the variation of *D*<sup>2</sup> along with increasing Δ*h* from 0 m to 7 m.

As can be seen from Figure 17, the *D*<sup>2</sup> decreases with the progressive scouring until the Δ*h* reaches the increment of 4.5 m. Thereafter, the *D*<sup>2</sup> turns to increase as the Δ*h* continues going deeper. In other words, the lowest value of *D*<sup>2</sup> is obtained at the 10th sub-step of the iteration when the Δ*h* is 4.5 m. At this moment, the difference between the simulated and measured natural frequency changes of the 2nd, 3rd, 4th, and 5th vibration modes reaches its minimum value. Therefore, the scouring of the Hangzhou Bay Bridge from 2013 to 2016 was successfully identified as the increment of 4.5 m. It is clearly observed from Table 9 that after model updating the frequency changes by adding 4.5 m scour depth in the FE model, which is very close to the measured changes. The proposed scour identification method worked very well in the present case study.


**Table 8.** Values of *D*<sup>2</sup> for different Δ*h*.

**Figure 17.** Variation of *D*<sup>2</sup> along with the increasing Δ*h*.



#### **6. Verification by Results from Underwater Terrain Map**

In this section, the documented results from the underwater terrain map were used to verify the accuracy of the scour identification by the proposed method based on the ambient vibration measurements.

Since a 6 m-deep local scour hole was observed at the Hangzhou Bay Bridge, an underwater terrain scanning measurement was conducted every year. A multiple-wave depth survey system called "Atlas FanSweep20 (FS20)" was applied for the scanning, which included the depth measurement meter, global navigation satellite system (GNSS) receiver, sound velocity meter, differential global positioning system (DGPS) device, acoustic doppler current profiler (ADCP), electronic total station, and survey boat. Each scanning can draw a digitized underwater terrain map for the region around the

Hangzhou Bay Bridge, as shown in Figure 18. The scour developments can be quantitatively obtained by comparing the scanning results between different years. By doing this, Table 10 lists the increments of the scour depths from 2013 to 2016 at the side pier (B9) and pylon (B10) [6]. It should be noted that all the elevation values in Table 10 were measured in the deepest positions of the local scour holes. To make sure the underwater scan results and the vibration-based identification results are comparable, the months of conducting the vibration measurements in 2013 and 2016 were selected according to the time of the underwater scan.

**Figure 18.** Underwater terrain map of the Hangzhou Bay Bridge.



An obvious scour development can be observed from Table 10 at either the side pier or pylon during the time from 2013 to 2016. The maximum increment was 5.2 m, locally located at the pylon B10, which was very similar to the above identified scouring of 4.5 m. The disparity between the identification by the superstructure vibrations and the result by the direct underwater measurements is only 13%. If considering the scouring of 5.2 m could include the interference of sediment refilling in scour holes, the real scour development at the pylon should be less than 5.2 m. Moreover, the scour may quickly vary in a very short-term period even though the underwater scan and vibration measurements were conducted in the same season. That is also one of the possible reasons for the disparity (4.5 m versus 5.2 m). Therefore, it was verified that the proposed method by the ambient vibrations of the superstructure can assure an accurate and quantitative result for the practical scour identification.

Although the dynamic nature of live bed scour condition (erosion and refilling) would continuously change the underwater scanning results of scour over time, it is actually not significant compared to the final scour depths and total lengths of piles. Such a measuring difference from real scour depths should only result in slight influence on the assessment of bridge safety and stability. Moreover, only the underwater terrain measurements were conducted as the yearly routine scour inspection for the Hangzhou Bay Bridge. Therefore, using the data from the underwater terrain measurements becomes the best choice at present to verify the accuracy of the proposed method.

In addition to the good accuracy, the proposed method only needs to trace the vibration signals of the superstructure without any underwater operations or devices. The acceleration sensors on the superstructure and their signal receiver are the only instruments installed during the identification. By doing this, the common issues of traditional scour inspections can be well avoided or solved, such as the high expense of operations, high maintenance of devices, and difficulty of long-term and high-frequency assessments. Therefore, this vibration-based scour identification could be easily integrated to a routine and long-term assessment task for bridges.

#### **7. Concluding Remarks**

This study shows the great potential for the use of the ambient vibration of the superstructure in identifying the bridge scour. The improvements in the present study compared to other existing studies are concluded from following two aspects.


The following conclusions can also be drawn.


While the current research led to several new and interesting conclusions regarding the application of identifying bridge scour based on the ambient vibrations of the superstructure, additional research would also be beneficial. For example, the reliability in the practice of this method needs more validations, especially when other local damages besides the foundation scour pre-exist in the superstructure. Since the measurements of vibration modes (e.g., scour-sensitive and -insensitive) significantly affect the accuracy of the identification, an investigation on effective arrangement of sensors is also needed. More applications on different types of bridges are still required to fully confirm the applicability of the proposed vibration-based method.

**Author Contributions:** W.X. managed the whole research plan, took care all the data analysis, and did the manuscript writing. C.S.C. did the quantitative scour identification and model updating. B.K. did part of the simulation and the corresponding data analysis. X.Z. provided some of the original data. P.T. did the qualitative scour identification analysis.

**Funding:** This research was funded by the Natural Science Foundation of Jiangsu Province of China (grant number BK20161417), Fundamental Research Funds for the Central Universities (grant number 2242016R30023), and Highway Science and Technology Project of Zhejiang Province of China (grant number 2018H10).

**Acknowledgments:** The financial support for this work from the Natural Science Foundation of Jiangsu Province of China (Project No. BK20161417), Fundamental Research Funds for the Central Universities (2242016R30023), and Highway Science and Technology Project of Zhejiang Province of China (2018H10) is gratefully acknowledged. The opinions and statements do not necessarily represent those of the sponsors.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **2D Numerical Study of the Stability of Trench under Wave Action in the Immersing Process of Tunnel Element**

#### **Wei-Yun Chen 1, Cheng-Lin Liu 1, Lun-Liang Duan 2, Hao-Miao Qiu <sup>3</sup> and Zhi-Hua Wang 1,\***


Received: 16 January 2019; Accepted: 14 February 2019; Published: 27 February 2019

**Abstract:** The evaluation of the trench stability under the action of ocean waves is an important issue in the construction of an immersed tunnel. In this study, a two-dimensional coupling model of a wave-seabed-immersed tunnel is proposed for the dynamic responses of a trench under wave action in the immersing process of tunnel elements. The porous seabed is characterized by Biot consolidation equations. The *k* − *ε* model and RANS equation are adopted to achieve the flow field simulation, and the level set method (LSM) is used to capture the free surface between the water and air. The proposed numerical model is verified using the experimental data and analytical results. Then, the transient liquefaction and shear failure in the vicinity of the trench are discussed at two different conditions, namely, after the foundation groove is excavated and after the tunnel element is placed. The pore pressure amplitude on the weather side slope is demonstrated to be significantly smaller than that on the lee side slope. Also, the distribution of the surrounding flow field and pressure field change dramatically after the tunnel element is settled, leading to the significant changes of seabed stability.

**Keywords:** immersed tunnel; wave action; trench; numerical study; porous seabed

#### **1. Introduction**

With the development of ground transportation, inland river transportation and sea transportation, there is an increasing demand for channels across the river and sea. An immersed tube tunnel, as a kind of underwater tunnel construction method, has been developed since the beginning of the 20th century. Its applicability and reliability have been verified through many successful cases of immersed tube tunnels around the world such as the Oresund tunnel between Denmark and Sweden, as well as the Hong Kong-Zhuhai-Macao immersed tunnel in China. By excavating a trench of fixed geometry into the soil surface, the prefabricated pipe sections are laid successively in the trench. The trench soil is usually soft and very sensitive to the contact deformation and load. Moreover, the structure is also sensitive to the uneven settlement of the foundation as the immersed tube tunnel could be regarded as nearly infinite in length direction compared with its section size. Thus, the stability of the excavated trench soil around the immersed pipe tunnel is an important issue for the safety of the tunnel.

Extensive research has been conducted to the dynamic interaction mechanism between wave/flow, seabed and structures [1–11]. In general, when the wave propagates over the seabed, the dynamic wave pressure will induce pore water pressure in the seabed. When the pore pressure reaches the limit value, the effective stress in the soil disappears and is always accompanied by instability or even liquefaction. According to the available literatures, the wave-induced liquefaction can be divided into two forms according to the generation mechanisms of the pore pressure. One is due to the accumulative effect of pore pressure, caused by the volume deformation of the soil under cyclic loading [12,13]. Another is induced by transient pore water pressure, which usually happens under wave troughs, with the reduction of pore pressure amplitude and phase lag [14–16].

In recent years, many researchers used various analytical formulas, which assume that the seabed is flat, to study the dynamic response of the porous seabed under linear waves [17–19]. However, this method is difficult to consider the existence of the structure. Meanwhile, a numerical simulation is also used to study the response mechanism of the wave-sea-floor structure system. Jeng and Cheng [20] used the finite difference method to study the effect of the wave on the soil around a buried pipe. Jeng et al. [6] studied the coupled response of the wave-seabed-breakwater system by using the finite element method. Zhao et al. [21] used the combined FVM-FEM (Finite Volume Model-Finite Element Model) to study the effect of the flow field on the submerged rubble mound breakwater. Kasper et al. [22] used the FEM method in a practical engineering and studied the stability of an immersed tunnel under deep water wave impact. Liao et al. [23,24] adopted a two-step numerical scheme combined CFD (Computational Fluid Dynamics) model to study the oscillatory response around slope breakwater heads.

The immersed tunnel, however, is rarely concerned in the existing literature. However, the dynamic response of seabed soil in fluid field could provide a reference for this study. The size of an immersed tube tunnel is relatively large (e.g., the one-way three-lane tunnel designed in the Hong Kong-Zhuhai-Macao bridge has a single-hole span of 14.55 m). When the tunnel section is placed in the trench, the distribution of the surrounding flow field will be inevitably changed. The flow field around the tunnel could then greatly affect the stability of the surrounding soil. However, the pressure boundary setting method, associated with the analytical solution, cannot capture the local flow state or perform the bidirectional coupling process.

In this study, a coupling model of the wave-seabed-tunnel is proposed to study the transient response of the soil in the vicinity of a trench and the flow field distribution nearby. The wave field is simulated by solving the RANS equation and *k* − *ε* turbulence model. The dynamic response of the soil is solved by the Biot porous elastic theory, and the deformation and displacement of the tunnel are ignored in this study. In Section 2, the wave model and the soil model are respectively verified in detail. In Section 3, the dynamic response and failure of the trench soil are analyzed; the key parameters affecting the soil instability around the tunnel are also discussed.

#### **2. Numerical Model**

The numerical model mainly includes two parts: a wave model and a seabed model with an immersed tube tunnel. The wave model is used to simulate the flow field, and the seabed model is used to study the oscillatory response in the trench around the tube tunnel in the flow field. Figure 1 describes a sketch of the 2-D wave-seabed-tunnel interaction, where *h* is the seabed depth, *d* is the water depth, A1 and A2 are two asymmetric points on two slopes, B1 and B2 are 1 m below the positions of two bottom corners of the tunnel and *θ* is the slope ratio of the trench. The proposed numerical model mainly focuses on the comparison between the case of the free field and the case after the tunnel is settled. Thus, there is no backfill material in the trench. The detailed cross-sectional dimensions of the tunnel are depicted in the following section.

**Figure 1.** A sketch of the model for the wave-seabed-tunnel interaction.

#### *2.1. Fluid Dynamic Model*

In this part, the momentum source function is used to make waves by replacing the source term with the momentum source function in the momentum conservation equation [25,26]. Moreover, the level set method (LSM) is used to capture the free surface between water and air, and this free contact surface is a time-dependent variable varying with time as the wave surface changes.

The *k* − *ε* model and RANS equation are adopted to achieve a flow field simulation. The standard turbulence model mainly includes two transport equations and two independent variables: turbulent kinetic energy *k* and turbulent dissipation rate *ε* [27]. Turbulence is a complex, unsteady, irregular flow with rotation, which is caused by viscous forces. When the Reynolds number is large enough and the inertial force dominates, the flow becomes turbulent. The eddy viscosity is

$$
\mu\_T = \rho \mathbb{C}\_{\mu} \frac{k^2}{\varepsilon} \tag{1}
$$

where *Cμ* is a model constant and *ρ* is the fluid density.

The transport equation for turbulent kinetic energy *k* is written as

$$
\rho \frac{\partial k}{\partial t} + \rho u \cdot \nabla k = \nabla \cdot \left( \left( \mu + \frac{\mu\_T}{\sigma\_k} \right) \nabla k \right) + P\_k - \rho \varepsilon \tag{2}
$$

where *u* is the instantaneous velocity in vector notation, ∇ is the Laplace operator, *t* is time, *μ* is the molecular viscosity and the production term can be expressed as

$$P\_k = \mu\_T \left(\nabla u : \left(\nabla u + \left(\nabla u\right)^T\right) - \frac{2}{3} (\nabla \cdot u)^2\right) - \frac{2}{3} \rho k \nabla \cdot u \tag{3}$$

The transport equation for dissipation rate *ε* is written as

$$
\rho \frac{\partial \varepsilon}{\partial t} + \rho u \cdot \nabla \varepsilon = \nabla \cdot \left( \left( \mu + \frac{\mu\_T}{\sigma\_\varepsilon} \right) \nabla \varepsilon \right) + \mathbb{C}\_{\varepsilon 1} \frac{\varepsilon}{k} P\_k - \mathbb{C}\_{\varepsilon 2} \rho \frac{\varepsilon^2}{k} \tag{4}
$$

The aforementioned equations include closure coefficients *Cμ*, *Cε*1, *Cε*2, *σ<sup>k</sup>* and *σε* which are 0.09, 1.44, 1.92, 1.0 and 1.3, respectively.

The stress tensor, including viscous stress and Reynolds stress, can be expressed as

$$\pi\_{i\bar{j}} = \upsilon \left[ \frac{\partial \overline{u}\_i}{\partial \mathbf{x}\_{\bar{j}}} + \frac{\partial \overline{u}\_{\bar{j}}}{\partial \mathbf{x}\_{\bar{i}}} \right] - \rho \overline{u}\_i \overline{u}\_{\bar{j}} \tag{5}$$

where *υ* is the dynamic viscosity and −*ρuiuj* is the Reynolds stress term; under the eddy-viscosity assumption [27,28], the Reynolds stress term can be expressed as

*J. Mar. Sci. Eng.* **2019**, *7*, 57

$$
\rho \overline{u}\_i \overline{u}\_j = -2\mu\_T \mathbb{S}\_{ij} + \frac{2}{3} \rho k \delta\_{ij} \tag{6}
$$

where *δij* is Kronecker delta and *Sij* is the strain-rate tensor.

For this 2-D problem in this model, the mass conservation equation and the momentum conservation equation can be expressed as

$$\frac{\partial \overline{u}\_i}{\partial x\_i} = 0 \tag{7}$$

$$\frac{\partial \overline{u\_i}}{\partial t} + \overline{u\_j} \frac{\partial \overline{u\_i}}{\partial x\_j} = -\frac{1}{\rho} \frac{\partial \overline{p}}{\partial x\_j} + \frac{1}{\rho} \frac{\partial \tau\_{ij}}{\partial x\_j} + g + S\_i \tag{8}$$

where, *ui*, *uj* and *p* are the average velocity in the *x* and *z* direction and the wave pressure in the flow field, respectively; *xi* and *xj* are the cartesian coordinates that are orthogonal to each other; *t* is time; *ρ* is the density of the fluid; *g* is the acceleration of gravity; *τij* is the shear stress tensor; and *Si* = (*Sx*, *Sz*) is the momentum source function. In this study, the momentum source function can be expressed as

$$S\_x = -g(2\beta x) \exp(-\beta x^2) \frac{D\_s}{\omega} \sin(k\_w z - \omega t) \tag{9}$$

$$S\_z = \text{g} \exp(-\beta \mathbf{x}^2) \frac{kD\_s}{\omega} \cos(k\_w z - \omega \mathbf{t}) \tag{10}$$

where *kw* denotes the wave number; *ω* is the wave angular frequency; and *β* = 80/*δ*2/*L*<sup>2</sup> is the width of the wave source region, in which *L* is the wave length and *δ* is a parameter that indicates the wave source region. *Ds* denotes amplitude of source region and is expressed as

$$D\_s = \frac{2A\left(\omega^2 - \alpha\_1 \text{g} k\_w \,^4 d^3\right) \cos \theta\_w}{\omega I\_1 k\_w \left[1 - a\left(k\_w d\right)^2\right]} \tag{11}$$

where *A* is the wave amplitude, *d* is the water depth, *θ<sup>w</sup>* is the angle between the direction of wave propagation and the horizontal line (set as zero in the model), *<sup>I</sup>*<sup>1</sup> = *<sup>π</sup>*/*<sup>β</sup>* exp(−*k*2/4*β*), *<sup>α</sup>*<sup>1</sup> = *<sup>α</sup>* + 1/3 and *α* is another wave parameter expressed as

$$\alpha = \frac{z\_a}{d} \left(\frac{z\_a}{2d} + 1\right), \ z\_a = -0.53d \tag{12}$$

Detailed parameters of other related parameters can be found in Wei and Kirby (1999) [25] and Choi and Yoon (2009) [26].

The LSM method is used to capture the free surface between water and air, which can be expressed as

$$\frac{\partial \phi}{\partial t} + \boldsymbol{\mu} \cdot \nabla \phi = \gamma \nabla \cdot \left( \varepsilon\_t \nabla \phi - \phi (1 - \phi) \frac{\nabla \phi}{|\nabla \phi|} \right) \tag{13}$$

where *φ* is the level set function; *ε<sup>t</sup>* is the parameter controlling the interface thickness, and its default value is half of the maximum mesh element size in which the interface passes through; *γ* is the reinitialization parameter, and the default value is 1 m/s; and *u* is the velocity field component, and the applied velocity field transports the level set method through convection. A detailed introduction about the LSM method could be found in Olsson et al. [29], Olivier et al. [30] or Sheu et al. [31].

#### *2.2. Seabed Model*

In the seabed model, the pore pressure changes periodically due to the wave pressure obtained from the upper wave model. Two main stages during the construction of the immersed tunnel are considered. One is the state when the foundation groove excavation is completed, and another is the unfilled state when the tunnel subsidence is completed. Due to the short time between the placement of the immersed tube tunnel and backfilling, the transient response caused is considered in this paper. The 2-D seabed model is established by adopting the Biot consolidation equation, which ignores the accelerating inertia term of flow and soil particles [21,23,24].

Assuming the seabed to be homogeneous and isotropic, the permeability coefficient in all directions are the same. The mass conservation equation of pore fluid can be expressed as

$$k\_s \nabla^2 p - \gamma\_{\text{uv}} n\_s \beta\_s \frac{\partial p\_s}{\partial t} = \gamma\_{\text{uv}} \frac{\partial \varepsilon\_s}{\partial t} \tag{14}$$

where *ks* is the Darcy permeability coefficient, <sup>∇</sup><sup>2</sup> = ( *<sup>∂</sup> ∂x* ) 2 + ( *<sup>∂</sup> ∂z* ) <sup>2</sup> is the Laplace operator, *<sup>γ</sup><sup>w</sup>* is the unit weight of pore water, *ns* is the soil porosity and *ε<sup>s</sup>* = *<sup>∂</sup>us <sup>∂</sup><sup>x</sup>* <sup>+</sup> *<sup>∂</sup>vs <sup>∂</sup><sup>z</sup>* is the soil elastic volume strain. The compressibility of pore water could use *β<sup>s</sup>* expressed as [32]

$$\beta\_s = \frac{1}{K\_f} = \frac{1}{K\_w} + \frac{1 - S\_r}{p\_0} \tag{15}$$

where *Kf* is the measured volume modulus of pore water, *Kw* is the true volume modulus of water, *p*<sup>0</sup> is the absolute water pressure (compared to zero), *p*<sup>0</sup> = *γwd* is defined in the model, *d* is the water depth and *Sr* is the soil saturation.

The governing equation for the porous seabed can be expressed as

$$\frac{\partial \sigma'\_{\text{xx}}}{\partial \mathbf{x}} + \frac{\partial \tau\_{\text{xz}}}{\partial \mathbf{z}} = \frac{\partial p\_{\text{s}}}{\partial \mathbf{x}} \tag{16}$$

$$\frac{\partial \tau\_{xz}}{\partial x} + \frac{\partial \sigma\_{zz}^{\prime}}{\partial z} + \rho \mathbf{g} = \frac{\partial p\_s}{\partial z} \tag{17}$$

where *σ xx* and *σ zz* are the horizontal and vertical effective normal stresses; *τxz* is the shearing stress; the actual density of seabed is *ρ* = (1 − *n*)*ρ<sup>s</sup>* + *nρw*, in which *ρ<sup>s</sup>* is the density of soil particles and *ρ<sup>w</sup>* is the water density.

The aforementioned stress can be derived in the poro-elastic theory; the relationship between effective stress and deformation can be expressed as

$$
\sigma'\_{xx} = \frac{2G\_s}{1 - 2\nu\_s} \left[ (1 - \nu\_s) \frac{\partial u\_s}{\partial x} + \nu\_s \frac{\partial v\_s}{\partial z} \right] \tag{18}
$$

$$
\sigma\_{zz}' = \frac{2G\_s}{1 - 2\nu\_s} \left[ (1 - \nu\_s) \frac{\partial \upsilon\_s}{\partial z} + \nu\_s \frac{\partial \mu\_s}{\partial x} \right] \tag{19}
$$

$$\mathbf{r}\_{\mathbf{x}\mathbf{z}} = \mathbf{r}\_{\mathbf{z}\mathbf{x}} = \mathbf{G}\_{\mathbf{s}} \left( \frac{\partial v\_{\mathbf{s}}}{\partial \mathbf{x}} + \frac{\partial u\_{\mathbf{s}}}{\partial \mathbf{z}} \right) \tag{20}$$

where *Gs* is the soil shear modulus, which can be expressed through Young's modulus *Es* and Poisson's ratio *ν<sup>s</sup>* as *Gs* = *Es*/[2(1 + *νs*)].

#### *2.3. Tunnel Model*

Herein, the immersed tunnel placed in an underwater trench is considered to be an impermeable uniform elastic medium with little deformation. Based on Hooke's law, the governing equation for the immersed tunnel can be derived:

$$\frac{1}{2}G\_t\left(\frac{\partial^2 u\_t}{\partial \mathbf{x}^2} + \frac{\partial^2 u\_t}{\partial z^2}\right) + \frac{G\_t}{1 - 2\nu\_t}\frac{\partial}{\partial \mathbf{x}}\left(\frac{\partial u\_t}{\partial \mathbf{x}} + \frac{\partial v\_t}{\partial z}\right) = 0\tag{21}$$

$$G\_{l}\left(\frac{\partial^{2}v\_{l}}{\partial\mathbf{x}^{2}}+\frac{\partial^{2}v\_{l}}{\partial\mathbf{z}^{2}}\right)+\frac{G\_{l}}{1-2v\_{l}}\frac{\partial}{\partial\mathbf{z}}\left(\frac{\partial u\_{l}}{\partial\mathbf{x}}+\frac{\partial v\_{l}}{\partial\mathbf{z}}\right)+\rho\_{l}g=0\tag{22}$$

where *ut* and *vt* are displacements in the *x* direction and *z* direction respectively, *ρ<sup>t</sup>* is the density of the tunnel, and *Gt* and *ν<sup>t</sup>* represent the shear modulus and Poisson's ratio of the immersed tunnel, respectively.

#### *2.4. Boundary Conditions*

The wave field is simulated numerically using the wave model. Under wave loading, the wave pressure is transmitted to the porous seabed with a finite thickness and the bottom of the seabed is assumed to be rigid and impermeable. The still water surface is located at *z* = 0, the waves propagate in the positive *x* direction, the vertical *z*-axis starts at the still water surface and upward is positive. In the wave field, *H* is the wave height, *L* is the wave length, the distance between the still water surface and seabed surface is *d*, and the seabed thickness is *h*. Herein, the oscillatory pore water pressure and seabed deformation could be obtained through solving Equations (16)–(20). Thus, the boundary conditions should be defined clearly, including the seabed conditions, tunnel surface boundary conditions and free water surface boundary conditions.

#### 2.4.1. Seabed Boundary Conditions

At the seabed surface, the vertical effective stress and shear stress become zero, and the pore water pressure is equal to the wave pressure in a wave field.

$$
\sigma'\_{zz} = 0, \ \tau'\_{xz} = 0, \ p\_s = p\_b \text{ (at seabed surface)} \tag{23}
$$

where *pb* is the wave pressure at the seabed surface.

The bottom of the seabed is impermeable bedrock; there is no displacement and vertical seepage at the bottom of the seabed, which is expressed as

$$u\_s = v\_s = 0, \ \frac{\partial p\_s}{\partial z} = 0 \text{ (at the bottom of seabed)}\tag{24}$$

Both sides of the seabed boundary are impermeable, and there is no horizontal displacement, which is expressed as

$$
\mu\_s = 0, \ \frac{\partial p\_s}{\partial z} = 0 \tag{25}
$$

#### 2.4.2. Tunnel Boundary Conditions

The tunnel is assumed to be impermeable and elastic material with large stiffness; the pore water pressure gradient at the tunnel's outside surface is zero:

$$\frac{\partial p\_s}{\partial n} = 0\tag{26}$$

Meanwhile, no relative displacement is assumed to occur between the tunnel and seabed.

#### 2.4.3. Free Water Surface Boundary Conditions

Owing to the pressure continuity condition of the two-phase flow in the flow field, the pressure at the water free surface is equal to the atmospheric pressure.

#### *2.5. Integration of Fluid Dynamic Model and Seabed Model*

In this model, FEM (Finite Element Model) codes are constructed within the COMSOL Multiphysics by designing the user's own governing equations and boundary conditions in a certain physical model. To figure out the dynamic response of seabed around the immersed tunnel under the action of waves, the RANS equation with momentum source function source terms is solved to realize a wave simulation and the PDE (partial differential equation) module is defined to realize a finite element calculation for the seabed soil. In this model, the tunnel is considered to be an impermeable, uniform, elastic medium with great stiffness; the interaction between the soil and the tunnel structure is not discussed in detail. Moreover, the dynamic interaction between the fluid and the tunnel might affect the stability of the tunnel. This paper, however, focuses on the stability of the soil trench under wave action in the immersing process of the tunnel element. Thus, the interaction between the fluid and the tunnel is also not emphasized.

First, MUMPS (a multifrontal massively parallel sparse direct solver) is adopted to solve the governing equations for the wave field and seabed, and the pressure fields and displacement fields can be obtained. Then, PARDISO (a nested dissection multithreaded solver) is used to get the turbulence energy and turbulent dissipation rate due to the large amount of computation in solving these two turbulence variables. The whole wave field is simulated to realize the flow field and wave pressure field simulation around the seabed and immersed tunnel. At last, two sub-modules (i.e., the wave model and seabed model) are coupled to achieve physical field coupling. Therefore, in every time step, the soil response under the wave field could give feedback to the wave sub-model.

The meshes of the model are divided into two types. The meshes are divided into triangles automatically in the wave flow field with a soil and tunnel part. Meanwhile, the automatic subdivision method is used to match the grids well with physical fields. The meshes around the foundation trench of the tunnel are also refined, which could make the simulation results of the flow field more reliable. The moving mesh method is used to capture the interface between the free surface and air, and the Lagrange smooth type is used in the freely deformed mesh area. The wave surface at 1 m outside the source region is selected for the comparison with the analytical solution, as shown in Figure 2.

**Figure 2.** The comparison of the water surface elevation in the present wave model and analytical solution.

#### **3. Model Validation and Numerical Results**

#### *3.1. Model Validation*

In this section, the calculated results of the proposed model are compared with the experimental values or analytical solutions from previous studies. At first, the wave verification is carried out to ensure the accuracy of the simulated waves. The wave parameters in Liu et al. [33] are adopted: wave height *H* = 0.143 m, wave period *T* = 1.75 s, water depth *d* = 0.533 m and wave length *L* = 3.53 m. The wave surface at 1 m outside the source region is selected for the comparison with the analytical solution, as shown in Figure 2. The results in the present wave model overestimate the wave surface at first; however, the overall result is basically consistent with the results. Meanwhile, the wave motion pattern at the free wave surface is compared with the analytical solution, as shown in Figure 3. The result of the present model tends to be consistent with the analytical solution. As a whole, the results of the wave simulation agree well with the expected analytical solution.

**Figure 3.** A comparison of the free surface of the present wave model with the analytical solution.

Then, the dynamic response in the seabed is compared with the analytical solutions [34] and experimental data [33]. Figure 4 shows the variation of the maximum pore pressure (*ps*/*p*0) with soil depth (*z*/*h*), in which *ps* denotes the wave-induced maximum pore pressure and *p*<sup>0</sup> is the pressure amplitude at the seabed surface. The soil parameters are given in Table 1. It is shown that the simulated result from the present numerical model agrees well with the analytical solution in Hsu and Jeng [34], while the gap between the present model and the experimental data becomes more significant near the bottom of the soil. However, this paper focuses on the dynamic response of the soil near the seabed surface which has significant influence on the stability of the trench. Therefore, the reliability of the proposed model in a soil surface layer could make the following analysis realized with good efficacy.

**Figure 4.** The vertical distributions of the maximum pore pressure (*ps*/*p*0) versus the seabed depth (*z*/*h*).

**Table 1.** The soil parameters for the present study.


#### *3.2. Consolidation of the Seabed*

In a natural marine environment, the seabed soil will reach a new state of consolidation after the construction of the structure, with the dissipation of the pore pressure, the increase of effective stress and the settlement of the foundation. The stress distribution around marine structures may be significantly affected, so the initial consolidation state and stress distribution should be clearly defined. In this section, the initial consolidation of the porous seabed is predetermined under dead weight and tunnel load. The tunnel size and material parameters are given in Figure 5 and Table 2.

**Figure 5.** The main section dimensions of an immersed tube tunnel (mm).


**Table 2.** The input data of a standard case for parametric study.

When the immersed tunnel is arranged, its sinking velocity is basically stable by controlling the irrigation amount; thus, the self-weight of the tunnel is basically equivalent to the buoyancy. The effect of the tunnel placement on the internal force of the seabed soil is not considered in this model. The initial effective stress distribution is shown in Figure 6. This pre-consolidation is considered as the initial circumstance in the following analysis.

**Figure 6.** The initial stress distribution after the consolidation of the seabed.

#### *3.3. Dynamic Responses of the Seabed*

Figure 7 is given to compare the distributions of pore pressure and various stresses in two cases: *t* = 29 s after the foundation groove is excavated and *t* = 29 s after the tunnel element is placed. The horizontal/vertical stresses and shear stress, as well as the pore pressure, are respectively depicted. These response variables are nondimensionalized using the wave-induced pressure at the seabed surface. It is shown that the horizontal and vertical stresses are right below the wave crest and wave trough, and the effective horizontal stress under the wave crest is negative while the effective vertical stress is positive. The shear stress is shown to be concentrated at the joint where the crests and troughs meet (i.e., the position of wave node where the wave amplitude is zero). From the comparison between the two cases, the horizontal and vertical effective stresses in the lower part of the tunnel tend to be concentrated. The negative pore pressure region on the right side of the tunnel expands, which is consistent with the expansion of a liquefaction trend which will be discussed in the subsequent section.

Figure 8 shows the variation of pore pressure with time at 1 m below the middle point in both the lee (left) side and weather (right) side slopes. Due to the existence of the trench, the pore pressure amplitude on the weather side slope is significantly smaller than that on the lee side slope. By comparing Figure 8a with Figure 8b, the amplitude of pore pressure decreases in the case when the tunnel is placed in the trench.

Figure 9 shows the variation of the pore pressure at 1 m below the bottom two corners of the tunnel with time. The pore pressure amplitude on the lee side is shown to be larger than that on the weather side, which may be caused by the changes of water depth. While in Figure 9b, the amplitude of the pore pressure on the weather side is larger than that on the lee side. Compared with Figure 9a, the amplitude in Figure 9b on the lee side is reduced slightly. The violent change of pore pressure may lead to a more serious transient liquefaction.

**Figure 7.** The distribution of (**a**) *σx*/*p*0, (**b**) *σy*/*p*0, (**c**) *τxz*/*p*0, (**d**) *ps*/*p*<sup>0</sup> in two cases: after the foundation groove is excavated (*t* = 29 s) and after the tunnel element is placed (*t* = 29 s). (*p*<sup>0</sup> is the water pressure at the seabed surface.)

**Figure 8.** The pore pressure at two asymmetric points on the two-side slopes: (**a**) without immersed tunnel and (**b**) with immersed tunnel.

**Figure 9.** The pore pressure at 1 m below the positions of two bottom corners of the tunnel: (**a**) without immersed tunnel and (**b**) with immersed tunnel.

#### *3.4. Wave-Induced Liquefaction*

The wave-induced liquefaction of soil is important for the design and construction of offshore engineering structures. Meanwhile, several wave-induced liquefaction criterions have been put forward [35–37]. Zen and Yamazaki [35] proposed a two-dimensional liquefaction criterion:

$$
\sigma\_{v0}' < p\_s - p\_b \tag{27}
$$

in which *pb* is the wave pressure acting on the seabed surface and *ps* is the wave-induced oscillatory pore pressure in the seabed.

Figure 10 presents the liquefaction in the vicinity of the trench at the same time (*t* = 36 s) in two cases: (a) after the foundation groove is excavated and (b) after the tunnel is settled down. The maximum liquefied depth in case (b) is 0.95 m, smaller than that in case (a) with 1.75 m. In the wave field, the maximum wave pressure in case (b) is 1.90 × <sup>10</sup><sup>4</sup> Pa, smaller than the 2.01 × <sup>10</sup><sup>4</sup> Pa in case (a). This may be due to a certain resistance to the wave propagation from the tunnel.

**Figure 10.** The liquefied area in the trench in two cases: (**a**) after the foundation groove is excavated (*t* = 36 s) and (**b**) after the tunnel element is placed (*t* = 36 s).

#### *3.5. Wave-Induced Shear Failure*

Generally speaking, when the shear stress is greater than the shear strength of the soil, the shear failure occurs and the relative displacement of the soil particles leads to instability failure [38]. Although the Mohr–Coulomb criterion is based on the linear elastic theory, it still is a basic and convenient tool for engineering reference and is used here to determine the shear failure of the seabed. First, it is necessary to comprehend the effective stress path of the soil. The *p* − *q* plane is adopted to analyze the effective stress path; *p* and *q* are respectively defined as

$$p' = (\sigma\_1' + \sigma\_3')/2 \tag{28}$$

$$q' = (\sigma\_1' - \sigma\_3')/2 \tag{29}$$

where *σ* <sup>1</sup> and *σ* <sup>3</sup> are the effective principal stresses in the seabed, respectively. The effect of intermediate principal stress is not considered.

Thus, considering the initial consolidation state of the seabed, the initial stress without a wave load can be expressed as

$$
\sigma\_{z0}' = \gamma' z \tag{30}
$$

$$
\sigma'\_{x0} = \mathbb{K}\_0 \gamma' z \tag{31}
$$

where *K*<sup>0</sup> is the coefficient of the lateral earth pressure, which is related to Poisson's ratio; *K*<sup>0</sup> = *μ*/(1 − *μ*); and *γ* is the effective unit weight of soil.

Herein, after being subjected to the wave load, the total effective stress of soil can be expressed as

$$
\overline{\sigma}'\_{z} = \sigma\_{z0}' + \sigma\_{zz}'\tag{32}
$$

$$
\overline{\sigma}'\_{\underline{x}} = \sigma'\_{\underline{x}0} + \sigma'\_{\underline{x}\underline{z}} \tag{33}
$$

where *σ zz* and *σ xz* denote the vertical and horizontal effective stresses in soil under wave loads.

Since the shear stress under the initial dead weight is equal to 0, the final total shear stress *τ xz* is induced by the wave load, which is expressed as

$$
\overline{\tau}'\_{xz} = \tau'\_{xz} \tag{34}
$$

The effective principal stresses *σ* <sup>1</sup> and *σ* <sup>3</sup> are expressed as

$$
\sigma\_1' = \frac{\overline{\sigma}\_x' + \overline{\sigma}\_z'}{2} + \sqrt{\left(\frac{\overline{\sigma}\_x' - \overline{\sigma}\_z'}{2}\right)^2 + \left(\overline{\tau}\_{xz}'\right)^2} \tag{35}
$$

$$
\sigma\_3' = \frac{\overline{\sigma}\_{\ge}' + \overline{\sigma}\_z'}{2} - \sqrt{\left(\frac{\overline{\sigma}\_{\ge}' - \overline{\sigma}\_z'}{2}\right)^2 + \left(\overline{\tau}\_{\ge \overline{\tau}}'\right)^2} \tag{36}
$$

Then, the stress statement at one point in the soil part can be expressed as

$$\sin \phi = \frac{\sigma\_1' - \sigma\_3'}{\sigma\_1' + \sigma\_3' + 2c/\tan \phi\_f} \tag{37}$$

where *φ* is the stress angle, and *c* and *φ* are the cohesion and internal friction angles, respectively. Based on the Mohr–Coulomb criterion, the shear strength of the soil can be expressed as

$$
\pi\_f = \sigma\_f \tan \phi\_f + c \tag{38}
$$

where *σ<sup>f</sup>* and *τ<sup>f</sup>* are the normal stress and shear stress on the failure surface, separately.

When the stress in the soil reaches the strength envelope, the soil reaches the ultimate equilibrium state. Therefore, the discriminant formula of shear failure at one point in the soil can be written as

$$
\Phi \ge \Phi\_f \tag{39}
$$

Substitute Equation (39) into Equation (37), and the discriminant formula of shear failure is obtained as

$$\alpha \pm \alpha \phi \ge \arcsin\left(\frac{\frac{\sigma\_1' - \sigma\_3'}{2}}{\frac{\sigma}{\tan \phi} + \frac{\sigma\_1' + \sigma\_3'}{2}}\right) \ge \phi\_f \tag{40}$$

For a more detailed introduction about shear failure, readers can refer to Zen et al. (1998) [39] and Jeng (2001) [38].

Figure 11 shows the distribution of the shear failure area in the trench in two cases: (a) after the foundation groove is excavated and (b) after the tunnel element is placed. The results at *t* = 36 s are presented as the depth of the shear failure area when it reaches the maximum at this moment. The maximum depths of shear failure at the left toe are almost the same (at *z* = −33.5 m) in these two cases. Due to the existence of the tunnel, the depth of the shear failure area below the tunnel reduces. Moreover, the shear failure is more likely to occur near two corners at the bottom of the tunnel because the stress is concentrated here.

**Figure 11.** The shear failure area in the trench in two cases: (**a**) after the foundation groove is excavated (*t* = 36 s) and (**b**) after the tunnel element is placed (*t* = 36 s).

#### *3.6. Influence of Wave Characters on Liquefaction*

In this section, based on the proposed model, the influences of the wave parameters on the dynamic response of soil around the foundation trench are studied. The standard wave conditions are *T* = 8 s, *H* = 4 m and *d* = 16 m. In the parametric analysis, the wave periods are *T* = 6 s, 7 s and 8 s; the wave heights are *H* = 3 m, 4 m and 5 m; and the water depths are *d* = 12 m, 16 m and 20 m. The calculation domain is 100 < x < 200 m and −46.5 < y < −16 m, and the maximum depth of liquefication, the area of liquefication zone and the width of liquefication at the seabed surface are calculated.

In general, the waves with a large wave length and a large wave height in shallow water tend to induce liquefaction in the seabed easily [6]. As shown in Figure 12, with the increases of wave height *H*, the increases of wave period *T* or the decreases of water depth *d*, the liquefaction depth, the liquefaction area and the liquefaction width increase. However, the increase of the liquefaction area is mainly associated with the increase of the liquefied depth. Moreover, the increase of water depth can effectively restrain the dynamic pressure generated by waves, and thus, the dynamic response would be much smaller.

**Figure 12.** The liquefied conditions around the trench with different wave parameters.

#### *3.7. Influence of Slope Rate on Liquefaction*

Then, the influence of the slope side angle to the stability of the foundation trench is further investigated. It is obvious that once the bottom width and the excavation depth of the trench are fixed, the factor affecting the amount of excavation and backfill is exactly the slope angle *θ*. The suitable *θ* could ensure the stability of the excavated trench, while meeting the economic requirements. In this section, the wave adopts the standard condition from above and the slope ratios are taken to be 1:3, 1:2 and 1:1.5, respectively. As discussed previously, the maximum liquefaction depth, the liquefaction area and the liquefaction width are calculated, and the latter two are expressed in the forms of percentages (the liquefied area is nondimensionalized using the area of the trench and the liquefied width using the width of trench bottom). As shown in Figure 13, with the increase of the slope ratio, the maximum liquefaction depth, the liquefaction area and the liquefaction width decrease at first and then increase without exception. That is to say, under certain wave conditions, there is an optimal value for the slope ratio.

**Figure 13.** The liquefied conditions around the trench with different slope angles: (**a**) depth of liquefied seabed, (**b**) area of liquefied seabed, (**c**) width of liquefied seabed.

#### **4. Conclusions**

In this paper, a two-dimensional coupling model of a wave-seabed-immersed tunnel is proposed, which aims to study the dynamic response of the soil around the foundation trench and immersed tube tunnel under wave load. The model is verified using the analytical and experimental results. The influences of the wave characteristics, slope ratio and soil parameters on the soil responses are analyzed. The influence of soil failure on the stability of the tunnel is also discussed. Based on the numerical results, the main conclusions can be drawn as follows:


**Author Contributions:** Conceptualization, W.-Y.C. and Z.-H.W.; software, L.-L.D.; data curation, H.-M.Q.; writing—original draft preparation, C.-L.L.; writing—review and editing, W.-Y.C.; project administration, Z.-H.W.; funding acquisition, W.-Y.C.

**Funding:** The authors are grateful for the support from the National Natural Science Foundation of China (Grant Nos. 41877243 and 41502285), the Natural Science Foundation of Jiangsu Province (No. BK20150952), and the Postgraduate Research & Practice Innovation Program of Jiangsu (No. KYCX18\_1060).

**Acknowledgments:** The authors are grateful for the financial support from the National Science Foundation of China (Grant No. 41877243, No. 41502285), the Natural Science Foundation of Jiangsu Province (Grant No. BK20150952), and the Postgraduate Research & Practice Innovation Program of Jiangsu (Grant No. KYCX18\_1060).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Field Test on Buoyancy Variation of a Subsea Bottom-Supported Foundation Model**

#### **Tianyi Fang 1, Guojun Liu 2, Guanlin Ye 1,\*, Shang Pan 1, Haibin Shi <sup>2</sup> and Lulu Zhang <sup>1</sup>**


Received: 29 March 2019; Accepted: 10 May 2019; Published: 13 May 2019

**Abstract:** The bottom-supported foundation is the most important component of offshore platforms, as it provides the major support to the upper structure. The buoyancy of the bottom-supported foundation is a critical issue in platform design because it counteracts parts of the vertical loads. In this paper, a model box was designed and installed with earth pressure transducers and pore pressure transducers to simulate the sitting process of the bottom-supported foundation. The buoyancy acting on the model box was calculated on the basis of two different methods, i.e., the water pressure difference between top and bottom surface and the effective stress at the bottom of the model. Field tests with different sitting times were carried out on the saturated soft clay seabed. Numerical coupled analysis was performed to verify the dissipation of the excess pore pressure at the bottom of the model. The results showed that the buoyancy of the model could reach twice the calculated value of Archimedes' law in the initial stage, however, it eventually stabilized near the theoretical value as the excess pore pressure dissipated. There was a slight fluctuation in buoyancy due to the phase lag of the pore pressure response caused by the low permeability of the seabed.

**Keywords:** buoyancy; pore pressure; bottom-supported foundation; field test; numerical analysis

#### **1. Introduction**

The offshore platform has good applicability in shallow sea areas with flat clayey seabed because of its relatively low requirement of soil-bearing capacity. Generally, such platform can be divided into an upper structure and a bottom-supported foundation. The latter is usually designed to be hollow, with a larger bottom area to provide sufficient support. The buoyancy of the bottom-supported foundation is an important part of the support force, since it counteracts parts of the vertical loads. Therefore, it is very important for the design of offshore platforms to investigate the buoyancy variation of the bottom-supported foundation during the whole operation process.

The existing researches regarding the buoyancy of structures are basically focused on underground structures embedded in saturated soils. The buoyancy acting on the structure varies depending on the type of the soil (clay or sand). It is proved that the buoyancy of a structure in saturated sand is almost equal to the calculated value of Archimedes' law [1,2]. When liquefaction occurs, the buoyancy is much greater than that in hydrostatic state [3–5]. The problem of buoyancy in weakly permeable soils like clay is usually investigated with indoor model test. The bucket model is usually used to simulate the underground structure, and the additional weight is adjusted to control whether the bucket is floating or not. The results of these researches show that the measured buoyancy is less than the theoretical value based on Archimedes' law, especially in clays [6,7]. The phenomenon of reduced buoyancy is explained by some microscopic-level studies [8,9]. Furthermore, the buoyancy acting

on the underground structure is also affected by the hydraulic gradient depending on the soil type. Indoor model tests and numerical analysis were carried out to study the effect of seepage on buoyancy. The results show that the buoyancy of a structure is greater than the theoretical value, considering the vertical seepage [10]. In the aforementioned studies, the buoyancy measurements were carried out only after the structure was embedded in the soil. For offshore platforms, attention should be paid to the point of the contact between the bottom-supported foundation and the seabed.

In this paper, field tests based on the characteristics of the bottom-supported foundation were carried out. A model box was designed with earth pressure and pore pressure transducers to simulate the entire operation process of such foundation. The buoyancy acting on the model box can be calculated on the basis of two different methods, i.e., the water pressure difference between top and bottom surface and the effective stress at the bottom of the model. Five tests were performed, with a sitting time of 3 h, 6 h, 22 h, 2.5 days, and 5 days, and the sitting time was calculated from the time the model contacted the seabed. The buoyancy variation was recorded during the entire sitting time. The effect of tides on buoyancy was also considered during the tests. The test revealed the variation of buoyancy acting on the model and the excess pore pressure dissipation at the bottom of the model. A coupled finite-element analysis was performed to verify the dissipation of the excess pore pressure using the Modified Cam Clay (MCC) model [11,12], and the comparison of buoyancy was made between the simulation and the measured values.

#### **2. Field Tests**

The site of this field test is located in the East China Sea, as shown in Figure 1. It is characterized by a soft clay seabed, with water depth of about 5 m and tidal difference of about 2.5 m. The test was executed in the industrial area of Changxing Island.

**Figure 1.** Schematic diagram of the test site.

#### *2.1. Test Model*

In the field test, a hollow cuboid sealing box model was used to simulate the bottom-supported foundation. The size of the model box was 2 m × 2 m × 1 m, and the weight was 10 tons, which can be considered to be proportional to the real field condition. It was connected to a crane by a dynamometer, whose real-time readings were recorded using a computer. A total of seven pore pressure transducers

were used to measure the pore water pressure, one on the top and six at the bottom of the model. Five earth pressure transducers were installed at the bottom of the model to measure the total stress. The readings of all the transducers were recorded in the data acquisition instrument at a defined time interval. The theoretical buoyancy of the model was 40 kN when it was completely submerged in water. Figure 2 shows the overall and bottom schematic diagram of the aforementioned box model.

**Figure 2.** (**a**) Overall and (**b**) bottom schematic diagram of the bottom-supported model foundation.

#### *2.2. Test Principles*

The purpose of this test was to measure the buoyancy variation of the model box during the entire sitting time. The test model can calculate the buoyancy on the basis of the water pressure difference between top and bottom surface and the effective stress at the bottom of the model. Since the model box was a rectangular parallelepiped, the pressure on the side walls was offset reciprocally by the walls. Figure 3 shows the principle of this test. Because of the relatively large bottom area of the model box, only the lower part was embedded in the seabed. The top surface of the model box was subjected to water pressure. The bottom surface of the model box was subjected to water pressure and to the support force of the seabed. In the first method (Figure 3a), only the water pressure on the top and bottom surface was considered, and the buoyancy was equal to the water pressure difference between them. The buoyancy can be expressed as follow:

$$F = (\mu\_1 - \mu\_2) \times A \tag{1}$$

where *F* is the buoyancy of the model box, *u*<sup>1</sup> is the pore water pressure measured by the bottom pore pressure transducer, *u*<sup>2</sup> is the pore water pressure measured by the top pore pressure transducer, and *A* is the bottom area of the model.

In the second method (Figure 3b), the concept of effective stress was introduced. It is considered that the self-weight of the model is balanced by the buoyancy and the effective support force from the seabed. Hence, the buoyancy can be expressed as follow:

$$F = G - P\tag{2}$$

$$P = (w - u\_1) \times A \tag{3}$$

where *G* is the self-weight of the model, *P* is the effective support force of seabed, and *w* is the total stress measured by the bottom earth pressure transducer.

In order to eliminate the influence of uneven pressure on the bottom surface, when calculating the buoyancy, the readings of transducers installed at the bottom were averaged.

**Figure 3.** Schematic diagram of the test principle for (**a**) Formula 1 and (**b**) Formula 2.

#### *2.3. Test Plan*

The steps of the field tests were as follows: (1) number the transducers and calibrate their initial values before the start of the test; (2) connect all the parts correctly (crane, dynamometer, model box, transducers, data collector, and computer) and record the initial reading of the dynamometer (the weight of the model); (3) slowly sink the model into the sea until the dynamometer reading approaches zero; (4) continuously record the readings of the transducers until the preset sitting time; (5) pull up the steel box for the next test. Five tests with different sitting times were carried out to investigate the buoyancy acting on the model, as listed in Table 1. Figure 4 shows some photographs of the test.

**Figure 4.** Photographs taken during the test: (**a**) numbering of the transducers and calibration of their initial values; (**b**) dynamometer used in the test; (**c**) lowering of the model to start the test; (**d**) test site.

**Table 1.** Test plan.


#### *2.4. Tests Results*

Figure 5 shows the total stress and pore water pressure results during the sitting time of 3 h, 6 h, 22 h, 2.5 days, and 5 days. In Figure 5a,b, the sampling frequency was every 1 min, and in Figure 5c–e, the sampling frequency was every 10 min. The average total stress (*w*) represents the mean of the readings of five bottom earth pressure transducers. The average bottom pore water pressure (*u*1) represents the mean of the readings of six bottom pore pressure transducers. The top pore water pressure (*u*2) was obtained from the top pore pressure transducer. The test results in Figure 5 are fluctuating because of the influence of tides. The state of the tide at the beginning stage of each test can be obtained according to *u*2. At the beginning of Case a, the tide was rising, and at the beginning of Cases b, c, d, e, the tide was falling. The initial phase of the tide had an influence on the readings of the transducers (as shown by the difference between Case a and Case b) but had no effect on the variation of the buoyancy, since both methods of calculating buoyancy are based on subtraction. As the sitting time increased, the result of *u*<sup>1</sup> showed a decreasing trend, which is shown in Figure 5e (the dotted red line is the trend line for *u*1). On the basis of the tests results, the buoyancy variation of the model was determined during a long sitting time. Therefore, further discussion on pore pressure difference, effective stress, and buoyancy is based on the sitting time of 5 days.

**Figure 5.** *Cont*.

**Figure 5.** Field test results of total stress and pore water pressure during the sitting time of (**a**) 3 h, (**b**) 6 h, (**c**) 22 h, (**d**) 2.5 days, and (**e**) 5 days.

Figure 6 shows the variation of the pore pressure difference and effective stress during 5 days, calculated from the aforementioned field test results. The pore pressure difference is obtained by the difference between the average bottom pore water pressure and the top pore water pressure (*u*<sup>1</sup> <sup>−</sup> *<sup>u</sup>*2). When the model was completely submerged in the sea, the theoretical value of the pore pressure difference was 10 kPa, since the height was 1 m. The sitting process of the model is considered to be an undrained compression. The external load was almost completely borne by the pore water because of the low permeability of the seabed at the initial stage. Significant excess pore pressure was generated at the bottom of the model. As the sitting time increased, the excess pore pressure gradually dissipated and eventually stabilized at the theoretical value. When calculating the difference, it appeared that the influence of the tide on the pore pressure difference was almost eliminated, but a slight fluctuation was still visible. This phenomenon can be attributed to the phase lag of the pore pressure response caused by the low permeability of the seabed. When the water level changed due to the tide, the response of the top pore pressure transducer was instantaneous. However, the response of the bottom pore pressure transducer was very slow compared to that of the top one, since the lower half of the model was embedded in the lowly permeable seabed. The phase lag in the pore pressure response between the two places has also been reported in many studies [13,14]. The effective stress is the average total stress minus the average bottom pore water pressure (*<sup>w</sup>* <sup>−</sup> *<sup>u</sup>*1). The variation trend of the effective stress was completely opposite to that of the pore pressure difference. At the initial stage of the sitting time, the effective stress of the soil was at a low level because of the significant excess pore pressure. With the dissipation of the excess pore pressure, the effective stress gradually accumulated and eventually stabilized at about 15 kPa

**Figure 6.** Variation of the pore pressure difference and effective stress during the sitting time of 5 days.

Figure 7 shows the variation of buoyancy during 5 days, which was based on the pore pressure difference and the effective stress. The trend of the buoyancy results obtained by the two methods was consistent. The results indicated that the model was subjected to buoyancy during the entire sitting time because of the connectivity of the pore water in the seabed to the outside seawater. At the beginning of the sitting time, the buoyancy could reach twice the theoretical value as a result of the significant excess pore pressure at the bottom of the model. With the dissipation of the excess pore pressure (increase of the effective stress), the buoyancy decreased and eventually stabilized near the theoretical value. The fluctuation of the buoyancy was consistent with that of the pore pressure difference and effective stress.

**Figure 7.** Variation of buoyancy (based on two calculation methods) during the sitting time of 5 days.

#### **3. Numerical Simulation**

In order to exclude the contingency of field results, since tests were conducted only in one place, the numerical coupled analysis was performed to verify the dissipation of the excess pore pressure at the bottom of the model foundation. Furthermore, the numerical result of buoyancy based on the pore pressure difference was also obtained, but the effect of tides was not taken into account. Some scholars have investigated the pore pressure response of seabed soil by fluid–solid coupling numerical analysis [15–18]. The dissipation process of the excess pore pressure in the field test was due to the consolidation of the soil after being subjected to external loads [19,20]. The analysis was performed with the software Abaqus.

#### *3.1. Soil Property*

The MCC model was used to simulate the behavior of the seabed soil, and the soil was considered homogeneous and normally consolidated. Such an approximation is appropriate for simulating the pore pressure dissipation. The parameters of the MCC model are summarized in Table 2. These parameters are empirical values based on existing Shanghai clay parameters [21–23].


**Table 2.** Parameters of the Modified Cam Clay (MCC) model.

According to the theory of the MCC model, the initial size of yield surface can be expressed as:

$$p\_0' = \frac{q^2}{M^2 p'} + p'\tag{4}$$

where *<sup>M</sup>* is the slope of the critical state line in *p'* <sup>−</sup> *<sup>q</sup>* space, and *<sup>p</sup>* and *<sup>q</sup>* are mean effectives stress and deviatoric stress, respectively. The initial void ratio varies with depth, which is expressed by:

$$
\varepsilon\_0 = \varepsilon\_1 - \lambda \ln p'\_{0} + \kappa \ln \frac{p\_0'}{p'} \tag{5}
$$

where *e***<sup>1</sup>** is the void ratio at *p'* = 1 kPa on the normally consolidated line, λ is the slope of the normally consolidated line in *e* − ln *p*' space, and κ is the slope of the swelling and recompression line in *e* − ln *p*' space.

$$\mathbf{e}\_1 = \mathbf{e}\_{\mathfrak{c}\mathfrak{s}} + (\lambda - \mathfrak{x})\ln(\mathfrak{z}) \tag{6}$$

#### *3.2. Model and Mesh*

A two-dimensional model was built to demonstrate the dissipation of the excess pore pressure. As shown in Figure 8, the model foundation was considered to be a rigid body with a width of 2 m. The soil was 20 m and 10 m long in the horizontal and vertical directions, respectively, to eliminate boundary interference. The CPE4P (four-node plane strain quadrilateral, bilinear displacement, bilinear pore pressure) elements were used in the model. Finer meshes were used on the upper part of the soil to improve the analysis accuracy. The structure and the soil were bound, since the contact between them was considered rough during the whole sitting process [24,25]. Of all the boundaries, only the upper surface of the soil was considered to be permeable. The sitting process was considered to be an undrained compression. Force-controlled analysis was used in the simulation, and the external load was the weight of the model box minus the theoretical buoyancy. The consolidation time was 5 days.

**Figure 8.** Model and mesh of the numerical analysis.

#### *3.3. Comparison between Numerical Analysis and Field Test*

The dissipation of the excess pore pressure at the bottom of the model foundation was obtained from the numerical analysis. Figure 9 presents the distribution of the excess pore pressure within the 4 m depth after different sitting times. At the initial stage of the sitting time, the excess pore pressure was significant and concentrated at the corner of the model foundation, as shown in Figure 9a. With the increase of the sitting time, the excess pore pressure of the seabed surface dissipated first, while the excess pore pressure in the deep was still significant, as shown in Figure 9d. The peak value of the excess pore pressure decreased as the sitting time increased. The simulation value of the pore pressure difference was composed of the average excess pore pressure at the bottom of the model foundation and a fixed value of 10 kPa, which was determined by the height of the model foundation. The simulation result was a smooth curve, since the influence of tides was not considered in the numerical analysis. In Figure 10, two buoyancy curves based on the pore pressure difference are presented (one is the measured value and the other is the simulation value). The agreement between the simulated results and the measured results was acceptable. Therefore, the pore pressure obtained in the field test and the calculated buoyancy were considered reasonable. According to the results from the numerical analysis, the deep excess pore pressure had little effect on the buoyancy acting on the model foundation.

**Figure 9.** Distribution of the excess pore pressure after sitting for (**a**) 0.5 day, (**b**) 1 day, (**c**) 2.5 days, and (**d**) 5 days.

**Figure 10.** Comparison of buoyancy between numerical analysis and field test.

#### **4. Conclusions**

A model box comparable to the real field condition was designed to simulate the bottom-supported foundation, and transducers were installed on both top and bottom surfaces of the box. Field tests with different sitting times were carried out to investigate the buoyancy acting on the model box. Two different methods were used to calculate the buoyancy based on the test results. The main focus of this paper was on the dissipation of excess pore pressure at the bottom of the model and on the variation of buoyancy during the entire sitting time. Moreover, a numerical coupled analysis based on normal consolidation soil was carried out to verify the field tests results. The following conclusions can be drawn:


During operation, a bottom-supported foundation is mainly subjected to two processes, that is, the sitting process and the uplifting process. The research of this paper mainly focused on the sitting process of the foundation. Further work will focus on the uplifting process of the foundation. During the uplifting process, the foundation is subject to resistance mainly caused by the negative pore pressure, which we call the bottom separation force. Both experimental and numerical studies will be conducted to explore (i) the formation mechanism of the bottom separation force, (ii) the factors affecting the bottom separation force, and (iii) the measures necessary to reduce the bottom separation force. This will provide a reference for practical engineering.

**Author Contributions:** Conceptualization, G.Y. and L.Z.; Data curation, S.P.; Formal analysis, T.F.; Methodology, G.Y.; Project administration, G.L. and H.S.; Writing—original draft, T.F.; Writing—review & editing, G.Y.

**Funding:** The authors are grateful for the financial support from the National Natural Science Foundation of China (Grant No. 41727802, 41630633) and the Science and Technology Commission of Shanghai Municipality (STCSM) (Grant No. 18DZ1100100).

**Acknowledgments:** The valuable discussions with Chencong Liao during the tests are appreciated.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Giant Submarine Landslide in the South China Sea: Evidence, Causes, and Implications**

**Chaoqi Zhu 1,2, Sheng Cheng 1, Qingping Li 3, Hongxian Shan 1,2,4, Jing'an Lu 5, Zhicong Shen 1, Xiaolei Liu 1,2,4 and Yonggang Jia 1,2,4,\***


Received: 6 April 2019; Accepted: 13 May 2019; Published: 17 May 2019

**Abstract:** Submarine landslides can be tremendous in scale. They are one of the most important processes for global sediment fluxes and tsunami generation. However, studies of prodigious submarine landslides remain insufficient. In this review paper, we compile, summarize, and reanalyze the results of previous studies. Based on this reanalysis, we discover the giant Baiyun–Liwan submarine slide in the Pearl River Mouth Basin, South China Sea. We describe three concurrent pieces of evidence from ~23 Ma to 24 Ma, the Oligocene–Miocene boundary, for this landslide: the shoreward shift of the shelf break in the Baiyun Sag, the slump deposition to the southeast, and the abrupt decrease in the accumulation rate on the lower continental slope. This landslide extends for over 250 km, and the total affected area of the slide is up to ~35,000–40,000 km2. The scale of the landslide is similar to that of the Storegga slide, which has long been considered to be the largest landslide on earth. We suggest that strike–slip movement along the Red River Fault and ridge jump of the South China Sea caused the coeval Baiyun–Liwan submarine slide. The identification of the giant landslide will promote the understanding of not only its associated geohazards but also the steep rise of the Himalayan orogeny and marine engineering. More attention needs to be paid to areas with repeated submarine landslides and offshore installations.

**Keywords:** giant submarine landslides; shelf break; South China Sea; Himalayan orogeny; repeated submarine landslides

#### **1. Introduction**

Terrestrial landslides are efficient agents for the transport of rock, sediment, and carbon [1]. Alarmingly, submarine landslides on continental slopes can be more tremendous in scale—up to three orders of magnitude larger than terrestrial landslides in terms of total volume [2–4]. The volume of a single prodigious submarine landslide can be hundreds of times larger than the annual sediment flux from all the world's rivers to the ocean. An example is the Storegga slide, with over 3000 km3 of sediment—long considered to be the largest submarine slide on earth [5,6]. Recently, a similar giant submarine landslide caused by the reactivation of major intra-plate faults, the Halibut Slide, was found in the North Atlantic [7]. The latest large submarine landslide triggered by the 2011 Tohoku earthquake covered an area of ~27.7 km<sup>2</sup> [8]. Prodigious submarine landslides are rare, but can dramatically alter seafloor morphology and sedimentary structure, generate huge tsunamis, and damage offshore

infrastructures [3,9–11]. The Grand Banks landslide in 1929 broke 12 telegraph cables and generated a tsunami with runup of 13 m [12]. A submarine landslide with a volume of ~5 km3 in Papua New Guinea generated a tsunami killing 2200 people in 1999 [13]. The 2011 Tohoku tsunami with runups of 20 m was also exemplified by the part contribution of a submarine landslide [14]. Exploring giant submarine landslides on a long time scale will provide clues for global change and tectonic events. Giant submarine landslides attract immense research interest on the global level due to their geological, ecological, societal, and economical significance.

Submarine landslides are widespread in the Pearl River Mouth Basin (PRMB, Figure 1, [15]), which is the main reservoir of oil, gas, and gas hydrate in the northern South China Sea. However, many submarine landslides have been buried or obscured. It is difficult to recognize and map them accurately. Apart from numerous small-scale submarine landslides, 142 mass transport complexes with an average area of over 7 km2 can be found in the submarine canyons of PRMB. In deeper waters, a series of Quaternary mass transport complexes with a total areal extent of ~11,000 km2 and a conservative volume estimate of ~1035 km<sup>3</sup> are recognized [16,17]. Alarmingly, two large submarine landslides or submarine creep zones have been discovered in the PRMB, including the Baiyun slide (>13,000 km2; ~0.3 Ma; Figure 1) and the Dongsha creep zone (>800 km2; active at present; Figure 1) [18–20]. Here, we summarize several pieces of evidence and report the world's potentially largest landslide (the Baiyun–Liwan submarine slide, Figure 1) in the South China Sea, as well as its scale, timing, and triggering mechanism. It is proposed that the recognition and study of the giant Baiyun–Liwan submarine slide will promote the understanding of not only its associated geohazards but also the spreading of the South China Sea, the steep rise of the Himalayan orogeny, and marine engineering.

**Figure 1.** Multi-beam submarine geomorphology shadow map showing the Baiyun–Liwan submarine slide, the Baiyun slide, the Dongsha creep zone, and the shelf break line (SBL). The white dotted line is a scarp of the Baiyun–Liwan submarine slide. The brown dotted lines are the scarp of the Baiyun slide and range of the Dongsha creep zone, respectively.

#### **2. Geological Setting**

The Baiyun–Liwan submarine slide is located mainly at the Baiyun Sag and Liwan Sag, in the southern part of the PRMB and close to the continent–ocean boundary of the South China Sea [15]. The PRMB is the largest basin in the northern continental margin of the South China Sea. From north to south, the PRMB includes three uplifts and two depressions: the Northern Terrace, the Northern Depression (Zhu I Depression and Zhu III Depression), the Central Uplift (Shenhu Uplift, Panyu Low Uplift, and Dongsha Uplift), the Southern Depression (Zhu II Depression, and Chaoshan Depression), and the Southern Uplift (Figure 1) [21]. From bottom to top, eight formations are identified in the PRMB, namely, the Shenhu, Wenchang, Enping, Zhuhai, Zhujiang, Hanjiang, Yuehai, and Wanshan Formations. The PRMB experienced three major tectonic evolution stages: rifting (65–30 Ma), transition (30–23.8 Ma), and thermal subsidence (23.8 Ma–present) [15,22].

The Baiyun Sag is bordered by the Panyu Low Uplift to the north and the Liwan Sag and the Southern Uplift to the south (Figures 1 and 2). Covering an area of more than ~30,000 km<sup>2</sup> and exceeding 11 km in thickness, the Baiyun Sag and the neighboring Liwan Sag are the largest and deepest sags in the PRMB [23,24]. In addition, Liwan 3-1, the largest offshore platform in Asia, is built in the Baiyun Sag. Gas hydrate production tests were also conducted there in 2017 [25].

**Figure 2.** Geological map showing the Pearl River Mouth Basin (PRMB).

Our study suggests that the head scarp of the Baiyun–Liwan submarine slide is located at the northern Baiyun Sag and that the main slide body blankets the Baiyun Sag and the Liwan Sag. The landslide masses moved southeastwards, and some masses were transported to Ocean Drilling Program (ODP) Site 1148 and even further. The landslide extends for over ~250 km, and the total affected area of the slide is up to ~35,000–40,000 km2.

#### **3. Evidence of the Baiyun–Liwan Submarine Slide**

#### *3.1. Shoreward Migration of the Shelf Break*

The shoreward shift of the shelf break in the Baiyun Sag provides significant clues about the giant Baiyun–Liwan submarine slide. The present shelf break is aligned with the boundary between the Panyu Low Uplift and the Baiyun Sag, namely the northern Baiyun Sag (Figure 1). The Neogene sequence stratigraphy of the Baiyun Sag reveals that the shelf break has been located in its present position since ~23.8 Ma, and that the Baiyun Sag was in a deep-water slope environment at that time [26–28]. In Han et al. (2016), detailed results for the evolutionary history of the shelf break showed that the shelf break from 23.8 Ma swung back and forth around the boundary, although it stayed quite close to the present location [22]. However, continental shelf deltaic deposition was found in the Zhuhai Formation (32–23.8 Ma) and was characterized by southward progradational reflections with sigmoid-oblique configurations, while deep-water slope depositional systems from 23.8 Ma were observed [26,29]. In addition, the shelf deltaic deposition extended to the southern Baiyun Sag, and deepwater fan facies were recognized there [22,30]. However, there was a progressive seaward migration from 30 Ma to 23.8 Ma when the shelf breaks were approximately situated in the southern Baiyun Sag. The shelf break migration from the early Oligocene period to the present shows a giant sudden shoreward shift from south to north of the Baiyun Sag at ~23.8 Ma (Figure 3).

**Figure 3.** Schematic diagram showing the sequence-stratigraphic framework of the Baiyun Sag and the migration of shelf breaks from 30 Ma to the present in the PRMB. The location of the shelf break was acquired from Han et al. (2016) [22].

The mechanism for shelf break migration is ascribed to sea level changes, sedimentary processes, and geological processes [31,32]. The progressive seaward migration in the southern Baiyun Sag during 30 Ma to 23.8 Ma was proven to be mainly controlled by sediment supply and sea level [22,33]. We attributed the giant sudden shoreward shift from the south to north of the Baiyun Sag ~23.8 Ma ago to a prodigious submarine landslide, namely the Baiyun–Liwan submarine slide. Before the landslide, the Baiyun Sag was part of the continental shelf and in a neritic depositional environment. Therefore, the shelf break was in the south of the Baiyun Sag. After the landslide, a large volume of material was moved from the Baiyun Sag and Liwan Sag to deep water. As a direct consequence, the Baiyun Sag changed from a neritic depositional environment to a deep-water depositional environment, and the shelf break migrated from south to north of the Baiyun Sag.

#### *3.2. Slump Deposition to the Southeast*

Slump deposition, found in ODP Core 1148 at ~458–472 mcd (meter composite depth), provides another important piece of evidence for the Baiyun–Liwan submarine slide. ODP Site 1148 (18◦50.17 N, 116◦33.94 E; water depth 3294 m) is located on the lower continental slope to the southeast of the Baiyun Sag. There, an ~860 m long composite section spanning more than the last 32 Ma was recovered. It was the most offshore site, with the longest core drilled during ODP Leg 184. The dominant lithology at ODP Site 1148 is clay with a variable proportion of nannofossils [34]. The sediment sequence is mainly characterized by three sections: a rapidly deposited Oligocene section (Section 1) and a relatively slowly deposited Miocene to Pleistocene section (Section 3), divided by slumped and faulted intervals at 458–472 mcd (Section 2) [35,36]. Section 2, though similar in composition to the overlying and underlying sections, is dominated by episodic gravitational redeposition, including mass flows and slumping. The Oligocene–Miocene boundary sediments represent massive convolute bedding, soft-sediment plastic deformation, and light-colored carbonate mud clasts of nannofossil chalk, especially at 458–460 mcd (Figure 4). Normal microfaults are common at 460–472 mcd (Figure 4).

**Figure 4.** (**A**) Seismic profile through ODP Site 1148, showing a prominent reflector at Section 2, ~5.0 two-way travel time (TWT). (**B**) Section 2 is characterized by plastic deformation (pd), displaced chalk (c), and microfaults (f). Data from Wang et al. (2000) [34].

Coincidentally, the slump deposition is constrained between 23.5 and 24.5 Ma [37]; this epoch is quite close to 23.8 Ma, when the shelf break in the Baiyun Sag shifted from south to north. Given the concurrent events (shift of the shelf break in the Baiyun Sag and the slump deposition at the ODP Site 1148) and the relative elevation (high in the Baiyun Sag and low in ODP Site 1148), we conclude that the Baiyun–Liwan submarine slide does exist and occurred in the Oligocene–Miocene boundary. When the slide occurred, the submarine blocks were removed and transported to the deeper water. Some of these blocks were deposited at ODP Site 1148, forming the slump deposition at 458–472 mcd.

#### *3.3. Decrease in Accumulation Rate*

Another noteworthy piece of sedimentary evidence in ODP Core 1148 is the abrupt decrease in the accumulation rate at ~23–24 Ma. The average mass accumulation rates (MAR) decreased from ~7–20 g/cm2/ky in the early Oligocene to ~1–3 g/cm2/ky in the Miocene (Figure 5). The distance from the sediment sources may be responsible for the accumulation rates at this location. The higher accumulation rates of the Oligocene sediments imply active downslope sediment transport from the continental shelf and upper continental slope. After the Baiyun–Liwan submarine slide, the shelf break moved shoreward, far away from the site. Consequently, the greater distance from the sediment sources led to lower accumulation rates at this site, at least within the Neogene. The lower accumulation rates are representative of hemipelagic or pelagic sedimentation.

**Figure 5.** The mass accumulation rates (MAR) vs. depth and the age-depth model for ODP Site 1148. Data from Wang et al. (2000) [34].

#### **4. Discussion**

Although many triggering mechanisms can cause landslides, either solely or concurrently, prodigious landslides are more likely to be trigged by tectonic events and global change [6,38–41]. The Storegga slide is attributed to earthquake activity-associated gas hydrate dissociation during postglacial isostatic rebound [42,43]. Volcanic growth is another tectonic activity that causes gigantic landslides both in subaerial environments (e.g., the Markagunt gravity slide—the largest subaerial landslide) [44] and in submarine environments [45].

The Baiyun–Liwan submarine slide seemed to be triggered by other tectonic events e.g., strike–slip movement along the Red River Fault and the ridge jump of the South China Sea. The strain observed by Tapponnier et al. [46] reveals that the penetration of India into Asia extrudes Indochina by more than 500 km southeastwards relative to South China (Figure 6). This strike–slip movement along the Red River Fault occurs at ~22 to 24 Ma at the Oligocene–Miocene boundary. In response to the strike–slip movement along the Red River Fault, the ridge of the South China Sea jumped to the south and changed orientation from nearly E–W to NE–SW at ~24 Ma, which opened the South China Sea [47]. The ridge orientation is exactly perpendicular to the direction of mass movement of the Baiyun–Liwan submarine slide. Our research suggests that the strike–slip movement along the Red River Fault and the ridge jump of the South China Sea contributed to the giant Baiyun–Liwan submarine slide. A steep rise of the Himalayan orogen in the Oligocene–Miocene boundary [48] provides another circumstantial piece of evidence for a triggering mechanism. These coeval events suggest active tectonic activity in the South China Sea and its vicinity in the Oligocene–Miocene boundary. Interestingly, the reversion of strike–slip along the Red River Fault (~5.5 Ma) triggered another prodigious submarine landslide with area of about 18,000 km2 [49]. More supporting evidence is needed to deepen the understanding of the prodigious Baiyun–Liwan submarine slide and its origin.

**Figure 6.** Simplified map showing the triggering mechanism of the Baiyun–Liwan submarine slide.

Given the numerous slope failures subsequent to the Baiyun–Liwan submarine slide in the same place, this area appears to be prone to failure. Nineteen migrating submarine canyons (Figures 1 and 7) have developed on the head scarp of the Baiyun–Liwan submarine slide; and 142 mass transport complexes, each with an average area of over 7 km2, can be found in the canyons [16,50]. Furthermore, the giant Baiyun slide and the active Dongsha creep, as mentioned in the introduction, are recognized in the affected area of the Baiyun–Liwan submarine slide. Repeated submarine landslides have also been found in other places such as the Gela Basin [51], the Gioia Basin [52], the Eivissa Channel [53], and the Yellow River subaqueous delta [54]. Given the main reservoir of oil, gas, and gas hydrate in the northern South China Sea, more attention needs to be paid to areas of repeated submarine landslides, especially those with many offshore installations.

**Figure 7.** Multi-beam submarine geomorphology shadow map showing the head scarp of the Baiyun–Liwan submarine slide, where the migrating submarine canyons and the Baiyun slide developed on the head scarp of the Baiyun–Liwan submarine slide. I = continental shelf; II1 = upper continental slope; II2 = submarine canyons; II3 = lower continental slope; III = Baiyun slide.

#### **5. Conclusions**

Three concurrent events (the shoreward shift of the shelf break in the Baiyun Sag, the slump deposition, and the abrupt decrease in the accumulation rate on the lower continental slope) indicate the giant Baiyun–Liwan submarine slide in the PRMB, South China Sea, at ~23 to 24 Ma, in the Oligocene–Miocene boundary. This landslide extends for over ~250 km, with the total affected area of the slide up to ~35,000 to 40,000 km2. The scale of the Baiyun–Liwan submarine slide is similar to that of the Storegga slide, which has long been considered to be the largest submarine slide on earth. Our research suggests that coeval events (the strike–slip movement along the Red River Fault and the ridge jump of the South China Sea) in the Oligocene–Miocene boundary triggered the Baiyun–Liwan submarine slide. Attention needs to be paid to areas with repeated submarine landslides and offshore installations.

**Author Contributions:** Authorship must be limited to those who have contributed substantially to the work reported. Conceptualization, C.Z. and Y.J.; Data curation, C.Z. and Q.L.; Formal analysis, C.Z., S.C. and H.S.; Funding acquisition, Y.J.; Project administration, Y.J.; Visualization, Z.S.; Writing – original draft, C.Z.; Writing – review & editing, S.C., H.S., J.L., X.L. and Y.J.

**Funding:** This research was funded by the National Natural Science Foundation of China, grant number 41831280 and 41427803, National Key R&D Program of China, grant number SQ2018YFC030044.

**Acknowledgments:** Authors are indebted to the Ocean Drilling Program Leg 184.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **An Approach to Assess the Stability of Unsaturated Multilayered Coastal-Embankment Slope during Rainfall Infiltration**

#### **Jian-feng Zhu 1, Chang-fu Chen <sup>2</sup> and Hong-yi Zhao 3,4,\***


Received: 5 May 2019; Accepted: 23 May 2019; Published: 29 May 2019

**Abstract:** This study aims to develop a simple but effective approach to investigate the stability of an unsaturated and multilayered coastal-embankment slope during the rainfall, in which a Random Search Algorithm (RSA) based on the random sampling idea of the Monte Carlo method was employed to obtain the most dangerous circular sliding surface, whereas the safety factor of the unsaturated slope was calculated by the modified Morgenstern–Price method. Firstly, two typical distributions of matric suction were illustrated and the associated methods for determining the strength parameters of unsaturated soil were developed. Based on this, the Morgenstern–Price method was further modified to calculate the safety factor, and RSA was adopted to locate the most dangerous sliding surface of the unsaturated multilayered coastal-embankment slope. Finally, the slope breaking process under rainfall infiltration was simulated through continuously searching the critical slip surfaces under different groundwater levels by RSA. The results indicated that the stability of the unsaturated embankment slope was gradually deteriorated with the increase of rainfall infiltration. It was also found that both of the distributions of the matrix suction (*u*a-*u*w) and the suction angle (ϕb) had significant effects on the safety factor of the embankment slope. Basically, linear distribution of (*u*a-*u*w) along the depth and linear relationship between ϕ<sup>b</sup> and (*u*a-*u*w) should be adopted in assessing the stability of the unsaturated multilayered coastal-embankment slope.

**Keywords:** coastal-embankment slope; stability; unsaturated soil; multilayered; matric suction; random searching algorithm; rainfall infiltration

#### **1. Introduction**

As one of the effective defensing coastal structures, embankment slope or breakwater is widely used to protect ashore human life and property from severe environment (storm, rainfall, long-term wave loading, etc.). Therefore, the instability of the embankment slope has attracted more and more attention in coastal engineering with the failure modes of scour [1–4] and shear failure [5–8]. Moreover, most of the above literature has mainly focused on wave-induced seabed instability, in which the soil was considered as saturated material [4,6,9–13]. However, in practical coastal engineering, the coastal-embankment slope is always unsaturated and multilayered. Its stability suffers from huge challenges, especially during rainfall, which has proven to be the key factor linked to landslides [14–18].

As shown in Figure 1, rainfall usually results in a rising of the water table, and a decreasing in matric suction [19–22]. In addition, as the moisture content increases, the unit weight increases. The combined factors mentioned above may reduce the stability of an unsaturated-soil slope. The occurrence of instabilities of unsaturated slopes during wet periods is quite common all over the world [23,24] and has an expectable increasing trend in intensity and frequency due to the warming climate [22,24]. Therefore, it becomes more and more urgent to develop a simple but effective method to assess the stability of the unsaturated-soil slope during rainfall infiltration.

To comprehensively investigate the stability of an unsaturated-soil slope, three key issues should be addressed, as follows: (1) an appropriate method to determine the matric suction and its associated strength parameters; (2) a reasonable approach to evaluate the stability of a trial unsaturated-soil slope; (3) an effective algorithm to search for the critical slip surface. Fredlund and Rahardjo [25] pointed out that the presence and the magnitude of matric suction (*u*a-*u*w), which were critical to the stability of unsaturated-soil slopes, were dependent on the practical environmental conditions. For the sake of simplicity, the distribution of matric suction in general is considered to be uniformed or linearly decreased along the buried depth [26,27]. Recently, Zhang et al. [28] investigated the effect of the different distributions of matric suction on the overturning stability of the retaining wall with homogeneous, continuous and non-layered surrounding soils. It was found that the influence of uniformed suction was more remarkable than that of linear suction. Xu and Yang [21] found that there is a little difference between the effect of the distribution pattern of matric suction on stability of three-dimensional unsaturated, homogeneous, continuous and non-layered soil slope. However, in practice, the ground conditions in the unsaturated region are generally multilayered, which results in the uncertainty of influence of the matric suction distribution patterns on the stability of unsaturated-soil slope. Moreover, the existence of the multilayered behavior will significantly increase the difficulty in the determination of strength parameters when the rigorous limit equilibrium method was adopted to investigate the safety of the unsaturated-soil slope, which will be analyzed in detail in a later section.

In practical slope engineering design, the limit equilibrium method (LEM) has proven to be an efficient tool to investigate the stability of a saturated soil slope under two-dimensional (2-D) plane strain conditions by computing the safety factor of a given slip surface with some calculation models (Simplified Bishop method [29], Spencer method [30], Morgenstern–Price method [31], etc.). As the most rigorous theoretical approach in LEM under 2-D conditions, Morgenstern–Price (represented by M–P for short) method was firstly developed by Morgenstern and Price in 1965 with the integral form, which resulted in difficulty in programming. Then, M–P method was improved for the convenience of programming by Chen et al. [32], Chen et al. [33] and Zhu and Chen [34] with the discrete pattern. However, both of the improved M–P methods were applicable to compute the safety factor of the saturated soil slope, which should be modified to investigate the stability of the unsaturated-soil slope by combined with the failure criterion of unsaturated soil as suggested by Fredlund and Rahardjo [25].

After determining the safety factor of a given slip surface, another work to investigate the stability of an unsaturated multilayered coastal-embankment slope is to adopt an effect global optimization algorithm to locate the most dangerous slip surface. In the last few decades, nonlinear programming approaches were usually adopted to address the failure surface of saturated slope both in practice and in academic research [35,36]. However, their efficiencies in locating the most dangerous sliding surface have proven to be poor in the case of multilayered slopes with discontinuous soil properties [37]. The Random Search Algorithm (RSA), developed based on the Monte Carlo principle, behaves much better in solving complicated optimization problems in comparison to nonlinear programming approaches [38]. Within the framework of RSA, the global optimal solution can be addressed without iteration by generating a series of trial solutions randomly and comparing them with current optimum solution one by one until reaching the maximum number of locating. The main shortcoming of RSA is the generation of the large number of trial solutions which reduces the calculation efficiency in the past and has become of secondary importance due to the rapid development of computational technology. Currently, it has been widely used in determining the most dangerous sliding surface by search adequate times [39,40].

**Figure 1.** Cross section of unsaturated multilayered coastal-embankment slope under rainfall infiltration.

**Figure 2.** Framework of the present study.

It should be noted that the stability of coastal-embankment slope with the unsaturated and multilayered feature during rainfall were greatly different from those of dry or saturated soils and should be investigated by a simple and effective approach in consideration of effect of matric suction distributions and underground water levels. As a consequence, in the framework of this research (shown in Figure 2), the influence of rainfall infiltration on unsaturated-soil slope was simulated by the gradual rise of the groundwater level and strength parameters under different distributions of matric suction were determined. Then, the M–P method was modified to investigate the safety of the unsaturated multilayered coastal-embankment slope by combined the rigorous limit equilibrium theory with failure criterion of unsaturated soil. Furthermore, the RSA developed by Malkawi's method [40] was adopted to locate the critical slip surface of the unsaturated multilayered coastal-embankment slope. Finally, the failure process under the rainfall infiltration was reproduced by searching for the critical slip surface and the associated safety factor, on which the effect of distribution of the matric suction was also investigated. In addition, the influence of the determined method of suction angle on the safety factor is discussed in detail.

#### **2. A Simple Approach to Simulate Rainfall Infiltration**

When the rainfall is heavy enough to overcome the hydraulic conductivity of the sub-layer, the groundwater table will move upward gradually; however, assuming capillarity rising to the ground surface, the distribution depth of matric suction will decrease until the soil becomes fully saturated, as depicted in Figure 1. Moreover, the density of Layers 2–3 will lose the effective density from its natural pattern and the pore-water pressure will decrease with the associated reduction in the effective stress. Then, the shear strength of soil will drop and the stability of the soil slope will in turn deteriorate. Actually, the effect of rainfall on stability of unsaturated-soil slope is dependent on many of factors (rainfall amount, average intensity, duration, pattern, vegetation) [16,19,41]. For more detailed investigations, it is necessary to address the soil-water profile with 1-D or 2-D infiltration model to address the effect of infiltration on unsaturated-soil slope's stability [42], which will make the problem become more complicated, laborious and computationally expensive [43]. Since the groundwater level will gradual rise during long-term rainfall infiltration, the effect of rainfall on the unsaturated coastal-embankment slope can be alternatively reproduced by the progressive elevation of the groundwater level without considering the interaction between soil and water. For the sake of simplicity, the phreatic line is assumed to be horizontal in this study as shown in Figure 1.

#### **3. Limit Equilibrium Method of Unsaturated-Multilayered-Soil Slope**

#### *3.1. Failure Criterion of Unsaturated-Soil*

Fredlund and Rahardjo [25] developed the Mohr–Coulomb criterion to evaluate the failure of the unsaturated soil through introducing matric suction:

$$
\pi\_{\mathfrak{u}} = \mathfrak{c}\prime + (\sigma\_{\mathfrak{n}} - \mathfrak{u}\_{\mathfrak{w}})\tan q\nu + (\mathfrak{u}\_{\mathfrak{a}} - \mathfrak{u}\_{\mathfrak{w}})\tan q\nu \tag{1}
$$

where τ*<sup>u</sup>* = the shear strength; *c*' = the effective cohesion; σ<sup>n</sup> = the total normal stress; *u*<sup>w</sup> = the pore-water pressure; ϕ' = the effective internal friction angle associated with (σn-*u*w); (*u*a-*u*w) = the matric suction; ϕ<sup>b</sup> = the suction angle. (*u*a-*u*w) and ϕ<sup>b</sup> are the key parameters of unsaturated-soil failure criterion. Furthermore, the value of ϕ<sup>b</sup> is always dependent on (*u*a-*u*w) [44], the (*u*a-*u*w) has become a crucial issue unsaturated soil theory [25–28].

#### *3.2. Distributions of Matric Suction*

As shown in Equation (1), (*u*a-*u*w) and ϕ<sup>b</sup> are the key parameters of unsaturated-soil failure criterion. Furthermore, the value of ϕ<sup>b</sup> is always dependent on (*u*a-*u*w) [44], the accurate determination of (*u*a-*u*w), therefore, is of great important to the failure criterion of unsaturated soil, which has become a crucial issue in unsaturated-soil theory.

Although the matric suctions at the ground surface (*u*a-*u*w)g can be easily tested by tensiometer, the underground matric suctions are difficult to test in practical engineering. For the sake of simplicity, the matric suction is assumed to be distributed along the buried depth by following two patterns [21,25–28]:

(1) Distribution I

As shown in Figure 3a, the matric suction is uniform along the depth and its value can be half of the matric suction at the ground surface in a general stratified unsaturated-soil slope.

(2) Distribution II

As shown in Figure 3b, the matric suctions decrease linearly along the depth and attained their minimum value at the groundwater level.

#### *3.3. Determination of the Strength Parameters under Di*ff*erent Distributions of Matric Suction*

For the sake of simplicity, the total cohesion *C* of unsaturated soil is always considered to be composed of the effective cohesion and the contribution of suction to strength. Then, Equation (1) can be reduced to

$$
\tau\_u = \mathcal{C} + (\sigma\_n - u\_w) \tan \varphi \nu. \tag{2}
$$

where *C* is the total cohesion and *C* = *c*'+(*u*a-*u*w)tanϕb.

The associated soil strength parameters at different depths can be calculated as follows:

(1) For Distribution I

As shown in Figure 3a, assume that the soil slope is composed of three layers (Layer I, Layer II, Layer III) and for each layer the strength parameters are *c*'*i*, ϕ*'i* (*i* = 1,2,3). Because of the existence of (*u*a-*u*w) in the soil layers above the groundwater level, the correspondent strength parameters can be determined as follows:

$$\mathbf{C}\_{i} = \mathbf{c}'\_{\,\,\,\! } + 0.5(\mathbf{u}\_{a} - \mathbf{u}\_{w})\_{\,\,\,\! } \tan q^{b}; \mathbf{q}'\_{\,\,\,\,\,} = \mathbf{q}'\_{\,\,\,\,\,\,'} (i = 1, 2; \,\,\mathbf{j} = \mathbf{I}, \Pi); \tag{3}$$

$$\mathbf{C}\_{i} = \mathbf{c}'\_{\,\,\,\forall} \mathbf{q}'\_{\,\,i} = \mathbf{q}'\_{\,\,\,\,\mathbf{j}'} \,(i = 3, 4; \, j = \text{II}, \text{III}) . \tag{4}$$

(2) For Distribution II

As shown in Figure 3b, the soil layer above the groundwater level can be divided into (*m*+*n*) small soil layers. In each small layer, the strength parameters are supposed to be uniform and to be determined as follows,

$$\mathbf{C}\_{i} = \mathbf{c}\_{\,1}^{\prime} + (\mathbf{u}\_{a} - \mathbf{u}\_{w})\_{\,\beta} \tan q^{b} \times (H - h\_{i}) / H; \\ \mathbf{q}^{\prime}\_{\,i} = q^{\prime}\_{\,1\prime} \, (\mathbf{i} = 1, 2, \cdots, m); \tag{5}$$

$$\mathbf{C}\_{i} = \mathbf{c}\_{\text{II}}^{\prime} + 0.5(\mathbf{u}\_{\text{a}} - \mathbf{u}\_{\text{w}})\_{\text{g}} \tan \boldsymbol{\varphi}^{\text{b}} \times (\mathbf{H} - \mathbf{h}\_{i}) / H;\\ \boldsymbol{\varphi}^{\prime}\_{\text{i}} = \boldsymbol{\varphi}^{\prime}\_{\text{II}} \left(\mathbf{i} = m + 1, m + 2, \cdots, m + n\right); \tag{6}$$

$$\mathbf{C}\_{i} = \mathbf{c}'\_{\Pi}; q\mathbf{y}'\_{i} = q\mathbf{y}'\_{\Pi'} (i = m + n + 1); \tag{7}$$

$$\mathbf{C}\_{\mathbf{i}} = \mathbf{c'}\_{\mathbf{II}}; \mathbf{q'}\_{\mathbf{i}} = \mathbf{q'}\_{\mathbf{III'}} (\mathbf{i} = m + n + 2); \tag{8}$$

where *c*'*i*, ϕ*'i*(*j* = I, II, III) = the strength parameters of the original soil layers; (*u*a-*u*w)g = the matric suction at the ground surface; *Ci*, ϕ'*<sup>i</sup>* (*i* = 1,2, ... ,*m*+*n*+2) = the strength parameters of the divided small layers; *H* = the distance between the groundwater level and the top of the slope, *hi* (*i* = 1,2, ... ,*m*+*n*+2) = the distance between the *i*th small soil layers and the top of the slope.

**Figure 3.** Distributions of the (*u*a-*u*w) along the depth in unsaturated-multilayered-soil slope (**a**) Distribution I: uniform distribution of (*u*a-*u*w); (**b**) Distribution II: linear distribution of (*u*a-*u*w).

#### *3.4. Modified M-P Method*

As shown in Figure 4a, a trial sliding mass was generally divided into several small slices based on the principle of slice method. The inter-slice forces imposing on the typical slice was further illustrated in Figure 4b. Within the framework of two-dimensional (2-D) limit equilibrium analysis, M-P method [31] and its improved pattern [32–34] had proven to be the most rigorous theoretical approach due to its excellent performance in satisfying both force and moment equilibrium conditions. However, the aforementioned M–P method and its improved pattern were developed within the framework of saturated soil and should be modified with reference to the unsaturated soil strength criterion to exactly address the stability of a two-dimensional unsaturated-soil slope.

**Figure 4.** Discrete model and inter-slice forces in the slip surface of an unsaturated-soil slope (**a**) discrete model of a slip surface, (**b**) inter-slice forces.

Based on both of the forces and moment equilibrium of a typical slice (as shown in Figure 4b), the safety factor with the iteration form can be addressed by the following equations [34]:

$$F\_s = \frac{\sum\_{j=1}^{n-1} \mathbb{R}\_j \prod\_{k=j}^{n-1} \mathbb{Y}\_k + R\_n}{\sum\_{j=1}^{n-1} P\_j \prod\_{k=j}^{n-1} \mathbb{Y}\_k + P\_n} \tag{9}$$

$$\lambda = \frac{\sum\_{j=1}^{n} \left[ b\_j (E\_j + E\_{j-1}) \tan \alpha\_j - 2Q\_j \sin \alpha\_j h\_j \right]}{\sum\_{j=1}^{n} b\_j \left( f\_j E\_j + f\_{j-1} E\_{j-1} \right)} \tag{10}$$

$$E\_j \Phi\_j = \Psi\_{j-1} E\_{j-1} \Phi\_{j-1} - F\_s P\_j + R\_j \tag{11}$$

$$P\_{\rangle} = \mathcal{W}\_{\rangle} \sin \alpha\_{\rangle} - Q\_{\rangle} \sin \left(\omega\_{\rangle} - \alpha\_{\rangle}\right) \tag{12}$$

$$R\_{\dot{j}} = \left[W\_{\dot{j}}\cos\alpha\_{\dot{j}} - Q\_{\dot{j}}\cos\left(\omega\_{\dot{j}} - \alpha\_{\dot{j}}\right) - u\_{\dot{j}}l\_{\dot{j}}\right] \tan q\nu\_{\dot{j}} + C\_{\dot{j}}l\_{\dot{j}}\tag{13}$$

$$\Phi\_{\hat{\jmath}-1} = \left(\sin a\_{\hat{\jmath}-1} - \lambda f\_{\hat{\jmath}-1} \cos a\_{\hat{\jmath}-1}\right) \tan \eta \prime\_{\hat{\jmath}-1} + \left(\cos a\_{\hat{\jmath}-1} + \lambda f\_{\hat{\jmath}-1} \sin a\_{\hat{\jmath}-1}\right) F\_{\hat{\wp}} \tag{14}$$

$$\Psi\_{j-1} = \frac{(\sin \alpha\_j - \lambda f\_{j-1} \cos \alpha\_j) \tan \alpha \nu\_j + \left(\cos \alpha\_j + \lambda f\_{j-1} \sin \alpha\_j\right) \mathbb{F}\_s}{\Phi\_{j-1}} \tag{15}$$

$$N\_{\dot{j}} = \left(Q\_{\dot{j}}\cos\omega\_{\dot{j}} + \mathcal{W}\_{\dot{j}} - \lambda f\_{\dot{j}-1}E\_{\dot{j}-1} + \lambda f\_{\dot{j}}E\_{\dot{j}}\right)\cos\alpha\_{\dot{j}} + \left(Q\_{\dot{j}}\sin\omega\_{\dot{j}} + E\_{\dot{j}-1} - E\_{\dot{j}}\right)\sin\alpha\_{\dot{j}} - u\_{\dot{j}}l\_{\dot{j}}\tag{16}$$

where *F*<sup>s</sup> = the safety factor of the trial sliding surface; λ = the scaling factor; *Ej*-1 and *Ej* = the inter-slice force for slice *j*-1 and *j*; *Wj* = the weight of *j*th slice; *Qj* = the external loading; *Sj* = the shear force; *Nj* = the normal pressure; *uj* = the pore-water pressure at slice-base; ϕ*'j* and *Cj* = the mobilized internal friction angle and total cohesion along; *fj*-1 and *fj* = the inter-slice force function for slice *j*-1 and *j* which can be determined by relating them with the horizontal coordinate. Other detailed information about the model parameters can be found in Reference [34]. The safety factor of a trial sliding surface in the unsaturated multilayered coastal-embankment slope can be performed as shown in Figure 5.

**Figure 5.** Flow diagram of modified M–P method.

#### **4. A Global Algorithm to Search for the Critical Slip Surface of Unsaturated Coastal-Embankment Slope**

As a typical failure mechanism in practical geotechnical engineering, circular sliding surfaces are widely employed in the stability analysis of saturated soil slopes with both single layer [45–47] and multilayer [34,48,49]. Then, the aforementioned problem of determining a trial circular failure surface can be represented by locating the two control intersections (*L*(*x*L, *y*L), *R*(*x*R, *y*R)) and a radius (*R*c) shown in Figure 6, in which the potential sliding surface should be addressed in order to satisfy the boundary conditions represented by *y* = *g*(*x*) and *y* = *r*(*x*), the hydraulic conditions *y* = *w*(*x*) and the discontinuity between the layers demonstrated by *y* = *lj*(*x*). Then, the potential sliding surface represented by *S* can be expressed as functions of *x*L, *y*L, *x*R, *y*<sup>R</sup> and *R*<sup>c</sup> in the following form:

$$\mathbf{S} = [\mathbf{x}\_{\mathrm{L}\prime} y\_{\mathrm{L}\prime} \mathbf{x}\_{\mathrm{R}\prime} y\_{\mathrm{R}\prime} \mathbf{R}\_{\mathrm{c}}] \tag{17}$$

On the basis of LEM, the safety factor (*F*s) with respect to the most dangerous sliding surface should be minimized among all of the trial failure surfaces. The issue in terms of locating the most dangerous slip surface can be mathematically represented by addressing the minimum of *F*<sup>s</sup> with regard to *S* as follows:

$$\min F\_s(S) \tag{18}$$

Theoretically speaking, the trial circular failure surface can be generated with any combination of intersections (*L*(*x*L, *y*L), *R*(*x*R, *y*R)) and *R*c. However, most of these are invalid due to their impossible occurrence in practical engineering. To improve the searching efficiency, the control parameters should be empirically constrained as:

$$\mathbf{x}\_{\text{L.min}} \le \mathbf{x}\_{\text{L.max}};\\\mathbf{x}\_{\text{R.min}} \le \mathbf{x}\_{\text{R}} \le \mathbf{x}\_{\text{R.max}};\\\mathbf{R}\_{\text{c.min}} \le \mathbf{R}\_{\mathbf{c}} \le \mathbf{R}\_{\mathbf{c}\text{max}} \tag{19}$$

where *x*Lmax and *x*Lmin = the upper and lower limits of horizontal ordinate of point L; *x*Rmax and *x*Rmin = the upper and lower limits of horizontal ordinate of point R; *R*cmax and *R*cmin = the upper and lower limits of radius. It should be noted that definitions of these parameters should be dependent on the experience of the researcher.

As shown in Figure 6, the following constraints are considered:

$$\mathbf{y}\_{\text{L}} = \mathbf{g}(\mathbf{x}\_{\text{L}}); \mathbf{y}\_{\text{R}} = \mathbf{g}(\mathbf{x}\_{\text{R}}); \mathbf{x}\_{\text{L}} < \mathbf{x}\_{\text{R}}; \mathbf{R}\_{\text{c}} \ge \|\mathbf{L}\mathbf{R}\|; H + \mathbf{y}\_{\text{c}} \ge \mathbf{R}\_{\text{c}} \tag{20}$$

where *<sup>H</sup>* <sup>=</sup> the distance between the bottom and the lower boundary; *LR*- = the length of *LR*; *y*<sup>c</sup> = the vertical ordinate of the center of the circular slip surface.

To resolve the problem mentioned above, a trial sliding surface should be firstly addressed by locating two points and the radius (represented by L, R and *R*c shown in Figure 6) within the range of [*x*Lmin, *x*Lmax], [*x*Rmin, *x*Rmax] and [*R*cmin, *R*cmax], in which points L and R can be located stochastically in the following forms:

$$\begin{aligned} \mathbf{x}\_{\text{L}} &= r\_1 \mathbf{x}\_{\text{Lmin}} + (1 - r\_1) \mathbf{x}\_{\text{Lmax}}; \ \mathbf{x}\_{\text{R}} = r\_2 \mathbf{x}\_{\text{Rmin}} + (1 - r\_2) \mathbf{x}\_{\text{Rmax}}; \ R\_{\text{c}} = r\_3 R\_{\text{cmin}} + (1 - r\_3) R\_{\text{cmax}}; \\ \mathbf{y}\_{\text{L}} &= \mathbf{g}(\mathbf{x}\_{\text{L}}); \mathbf{y}\_{\text{R}} = \mathbf{g}(\mathbf{x}\_{\text{R}}) \end{aligned} \tag{21}$$

where *r*1, *r*<sup>2</sup> and *r*<sup>3</sup> = stochastic numbers which can be obtained randomly from 0 to 1.

The horizontal and vertical ordinate of O' can be addressed by the following equations:

$$\left(\left(\mathbf{x}\_{\mathrm{c}} - \mathbf{x}\_{\mathrm{L}}\right)^{2} + \left(y\_{\mathrm{c}} - y\_{\mathrm{L}}\right)^{2} = R\_{\mathrm{c}}^{2}; \left(\mathbf{x}\_{\mathrm{c}} - \mathbf{x}\_{\mathrm{R}}\right)^{2} + \left(y\_{\mathrm{c}} - y\_{\mathrm{R}}\right)^{2} = R\_{\mathrm{c}}^{2} \tag{22}$$

where *x*<sup>c</sup> = the horizontal ordinate of O'. There are two groups of solution for Equation (22), and to make sure the trial slip surface to be reliable, the smaller value of *x*c and the lager value of *y*c are used.

The Random Search Algorithm (RSA) is adopted to obtain the most dangerous sliding surface and the associated minimum safety factor for the unsaturated-soil slope. The detailed procedure for the RSA can be schematically illustrated in Figure 7.

**Figure 6.** Cross section of a typical multilayered slope with circular sliding surface.

**Figure 7.** The flowchart of locating the critical slip surface.

#### **5. Application**

#### *5.1. Comparison with the Existing Model*

The feasibility of the developed model is assessed by comparison with the numerical results predicted by Zhang et al. [50], in which both the linear and nonlinear relationship between the unsaturated shear strength and matric suction determined by the soil-water characteristic curve (SWCC) were adopted to investigate the effect of matric suction on the slope stability, for a steep unsaturated-soil slope with 30 m high and 50◦ inclination angle, as illustrated in Figure 8. The parameters for both models are illustrated in Table 1. In addition, the inclined groundwater table in the research of Zhang et al. [50] is assumed to be horizontal with an average 5 m below the toe of the slope for simplicity.

As shown in Figure 8, the critical slip surfaces predicted by both models become gradually deeper and their associated minimum safety factors progressively decrease with the increase of ϕb. This is because of the amplification of ϕ<sup>b</sup> which significantly increases the total cohesion (shown in Equations (3)–(8), and enhances the shear strength (Equation (2)), which in turn improves the safety factor. Moreover, both the critical slip surfaces (marked by the solid line) and the associated safety factor always agree well with those (marked by other line type) predicted by Zhang et al. [50]. Therefore, the present method performs well in investigating the stability of the unsaturated-soil slope.

**Table 1.** Parameters of soil.

**Figure 8.** Comparison of the critical slip surface.

#### *5.2. Case Study*

As shown in Figure 9, an unsaturated coastal-embankment slope with the inclination of 1:1.5 and four layers is subject to long-term rainfall, the stability of which is facing a great challenge. The physical-mechanical parameters of each layer are illustrated in Table 2.

**Figure 9.** The geological profile of a high road fill.

**Table 2.** Physic-mechanical properties of soil.


To investigate the stability of the unsaturated coastal-embankment slope under rainfall, the failure surface under different groundwater levels and the associated minimum safety factor have been searched 100,000 times by RSA. Two matric suction distributions along the depth: namely the linear distribution (Distribution I); and uniformed distribution (Distribution II) are used in the analysis. Meanwhile, the most dangerous sliding surface and its associated minimum safety factor determined by the traditional saturated M–P method in which the matric suction along the depth is zero (Distribution III) are also addressed by RSA. Based on the comparison results shown in Figure 10, it can be summarized as follows:

(1) There is an apparent discrepancy among Distribution I, Distribution II and Distribution III with respect to the critical slip surfaces. The critical slip surfaces associated with the above three distributions do not coincide until the groundwater level reaches the top of slope in which the soil is fully saturated.

(2) The most dangerous sliding surface of Distribution III is always located at the shallow depth of the slope and it is independent on the fluctuation of the groundwater level. Unfortunately, these results do not coincide with the practical results. The location of critical slip surface of Distribution I and Distribution II are close and both of them change from deep slip to shallow slip with the rising of the groundwater level and these predicted results coincide well with the practical results.

(c) Groundwater table = -1 m

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(d) Groundwater table = 3 m

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(e) Groundwater table = 6 m

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(f) Groundwater table = 10 m

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(g) Groundwater table = 12 m

**Figure 10.** *Cont*.

(i) Groundwater table = 15 m

**Figure 10.** Critical slip surfaces associated with each distribution pattern of matric suction under rainfall infiltration.

#### *5.3. Discussion on the E*ff*ect of Distribution of (ua-uw) on Fsmin*

In this section, the minimum safety factor (*F*smin) under different groundwater level shown in Figure 10 was adopted to investigate the effects of the different distributions of matric suction (*u*a-*u*w) on the stability of unsaturated-soil slope. As shown in Figure 11, *F*smin is intensively related to the distribution of (*u*a-*u*w) along the buried depth. Overall, all of the three values of *F*smin corresponding to different distributions show gradual decrease with the rise of the groundwater level. However, the reduced rate of Distribution III without suction, which keeps constant (approaches to 0) under the condition that the groundwater level is below 6 m, is remarkable lower that of the other two distributions. Moreover, both of the amplitude and reduced rate of Distribution I are greater than those of Distribution II. When the groundwater level approaches to 15 m, *F*smin of the three distributions coincides with each other, in which the soil is fully saturated. The effects of distributions of (*u*a-*u*w) on *F*smin varies accordingly to the distributions. For Distribution I and II with consideration of (*u*a-*u*w), the reduction in *F*smin is due to the decrease of distributed depth of (ua-uw) (shown in Figure 1) and the increase of the pore-water pressure (Equations (13) and (16)) with the increase of groundwater level. However, the decrease of *F*smin for Distribution III regardless of (*u*a-*u*w) is mainly attributed to the increase of the pore-water pressure. Therefore, the reduced rate and amplitude of *F*smin for Distribution I and II are much greater than that for Distribution III.

Above all, it can be concluded that the Distribution III in traditional saturated limit equilibrium analysis is unexpected which may significantly increase the cost of solidified unsaturated-soil slope and the other two distributions are more rational. Moreover, the safety factor for uniformed distribution (Distribution I) of (*u*a-*u*w) is always greater than that for linear distribution (Distribution II), which may result in an unsafe design of the unsaturated-soil slope. With the comprehensive consideration of safety and economy, it is suggested that Distribution II can be used in addressing the stability of unsaturated coastal-embankment slope in practical engineering.

**Figure 11.** Effect of distribution pattern of matric suction on safety factor under rainfall infiltration.

#### *5.4. Discussion on the Relationship between* ϕ*<sup>b</sup> and (ua-uw)*

The suction angle ϕ<sup>b</sup> was conventionally supposed to be constant and independent on (*u*a-*u*w) in the previous literature [19,51–54]. However, the multistage direct shear test results [44] showed that the relationship between ϕ<sup>b</sup> and (*u*a-*u*w) could be depicted by Curve 3, as shown in Figure 12. When (*u*a-*u*w) is small, ϕ<sup>b</sup> is supposed to be the maximum value ϕ<sup>b</sup> max. With the increase of (*u*a-*u*w), ϕ<sup>b</sup> gradually decreases and when (*u*a-*u*w) increases to a certain value ϕ<sup>b</sup> reaches the minimum value ϕ<sup>b</sup> min. Some researchers [25,55,56] used the Curve 1 and Curve 2 to simulate the relationship between ϕ<sup>b</sup> and (*u*a-*u*w) as shown in Figure 12. Curve 1, Curve 2 and Curve 3 can be described by the following functions:

(1) For Curve 1

$$
\varphi^b = \left(\varphi^b\_{\text{min}} + \varphi^b\_{\text{max}}\right) / 2 \, (u\_a - u\_w) > 0 \tag{23}
$$

(2) For Curve 2

$$q^{b} = q^{b}\_{\text{max}} \qquad 0 \le (u\_{d} - u\_{w}) \le 50 \text{kPa} \tag{24}$$

$$q^{b} = q^{b}\_{\text{min}} + \left(q^{b}\_{\text{max}} - q^{b}\_{\text{min}}\right) \times \left[ (u\_{d} - u\_{w}) - 50 \right] / 450, \qquad 50 \text{kPa} < (u\_{d} - u\_{w}) \le 500 \text{kPa} \tag{24}$$

$$q^{b} = q^{b}\_{\text{min}'} \qquad 500 \text{kPa} < (u\_{d} - u\_{w})$$

(3) For Curve 3

$$\begin{aligned} \varphi^{b} &= \varphi\_{\text{max}}^{b}, 0 \le (u\_{d} - u\_{w}) \le 50 \text{kPa} \\ \varphi^{b} &= \left(\varphi\_{\text{max}}^{b} + \varphi\_{\text{min}}^{b}\right) / 2 + \left(\varphi\_{\text{min}}^{b} - \varphi\_{\text{max}}^{b}\right) \times \sin\left[2 \times (u\_{d} - u\_{w}) - 550\right] \pi / 900, \\ &50 \text{kPa} < (u\_{d} - u\_{w}) \le 500 \text{kPa} \end{aligned} \tag{25}$$

$$
\varphi^b = \varphi^b\_{\text{min}} \, \text{500kPa} \, \prec (u\_a - u\_w) \tag{26}
$$

where ϕ<sup>b</sup> max and ϕ<sup>b</sup> min are the upper and lower limit respectively. Herein, ϕ<sup>b</sup> max is 30◦and ϕ<sup>b</sup> min is 5◦.

**Figure 12.** Correlation curves between ϕ<sup>b</sup> and (*u*a-*u*w).

Now, two conditions in which the groundwater level reaches up to 6 m and 12 m are considered associated with the Distribution II of matric suction, respectively. Their correspondent critical slip surface is Surface 2 in Figure 10. The influence of the distribution of ϕ<sup>b</sup> on safety factor in different (*u*a-*u*w)g is shown in Figure 13. It can be seen that when ϕ<sup>b</sup> is supposed to be constant, the safety factor(*F*<sup>1</sup> s) associated with Curve 1 is almost linearly increased with the increasing of (*u*a-*u*w)g. When the distribution of ϕ<sup>b</sup> is linear (Curve 2) and nonlinear (Curve 3) both the safety factor (*F2* s) associated with Curve 2 and (*F*<sup>3</sup> s) associated with Curve 3 show the same change trend with the increase of (*u*a-*u*w)g. At first, *F<sup>2</sup>* <sup>s</sup> and *F*<sup>3</sup> <sup>s</sup> have a dramatic rise until reaching the maxima. This is because that (*u*a-*u*w)g is smaller in this stage and the matric suction on the slip surface is smaller than 50 kPa. Meanwhile, the correspondent value of ϕ<sup>b</sup> always equals to ϕ<sup>b</sup> max and the total cohesion on the slip surface increases with the increasing of (*u*a-*u*w)g. Therefore, *F2* <sup>s</sup> and *F*<sup>3</sup> <sup>s</sup> become larger and larger. When (*u*a-*u*w)g increases to a certain value (about 450 kPa in the example) the matric suction on the slip surface will exceed to 50kPa and the correspondent ϕ<sup>b</sup> will decrease accordingly to Equation (24) and Equation (25). The value of (*u*a-*u*w) will increase but the value of ϕ<sup>b</sup> will decrease on the slip surface and the total cohesion slightly decreases. Therefore, *F2* <sup>s</sup> and *F*<sup>3</sup> s decrease slightly at this stage. When (*u*a-*u*w)g is kept increasing to a certain value (about 750 kPa in the example), (*u*a-*u*w) in the slip surface always exceed to 500 kPa and the correspondent value of ϕ<sup>b</sup> no longer change. Then, the total cohesion keeps increasing with the increase of (*u*a-*u*w)g and *F<sup>2</sup>* <sup>s</sup> and *F*<sup>3</sup> s will increase at this stage.

Above all, it can be found that the safety factor is sensitive to the distribution of ϕ<sup>b</sup> and Curve 2 and Curve 3 are more rational than Curve 1 shown in Figure 13. For the sake of simplicity, it is more reasonable to adopt the linear relationship between ϕ<sup>b</sup> and (*u*a-*u*w) in addressing the unsaturated-soil slope's safety factor.

**Figure 13.** The influence of the distribution of ϕ<sup>b</sup> on the safety factor, (**a**) Underground water level = 6 m; (**b**) Underground water level = 12 m.

#### **6. Conclusions**

In this study, the modified Morgenstern–Price (M–P) method coupled with Random Search Algorithm (RSA) was developed to investigate the stability of an unsaturated and multilayered embankment slope considering the effects of matric suction distributions, determinations of suction angle and rainfall infiltration. A convenient programmable M–P method was developed to address the unsaturated-multilayered-soil slope's safety factor. Based on RSA without iteration, the critical slip surface of unsaturated-multilayered-soil slope could be located. Assuming that the effect of rainfall infiltration could be simulated by the gradual increase in groundwater level, the stability of an unsaturated-multilayered-soil slope under rainfall infiltration was evaluated by introducing different matric suction distribution patterns. Finally, the effect of the determined method of suction angle (ϕb) on the safety factor was investigated. It was found that the fluctuation of the groundwater level has a significant influence on the location of the most dangerous sliding surface. The associated minimum safety factor and the sliding modes of unsaturated-soil slope gradually change from deep sliding to shallow sliding with the rise of groundwater level. Moreover, the traditional slope stability method regardless of the matric suction is conservative to the predicted results. It is more reasonable to adopt the linear distribution of matric suction in practical calculation of the safety factor (*F*s). In addition, *F*s is sensitive to the distribution of ϕ<sup>b</sup> and the linear relationship between ϕ<sup>b</sup> and (*u*a-*u*w) is more beneficial in addressing the stability of the unsaturated multilayered coastal slope. It should be noted that the present approach is simple and a lot of factors such as the effect of soil layering on the matric suction and non-circular failure surface are not covered, which will be addressed in detail in future work.

**Author Contributions:** Methodology and Writing-original draft, J.-f.Z. Supervision, C.-f.C.; Writing-Review and Editing, H.-y.-Z.

**Funding:** This research was funded by National Natural Science Foundation of China (51879133, 51409142, 41572298), Natural Science Foundation of Zhejiang Province (LY17E080006). The APC was funded by K.C. Wong Magna Fund in Ningbo University of China.

**Conflicts of Interest:** The authors declare no conflicts of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Stability Analysis of Near-Wellbore Reservoirs Considering the Damage of Hydrate-Bearing Sediments**

**Xiaoling Zhang 1, Fei Xia 1, Chengshun Xu <sup>1</sup> and Yan Han 2,\***


Received: 4 March 2019; Accepted: 9 April 2019; Published: 13 April 2019

**Abstract:** The stability of hydrate-bearing near-wellbore reservoirs is one of the key issues in gas hydrate exploitation. In most previous investigations, the damage evolution process of the sediment structure and its effect on near-wellbore reservoir stability have been neglected. Therefore, the damage variable is introduced into a multi-field coupled model based on continuous damage theory and multi-field coupling theory. A thermo-hydro-mechanical-chemical (THMC) multi-field coupling mathematical model considering damage of hydrate-bearing sediments is established. The effects of damage of hydrate-bearing sediments on the thermal field, seepage field, and mechanical field are considered. Finally, the distributions of hydrate saturation, pore pressure, damage variable, and effective stress of a near-wellbore reservoir in gas hydrate exploitation by depressurization are calculated, and the stability of a hydrate-bearing near-wellbore reservoir is analyzed using the model. Through calculation and analysis, it is found that structural damage of hydrate-bearing sediments has an adverse effect on the stability of hydrate-bearing near-wellbore reservoirs. The closer to the wellbore, the worse the reservoir stability, and the near-wellbore reservoir stability is the worst in the direction of minimum horizontal ground stress.

**Keywords:** hydrate-bearing sediments; damage statistical constitutive model; multi-field coupling; wellbore stability

#### **1. Introduction**

As a new type of clean energy, natural gas hydrate has the advantages of large reserves, wide distribution, high energy density, and non-pollution [1]. However, the hydrate will dissociate during the process of exploitation, which will lead to a decrease in the mechanical strength of hydrate-bearing sediments, causing possibly shear failure of the hydrate-bearing reservoir and serious deformation of the ground. This will result in instability of hydrate-bearing near-wellbore reservoirs [2]. Therefore, it is of great theoretical significance and application value to evaluate the stability of hydrate-bearing near-wellbore reservoirs for ensuring safe gas hydrate exploitation.

The problems of reservoir stability such as wellbore instability and ground deformation during gas hydrate exploitation are essentially a multi-field coupling problem involving phase transition, fluid flow, heat transfer, and geomechanical deformation [3]. Freij-Ayoub et al. [4] established a hydrate-bearing wellbore stability analysis model considering the coupling of porous media deformation, heat transfer, fluid transport, and hydrate dissociation, but in this model the fluid was regarded as a single-phase flow without taking into account the effect of hydrate dissociation on reservoir permeability. Rutqvist et al. [5] proposed a multi-field coupled mathematical model considering thermal, fluid-flow, and geomechanical responses during hydrate dissociation, and the

Mohr–Coulomb strength criterion was used to judge the shear failure of a wellbore reservoir under two gas hydrate exploitation schemes of a vertical well and a horizontal well. Finally, the wellbore reservoir stability was analyzed from the perspective of stratum subsidence and shear failure. Sun et al. [6] studied the stability of hydrate-bearing near-wellbore reservoirs under the condition of drilling fluid invasion in the Shenhu area. The results showed that the change in pore pressure and the decrease of the mechanical strength of the stratum caused by hydrate dissociation were the key factors affecting the stability of the wellbore reservoir. Sánchez et al. [7] proposed a fully coupled thermo-hydro-mechanical framework to study different problems involving hydrate-bearing sediments from laboratory tests to field-scale simulations, and ice formation and thawing were considered in the model. Sasaki et al. [8] proposed a modelling methodology of the well construction process for unconsolidated hydrate-bearing reservoirs, and the effect of wellbore construction on the integrity of the unconsolidated hydrate-bearing reservoir in the Nankai Trough was investigated. Sun et al. [9] proposed a fully coupled thermal–hydraulic–mechanical–chemical model to investigate the response of hydrate-bearing sediments during gas production, and a new thermodynamics-based constitutive model was introduced into the coupled model to simulate the mechanical behavior of hydrate-bearing sediments. In addition, the differences between fully coupled and semi-coupled models were analyzed. Yoneda et al. [10] developed a coupled thermo-hydro-mechanical (THM) simulator to evaluate the mechanical stability of a hydrate-bearing reservoir and well completion at the eastern Nankai Trough. The results showed that the mechanical deformation occurred in a much wider area than the range of hydrate dissociation and large shear stress occurred near the production well. Zhou et al. [11] used a fully coupled THM numerical simulator to examine the stability of a hydrate-bearing reservoir during gas production, and the critical state constitutive model for hydrate-bearing sediments was used to evaluate the mechanical response of the formation.

However, hydrate-bearing sediments are a kind of composite material composed of soil particles, hydrate, gas, and water, and the hydrate mainly exists in the form of filling, cementing, or supporting between the pores of soil particles; this makes hydrate-bearing sediments have certain structural properties, and their mechanical properties are much more complex than those of ordinary sediments. Therefore, it is generally assumed in the existing multi-field coupling models that hydrate-bearing sediments are a kind of elastic or ideal elastoplastic medium without considering the damage evolution process of the hydrate-bearing sediments and its influence on the multi-field coupling process. Some continuous damage models based on damage mechanics have been proposed to simulate the failure process of hydrate-bearing sediments. For example, Wu et al. [12] established the constitutive model of hydrate-bearing sediment considering damage based on the construction method of the frozen soil constitutive model and composite material theory. Liu et al. [13] proposed a damage statistical constitutive model of hydrate-bearing sediments combined with the meso-mechanical mixed model for the equivalent elastic modulus. However, the existing studies on damage statistical constitutive models of hydrate-bearing sediments are established on the assumption that the sediment micro-elements will be damaged at the beginning of loading and the bearing capacity of sediment micro-elements would be lost completely after the damage, which is not consistent with the fact.

Based on the above, a damage statistical constitutive model of hydrate-bearing sediments considering the effects of damage threshold and residual strength is established by introducing a three-parameter Weibull distribution and residual strength correction coefficient, and then the damage constitutive model is introduced into the multi-field coupled model. A thermo-hydro-mechanical-chemical multi-field coupling model considering the influence of damage of hydrate-bearing sediments on temperature, seepage, and mechanics during hydrate dissociation is established. Finally, the near-wellbore reservoir stability during gas hydrate exploitation is analyzed using the coupling model.

#### **2. Theoretical Formulations**

#### *2.1. Damage Statistical Constitutive Model for Hydrate-Bearing Sediments*

As shown in Figure 1, the skeleton of hydrate-bearing sediments is a grain-reinforced composite material composed of soil particles and hydrate. The cementation, pores, and internal defects randomly distributed within the sediments are the main factors of damage, and the micro-element strength is also randomly distributed. According to Yun et al. [14], the appearance of structure yield stress in the sediment indicated that debonding or partial hydrate breakage began to occur between soil particles and hydrates. Therefore, there should be a damage threshold point during the deformation process of hydrate-bearing sediments.

**Figure 1.** Diagram of the structure of hydrate-bearing sediments.

With increased loading, the hydrate-bearing sediments begin to suffer damage when the sediment micro-element strength exceeds the damage threshold. In this paper, the sediment yield stress point is regarded as the sediment damage threshold point, and the micro-element strength of the sediments is assumed to follow the three-parameter Weibull distribution.

The density function of the three-parameter Weibull distribution is [15]

$$f(F) = \begin{cases} 0, & F < \mathcal{Y} \\ \frac{\mu \mathbb{I}}{F\_0} \left(\frac{F-\mathcal{Y}}{F\_0}\right)^{m-1} \exp\left[-\left(\frac{F-\mathcal{Y}}{F\_0}\right)^m\right], & F \ge \mathcal{Y} \end{cases} \tag{1}$$

where *m*, *F*0, and γ are Weibull distribution parameters, and *F* is the hydrate-bearing sediment micro-intensity random distribution variable.

The damage variable *D* is defined as the ratio of the number of broken micro-elements to the total number of micro-elements:

$$D = \frac{N\_f}{N} = \frac{\int\_{-\infty}^{F} N f(F) dF}{N} \tag{2}$$

where *Nf* is the number of broken micro-elements, and *N* is the total number of micro-elements.

By substituting Equation (1) into Equation (2), the evolution law of damage variable *D* when *F* > γ can be obtained as follows:

$$D = 1 - \exp\left[-\left(\frac{F-\gamma}{F\_0}\right)^m\right].\tag{3}$$

The micro-element strength *F* can be described by using the Drucker–Prager strength criterion:

$$F = a\_0 I\_1 + \sqrt{f\_2} \tag{4}$$

where α<sup>0</sup> is the material parameter, which is related to the internal friction angle ϕ; *I*<sup>1</sup> is the first invariant of the stress tensor; and *J*<sup>2</sup> is the second invariant of the deviator stress tensor.

According to the Lemaitre equivalent strain assumption [16],

$$
\varepsilon = \frac{\overline{\sigma}}{E} = \frac{\sigma}{\overline{E}} = \frac{\sigma}{E(1 - D)}\tag{5}
$$

where <sup>σ</sup> is the equivalent stress of the nominal stress <sup>σ</sup>, *<sup>E</sup>* is the elastic modulus of the non-damaged material, and *Ẽ* is the elastic modulus of the damaged material.

In the current research on the hydrate-bearing sediment damage constitutive model, it has been usually assumed that the bearing capacity of the material will be lost completely after its failure [12,13]. However, its bearing capacity will not be completely lost due to the influence of the confining pressure and friction after micro-elements in the sediments are completely damaged, and it still has a certain residual strength [17]. In order to describe the damage process of sediments, the residual variable correction coefficient δ [18] is introduced, and Equation (5) is modified as follows:

$$\varepsilon = \frac{\sigma}{E(1 - \delta D)}.\tag{6}$$

Combined with Equations (3) and (6), the damage statistical constitutive model of hydrate-bearing sediments considering the damage threshold and residual strength after sediment damage is obtained as follows:

$$\Delta \sigma = E \varepsilon\_1 \left\{ 1 - \delta \left\{ 1 - \exp \left[ - \left( \frac{F - \gamma}{F\_0} \right)^m \right] \right\} \right\} + (2v - 1) \sigma\_3 \qquad (F \ge \gamma) \,. \tag{7}$$

The starting point of sediment damage evolution is determined by parameter γ, namely, the damage threshold point (Δσ*d*, ε1*d*). It is assumed that the sediment damage variable *D* is equal to 0 at the damage threshold point, which can be obtained by *F* − γ = 0 as follows:

$$\gamma = \lim\_{\varepsilon\_1 \to \varepsilon\_M} (\alpha\_0 I\_1 + \sqrt{l\_2}). \tag{8}$$

#### *2.2. Multi-Field Coupling Model Considering Damage of Hydrate-Bearing Sediments*

#### 2.2.1. Mechanical Field Control Equations

In the initial stage of loading, the deformation of hydrate-bearing sediments is in the elastic stage, and the hydrate-bearing sediments' stress–strain relationship considering the influence of temperature can be expressed as [19]

$$
\sigma\_{i\dot{j}}\prime = \lambda \delta\_{i\dot{j}} \varepsilon\_{kk} + 2G\varepsilon\_{i\dot{j}} - K'a\_T \delta\_{i\dot{j}} (T - T\_0) \tag{9}
$$

where σ*ij* is the effective stress tensor, ε*ij* is the strain tensor, λ and *G* are Lame constants, *K* is the drainage bulk modulus for porous media, α*<sup>T</sup>* is the volumetric thermal expansion coefficient, and *T* is temperature.

Combining Equations (7) and (9) with the generalized Biot's effective stress principle [20], the small strain assumption theory, and the mechanical equilibrium equations of porous media, the governing equations of the mechanical field considering sediment damage can be expressed by the displacement *u*, the average pore pressure *P*, and the temperature *T* as follows:

$$\frac{\overline{E}}{2(1+v)}u\_{i,j\rangle} + \frac{\overline{E}}{2(1+v)(1-2v)}u\_{j,\overline{j}} + aP\_{\cdot i} + K'a\_{T}T\_{\cdot j} + F\_{\overline{i}} = 0 \tag{10}$$

where *v* is Poisson's ratio, α is Biot's consolidation coefficient, *Fi* and *ui* are the components of the volume force and displacement in the *i* direction, and *Ẽ* and *E* are the elastic modulus after the damage and before the damage, respectively.

According to the elastic damage theory, the elastic modulus of sediments after damage is as follows:

$$
\overline{E} = E(1 - D). \tag{11}
$$

In addition, the average pore pressure can be given by pore water pressure and pore gas pressure:

$$P = \frac{S\_{\text{uv}}}{S\_{\text{uv}} + S\_{\text{g}}} P\_{\text{uv}} + \frac{S\_{\text{g}}}{S\_{\text{uv}} + S\_{\text{g}}} P\_{\text{g}} \tag{12}$$

$$P\_{\mathcal{L}} = P\_{\mathcal{S}} - P\_{\mathcal{W}} = p\_c^{\xi} \left( \frac{\frac{S\_{\mathcal{W}}}{S\_{\mathcal{W}} + S\_{\mathcal{S}}} - S\_{\mathcal{W}\mathcal{T}}}{1 - S\_{\mathcal{W}\mathcal{T}}} \right)^{-0.65} \tag{13}$$

where *Pc* is the capillary pressure [21] and *Pce* is the nominal capillary pressure.

#### 2.2.2. Hydraulic Field Control Equations

Based on the continuity equation of fluid in porous media and Darcy's law, multiphase fluid flow equations considering the influence of sediment skeleton deformation and temperature gradient are established as follows:

$$\frac{\partial \rho\_{\varepsilon} S\_{l} \rho\_{l}}{\partial t} + \varphi\_{\varepsilon} S\_{l} \rho\_{l} \frac{\partial \varepsilon\_{\upsilon}}{\partial t} + \nabla \left( - \frac{K\_{l} K \rho\_{l}}{\mu\_{l}} \nabla P\_{l} - \rho\_{l} k\_{Tl} \nabla T \right) = \dot{m}\_{l} \tag{14}$$

where subscript *l* represents the gas phase *g* and water phase *w*, *Sl* is the gas and water saturation, *m*˙ *<sup>l</sup>* is the gas and water production rate, and the gas production rate *m*˙ *<sup>g</sup>* during hydrate dissociation can be obtained from the Kim–Bishnoi hydrate dissociation kinetic model [22]. *Pl* is the pore gas pressure and pore water pressure, ε*<sup>v</sup>* is the volume strain, μ*<sup>l</sup>* is the dynamic viscosity coefficient of gas and water, ϕ*<sup>e</sup>* is the effective porosity, ϕ*<sup>e</sup>* = ϕ0(1 − *Sh*), ϕ<sup>0</sup> is the porosity of porous media without hydrate, *kTl* is the diffusion rate of fluid under a temperature gradient, *K* is the absolute permeability of the porous media, *K* = *K*0(1 − *Sh*) *<sup>n</sup>* [23], *K*<sup>0</sup> is the absolute permeability of porous media without hydrate, *n* is the permeability decline index, and *Krl* is the relative permeability of the gas and water phase, described by the modified Corey model [24].

Relevant research has suggested that damage would change the permeability of hydrate-bearing sediments, which will have an impact on the seepage process. It is assumed that the permeability and the damage variable have an exponential relationship [25]. In this paper, it is also assumed that the effect of sediment damage on the absolute permeability *K* meets the following criterion:

$$
\overline{K} = K \exp(\alpha\_k D) \tag{15}
$$

where *K* is the absolute permeability of the porous media affected by the damage, *K* is the absolute permeability of the porous media, and α*<sup>k</sup>* is the influence coefficient of the damage on the permeability.

In addition, the mass conservation equation of the hydrate is expressed as follows:

$$\frac{\partial \rho\_c S\_{\hbar} \rho\_{\hbar}}{\partial t} = \dot{m}\_{\hbar}.\tag{16}$$

Moreover, the gas phase saturation, water phase saturation, and hydrate saturation in the porous media can satisfy the following equation:

$$S\_{\mathcal{S}} + S\_{w} + S\_{h} = 1.\tag{17}$$

#### 2.2.3. Energy Conservation Equation

As is known to all, the dissociation of gas hydrate depends on certain temperature and pressure conditions, and the dissociation of gas hydrate is an endothermic process. Temperature has a significant effect on hydrate dissociation. The energy conservation equation considering heat conduction, heat convection, hydrate dissociation endotherm, and soil skeleton deformation is established as follows:

$$\left(\sum q\_c \mathbf{S}\_{a} \rho\_{a} \mathbf{c}\_{a}\right) \frac{\partial T}{\partial t} = \nabla \cdot \left[\left(\sum q\_c \mathbf{S}\_{a} \lambda\_{a}\right) \nabla T\right] - \nabla \left[\left(\sum p\_{a} \mathbf{v}\_{ra} \mathbf{c}\_{a}\right) T\right] - K' \mathbf{a}\_{T} (T - T\_{0}) \frac{\partial \varepsilon\_{\upsilon}}{\partial t} + q\_{\parallel} + q\_{\parallel \text{in}} \tag{18}$$

where the subscript α denotes the gas phase *g*, the water phase *w*, or the hydrate phase *h*, and the soil skeleton phase *s*. *C*<sup>α</sup> is the specific heat, λα is the heat transfer coefficient, *qh* is the latent heat of hydrate phase change, and *qin* is the external heat supply.

After damage of hydrate-bearing sediments, micro-cracks and pores will appear in the hydrate-bearing sediment reservoir; the fluid with a lower temperature may then infiltrate into it, which will affect the heat conductivity coefficient of materials [25]. Therefore, it is assumed that the effect of structural damage of sediments on the heat conductivity coefficient of materials satisfies the following assumption:

$$
\lambda\_i = \lambda\_i \exp(a\_{\lambda i} D) \tag{19}
$$

where subscript *i* denotes hydrate phase *h* or soil skeleton phase *s*; λ*<sup>i</sup>* and λ*<sup>i</sup>* are the heat conductivity coefficients of the damaged and undamaged sediment skeleton, respectively; and αλ*<sup>i</sup>* is the damage influence parameter on the heat conductivity coefficient.

In addition, the latent heat for the hydrate phase change satisfies [26]

$$q\_{l\!\!\!} = -\frac{\dot{m}\_{\!\!\!\!} (56599 - 16.744T)}{M\_{\!\!\!\!\!\!}}.\tag{20}$$

#### **3. Model Verification**

#### *3.1. Verification of the Damage Statistical Constitutive Model of Hydrate-Bearing Sediments*

In order to verify the rationality of the damage statistical constitutive model established in this paper, a comparative analysis was carried out by citing the experimental data from Masui et al. [27]. There are the parameters *E*, *v*, ϕ, *m*, *F*0, γ, and δ in the constitutive model. By handling the experimental data from Masui et al., *E* can be obtained as shown in Table 1, *v* = 0.219, and ϕ = 30◦. The Weibull distribution parameters *m* and *F*<sup>0</sup> can be calculated according to the characteristic points of the stress–strain curve [12] and γ is determined according to Equation (8); these values are also shown in Table 1. The residual strength correction coefficient δ can be determined from the experimental results using the inversion trial method. The confining pressure values σ<sup>3</sup> in the experiments were 1.0 MPa, 2.0 MPa, and 3.0 MPa.


**Table 1.** Material parameters of hydrate-bearing sediments and Weibull distribution parameters.

Figures 2 and 3 show comparisons between the stress–strain theoretical curves of hydrate-bearing sediments and the experimental results by Masui et al. under different hydrate saturation and confining pressure values. As shown in the figures, with increased hydrate saturation and confining pressure, the strength and stiffness of the hydrate-bearing sediments also increased. It can be seen that the damage statistical constitutive model of hydrate-bearing sediments considering the damage threshold and residual strength proposed in this paper can better reflect the characteristics of the mechanical

properties of hydrate-bearing sediments changing with hydrate saturation and confining pressure, and the results of the theoretical model are in good agreement with the experimental results.

**Figure 2.** Comparisons of stress–strain theoretical curves and test curves of hydrate-bearing sediments under different hydrate saturations. (**a**) *Sh* = 0%; (**b**) *Sh* = 25.7%; (**c**) *Sh* = 40.7%; (**d**) *Sh* = 55.1%.

**Figure 3.** Comparison of stress–strain theoretical curves and test curves of hydrate-bearing sediments under different confining pressures.

#### *3.2. Verification of the Multi-Field Coupling Model*

In order to verify the validity of the multi-field coupling model established in this paper, the experimental results of hydrate dissociation by depressurization with a Berea sandstone core sample by Masuda et al. [28] were compared. The hydrate dissociation by the depressurization model from Masuda et al. is shown in Figure 4. The model parameters selected for verification in this paper are completely consistent with the work by Masuda et al. From Figure 4, the core length is 0.3 m and the diameter is 0.051 m. Three reference points A, B, and C were selected for comparison. Point A is 0.00375 m away from the outlet, Point B is 0.15 m away from the outlet, and Point C is 0.225 m away from the outlet.

**Figure 4.** Diagram of the model used in the hydrate dissociation experiment by Masuda et al.

Figure 5 shows a comparison of the experimental and numerical results of cumulative gas production. It can be seen from Figure 5 that the result of cumulative gas production of the outlet simulated by the coupling model established in this paper is in good agreement with the Masuda experimental results, and the final gas production obtained by the numerical method is also close to the experimental value.

**Figure 5.** Comparison of experimental and numerical results of cumulative gas production.

Figure 6 shows a comparison of the experimental and numerical results of the temperature at the three different positions A, B, and C of the sandstone core. It is clearly seen from Figure 6 that the temperature curves of the three measuring points A, B, and C simulated by the coupling model established in this paper are basically consistent with the overall trend of the Masuda experimental curves, which indicates that the multi-field coupling model established in this paper has good applicability and can be used for further analysis and calculation.

**Figure 6.** Comparisons of experimental and numerical results of the temperature at the three different positions A, B, and C of the sandstone core.

#### **4. Numerical Solution of the Multi-Field Coupling Model Considering Hydrate-Bearing Sediment Damage**

#### *4.1. Multi-Field Coupling Model Numerical Calculation Conditions*

In this paper, COMSOL Multiphysics was used to numerically solve the thermo-hydro-mechanical-chemical (THMC) multi-field coupling mathematical model considering damage of hydrate-bearing sediments, and the stability of the near-wellbore reservoir was analyzed. A simplified 1/4 axisymmetric model of a hydrate-bearing near-wellbore reservoir is shown in Figure 7. The geometric size of the finite element model was 20 m × 20 m, and the borehole radius was 0.2 m. Mechanical field boundary conditions were set as follows: The maximum horizontal in situ stress σ*<sup>H</sup>* and the minimum horizontal in situ stress σ*<sup>h</sup>* were applied to the sides of BC and CD, respectively. The vertical displacement and horizontal displacement were respectively restricted on the sides of AB and DE. Hydraulic field boundary conditions were set as follows: the initial pore pressure boundary on the sides of BC and CD, impermeable boundaries at AB and DE, a depressurization boundary at AE, and a constant pressure *P* = 2.84 MPa at the bottom hole were set in the simulation. Thermal field boundary conditions were set as follows: the boundaries of BC and CD were constant-temperature boundaries and the boundaries of AB and DE were adiabatic boundaries. The main model parameters were selected from Masui et al. [27] and Liu et al. [29]. The physical and mechanical parameters of the hydrate-bearing sediments are shown in Table 2, and the thermodynamic parameters are shown in Table 3.

**Figure 7.** Diagram of the hydrate-bearing near-wellbore reservoir.




**Table 3.** Thermodynamic parameters of hydrate-bearing sediments.

#### *4.2. Numerical Results and Analysis*

#### 4.2.1. The Evolution of Hydrate Saturation in the Reservoir

In order to analyze the stability of a near-wellbore reservoir in hydrate-bearing sediments, the distributions of results within 10 m from the wellbore are given. Figure 8 shows the spatial evolutions of hydrate saturation in the near-wellbore reservoir at different times. The hydrate dissociation in the initial stage of gas hydrate exploitation is mainly concentrated around the exploitation well. With the development of exploitation, the hydrate dissociation proceeds along a circular dissociation plane, and the dissociation range expands continuously. The farther the stratum away from the wellbore, the lowerthe degree of hydrate dissociation. After 72 h of hydrate exploitation, the hydrate dissociation front reaches approximately 6.492 m from the center of the wellbore. Figure 9 shows the curves of hydrate saturation with time at different positions. As the hydrate exploitation progresses, the hydrate saturation of the hydrate-bearing reservoir decreases continuously. For example, the point H starts to dissociate at about *t* = 7.5 h, and the point I begins to dissociate at about *t* = 12 h, which indicates that the hydrate will dissociate earlier when it is closer to the wellbore. Then, it can be seen that there is a clear dissociation front during the process of hydrate dissociation.

**Figure 8.** Spatial evolution of hydrate saturation of strata at different times. (**a**) Hydrate saturation in the reservoir at *t* = 0 h; (**b**) Hydrate saturation in the reservoir at *t* = 24 h; (**c**) Hydrate saturation in the reservoir at *t*= 48 h; (**d**) Hydrate saturation in the reservoir at *t* = 72 h.

**Figure 9.** Curves of hydrate saturation with time at different positions.

#### 4.2.2. The Evolution of Pore Pressure in the Reservoir

Figure 10 shows the spatial evolutions of the average pore pressure in the near-wellbore reservoir at different times. As the gas hydrate exploitation goes on, the area of depressurization is expanding, and there is a large pressure gradient at the pressure front. After 72 h of hydrate exploitation, the reservoir pressure front reaches approximately 6.577 m from the center of the wellbore. Figure 11 shows the curves of the average pore pressure with time at different positions. From Figure 11, the average pore pressure in the reservoir is continuously decreasing at the initial stage of hydrate dissociation but increasing gradually with the process of hydrate dissociation. Due to the continuous hydrate dissociation and structural damage of sediments, the bearing capacity of the stratum will decrease, and the stress in the stratum will transfer to water and gas gradually, resulting in increased average pore pressure in the reservoir.

**Figure 10.** Spatial evolution of the average pore pressure of strata at different times. (**a**) Average pore pressure in the reservoir at *t* = 0 h; (**b**) Average pore pressure in the reservoir at *t* = 24 h; (**c**) Average pore pressure in the reservoir at *t* = 48 h; (**d**) Average pore pressure in the reservoir at *t* = 72 h.

**Figure 11.** Curves of average pore pressure with time at different positions.

#### 4.2.3. The Evolution of the Damage Area in the Reservoir

Figure 12 shows the spatial evolutions of the damage area in the near-wellbore reservoir at different times. It can be seen from Figure 12 that with continuous hydrate dissociation, the damage area of the hydrate-bearing reservoir shows a trend of continuous expansion. When *t* = 72 h, the damage radius has expanded to 5.604 m, and the maximum damage variable has also increased to 0.73. With the effect of non-uniform horizontal in situ stress, the maximum value of the damage variable of the hydrate-bearing reservoir appears in the direction of minimum horizontal in situ stress of the wellbore, which is the priority position of wellbore instability. Figure 13 shows the curves of the average damage area with time at different positions. At the beginning of hydrate dissociation, the damage variable of hydrate-bearing sediments remains zero. When a certain time is exceeded, the damage variable increases gradually, and the closer the position to the wellbore, the more serious the damage of the reservoir. This is due to the fact that sediment damage will only occur when the micro-element strength of sediments reaches the damage threshold. In addition, due to the hydrate dissociation and drilling, the original equilibrium state of the reservoir is destroyed; stress concentration occurs in the hydrate-bearing near-wellbore reservoir, which will cause more serious damage of the hydrate-bearing near-wellbore reservoir.

**Figure 12.** Spatial evolution of the damage area of strata at different times. (**a**) Damage area at *t* = 0 h; (**b**) Damage area at *t* = 24 h; (**c**) Damage area at *t* = 48 h; (**d**) Damage area at *t* = 72 h.

**Figure 13.** Curves of the damage variable with time at different positions.

#### 4.2.4. Stability Analysis of the Near-Wellbore Reservoir

Figure 14 shows the distribution of shear stress in the hydrate-bearing near-wellbore reservoir. As shown in Figure 14, hydrate dissociation and drilling destroy the original equilibrium state of the formation. Under the effect of non-uniform in situ stress, stress concentration occurs in the hydrate-bearing near-wellbore reservoir, which easily leads to instability of the reservoir. The curves of the effective stress at Point F with and without considering damage of hydrate-bearing sediments are shown in Figure 15. According to the principle of effective stress, the average pore pressure of the hydrate-bearing near-wellbore reservoir decreases in the early stage of gas hydrate exploitation, and then the effective stress rises rapidly. However, the effective stress of the hydrate-bearing reservoir decreases with the continuous dissociation of gas hydrates. This is because of the hydrate in the sediments gradually dissociating and the structural damage of sediments; the skeleton stress gradually transfers to water and gas, which causes a reduction in the effective stress and a decline in the bearing capacity of the stratum. Therefore, the phenomenon of partial softening and stress release occurs in the near-wellbore reservoir. Furthermore, it can be seen from Figure 15 that the phenomenon of stress softening under the condition of considering sediment damage is more obvious than that without considering sediment damage.

Figure 16 shows the effective principal stress path with and without considering damage of hydrate-bearing sediments, and the Mohr–Coulomb strength criterion was used to judge the shear failure of the wellbore reservoir. It is observed from Figure 16 that the effective principal stress in the near-wellbore reservoir does not eventually reach the Mohr–Coulomb failure surface. However, the effective principal stress path is closer to the Mohr–Coulomb failure surface when considering the damage of hydrate-bearing sediments, which indicates that the damage of hydrate-bearing sediments has an adverse effect on the stability of the near-wellbore reservoir.

**Figure 14.** The distribution of shear stress of a hydrate-bearing near-wellbore reservoir.

**Figure 15.** Curves of the effective stress of Point F with and without considering damage of hydrate-bearing sediments.

**Figure 16.** Stress paths with and without considering damage of hydrate-bearing sediments.

Based on the Mohr–Coulomb failure criterion, the reservoir stability coefficient *Fb* was defined [30]. When *Fb* ≤ 1, the reservoir is unstable, and the smaller *Fb* is, the worse the reservoir stability is. When *Fb* > 1, the reservoir is in a stable state.

$$F\_b = \frac{2c\cos\varphi - (\sigma\_1\prime + \sigma\_3\prime)\sin\varphi - (\sigma\_1\prime - \sigma\_3\prime)\sin^2\varphi}{(\sigma\_1\prime - \sigma\_3\prime)\cos^2\varphi} \tag{21}$$

Figure 17 shows the distributions of polar coordinates of the reservoir stability coefficient with and without considering damage of hydrate-bearing sediments at different positions. It is clearly seen from Figure 17 that the hydrate-bearing near-wellbore reservoir is relatively unstable in the directions of θ = 90◦ and θ = 270◦ around the wellbore, and the reservoir in the directions of θ = 0◦ and θ = 180◦ is relatively stable. This is because under the effect of non-uniform in situ stress, the near-wellbore reservoir stress in the direction of the minimum horizontal in situ stress is the most concentrated. Coupled with the decrease in the formation strength caused by sediment structural damage and hydrate dissociation, the reservoir instability area in this direction, which is the priority position of mechanical instability, is further expanding.

Figure 18 shows the curves of the vertical deformation of Point F with and without considering damage of hydrate-bearing sediments. When *t* = 72 h, the vertical deformation of Point F is *uy* = −0.0011 m (the negative value indicates compression deformation) without considering the influence of sediment damage but *uy* = −0.0016 m with considering the influence of sediment damage. The vertical deformation is increased by approximately 45.5% compared to the case where the influence of sediment damage is not considered. This indicates that when the influence of sediment damage is taken into account, greater deformation of the near-wellbore reservoir is predicted.

**Figure 17.** The distributions of polar coordinates of the reservoir stability coefficient with and without considering damage of hydrate-bearing sediments at different positions.

**Figure 18.** Curves of the vertical deformation of Point F with and without considering damage of hydrate-bearing sediments.

On the one hand, with hydrate dissociation, the cementation strength of hydrate-bearing sediments gradually decreases, and its capacity for resisting deformation also decreases. On the other hand, with the development of soil deformation, the damage degree of hydrate sediments will be aggravated. Therefore, if damage of hydrate-bearing sediments during the dissociation of gas hydrates in the multi-field coupling process is ignored, the bearing capacity of the hydrate-bearing reservoir will be overestimated, which will bring certain safety risks to hydrate exploitation.

#### **5. Conclusions**

In most previous investigations, the damage evolution process of sediment structure and its effect on multi-field coupling models and near-wellbore reservoir stability have been neglected. Moreover, the existing studies on damage statistical constitutive models of hydrate-bearing sediments were established on the assumption that the sediment micro-elements are damaged at the beginning of loading and that the bearing capacity of sediment micro-elements would be lost completely after damage, which is not consistent with the fact. Based on continuous damage theory, a damage statistical constitutive model of hydrate-bearing sediments considering the influence of the damage threshold and residual strength was established, and the damage variable was introduced into a THMC multi-field coupling mathematical model. The effects of hydrate-bearing sediment damage on the thermal field, hydraulic field, and mechanical field were considered. Then, the distributions of hydrate saturation, pore pressure, damage variable, and effective stress of the near-wellbore reservoir were discussed in detail, and the stability of the hydrate-bearing near-wellbore reservoir was analyzed. The following conclusions are drawn:


**Author Contributions:** This research article was finished by X.Z., F.X., C.X., and Y.H. The statements of author contributions are as follows: conceptualization, X.Z. and F.X.; methodology, C.X.; writing—original draft preparation, F.X.; writing—review and editing, X.Z.; funding acquisition, Y.H.

**Funding:** The authors would like to appreciate the funding support from the National Natural Science Foundation of China (grant number 51778020) and the Innovative Research Group Project of National Natural Science Foundation of China (grant number 51421005).

**Conflicts of Interest:** All authors have read and approve this version of the article, and due care has been taken to ensure the integrity of the work. No part of this paper has been published or submitted elsewhere. No conflict of interest exists in the submission of this manuscript.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Experimental Study and Estimation of Groundwater Fluctuation and Ground Settlement due to Dewatering in a Coastal Shallow Confined Aquifer**

#### **Jiong Li 1, Ming-Guang Li 1, Lu-Lu Zhang 1, Hui Chen 2, Xiao-He Xia <sup>1</sup> and Jin-Jian Chen 1,\***


Received: 18 February 2019; Accepted: 25 February 2019; Published: 1 March 2019

**Abstract:** The coastal micro-confined aquifer (MCA) in Shanghai is characterized by shallow burial depth, high artesian head, and discontinuous distribution. It has a significant influence on underground space development, especially where the MCA is directly connected with deep confined aquifers. In this paper, a series of pumping well tests were conducted in the MCA located in such area to investigate the dewatering-induced groundwater fluctuations and stratum deformation. In addition, a numerical method is proposed for the estimation of hydraulic parameter, and an empirical prediction method is developed for dewatering-induced ground settlement. Test results show that groundwater drawdowns and soil settlement can be observed not only in MCA but also in the aquifers underneath it. This indicates that there is a close hydraulic connection among each aquifer. Moreover, the distributions and development of soil settlement at various depths are parallel to those of groundwater drawdowns in most areas of the test site except the vicinity of pumping wells, where collapse-induced subsidence due to high-speed flow may occur. Furthermore, the largest deformation usually occurs at the top of the pumping aquifer instead of the ground surface, because the top layer is expanded due to the stress arch formed in it. Finally, the proposed methods are validated to be feasible according to the pumping well test results and can be employed to investigate the responses of groundwater fluctuations and stratum deformations due to dewatering in MCA.

**Keywords:** pumping well test; groundwater fluctuation; stratum deformation; micro-confined aquifer

#### **1. Introduction**

Shanghai is located at the riverfront and coastal plain of the Yangtze River deltaic deposit, where soft Quaternary deposits with a thickness of about 300 m are widely distributed [1,2]. In this region, an alternated multi-aquifer-aquitard system (MAAS) is formed due to complicated palaeoclimatic and palaeogeographic conditions as well as frequent transgressions and regressions in history [3–5]. The groundwater system in Shanghai is a part of the deltaic groundwater system of the Yangtze River [4] and mainly includes a phreatic aquifer group and five confined aquifers (labeled as Aq I to Aq V). Specifically, the phreatic aquifer group is composed of a phreatic aquifer (labeled as Aq0) and a micro-confined aquifer (labeled as MCA). All the aquifers are separated by seven aquitards (labeled as Ad0 to Ad VI). According to the distribution of MCA and its hydraulic connection with adjacent aquifers, the MAAS in Shanghai central city can be divided into five types (labeled as Type I to Type V), as depicted in Figure 1d. A geological survey shows that the MAAS keeps abundant and high artesian groundwater in aquifer layers [6], easily causing adverse effects on the construction of the underground facilities as well as deep excavations in these soft deposits [7–10], especially the

groundwater stored in MCR, Aq I and Aq II. In past few decades, the influence of Aq I and Aq II has been deeply concerning and widely studied by both researchers and engineers [7,9–13], whereas literatures concentrated on the influence of MCA are rare.

**Figure 1.** Typical distribution and hydro-geological profile of multi-aquifer-aquitard system (MAAS) in Shanghai central city: (**a**) location of Shanghai; (**b**) plan view for the location of Shanghai Administration Region; (**c**) type distribution of MAAS in Shanghai central city; (**d1**) schematic diagram for MAAS of Type I and Type II; (**d2**) schematic diagram for MAAS of Type III; (**d3**) schematic diagram for MAAS of Type IV; (**d4**) schematic diagram for MAAS of Type V.

The micro-confined aquifer in Shanghai is located at the top of MAAS and is the lower sublayer, underneath the phreatic aquifer, of Holocene phreatic aquifer group. The aquifer is discontinuously distributed in the horizon direction and is usually buried at a depth of only 15 to 22 m [4]. For this reason, the piezometric head of the groundwater in MCA is easily affected by meteorological and hydrological conditions [3] and periodically varies from 3 to 11 m below the ground surface [14,15]. The thickness of the aquifer generally varies from 5 to 20 m in most regions, whereas in the region of Type V, MCA is directly connected with Aq I due to the hiatus of Ad I and its thickness can reach 40 to 50 m [4,16,17]. Additionally, the MCA in Shanghai is mainly composed of silty sand and silty clay and its hydraulic conductivity is relatively lower than those of the deep confined aquifers. The hydraulic conductivity of MCA is variable between (3~6) × <sup>10</sup>−5~10−<sup>3</sup> cm/sec while those of Aq I and Aq II usually vary between (3~6) × <sup>10</sup>−4~10−<sup>3</sup> cm/sec and (2~6) × <sup>10</sup>−3~10−<sup>2</sup> cm/sec, respectively [14]. In addition, the maximum specific discharge capacity of MCA is about 43.2 m3/day-meter, which is also smaller than those of deep confined aquifers with the maximum being greater than 720 m3/day-meter, e.g., Aq II and Aq IV [4].

As aforementioned, the artesian head in the MCA is high, and the burial depth of MCA is small, causing deep excavations to be more easily affected by the MCA, especially the MCA in Type V. Moreover, with rapid development of ocean economy and coastal industry in recent decades, an increasing number of municipal and commercial infrastructures, e.g., metro tunnels and stations, were or are being constructed in the coastal soft deposits of Shanghai [4,18–21], resulting in the increase of excavation scale and depth [5,22–24]. In some projects, the depth of excavation reaches the top of MCA [15]. As excavation depth is increased, the remaining bottom soil is insufficient to counteract the artesian head underneath the excavation, leading to seepage and inrushing damage to the excavation [9,25,26]. This is particularly true for the excavations above the coastal MCA, for which

the critical excavation depth is usually less than 10 m according to the Shanghai design specification for the requirement of surge resistance [14]. Consequently, dewatering measurements should be adopted in excavations to reduce or even eliminate the adverse effects due to artesian groundwater [8,10,26–28].

However, regardless of many advantages of dewatering for protecting excavations, it has been widely accepted that groundwater extraction can induce stratum deformation in the vicinity of the excavations [5,7,8,18,19,29–33] as well as secondary hazards to surrounding structures [34–37]. Many scholars and engineers have been devoted to investigating the dewatering-induced environment influence in Shanghai. As a matter of fact, these studies were primarily concentrated on the influence of groundwater extraction in deep confined aquifers [10–12], such as Aq I and Aq II, whereas few literatures were dealt with that in the shallow-buried MCA. As above mentioned, the hydrological and geological conditions (e.g., burial depth, compressibility, hydraulic conductivity, and specific discharge capacity) of the MCA are distinguished from those of deep confined aquifers. Additionally, the MCA is more easily affected by meteorological and hydrological conditions than deep confined aquifers due to its shallow burial depth [3], resulting in complex hydrological conditions in the MCA. Moreover, hydrological conditions can be more complex if there is aquitard hiatus, e.g., in the MAAS of Type V, where the MCA is directly connected with Aq I due to the hiatus of Ad I. Considering these factors, the environmental risk from excavation dewatering in a coastal MCA is still uncertain.

The objective of this study is to investigate the response of groundwater fluctuations and ground settlement induced by dewatering in the coastal MCA of Shanghai. To achieve this aim, a series of field pumping well tests were conducted at a construction site located at Pudong New Area District, where the MCA is connected directly with Aq I. To help analyze the responses, the following are addressed:


#### **2. Study Area**

To investigate the responses of groundwater fluctuations and stratum deformation as well as the influence on excavation, three groups of single-well pumping test and a group of multi-well pumping test were conducted.

#### *2.1. Engineering Geology*

The test site is located in the northwest of Pudong New Area District (as shown in Figure 1c), and the elevation at the test site varies from 4.57 m to 5.66 m. The soil distributed in the influence depth of the foundation is characterized as a depositional soil layer of the coastal plain from Quaternary Holocene to Pleistocene, mainly including clay, silty soil, and silty sand. The columns in the left part Figure 2 plot the soil profile of the construction site. The first layer is an artificial layer in the upper about 3.76 m below the land surface, underlain by silty clay, mucky silty clay, mucky clay, and clay to the depth of about 19.85 m. The following layer is sandy silt to the depth of about 25.65 m overlying the layer of silty clayey silt to the depth of about 30.17 m. The next layer is a silt layer extending to a depth of about 41.42 m, followed by sandy silt mixed up with silty clay to a depth of about 64.90 m. Underneath the above layers is silty sand to a depth of about 74.98 m. Vertically below all these layers is the interbedded strata of silty clay and silty sand until the termination depth of about 82.14 m.

Figure 3 presents the geotechnical parameters for each layer at the test site. The grain size distribution indicates that the content of silty sand and sand in deep confined aquifers is higher than that in the MCR. The initial void ratio, *e*0, was determined based on the physical properties of the soils at different depths tested from the laboratory tests. The vertical and horizontal hydraulic conductivity, *kv* and *kh*, of soils were obtained based on laboratory tests and injection tests. The compression index, *a*0.1−0.2, was determined by laboratory oedometer tests. The water content of each stratum was usually close to its liquid limit, whereas the plastic limit varied little along with the depth and was approximately 21%.

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**Figure 2.** Profile of geological and hydrogeological section and well structure.

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**Figure 3.** Soil profile and properties at the construction site.

#### *2.2. Hydrogeology*

There are mainly three types of groundwater stored in the influence depth of the proposed project, and they are phreatic water referred to as phreatic aquifer, feeble confined water known as MCA, and confined water dubbed as confined aquifer (Aq I), respectively. As can be seen in Figure 2, the phreatic aquifer and the MCA are separated by an aquitard (Ad0) while MCA is adjacent to the confined aquifer directly and there is a certain hydraulic connection between the two aquifers. The phreatic aquifer is mainly composed of the silty clay with a thickness of about 9 m, and the water head of it is variable from 0.75 m to 1.70 m below the ground surface. The MCA is primarily stored in the sandy silt (Layer <sup>5</sup> 2), the silty sand (Layer <sup>5</sup> 3-2), and the silty clay (Layer <sup>5</sup> 3-3) with an aggregate thickness of about 44.2 m. The artesian head in these layers varies from −5.1 m to −6.8 m compared to the ground surface. In addition, the confined aquifer is mainly composed of silty sand (Layer <sup>7</sup> ) with a thickness of about 9.8 m and the water level varies from −8.9 m to −9.7 m.

#### **3. Pumping Well Tests**

#### *3.1. Well Installation*

In the tests, twelve test wells, including seven pumping wells and five observation wells, were employed. The layout of the wells is plotted in Figure 4, and the distance between each well is also labeled in the figure. Each test well was composed of a steel tube, screen pipe, and sedimentary pipe. The steel pipe was installed at the upper part of a well and outside the steel pipe were sealed with clay, high-quality clay, and gravel pack from top-down to prevent the groundwater from the upper aquifers flowing into test wells. The screen was installed underneath the steel pipe and outside the screen was backfilled with gravel to ensure the groundwater flowed into the well smoothly. The sedimentary pipe with a length of 1 m was installed at the bottom of the well to prevent the screen pipe being clogging by the sediment in the groundwater. The structures of the test well are depicted in Figure 2, and the associated parameters are also presented in the figure.

**Figure 4.** Layout of test wells and ground settlement monitoring points: (**a**) Layout of test wells; (**b**) Layout of ground settlement monitoring points; (**c**) profile of ground settlement monitoring points.

In the tests, there were four pumping wells (labelled C5-2-1~C5-2-4, at a depth of 35 m, 38 m, 35 m, 38 m, respectively) and three observation wells (labelled G5-2-1~G5-2-3, at a depth of 35 m) installed in Layers <sup>5</sup> 2~<sup>5</sup> 3-2, a pumping well (labelled C5-3-1, at a depth of 56 m) and an observation well (labelled G5-3-1, at a depth of 56 m) installed in Layer <sup>5</sup> 3-3, a pumping well (labelled C7-1, at a depth of 73 m) and two observation wells (labelled G7-1 and G7-2, at a depth of 73 m) installed in Layer <sup>7</sup> . The detailed structural parameters for each well are listed in Table 1. The external radius for pumping well was identical to that for observation well and was 650 mm, whereas the internal radius for pumping well was 273 mm while that for observation well was 168 mm. The length of the screen for C5-2-1~C5-2-4 was 14 m, 17 m, 14 m, and 12 m, respectively, for G5-2-1~G5-2-3 it was 12 m, for C5-3-1 and G5-3-1 it was 9 m, for C7-1 and G7-1~G7-2 it was 6 m.


**Table 1.** Detailed characteristics of test wells.

#### *3.2. Test Scheme*

Three single-well pumping tests and a multi-well pumping test were performed successively from July 26, 2014, to August 26, 2014. Table 2 shows the detailed process of the tests. The single-well pumping tests were performed in Layers <sup>5</sup> 2~<sup>5</sup> 3-2 using well C5-2-1 at a rate of 9.96 m3/h from 9:00, July 26 to 18:37, July 27 lasting for 2017 min, in Layer <sup>5</sup> 3-3 using well C5-3-1 at a rate of 9.77 m3/h from 12:00, August 5 to 13:00, August 7 lasting for 2940 min and in Layer <sup>7</sup> using well C7-1 at a rate of 26.7 m3/h from 8:00 August 9 to 10:00 August 10 lasting for 980 min. The multi-well pumping test was conducted in Layers <sup>5</sup> 2~<sup>5</sup> 3-2 from 12:00, August 12 to 15:00, August 20 and consumed 10930 min using well C5-2-1~C5-2-4 at the discharge rate of 7.28 m3/h, 15.64 m3/h, 11.68 m3/h, and 10.87 m3/h, respectively. It should be noted that enough time should be left for groundwater recovery between each test and in this test they were 2920 min, 2828 min,1020 min, and 8120 min, respectively.

**Table 2.** Process of the pumping well tests.


Note: *sw* = drawdown in pumping well; *t*<sup>0</sup> = start time; *te* = end time; *tp* = pumping time; *tr* = recovery time; *Q* = discharge rate.

#### *3.3. Stratum Deformation*

To obtain the responses of the ground settlement and deep stratum deformation induced by multi-well dewatering, 30 ground settlement monitoring points (labelled D1 to D30) and four deep

soil settlement monitoring holes (labelled X1 to X4) were installed at the test site. The layout of all the monitoring points is shown in Figure 4. The ground settlement monitoring points were arranged in a radial shape with the center of pumping area (D1) as its endpoint, and the distance between two adjacent points was 10 m. The burial depth of each monitoring point was 1.3 m to protect the point from external disturbance. The deep soil settlement monitoring holes were laid close to the monitoring point D1, D2, D3, and D5. The profile of the four monitoring holes is plotted in Figure 2. Each of the monitoring holes was 68 m deep and had four deep soil settlement monitoring points installed from Layer <sup>5</sup> 2 to Layer <sup>7</sup> at a depth of 22 m, 35 m, 50 m, and 68 m, respectively.

#### **4. Results and Analysis**

#### *4.1. Responses of Groundwater Level*

As aforementioned, the MCA concerned in this study is directly connected with Aq I. For this reason, dewatering in the MCA can also induce the groundwater drawdowns in its adjacent aquifers. To investigate the hydraulic connection between MCA and Aq I as well as its influence on groundwater fluctuations, the responses of groundwater level in different aquifers during the field tests are analyzed in this section.

#### 4.1.1. Test Results

Figure 5 presents the discharge rates of the pumping wells and the groundwater level obtained in the observation wells. In the C5-2-1 pumping test, the groundwater head in Layers <sup>5</sup> 2~<sup>5</sup> 3-2 had a rapid decline at the beginning and then decreased slowly until it reached a steady value, whereas the drawdowns observed in Layer <sup>5</sup> 3-3 and Layer <sup>7</sup> were relatively small. Specifically, the maximum drawdowns obtained using G5-2-1~G5-2-3 in Layers <sup>5</sup> 2~<sup>5</sup> 3-2 were 1.75 m, 1.18 m and 0.74 m, respectively, while those monitored by G5-3-1 in Layer <sup>5</sup> 3-3 and G7-1 and G7-2 in Layer <sup>7</sup> were almost negligible, with the maximum drawdown of 0.14 m, −0.22 m, and −0.1 m, respectively, where minus meant head increment. A similar phenomenon was also observed from the C5-3-1 and the C7-1 pumping well test. The maximum drawdowns monitored by G5-2-1~G7-2 in the C5-3-1 pumping well test were 0.44 m, 0.23 m, 0.12 m, 2.04 m, 0.56 m, and 0 m, respectively, while those in the C7-1 pumping well test were 0.01 m, 0.02 m, −0.01 m, 0.03m, 1.23 m and 0.94m, respectively. In addition, once the dewatering was interrupted, the groundwater head recovered immediately.

When the multi-well pumping test was conducted in Layers <sup>5</sup> 2~<sup>5</sup> 3-2 using well C5-2-1~C5-2-4, besides obvious drawdowns observed in the pumping aquifer, maximum drawdowns of 8.18 m monitored in well G5-2-1, 7.25 m in well G5-2-2, and 6.26 m in well G5-2-3, an inconspicuous but not negligible drawdown of 1.59 m monitored in well G5-3-1 was also observed in the underneath Layer <sup>5</sup> 3-3 as well as a 0.05-m-deep drawdown monitored in G7-1 and a 0.13-m-deep drawdown monitored in G7-2 traced in Layer <sup>7</sup> . The groundwater drawdown rate in Layers <sup>5</sup> 2~<sup>5</sup> 3-2 was large at the beginning and then declined gradually until it reached a steady level. The groundwater head in Layer <sup>5</sup> 3-3 and Layer <sup>7</sup> also followed the same development law. After the multi-well pumping test was shut down, the groundwater head in each aquifer recovered immediately. In addition, the groundwater head in Layers <sup>5</sup> 2~<sup>5</sup> 3-2 increased unexpectedly after the test continued for about 65 h due to power failure.

**Figure 5.** The time-history curves for discharge rate and groundwater level: (**a**) curves for discharge rate of pumping well; (**b**) curves for groundwater level in Layer -<sup>5</sup> 2~-5 3-2; (**c**) curves for groundwater level in Layer -<sup>5</sup> 3-3; (**d**) curves for groundwater level in Layer -7 .

#### 4.1.2. Analyses

As aforementioned, in the single-well pumping test, remarkable groundwater drawdowns could be observed in the pumping aquifers, while small but not negligible drawdowns could also be detected in other aquifers especially in the adjacent aquifers, indicating that there were hydraulic connections and a leakage effect among each aquifer. The hydraulic connection and leakage effect were more easily observed in the multi-well pumping test because the accumulated discharge rate and operation time were much larger than those in the single-well pumping tests. In real field work, the dewatering is usually performed using multi-well for several months or even more than a year [7], consequently causing apparent drawdowns in the pumping aquifer as well as its adjacent aquifers due to the hydraulic connection and leakage effect. Hence, during the excavation dewatering, although the pumping wells were only installed in Layers <sup>5</sup> 2~<sup>5</sup> 3-2, the groundwater drawdowns and stratum deformation in Layer <sup>5</sup> 3-3 and Layer <sup>7</sup> should also be considered because of the hydraulic connection and leakage effect. Otherwise, an underestimate of water inflow and land subsidence may emerge, resulting in an adverse effect on the safety of the excavation and the surroundings.

#### *4.2. Responses of Ground Settlement*

Responses of ground settlement induced by dewatering in deep confined aquifers [8–12] has been widely investigated, whereas few literatures focus on those due to groundwater extraction in shallow confined aquifers. In this section, the ground settlement due to dewatering in the MCA as well as the

correlation between ground settlement and groundwater drawdowns is analyzed to investigate the responses of ground settlement induced by dewatering in shallow confined aquifers.

#### 4.2.1. Results

In Figure 6, the distributions of ground settlements obtained by the 30 monitoring points (D1 to D30) during the multi-well pumping test are depicted. As can be seen in Figure 6, the ground settlement increased gradually with the proceeding of the pumping test and reached its maximum value at the end of the test (Aug. 20th). After that, a remarkable rebound of ground settlement could be observed. Besides, it is notable that the distributions of ground settlement at different stages of the test were similarly shown in bell-shaped distribution. The settlement was larger when the distance of monitoring point to the pumping center was smaller and reached its maximum value at the monitoring point D24. Additionally, the settlement at the left side of D1 was apparently larger than that at the right side, which was inconsistent with the groundwater drawdown. The possible reason responsible for this may be the soil erosion and ground collapse induced by high-speed groundwater flow.

**Figure 6.** The distributions of ground settlement: (**a**) curves for ground settlement of D1–D9 and D24–D30; (**b**) curves for ground settlement of D1, D10–D16, and D17–D23.

Furthermore, the time-history curves for the settlement of D1, D3, and D5 as well as the drawdowns in G5-2-1, G5-2-2, and G5-2-3 (next to D1, D3, and D5, respectively) are observed in Figure 7. As can be seen, the development of the settlement was similar to that of the drawdown observed in the adjacent well. The rate of ground settlement varied in proportion to that of groundwater drawdowns, and a larger ultimate value of groundwater drawdown would cause a greater ultimate value of ground settlement. Once the multi-well pumping was terminated, the groundwater level recovered immediately, and the surface subsidence rebounded subsequently. As can be seen in Figure 7, slight hysteresis can be observed between the development of groundwater drawdowns and ground settlement. The possible reason responsible for this phenomenon could be that the aquitard and phreatic aquifer overlying the pumping aquifer limit the delivery and accumulation of the stratum deformation.

**Figure 7.** The time-history curve of ground settlement and groundwater drawdowns.

#### 4.2.2. Analyses

Groundwater extraction would reduce the artesian head and pore pressure in the pumping aquifer as well as its adjacent aquifers, causing the increase of effective stress and finally resulting in the compression of pumping aquifers, which is the primary reason for ground settlement [3]. Thus, needless to say, the distributions of the ground settlement should be parallel to those of drawdowns. According to test results, the conclusion is tenable in the most areas of the test site except the immediate vicinity of pumping wells, where the ground settlement was a little larger, e.g., the settlement of D24. As aforementioned, the possible reason may be soil erosion and ground collapse. In fact, the average discharge rate of pumping well C5-2-2 was 15.64 m3/h with the maximum value of over 25 m3/h, making the groundwater flow to C5-2-2 at a higher speed. This high-speed flow, carrying along plenty of soil particles, flowed out of underground through pumping wells, consequently causing ground collapse and unexpected settlement. Moreover, the collapse-induced settlement was unrecoverable. Thus, the settlement could not rebound sufficiently as the drawdown did in the recovery stage (see Figure 7).

#### *4.3. Responses of Deep Soil Deformation*

Both engineering practices and theoretical researches have revealed that the dewatering-induced settlement of deep soil can be larger than that of the ground surface [8]. In this section, to investigate the responses of deep soil deformation, the soil settlement at different depths as well as its correlation to the groundwater fluctuations is analyzed.

#### 4.3.1. Results

On the left part of Figure 8 are the history curves for deep soil settlement monitored by X1~X3. The data of X4 were absent due to technical failures. As can be seen, during the pumping stage, the soil at various depths and positions firstly subsided gradually until the pumping was shut down and shortly afterwards rebounded progressively with groundwater recovery. At the same depth, the subsidence of the monitoring point was larger as its distance to the pumping center became closer. In the same monitoring hole, the soil at a depth of 22 m suffered the largest subsidence for most of the time, followed by that at a depth of 35 m (in X1 and X2) or the surface soil (in X3), whereas that at a depth of 68 m held the smallest deformation, smaller than that at a depth of 50 m.

Subsequently, the stratum is divided into five layers by the deep soil settlement monitoring points and they are Layer A (0 m to −22 m), Layer B (−22 m to −35 m), Layer C (−35 m to −50 m), Layer D (−50 m to −68 m) and Layer E (≤−68 m), respectively, as can be seen in Figure 2. Here, Layer A refers to the soil layers overlying the MCA, Layers B, C, and D refer to the upper, middle, and lower part of the MCA, respectively, while Layer E refers to the soil layers underlying the MCA. The deformation of each layer can be obtained by subtracting the displacement at its bottom by that of its top, and positive values mean soil expansion while negative values mean soil compression. The results are depicted in the right part of Figure 8. As can be seen, there were usually four layers, including Layer B, C, D, and E, compressed to varying degrees and one layer, Layer A, expanded and the expansion decreased as the distance to the pumping center increased. Among the compressed layers, Layers B, C, and D located in the pumping aquifer (MCA) usually had relatively larger deformation while layer E in Layer <sup>7</sup> suffered the smallest compression.

**Figure 8.** The time-history curves of deep soil settlement and stratum deformation.

Additionally, the time-history curves for the soil compression of Layers B, D, and E in X3 and the corresponding drawdowns monitored by G5-2-2, G5-3-1, and G7-1 (next to X3) are depicted in Figure 9. As demonstrated, the development of soil compression in the pumping stage was irregular, whereas that in the recovery stage showed good correlation with the development of the groundwater fluctuations in the same layer. The larger the drawdown was, the more severely the soil compressed, also indicating that dewatering-induced drawdown was an important reason for ground settlement.

**Figure 9.** The time-history curves of deep soil compression and groundwater drawdowns.

#### 4.3.2. Analyses

As aforementioned, there was expansion in Layer A, and that was why the maximum subsidence usually occurred at the top of the pumping aquifer instead of the ground surface. The reason for soil expansion could be attributed to dewatering-induced differential drawdowns and compression in the pumping aquifer. During the test, although there were no obvious drawdowns in Layer A, whereas to satisfy deformation coordination, uneven downward displacement took place at the bottom of Layer A, which induced the rotation of principal stress and the formation of stress arch in Layer A due to surrounding constraint and caused the expansion of Layer A. In addition, as observed in Figures 8 and 9, the distributions and development of the stratum deformation were relatively irregular. The possible reasons, including compressibility, body force, and stress history of soil, piping erosion, and ground collapse due to high-speed flow as well as the influence of partially penetrating well, etc., are various and complex. Therefore, further study on this issue is still imminently required.

#### **5. Back Analysis of Groundwater Fluctuations and Ground Settlement**

#### *5.1. Hydrogeological Parameter Estimation Based on Pumping Well Test*

#### 5.1.1. Limitations of the Analytical Methods

For the prediction model of groundwater drawdown and stratum deformation, in addition to accurate mechanisms of groundwater seepage, the precise parameters of pumping aquifers play significant roles for the final results. Several mathematical models and corresponding analytical or semi-analytical solutions have been proposed for aquifer parameter estimation [38–46]. These models can consider the influence of leaky aquifers [40,41] or variable discharge [42,45] to a certain degree. However, the preceding models assume homogeneous, isotropic, and laterally-unbounded aquifers. Nevertheless, in reality, the aquifer is much more complicated than the previously-assumed aquifer aforementioned. Real aquifers are generally characterized by anisotropy and leakage effect, resulting in the above models being not accurate enough for the description of groundwater flow and estimation of aquifer parameters.

#### 5.1.2. Parameter Estimation using Numerical Method

This research mainly focuses on an aquifer system consisting of several micro-confined and confined aquifers. Considering the complexity of the hydrogeology and the limitations of the analytical methods, the numerical method is more reliable and recognized for parameter estimation [47–50]. In this study, a three-dimensional numerical model is developed in Visual Modflow [51] for groundwater drawdown calculation. In the model, the leaky aquifers are considered by simulating the real site conditions, whereas the variable injection rate is considered by setting a pumping schedule in the software. Moreover, the parameter estimation program PEST of Visual Modflow has been adopted for hydrogeological parameter estimation [51]. The following steps are undertaken for the parameter estimation using numerical method:

Step 1: Develop a numerical model based on the site condition and calculate the groundwater drawdowns in observation wells due to single-well pumping with the parameters obtained using the Hantush–Jacob solution.

Step 2: Call PEST for the parameter optimization of Layers <sup>5</sup> 2~<sup>5</sup> 3-2 and update the parameters.

Step 3: Call PEST for the parameter optimization of Layer <sup>5</sup> 3-3 and update the parameters.

Step 4: Recalculate the groundwater drawdowns in Layers <sup>5</sup> 2~<sup>5</sup> 3-2 and compare the discrepancy between the results of Step 4 and Step 2. Repeat step2 and step 3 until the discrepancy is sufficiently small.

Step 5: Call PEST for parameter optimization for Layer <sup>7</sup> and check the influence of parameter optimization on groundwater level in other aquifers. Repeat Step 2 and Step 3 until the influence can be negligible.

#### 5.1.3. Results and Analyses

In this section, the proposed numerical method is employed for parameter estimation as well as an analytical method based Theis solution. Table 3 lists the results for parameter estimation. The estimation value using the analytical method is larger in hydraulic conductivity and smaller in storage than that using the numerical method in Layer <sup>5</sup> 3-3 and Layer <sup>7</sup> , while it is the opposite in Layers <sup>5</sup> 2~<sup>5</sup> 3-2. Moreover, the drawdowns are calculated in Visual Modflow using the two group of parameters, and the results are depicted in Figure 10 as well as the observed drawdowns. As can be seen, the calculated drawdowns using the proposed numerical method match the observed drawdowns well except at the early stage of pumping when the discharge rate is unstable, whereas those using the analytical method present significant discrepancy with the observed drawdowns, indicating that the proposed numerical method is more reliable for parameter estimation and drawdown calculation in complex aquifer systems.

As mentioned earlier, there are significant discrepancies between the numerical results and the analytical ones. The primary reason for this is the hydraulic connection among each aquifer. When pumping in Layer <sup>5</sup> 3-3 or Layer <sup>7</sup> , the head in the pumping aquifer decreased rapidly, and head difference was formed, causing the groundwater from the adjacent aquifers to flow into the pumping aquifer. When pumping in Layers <sup>5</sup> 2~<sup>5</sup> 3-2, significant drawdowns could be widely observed in the pumping aquifer. However, the natural head in this layer was relatively larger than that in Layer <sup>5</sup> 3-3 before dewatering, and this held true for most areas during dewatering, resulting in the groundwater flowed into Layer <sup>5</sup> 3-3. In analytical methods, the aquifers were assumed to be entirely isolated. Thus, the groundwater flowing into or out of the pumping aquifer is oversimplified, causing the hydraulic conductivity in Layer <sup>5</sup> 3-3 and Layer <sup>7</sup> was overrated and that in Layers <sup>5</sup> 2~<sup>5</sup> 3-2 was underestimated. Whereas, in the numerical method this characteristic could be considered by developing a unified numerical model according to the site condition. Consequently, the numerical method is more accurate for parameter identification in complex aquifer systems.


**Table 3.** Comparison between the parameter obtained by the analytical and numerical method.

**Figure 10.** Comparison between observed and calculated drawdowns: (**a**) drawdowns in observation well G5-2-2; (**b**) drawdowns in observation well G5-3-1; (**c**) drawdowns in observation well G7-1;

#### *5.2. Ground Settlement Prediction Induced by Dewatering*

#### 5.2.1. Basic Assumptions

In this section, a simple prediction method for ground settlement induced by dewatering is proposed and discussed. In this method, the conventional Theis Model is employed to calculate the drawdowns due to dewatering, and the unidirectional compression formula is utilized to estimate the soil compression. Hence, the assumptions used in this method are identical to those in the Theis Model except that the compression of the pumping aquifer is assumed to be completed instantly and equal to the ground settlement. This assumption is conservative considering the expansion of the overlying non-pumping layers. Besides, the assumptions above cannot consider many factors related to the geological condition, such as the body force and stress history of soil. However, these factors can be considered indirectly by calibrating the predicting result using the observation data.

#### 5.2.2. Ground Settlement Prediction Based on Pumping Well Test

In application, the following steps may be taken to calculate the dewatering-induced ground settlement:

Step 1: Calculate the groundwater drawdown induced by single-well pumping based on Theis Formula.

Step 2: Acquire the groundwater drawdown caused by multi-well pumping using the superposition principle.

Step 3: Obtain the additional effective stress due to groundwater drawdown by the theory of effective stress.

Step 4: Develop the prediction expression for settlement utilizing unidirectional compression formula.

Step 5: Compute the undetermined coefficients in the expression with the observation data.

According to the aforementioned, the expression for ground settlement prediction can be described as follows (details can be found in the Appendix A):

$$\Delta s = \frac{\gamma\_w H}{E\_s} \sum\_{i=1}^{n} \left( \frac{2.3 Q\_i}{4 \pi T} \lg \frac{2.25 T t}{S} - \frac{22.3 Q\_i}{4 \pi T} \lg r\_i \right) = \sum\_{i=1}^{n} f\_i \lg t - K\_i \lg r\_i + L\_i \tag{1}$$

where Δ*s* is the ground settlement [L] at distance *r* [L] and time *t* [T]; *Es* is the compressing modulus [ML−1T−2]; *H* is the thickness [L] of the pumping aquifer; *γ<sup>w</sup>* the bulk density [ML−2T−2] of water; *Q* is the discharge rate [L3T−1]; *T* is the transmissivity [L2T−1], and *S* is the storage coefficient [dimensionless]. Besides, *Ji*, *Ki* and *Li* are undetermined coefficients and can be expressed as *Ji* = 2.3*γwHQi* 4*πTEs* , *Ki* <sup>=</sup> <sup>2</sup>·2.3*γwHQi* 4*πTEs* and *Li* <sup>=</sup> 2.3*γwHQi* 4*πTEs* ·lg(2.25*T*/*S*). The undetermined coefficients follow the relationships: *J*<sup>1</sup> : *J*<sup>2</sup> : ··· : *Jn* = *K*<sup>1</sup> : *K*<sup>2</sup> : ··· : *Kn* = *L*<sup>1</sup> : *L*<sup>2</sup> : ··· : *Ln* = *Q*<sup>1</sup> : *Q*<sup>2</sup> : ··· : *Qn*, *Ki* = 2*Ji*, and can be determined using the nonlinear curve fitting function of the Origin software based on the ground settlement observation data.

#### 5.2.3. Validation and Analyses

During the test, there were four pumping wells employed. Thus, the total number of the undetermined coefficient was 12. Subsequently, these undermined coefficients were determined by performing the nonlinear fitting according to Equation (1) based on the ground settlement observation data of D1~D23 from Day 4 to Day 7. The results are shown as: *J*<sup>1</sup> = 0.00128, *J*<sup>2</sup> = 0.00275, *J*<sup>3</sup> = 0.00205, *J*<sup>4</sup> = 0.00191, *K*<sup>1</sup> = 0.00256, *K*<sup>2</sup> = 0.00550, *K*<sup>3</sup> = 0.00410, *K*<sup>4</sup> = 0.00382, *L*<sup>1</sup> = 0.00228, *L*<sup>2</sup> = 0.00490, *L*<sup>3</sup> = 0.00366, *L*<sup>4</sup> = 0.00340, and the goodness of fitting is 0.870. Further, the predicting formula is employed to predict the ground settlement on Day 8, and the predictive values and observation values, as well as the error analyses, are depicted in Figure 11.

**Figure 11.** Comparison of the observation value and prediction value.

As illustrated in Figure 11, the percentage errors vary from 0.428% to 70.5% with an average value of 25.57%. The prediction method functioned well for a majority of the monitoring points with a percentage error of less than 25%. However, for the points at the far-field (such as D9 and D15) of the test site, the error was much larger and even exceeded 50%. At the far-field, the ground settlement induced by groundwater extraction was very small, usually 2~3 mm. For this reason, the error caused by human activities and measurements was inevitable and considerable. In addition, at the central part of the test site, the measured value of subsidence was usually larger than the prediction value. The possible reason may be the aforementioned ground collapse due to high-speed groundwater flow. In general, the prediction values match well with the observation data, indicating that the prediction method proposed in this paper is feasible.

#### **6. Conclusions**

In this paper, field pumping well tests were performed in Pudong New Aera to investigate the responses of the groundwater level and stratum deformation due to dewatering in the MCA. On this basis, practical methods for hydrological parameter estimation and ground settlement prediction were proposed and discussed. Following conclusions can be drawn:

(1) Both the single-well and multi-well pumping tests indicate that there is a close hydraulic connection between MCA and Aq I. Hence, even if dewatering measurements are only performed in MCA, the groundwater drawdowns and stratum deformation in Aq I should be considered to avoid underestimating water inflow and ground settlement.

(2) The distributions and development of ground settlement are similar to those of groundwater drawdowns. It is tenable for most areas except the immediate vicinity of the pumping wells, where the subsidence is larger and cannot rebound sufficiently with groundwater recovery due to ground collapse induced by high-speed groundwater flow.

(3) During the pumping well test, soil settlement and stratum compression can be observed not only in the pumping aquifer but also in its underlying aquifers and their distributions and development show correlation with those of groundwater fluctuations, which also indicates there is a close hydraulic connection among each aquifer.

(4) During the pumping well test, because a stress arch is formed in the top layer due to uneven deformation and surrounding constraint, the top layer is expanded, and the largest subsidence usually occurs at the top of the pumping aquifer instead of the ground surface.

(5) For the parameter estimation under complex hydrogeology conditions, especially when the micro-confined aquifer is directly connected with the deep confined aquifer, the proposed numerical method can consider the effect of hydraulic connection, and the results are more reliable and accurate compared with those of the conventional analytical methods.

(6) The proposed prediction method for the dewatering-induced ground settlement functions well at most parts of the test site except at the far-field and the central parts, indicating its feasibility. Moreover, the parameters used in the method can be obtained by performing fitting with observation data, avoiding the dependence on precise hydrogeological parameters.

**Author Contributions:** J.L. analyzed the experimental data and wrote the original draft paper; M.-G.L. provided analytical procedures and conducted the formal analysis; H.C. conducted the series of pumping well tests and provided experimental materials. X.-H.X., L.-L.Z. and J.-J.C. motivated this paper with good ideas and funding support.

**Funding:** This research was funded by the National Natural Science Foundation of China, grant numbers 41602283, 41472250.

**Conflicts of Interest:** The authors declare no conflicts of interest

#### **Appendix A**

To develop the prediction method for dewatering-induced ground settlement, several classic theories, including Theis solution and Jacob solution for unsteady flow to a pumping well, superposition principle, theory of effective stress as well as unidirectional compression formula, are employed here. The following steps may be taken to calculate the ground settlement induced by the groundwater extraction:

Step 1: Calculate the groundwater drawdown induced by single-well pumping based on Theis Formula.

Step 2: Acquire the groundwater drawdown caused by multi-well pumping using superposition principle.

Step 3: Obtain the additional effective stress due to groundwater drawdown by theory of effective stress.

Step 4: Develop the prediction expression for settlement utilizing unidirectional compression formula.

Step 5: Compute the undetermined coefficients in the expression with the observation data.

In Step 1, the groundwater drawdown induced by single-well pumping can be obtained by Theis formula, a classic solution for unsteady groundwater flows to a pumping well in a homogeneous, horizontally isotropic, laterally unbounded confined aquifer with a constant discharge rate. And it is expressed as follows:

$$s(r,t) = \frac{Q}{4\pi T} \mathcal{W}(u) = \frac{Q}{4\pi T} \int\_{\frac{r^2 S}{4Tt}}^{\infty} \frac{e^{-u}}{u} du\tag{A1}$$

where *s* is the groundwater drawdown [L] at distance *r* [L] and time *t* [T], *Q* is the discharge rate [L3T−1], *T* is the transmissivity [L2T−1] and *S* is the storage coefficient [dimensionless], *W*(*u*) is the well function, *<sup>u</sup>* can be expressed as *<sup>u</sup>* <sup>=</sup> *<sup>r</sup>*2*<sup>S</sup>* <sup>4</sup>*Tt* and is a dummy variable of integration [dimensionless]. Specially, when *u* ≤ 0.01, Theis solution can be simplified to Jacob solution and can be expressed as follows:

$$s(r,t) = \frac{2.3Q}{4\pi T} \text{lg}\frac{2.25Tt}{S} - \frac{2\cdot 2.3Q}{4\pi T} \text{lg}r \tag{A2}$$

In Step 2, the groundwater drawdown caused by multi-well pumping is equal to the sum of drawdown induced by each single-well pumping utilizing the superposition principle and can be calculated as follows:

$$s(r,t) = \sum\_{i=1}^{n} \left(\frac{2.3Q\_i}{4\pi T} \lg \frac{2.25Tt}{4\pi T} - \frac{22.3Q\_i}{4\pi T} \lg r\_i\right) \tag{A3}$$

where *Qi* is the discharge rate [L3T−1] of the *i*th well and *ri* is the distance [L] between the monitoring point and the *i*th well.

In step 3, the increment of effective stress due to groundwater drawdown is equal to the decline of pore pressure according to principle of effective stress and the additional effective stress can be calculated as follows:

$$
\Delta p = \gamma\_w \cdot \sum\_{i=1}^{n} \left( \frac{2.3Q\_i}{4\pi T} \lg \frac{2.25Tt}{4\pi T} - \frac{22.3Q\_i}{4\pi T} \lg r\_i \right) \tag{A4}
$$

where Δ*p* is the additional effective stress [ML<sup>−</sup>1T−2] and *γ<sup>w</sup>* is the water bulk density [ML<sup>−</sup>2T−2].

In Step 4, the unidirectional compression formula is employed to compute the compression of the dewatered confined aquifer, which is assumed to be as large as the ground settlement, and can be expressed as follows:

$$\Delta s = \frac{\Delta \mathbf{p}H}{E\_s} = \frac{\gamma\_w H}{E\_s} \sum\_{i=1}^{n} \left( \frac{2.3 Q\_i}{4 \pi T} \lg \frac{2.25 T t}{4 \pi T} - \frac{22.3 Q\_i}{4 \pi T} \lg r\_i \right) = \sum\_{i=1}^{n} f\_i \lg t - K\_i \lg r\_i + L\_i \tag{A5}$$

where Δ*s* is the ground settlement [L], *Es* is the compressing modulus [ML−1T−2], *H* is the thickness [L] of pumping aquifer.

In Step 5, fitting method is performed on the ground settlement observation data to calculate the undetermined coefficients in Equation (A5). It should be noted that the observation data adopted here should meet the requirement *u* ≤ 0.01 to reduce the error in the simplification from Theis Formula to Jacob Formula.

#### **References**


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