Nonlinear Wave-Induced Uplift Force onto Pipelines Buried in Sloping Seabeds

Abstract

In this paper, a two-dimensional numerical model for wave-seabed-pipeline interaction is developed to examine the wave-induced uplift force onto pipelines buried in sloping seabeds. The Reynolds-averaged Navier stokes equation and the poro-elastic equation are used to simulate the wave motion and seabed response, respectively. Meanwhile, the pipeline is considered to be elastic. Firstly, three laboratory experiments are taken to verify the effectiveness of the numerical model. Then, the effects of pipeline characteristics, soil properties and wave parameters on the nonlinear wave-induced uplift force onto a pipeline buried in a sloping seabed are analyzed. Finally, an empirical formula for predicting the nonlinear wave-induced uplift force onto buried pipelines under different slope angles is proposed. It can be found that the slope angle can greatly affect the nonlinear wave-caused pore pressure response, as well as the uplift force onto the pipeline. Moreover, the simple method for predicting the uplift force proposed in this paper can facilitate engineering applications.

1. Introduction

Subsea pipelines are widely utilized for transporting gas and oil from offshore areas to land [1]. In order to maintain the stability of pipelines, some pipelines are buried in the seabed. However, this method cannot guarantee the stability of the pipeline completely due to the complexity of marine environments. One of the failure mechanisms is upheaval buckling due to temperature and pressure [2]. The hydrodynamic stability of pipelines buried in the seabed can be influenced by waves since waves can induce the sediment transport and pore pressure variation [3,4]. This study is devoted to studying the latter. Under the cyclic wave load, the seabed may get liquefied [4], which may further threaten the stability of pipeline.

Two wave-caused soil response mechanisms can be summarized from previous studies [5]. The first one relates to the formation process of pore pressure, which is called residual mechanism. Generally, the residual pore pressure occurs in the initial stage of the wave loading, which is induced by the contraction of soil particles. The other is an oscillatory mechanism and it is caused by the upward-directed vertical pressure gradient under the wave trough. Since lots of studies on wave-induced residual soil response have been performed [6,7,8,9,10,11,12], the present study focuses on the oscillatory soil response-caused uplift force applied to the pipeline.

A number of studies involving wave-seabed interaction have been conducted for investigating the oscillatory seabed response in the past few decades. Among these, Yamamoto et al. [13] used Biot consolidation theory [14] to research the wave-caused response of the porous elastic seabed, and the result shows that the seabed permeability and the rigidity ratio of soil particle shear modulus to pore water bulk modulus are the main influencing factors for the seabed response. Based on the aforementioned study, Hsu and Jeng [15] carried out a study under the assumption of a finite thickness seabed and proposed an analytical solution to the seabed response caused by waves. Ye and Jeng [16] studied the seabed liquefaction considering the combined current and wave load and found that the opposing current is able to relieve the liquefaction, while the following current is able to intensify it. Liu et al. [17] carried out a 1D experiment in which the hydrostatic pressure was included to research the influences of the soil properties as well as the wave parameters on pore pressure and liquefaction. By carrying out some flume experiments, Zhang et al. [18] researched the wave-caused pore pressure response along the soil depth, in which the phase lag of oscillatory pore pressure was found. On the basis of the dynamic stiffness matrix, Chen et al. [19] considered the combined load of current and wave to propose a semi-analytical solution for the dynamic response of multilayered transversely isotropic seabed and found an underestimated maximum liquefaction depth under the isotropic assumption. Zhang et al. [20] also proposed a semi-analytical solution using fully dynamic Biot’s theory and second-order Stokes theory and found that the stratification can greatly impact the seabed response caused by waves and that it is easy to liquefy when the surface layer is hard. Qin et al. [21] studied the seabed response around a dumbbell-shaped cofferdam under wave action using a three-dimensional numerical model built by OpenFOAM and found that the maximum liquefaction depth around the cofferdam occurred near the steel casing along the centerline of the cofferdam. Duan et al. [22] investigated the wave-induced soil response during caisson installation using the PORO-FSSI-FOAM model and found that the sequence of caisson installation has a large effect on the momentary liquefaction, and a formula is proposed for describing the variation of the liquefaction depth with the caisson submerged depth.

The above-mentioned studies laid foundations for further studies on wave-caused seabed response along the periphery of pipelines. Regarding the interaction between waves, seabed, and pipelines, Cheng and Liu [23] used the Biot model to research the seepage force applied by wave to the buried pipeline and studied the influences of water parameters as well as soil properties on the pore pressure response. Sumer et al. [24] performed an experiment for studying the sinking and floatation of pipelines subjected to progressive waves and obtained a drag coefficient of the sinking of pipelines in the liquefied seabed. Gao et al. [25] conducted some experiments using U-shaped oscillatory flow tunnel for investigating the effects of soil particle size on the lateral stability of pipelines, as well as the relationship for Froude number and pipeline weight. Dunn et al. [26] built a numerical model for researching the wave-caused seabed liquefaction next to the buried pipeline, evaluating the effects of pipeline diameter, boundary conditions as well as trench depth on the wave-caused liquefaction. Teh et al. [27] researched the failure mechanism of pipelines buried in the liquefied seabed and presented an analytical solution for predicting the depth of pipeline subsidence. Wen et al. [28] numerically examined the soil response caused by nonlinear wave and current and found that the trench layer had the function lowering the risk for the seabed liquefaction next to the pipeline. Zhou et al. [29] introduced cnoidal waves to research the seabed response next to the buried pipeline and found that the difference of vertical effective stress and maximum pore pressure between stokes waves and cnoidal waves is able to be 60–70%. Lin et al. [30] used an integrated finite element method model for studying the soil dynamic response and the stability of buried pipelines subjected to wave load. Gao et al. [31] proposed the pipeline-seabed interaction model considering the instability induced by current in downslope and upslope and found that the slope angle in the range from 15° to −15° had a significant influence on the lateral soil resistance around pipeline. Duan et al. [32] built a 2D fully coupled model for current and wave-seabed-pipeline interaction to research the soil response next to a buried pipeline in trench. Considering the influences of seepage flow on soil particle movement, Guo et al. [33] developed a numerical model by integrating the Shear-Stress Transport turbulence model into the porous seabed model. Sun et al. [34] performed some flume experiments for studying the soil dynamic response next to the partially buried pipeline in a trench, and the result shows that a deeper trench and thicker backfill can provide more seabed resistance.

The seabed response next to the pipeline directly affects the pressure applied by a wave to the buried pipeline, which may further affect the uplift force applied to the pipeline. Lots of previous research were performed to investigate the effects of wave-caused soil response on pipeline stability. Among these, Qi et al. [4] researched the uplift force applied by waves to a pipeline using a Mohr-Coulomb model and found that the uplift force depended largely on the amplitude of pore pressure applied by waves onto the surface of the seabed. Duan et al. [35] studied the uplift force applied by wave to a pipeline and got a relationship for the uplift force and influencing parameters using a fully coupled model considering the wave-soil-pipeline interaction. Yang et al. [36] further studied the uplift resistance of soil to pipeline considering both slope and wave and proposed a formula for the wave-induced reduction in the peak uplift resistance. Pisanò et al. [37] proposed a computational fluid dynamic model to predict the pipeline displacement in liquefied soil, where the large deformation and reconsolidation of soil are considered. Koley et al. [38] investigated the force applied by waves to a buried pipeline, in which the higher-order boundary element method was used to solve the related mathematical problem. The existing studies mainly focus on horizontal seabeds, and sloping seabeds are hardly involved although slopes are common in practical engineering.

Inspired by previous investigations, the present research aims to investigate the uplift force applied by nonlinear waves to a pipeline buried in the sloping seabed. In the present study, the pressure applied by a wave to the mudline is obtained by a wave sub-model firstly, and then this pressure is applied into an ABAQUS model using subroutines. After model validation with the previous experiments, the effects of soil properties, wave parameters as well as slope on the nonlinear wave-induced uplift force onto the pipeline are analyzed in detail. Finally, a simple formula describing the relationship of the uplift force and the influencing factors is proposed.

2. Numerical Model

The purpose of this research is to study the nonlinear wave-induced uplift force onto pipelines buried in sloping seabeds. Thus, a two-dimensional numerical model including three sub-models was developed. The wave motion is simulated using a wave sub-model and the soil dynamic response caused by nonlinear waves is simulated by a seabed sub-model. Moreover, the structure dynamic response can be simulated by the pipeline sub-model.

Figure 1 illustrates the sketch for this numerical model for the wave-seabed-pipeline interaction, where $d$ is the water depth of the incident wave; $\alpha$ is the slope angle of the seabed; $\mathrm{H}$ is the wave height; $\mathrm{L}$ is the wave length; $\mathrm{D}$ is the pipeline diameter; $\mathrm{e}$ is the burial depth.

2.1. Wave Sub-Model

As for the wave sub-model, the Reynolds-Averaged Navier–Stokes (RANS) equation and Renormalization-Group (RNG) $k-\epsilon$ turbulence model are adopted for simulating the fluid motions; at the same time, the Volume of Fluid (VOF) method is adopted for tracking the free water surface:

$$\frac{\partial }{\partial x_i}\left(u_i{A}_i\right)=0$$

(1)

$$\frac{\partial u_i}{\partial t}+\frac{u_j{A}_j}{V_F}\frac{\partial u_i}{\partial x_j}=-\frac{1}{\mathsf{\rho }}\frac{\partial p}{\partial x_i}+G_i+f_i$$

(2)

where $x_i$ is the Cartesian coordinates, $i$, $j$ are the directions; $A_i$ is the fractional area; $u_i$ is the velocity; $G_i$ is the body acceleration; $\mathsf{\rho }$ is the fluid density; $V_F$ is the fractional volume open to flow; $f_i$ is the viscous acceleration, which can be expressed as:

$$\mathsf{\rho }{V}_F{f}_i=-\frac{\partial }{\partial x_j}\left(A_j{\tau }_{ij}\right)$$

(3)

$${\tau }_{ij}=-2\mu \left(\frac{\partial u_i}{\partial x_i}-\frac{1}{3}\frac{\partial u_j}{\partial x_j}\right),i=j$$

(4)

$${\tau }_{ij}=-\mu \left(\frac{\partial u_j}{\partial x_i}+\frac{\partial u_i}{\partial x_j}\right),i\ne j$$

(5)

where $\mu$ is the dynamic viscosity and ${\tau }_{ij}$ is the stress tensor.

The RNG $k-\epsilon$ turbulence model is defined as:

$$\frac{\partial k_T}{\partial t}+\frac{u_j{A}_j}{V_F}\frac{\partial k_T}{\partial x_j}=P_T+{Diff}_k-{\epsilon }_T$$

(6)

$$\frac{\partial {\epsilon }_T}{\partial t}+\frac{u_j{A}_j}{V_F}\frac{\partial {\epsilon }_T}{\partial x_j}=\frac{CDIS1·{\epsilon }_T·P_T}{k_T}+{Diff}_{\epsilon }-\frac{CDIS2·{\epsilon }_T^2}{k_T}$$

(7)

where $k_T$ is the turbulent kinetic energy; $P_T$ is the turbulent kinetic energy production; ${\epsilon }_T$ is the rate of turbulent energy dissipation; ${Diff}_k$ is the diffusion term of $k_T$; ${Diff}_{\epsilon }$ is the diffusion term of ${\epsilon }_T$.

The governing equation for the VOF method is as follows:

$$\frac{\partial F}{\partial t}+\frac{1}{V_F}\frac{\partial }{\partial x_i}\left(F{A}_i{u}_i\right)=0$$

(8)

where $F$ is the fractional volume of fluid.

In actual marine environments, the wave height can reach tens of meters, and the wave exhibits a steep crest and flat trough. In this case, the linear wave theory is no longer applicable. Therefore, this paper focuses on the nonlinear wave-induced uplift force onto the pipeline buried in sloping seabed.

2.2. Seabed Sub-Model

As for the sub-model of the seabed, many studies [5,32,35] have demonstrated that the wave-induced pore pressure response can be well simulated using the poro-elastic model. Therefore, the Biot’s poro-elastic equation is used to simulate the seabed response in this study, which can be expressed as:

$$\Delta P_e-\frac{{\gamma }_w{n}_s\beta }{k_s}\frac{\partial P_e}{\partial t}+\rho \frac{{\partial }^2{\epsilon }_s}{\partial t^2}=\frac{{\gamma }_w}{k_s}\frac{\partial {\epsilon }_s}{\partial t}$$

(9)

where ${\gamma }_w$ is the unit weight of water; $P_e$ is the pore pressure caused by a wave; $k_s$ is the soil permeability; $n_s$ is the soil porosity; ${\epsilon }_s$ is the soil volume strain, which can be defined as:

$${\epsilon }_s=\frac{\partial u_s}{\partial x}+\frac{\partial w_s}{\partial z}$$

(10)

where $u_s$ and $w_s$ are the soil displacements along the x-and z-direction, respectively; $\beta$ is the compressibility of pore water, which can be defined as:

$$\beta =\frac{1}{K_w}+\frac{1-S_r}{P_{W0}}$$

(11)

where $S_r$ is the soil saturation; $K_w$ is the true elastic modulus of water; $P_{W0}$ is the hydrostatic pressure.

The overall equilibrium equations that take into account the soil displacements as well as the excess pore pressure are defined as:

$$G\Delta u_s+\frac{G}{1-2\nu }\frac{\partial {\epsilon }_s}{\partial x}=\frac{\partial P_e}{\partial x}+\rho \frac{{\partial }^2{u}_{si}}{\partial t^2}$$

(12)

$$G\Delta w_s+\frac{G}{1-2\nu }\frac{\partial {\epsilon }_s}{\partial z}=\frac{\partial P_e}{\partial z}+\rho \frac{{\partial }^2{w}_s}{\partial t^2}$$

(13)

where $G$ is the shear modulus, which can be defined as:

$$G=\frac{E}{2\left(1+\nu \right)}$$

(14)

where ν is the Poisson’s ratio, and $E$ is the Young’s modulus.

2.3. Pipeline Sub-Model

The pipeline is assumed to be isotropic and elastic, and the governing equations are defined as:

$$\frac{\partial {\sigma }_{px}}{\partial x}+\frac{\partial {\tau }_{pxz}}{\partial z}+f_{px}=0$$

(15)

$$\frac{\partial {\tau }_{pxz}}{\partial x}+\frac{\partial {\sigma }_{pz}}{\partial z}+f_{pz}=0$$

(16)

where ${\sigma }_{px}$ and ${\sigma }_{pz}$ are the effective stresses along the $x$ and $z$ directions, respectively; $f_{px}$ and $f_{pz}$ are the components of unit mass force along the $x$ and $z$ direction, respectively; ${\tau }_{pxz}$ is the shear stress of the pipeline.

2.4. Boundary Conditions

Under defined boundary conditions, the seabed governing equation is able to be solved to obtain the wave-caused seabed response. Figure 2 illustrates the boundaries of seabed sub-model. As for the lateral boundaries of seabed sub-model, the normal displacement is fixed, at the same time it does not allow the pore water to flow through. Additionally, as for the seabed bottom boundary, two translational degrees are fixed, and it also does not allow the pore water to flow through. Relatively complex boundary conditions occur on the seabed surface, in which all the components of displacement are free, and the wave pressure and pore pressure are assumed to be equal. The pore pressure gradient next to the interface of pipeline and soil is zero.

In the present model, two user subroutines are able to be adopted to apply the wave-caused response to the mudline. The pressure response applied by the wave to the mudline is numerically simulated first using the wave sub-model. Then, the pore pressure and wave pressure applied to the seabed surface are able to be loaded using the subroutines DISP and DLOAD, respectively.

In this study, the wave sub-model was built using the finite difference method and the seabed sub-model was established using the finite element method. In the wave simulation, the pressure solver is implicit while the viscous stress, the free liquid surface pressure, and the advection solver are explicit. Moreover, a first-order upwind differencing method is used to solve the momentum advection algorithm. In the seabed sub-model, the direct method and an unsymmetric matrix storage are used. In addition, the solution technique is Full Newton.

3. Model Validation

For present research, the pressure applied by waves to the mudline was first simulated by the wave sub-model. Then, the time-varying pressure was imported to the same location of seabed sub-model built by ABAQUS. To ensure the convergence of the grid, models with different grid sizes were built for the present section. The wave-caused pore pressure below the pipeline under different grid densities is illustrated in Figure 3.

To verify the accuracy of the present model to simulate wave propagation, the laboratory experiment from Umeyama [39] was selected and the parameters for this validation are listed on

Table 1. Parameters for verifying the wave propagation.

Parameters	Values
Wave period (T), s	1
Water depth (d), m	0.3
Wave height (H), m	0.0361

. Figure 4 shows the free water surface obtained from this numerical model and experiment. It can be concluded that the numerical result agrees well with the experiment (Umeyama [39]).

To validate the accuracy of the present model for simulating the wave-seabed interaction, an experiment from Liu et al. [17] was selected, and

Table 2. Parameters for verifying the wave-seabed interaction.

Parameters	Values
Wave period (T), s	9
Water depth (d), m	5.2
Wave height (H), m	3.5
Poisson’s ratio (ν)	0.3
Seabed porosity (n)	0.425
Shear modulus (G), pa	1.27 × 107
Saturation (Sr)	0.996
Soil permeability (ks), m/s	1.8 × 10−4

lists the parameters for this validation. The wave-caused oscillatory pore pressure obtained from this numerical model and experiment are shown in Figure 5. It can be concluded that the numerical result agrees well with the experiment (Liu et al. [17]).

To validate the accuracy of present model for simulating the pore pressure applied by waves to pipelines, an experiment from Turcotte et al. [40] was selected, and

Table 3. Parameters for verifying the pore pressure applied by wave to pipeline.

Parameters	Values
Wave height (H), m	0.143
Water depth (d), m	0.533
Wave period (T), s	1.75
Shear modulus (G), pa	6.62 × 105
Seabed porosity (n)	0.42
Poisson’s ratio (ν)	0.33
Soil permeability (ks), m/s	1.1 × 10−3
Pipeline diameter (D), m	0.168
Burial depth (e), m	0.107

lists the parameters for this validation. Figure 6 shows the pore pressure next to the pipeline obtained from the present model and experiment. And it is clear that the numerical result overall agrees with the experiment (Turcotte et al. [40]).

According to the three validations above, the wave-seabed-pipeline interaction is able to be effectively simulated using present numerical model.

4. Results and Discussions

To examine the influences of wave-induced uplift force onto the pipeline buried in sloping seabed, this study involves three aspects. Firstly, the characteristics of nonliear wave-caused oscillatory seabed response near pipelines buried in sloping seabeds is examined. Then, the influencing factors associated with the uplift force onto a pipeline are studied in detail, including wave parameters, soil properties, pipeline characteristics and slope characteristics. Finally, based on the nuemerical results, a simple formula describing the relationship between the wave-induced uplift force onto the pipeline and the influencing factors is proposed. The selection of wave parameters is based on existing literatures [4,36]. Unless otherwise emphasized, the parameters used in this section are taken according to

Table 4. Parameters used in present study.

Parameters	Values
Wave period (T), s	10
Water depth (d), m	30
Wave height (H), m	6
Poisson’s ratio (ν)	0.3
Seabed porosity (n)	0.8
Young’s modulus (E), pa	5 × 107
Soil permeability (ks), m/s	5 × 10−4
Saturation (Sr)	1
$\mathrm{Slope} \mathrm{angle} (\alpha$), °	6
Horizontal distance on the slope (S), m	100
Pipeline diameter (D), m	1.0
Burial depth (e), m	0.5

.

4.1. The Oscillatory Soil Response around the Pipeline Buried in Sloping Seabed

As mentioned previously, the key innovation for present research is examining the wave-caused oscillatory seabed responses next to the pipeline buried in the sloping seabed. Therefore, Figure 7 shows the comparison of wave trough-caused pore pressure response inside the sloping seabed with and without pipeline. As noted from Figure 7, the pipeline has a considerable impact on the pore pressure distribution within the seabed, especially in the area around the pipeline. In order to gain a more intuitive understanding of this difference, Figure 8 illustrates the wave trough and crest-caused pore pressure varying as the soil depth, and Figure 9 illustrates that around the pipeline. As shown in Figure 8a, under the wave trough, compared to the condition without pipeline, the pipeline will cause lower pore pressure at the corresponding area above the pipeline and higher pore pressure below it, while the opposite phenomenon occurs in Figure 8b. This phenomenon may be related to the “shielding effect” of pipelines. In addition, neither the surface soil nor the slightly deeper soil below the pipeline are affected by the pipeline under the wave trough or the wave crest. Based on Figure 9a,b, the pore pressure above the pipeline is less than that below the wave trough, while it displays an opposite phenomenon under the wave crest. This phenomenon may be related to the “shielding effect” of pipelines and the pore pressure gradient in the seabed. In addition, affected by the slope, the distribution of pore pressure next to the pipeline also shows an inclined state.

In fact, the uplift force applied by a wave to a pipeline correlates closely with the oscillating seabed response. As noted from the curve in Figure 8a, the pore pressure inside the seabed grows with the seabed depth growing under the wave trough. That is to say, below the pipeline, the pore pressure under the wave trough is always greater than that above. However, this phenomenon is opposite under the wave crest, which can be seen in Figure 8b. And the “shielding effect” of a pipeline exacerbates this difference between the pore pressure above and below the pipeline. Figure 8 also shows that the pore pressure gradient gradually decreases with the soil depth, especially when the depth exceeds 6 m, it decreases more rapidly. In other words, the deeper the pipeline is buried, the smaller the pore pressure difference between the upper and lower part of the pipeline.

The above pore pressure difference under the wave trough should be the reason for the uplift force applied by wave to the pipeline. To further quantify the influencing factors of this uplift force, this study obtained the uplift force onto the pipeline by integrating the pore pressure next to the pipeline. The following chapters will analyze the influencing factors of uplift force onto the pipeline in detail.

4.2. The Influences of Wave Parameters on the Uplift Force onto Pipelines Buried in the Sloping Seabeds

For the purpose of investigating the influences of wave parameters on the uplift force applied by waves to pipelines, some numerical models with different wave parameters were operated. In this section, the value of wave height is taken at 2 m intervals in the range of 2 m to 8 m. The value of wave period is taken at 1 s interval in the range of 9 s to 12 s. The value of water depth is taken at 3 m intervals in the range of 27 m to 36 m.

Figure 10 illustrates some simulated results of pore pressure distribution next to the pipeline using variable wave parameters. As noted from the distribution in Figure 10a, the pore pressure next to the pipeline increases significantly as the wave height grows; however, at the same time, the increase rate above the pipeline is higher than that below the pipeline. The greater the wave height, the stronger the pore water pressure under the wave trough. As the soil depth increases, the effect of waves on pore pressure decreases due to the energy absorption effect of the soil, which leads to different variation in pore pressure around the pipeline.

The reason for the different change rate of pore pressure above and below the pipeline can be attributed to the “shielding effect” of the pipeline on the pore water discharge. As Figure 10b shows, with wave period increasing, the pore pressure next to the pipeline gradually increases, but after a certain level, the effect of wave period is negligible. The growth in wave period increases wave length, which increases the range of pressure on the seabed surface. However, the soil response far from the pipeline cannot affect the pipeline, so the effect of increasing the range of pressure on the pore pressure around the pipeline is limited when the rest of the conditions remain unchanged.

Based on Figure 10c, as the water depth grows, the pore pressure next to the pipeline gradually reduces, while the decrease rate gradually decreases, and the decrease rate above the pipeline is always greater than that below. The phenomenon can be attributed to the fact that the greater the water depth, the less the wave effect the seabed, and this effect also diminishes with the soil depth.

The variation on the pore pressure around pipelines may result in a change in the uplift force applied to pipelines. Figure 11 illustrates some variations of the uplift force applied to the pipeline with the wave parameters. As noted from the curves in Figure 11, the higher wave height significantly enhances the uplift force applied to the pipeline. Meanwhile, the longer wave period is also able to increase the uplift force applied by wave to the pipeline in a certain range, once beyond this range, this influence becomes very small. In addition, the uplift force decreases as the water depth grows. The variation of the uplift force applied by wave to the pipeline in Figure 11 is explained by the varying distributions of the pore pressure next to the pipeline in Figure 10.

4.3. The Infjluences of Soil Properties on Uplift Force onto Pipelines Buried in Sloping Seabeds

For the purpose of investigating the influences of seabed properties on the uplift force applied by waves to pipelines buried in sloping seabeds; in present section, the permeability varies from 10⁻⁶ m/s to 10⁻² m/s, the Young’s modulus ranges from 10 MPa to 50 MPa and the Poisson’s ratio ranges from 0.2 to 0.35.

Considering that the uplift force applied to the buried structure may be affected by the pore pressure response inside the soil, the pore pressure distribution around the pipeline with various soil properties is illustrated in Figure 12. Figure 12a shows that the pore pressure increases as the permeability grows both above and below the pipeline, but the increase rate is decreasing. Moreover, the increase rate is greater above the pipeline than that below the pipeline within 10⁻⁴ m/s. As shown in Figure 12b, when the Young’s modulus is less than 40 MPa, the pore pressure next to the pipeline increases as the Young’s modulus grows, and the increase rate below the pipeline is greater than that above the pipeline. However, when the Young’s modulus exceeds 40 MPa, the pore pressure reduces as the Young’s modulus grows, and its decrease rate above the pipeline is slightly greater than that below the pipeline. As noted from the distribution in Figure 12c, the influence of Poisson’s ratio on the pore pressure response around the pipeline is very small, thus it can be ignored.

To analyze the influences of the soil properties on the uplift force applied by waves to pipelines buried in sloping seabeds, Figure 13 illustrates some variations of the uplift force applied to the pipeline with the soil properties. Figure 13a shows that the wave-caused uplift force increases with soil permeability increasing until it reaches 10⁻⁴ m/s, and the change trend is opposite after exceeding this value. From Figure 13b,c, it can be observed that both Young’s modulus and Poisson’s ratio have weakening effects on the uplift force onto the pipeline.

4.4. The Influences of Pipeline Characteristics on the Uplift Force onto Pipelines Buried in Sloping Seabeds

The pipeline burial depth and diameter may affect the boundary condition of the seabed next to the pipeline, which is able to further influence the wave-caused pore pressure around it. In order to explore how the pipeline characteristic affects the uplift force applied to the pipeline, for present study, the burial depth is set at an interval of 0.5 m in the range of 0.5 m to 2.0 m, and the pipeline diameter is set at an interval of 0.2 m in the range of 0.8 m to 1.4 m.

Figure 14 illustrates the pore pressure distributions next to the pipeline with different pipeline characteristics under the wave trough. As noted from the distribution in Figure 14a, as the burial depth grows, the pore pressure next to the pipeline reduces, and its decrease rate above the pipeline is always greater than that below. As noted from the distribution in Figure 14b, the pore pressure above the pipeline tends to gradually grow as the pipeline diameter increases, while an opposite phenomenon occurs below the pipeline. Figure 15 shows the variations of the uplift force applied to the pipeline with various pipeline characteristics. As noted from the curves in Figure 15, the uplift force applied by wave to the pipeline reduces as the buried depth grows, while it grows with the pipeline diameter increasing.

4.5. The Influences of Slope on Uplift Force Applied to Pipelines Buried in Sloping Seabeds

As well known, the slope angle can affect not only the wave transformations, but also the boundary condition on the surface of the seabed model. Thus, the effect of slope angle on the uplift force applied by waves to pipelines buried in sloping seabeds is examined, and the slope angles for the present study are set at an interval of 2° in the range of 2° to 8°. In addition, the horizontal distance of wave propagation on the slope is also taken into account, and the slope horizontal distances include 20 m, 60 m, 100 m and 140 m.

Figure 16 shows the distributions of pore pressure next to the pipeline for various water depths under the wave trough. As noted from the distributions in Figure 16, regardless of water depth, the pore pressure grows with slope angle increase, and the growth rate above the pipeline is greater than that below. Figure 17 shows the variations of the pore pressure distribution around the pipeline with the slope horizontal distance. As noted from the distributions in Figure 17, the relationship between the pore pressure around the pipeline and the horizontal distance of the slope is not monotonic, but first increases and then decreases with the horizontal distance increasing. The reason for this phenomenon can be attributed to two aspects. The first is that as the slope distance increases, the water depth decreases, resulting in enhanced wave action on the seabed, which is drawn from Section 4.2. Secondly, the slope has a lifting effect on the wave, which increases the height of the wave crest but reduces the height of the wave trough relative to the hydrostatic surface and further reduces the pressure acting on the surface of the seabed under the wave trough. If the slope angle is large enough, the wave may even break and its effect on the seabed will be smaller. When the wave enters the slope area, the former has a greater influence on it, so it leads to an increase in the pore pressure around the pipeline, while as the water depth becomes shallower, the latter influence gradually increases, leading to a decrease in the pore pressure around the pipeline. The variations of uplift force applied by waves to a pipeline with slope angles for various water depth are illustrated in Figure 18. As noted from the curves in Figure 18, regardless of the water depth, the uplift force increases as the slope angle increases. In other words, the bigger slope angle can enhance the uplift force applied by waves to the pipeline. Moreover, the shallower the water depth is, the more likely this enhanced effect is to be weakened at large slope angles. However, this effect of slope angle on the uplift force can be ignored when the water depth is large enough. Moreover, Figure 19 shows the relationship between the uplift force and the slope horizontal distance, which shows that the nonlinear wave-induced uplift force onto the pipeline increases first and then decreases as the horizontal distance increases.

4.6. Application for Engineering Practice

Through the above parametric study, the effects of the influencing factors on the nonlinear wave-induced uplift force onto pipelines were examined. Based on the numerical results, an empirical formula predicting the nonlinear wave-induced uplift force onto a pipeline is proposed here.

$$F_z=0.0798\zeta -c$$

(17)

$$\zeta ={\left[\frac{P{D}^{\frac{28}{15}}{a}^{\frac{3}{2}}{{\gamma }_w}^{\frac{25}{48}}}{{\left(G+\lambda \right)}^{\frac{1}{6}}{e}^{\frac{1}{5}}}\right]}^{\frac{4}{5}}$$

(18)

$$P=\frac{{\gamma }_{w}H}{\cosh \left(k{d}_i\right)}$$

(19)

$$d_i=d-S^{2}tan\left(\alpha \right)(-0.0002S+0.0317)$$

(20)

$$\lambda =\frac{\nu E}{\left(1+\nu \right)\left(1-2\nu \right)}$$

(21)

$$a=\left\{\begin{array}{c}10.5+\log {(k}_s\frac{T}{L}),k_s<{10}^{-4}\\ -\log {(k}_s\frac{T}{L}),k_s\ge {10}^{-4}\end{array}\right.$$

(22)

where $k$ is the wave number; $c$ is a coefficient and $c=-942.25$ in the range of this study.

To verify the effectiveness of this new formula, Figure 20 shows the comparison results of the uplift force between the formula and the numerical model. It can be clearly observed from Figure 20 that accuracy can be guaranteed when using the new formula to predict the nonlinear wave-induced uplift force onto the pipeline buried in sloping seabed. It should be noted that this empirical formula is obtained by fitting the numerical results of this study, and its applicable range is within the parameter used in this study.

5. Conclusions

In this paper, the nonlinear wave-induced uplift force onto pipelines buried in sloping seabeds is studied using a two-dimensional numerical model. Compared with previous investigations, the sloping seabed is considered in this study. Based on the numerical results, the following conclusions can be drawn:

(1)

The nonlinear wave-induced pore pressure distribution near pipelines buried in sloping seabeds is inclined, and the inclination direction of the pore pressure distribution corresponds to the seabed inclination.

(2)

The nonlinear wave-induced uplift force onto pipelines buried in sloping seabeds is positively correlated with wave height and period, but negatively correlated with water depth.

(3)

The nonlinear wave-induced uplift force onto pipelines has the trend of first positive and then negative correlation with soil permeability, while it keeps negative correlation with Poisson’s ratio and Young’s modulus.

(4)

A large burial depth can reduce the nonlinear wave-induced uplift force onto pipelines, while a large pipeline diameter can enhance it due to the “shielding effect” of the structure.

(5)

Large slope angles can enhance the nonlinear wave-induced uplift force onto pipelines, and the shallower the water depth, the more likely this enhancement effect is to weaken.