Analytical Calculation of Instantaneous Liquefaction of a Seabed around Buried Pipelines Induced by Cnoidal Waves

Abstract

Cnoidal wave theory perfectly describes nearshore wave characteristics. However, cnoidal wave theory is not widely applied in practical engineering because the formula for the wave profile involves a complex Jacobian elliptic function. In this paper, the approximate cnoidal wave theory is presented. Based on the Biot consolidation theory and the approximate cnoidal wave theory, an analytical solution for the pore water pressure around buried pipelines caused by waves is derived. In addition, based on the principle of effective stress, a theory of soil liquefaction around pipelines is proposed. The theoretical results were virtually identical to the results obtained in a practical flume test. Thus, the analytical method proposed in this paper is feasible. Further, the theory is applied to analyze the instantaneous liquefaction of the seabed around buried pipelines and the stability of the pipeline in the Chengdao oilfield.

1. Introduction

The assessment of wave-induced soil liquefaction plays a key role in submarine pipeline projects [1]. Wave-induced liquefaction in a porous seabed around submarine pipelines may result in catastrophic consequences such as large horizontal displacements of pipelines on the seabed, as well as the sinking or floatation of buried pipelines.

Wave-induced pore pressure exerts an upward seepage force onto the particles in the wave trough phase. If the seepage force is greater than the submerged weight of the soil particle, the seabed will be momentarily liquefied. In order to evaluate the stability of submarine pipelines under wave action rationally and ensure their normal operation, the effect of the pore pressure of the seabed around the pipelines induced by waves must be considered.

Cnoidal wave theory perfectly describes nearshore wave characteristics. However, it is not widely applied in practical engineering because of the Jacobian elliptic function in the wave profile formula. In this paper, the approximate cnoidal wave theory is presented. More specifically, based on previous research results and Biot’s consolidation theory, analytic solutions for the pore water pressure around buried pipelines are derived using the separation of variables method. Finally, the theory is applied to determine the stability of pipelines in the Chengdao oilfield.

Based on research into the dynamic response of the seabed, scientists have investigated the transient liquefaction of the seabed under wave loading. Ye et al. [2] analyzed the dynamic response of a sandy seabed under cyclic wave loading by numerical simulation based on an elastoplastic constitutive model that can uniformly describe the liquefaction behavior of sandy soils and the two-phase u-p theory of saturated soils. Qi and Gao [3] studied the wave-induced instantaneously liquefied soil depth of a non-cohesive seabed, based on the Biot’s consolidation equation, by modifying the instantaneous liquefaction criterion to derive an expression for the liquefaction depth. Liu et al. [4] developed a coupled mathematical model for the cumulative pore water pressure in a wave-induced seabed, considering the coupling effect between the development of pore water pressures and the evolution of seabed stresses. However, the above theories all adopt linear wave theory.

In recent years, many scholars have investigated the effect of nonlinear waves in shallow water [5,6]. Some scholars used the Stokes wave theory to study the dynamic response of the seabed and pipelines, and analyzed the cumulative pore pressure and residual liquefaction depth of an anisotropic, porous, elastoplastic seabed [7,8]. Jeng et al. [9] developed a meshless model to study the soil dynamic response around a submerged breakwater under both regular and irregular wave loads.

For the application of wave theories, Le [10] deemed that the Stokes wave theory was applicable for deep water, whereas the cnoidal wave theory was more suitable for shallow water. Zhou et al. [11] studied the cnoidal wave-induced excess pore pressure and liquefaction phenomenon based on numerical results. Xu et al. [12] studied the response of pore pressure on the seabed under the action of cnoidal waves by analyzing parameters. Hsu et al. [13] derived the seabed pressure solutions yielded by cnoidal waves up to the third order and proposed an analytical model to study the wave-induced seabed response under higher-order cnoidal wave theory.

2. Established Liquefaction Criterion

There are three main liquefaction determination criteria for the seabed.

Based on the principle of effective stress, Okusa [14] stated that if the vertical effective stress of soil at a certain depth was greater than its effective overburden pressure, the soil was liquefied.

$$-({\gamma }_s-{\gamma }_w)z-{\sigma }_z^{\prime }\le 0$$

(1)

where γ_s is the weight density of the seabed, γ_w is the weight density of the seawater, ${\sigma }_z^{\prime }$ is the vertical effective stress at depth z, and z is the height above the seabed.

The criterion of the three dimensions was further considered by Tsai and Lee [15],

$$-\frac{1}{3}(1+2K_0)({\gamma }_s-{\gamma }_w)z-\frac{1}{3}({\sigma }_x^{\prime }+{\sigma }_y^{\prime }+{\sigma }_z^{\prime })\le 0$$

(2)

where K₀ is the lateral pressure coefficient.

Zen and Yamazaki [16] stated that the soil was liquefied when the effective overburden pressure of the upper soil was less than its upward seepage force. They suggested that the soil liquefaction criterion under the action of 2D translator waves was given by

$$-({\gamma }_s-{\gamma }_w)z+\left(p_w-p\right)\le 0$$

(3)

where p_w is the wave pressure on the surface of the seabed in the same horizontal position, and p is the pore water pressure at depth z.

The criterion was promoted to three dimensions by Jeng [17]

$$-\frac{1}{3}(1+2K_0)({\gamma }_s-{\gamma }_w)z+\left(p_w-p\right)\le 0$$

(4)

Lin et al. [18] ignored horizontal shear stress and vertical lift, and defined the ratio of the pore water pressure gradient at z = 0 to the bulk density of soil as the liquefaction parameter LF. The soil was liquefied when LF ≥ 1.

$$LF=-{\left.\frac{dp}{dz}\right|}_{z=0}/({\gamma }_s-{\gamma }_w)$$

(5)

This method is only applicable for the determination of the liquefaction of the seabed surface. Moreover, the effect of shear stress on initial liquefaction is not taken into account, and so even if LF < 1, the seabed soil may still be liquefied.

The problem of instantaneous liquefaction of the seabed around buried pipelines in shallow water still needs further investigation, which is the main purpose of this paper.

3. The Approximate Cnoidal Wave Theory

3.1. Approximate Cnoidal Wave and Solution

The characteristics of waves must be considered before researching the wave-induced liquefaction of a seabed. When the cnoidal wave theory is studied, the following assumptions are adopted for simplicity and a clear description:

(a)

The fluid is an incompressible ideal fluid in a two-dimensional space.

(b)

The effect of gravity and the movement of the wave are irrotational.

(c)

The surface of the seabed is a horizontal impervious boundary.

(d)

The waves disappear in the infinite distance of the x direction.

(e)

The waves can retain their individual form for an indefinite period.

Based on the above assumptions, the solution to the cnoidal wave equation is given as follows [19]

$$\eta =z_s-d=H \mathrm{c}{\mathrm{n}}^2\left(\sqrt{\frac{3H}{4k^2{d}^2\left(d+\frac{3}{4}H\right)}}\left(x-t\sqrt{g\left(d+H\right)}\right), k\right)+\frac{16}{3}\frac{d^3}{L^2}K\left(K-E\right)-H$$

(6)

where η is the fluctuation-free surface, as defined in Figure 1; z_s is the height of the wave surface above the hydrostatic level, as defined in Figure 1; d is the still water depth; H is the wave height; L is the wavelength; K is the complete first-order elliptic integral; E is the complete second-order elliptic integral; cn(‧) is the Jacobi elliptic function; k is the modulus of the Jacobi elliptic function; x is the horizontal coordinate axis, as defined in Figure 1; t is time; and g is the gravitational acceleration.

Adopting the first-order approximation,

$$\eta \approx H{\mathrm{cn}}^2\left(\sqrt{\frac{3H}{4k^2{d}^2\left(d+\frac{3}{4}H\right)}}\left(x-t\sqrt{g\left(d+H\right)}\right), k\right)$$

(7)

The horizontal velocity of the water quality point in the flow field, u, can be written as

$$u=\frac{\partial \varphi }{\partial x}=\frac{\eta \sqrt{gd}}{d}$$

(8)

where ϕ is the velocity potential function.

According to Equations (6) and (7)

$$\frac{1}{\sqrt{gd}}\frac{\partial \varphi }{\partial x}=\frac{H}{d}\mathrm{c}{\mathrm{n}}^2\left(\sqrt{\frac{3H}{4k^2{d}^2\left(d+\frac{3}{4}H\right)}}\left(x-t\sqrt{g\left(d+H\right)}\right), k\right)+\frac{16}{3}\frac{d^2}{L^2}K\left(K-E\right)-\frac{H}{d}$$

(9)

The basic equation of wave motion is

$$\frac{p_f}{{\gamma }_w}=z_s-z$$

(10)

It can be seen from Figure 1 that

$$z_s=d+\eta$$

(11)

Substitute Equation (7) into Equation (11), then

$$z_s=d+H \mathrm{c}{\mathrm{n}}^2\left(\sqrt{\frac{3H}{4k^2{d}^2\left(d+\frac{3}{4}H\right)}}\left(x-t\sqrt{g\left(d+H\right)}\right), k\right)$$

(12)

According to Equations (10) and (12), the pressure of the fluid at different locations can be written as

$$p_f={\gamma }_{w}H \mathrm{c}{\mathrm{n}}^2\left(\sqrt{\frac{3H}{4k^2{d}^2\left(d+\frac{3}{4}H\right)}}\left(x-t\sqrt{g\left(d+H\right)}\right), k\right)+{\gamma }_w\left(d-z\right)$$

(13)

3.2. Calculation Results of Approximate Cnoidal Wave

In this study, the approximate cnoidal wave pressure is calculated using MATLAB. As shown in Figure 2, it can be found that the distance between two crests is about 53 m when the wave period is 8 s. Meanwhile, the distance between two crests arrives at 89 m when the wave period is 13 s. The distance between two crests increases as the wave period increases. From Figure 2 it can also be seen that the shape of troughs becomes longer and flatter, while the shape of the crests become more convex when the wave period T increases from 8 s to 13 s.

3.3. Comparison between Linear Wave and Approximation Cnoidal Wave

The comparison between the wave process curve, calculated with the theory of the linear wave, and the approximation cnoidal wave is shown in Figure 3. There is little difference between the results of the linear wave and the approximation cnoidal wave when the values of $T\sqrt{g/d}$ and $H/d$ are lower. However, when the values of $T\sqrt{g/d}$ and $H/d$ are large, there are significant differences in the wave surface process lines between the two theories, and the nonlinearity of the original results is more obvious. Due to the offshore waves having strong nonlinear characteristics, it is necessary to adopt nonlinear wave theories such as cnoidal wave.

3.4. Verification of the Approximate Cnoidal Wave Theory

First, to check the approximate cnoidal wave theory, the free surface profile (the results were calculated according to Equation (7)) is compared with the experimental data of Chang et al. [20] in Figure 4. The cnoidal wave conditions for laboratory experiments are as follows: the wave period T is 2.0 s; the wave height H is 3.6 cm; the water depth d is 24.0 cm; the wavelength L is 2.97 m. The free surface elevation of the cnoidal waves is measured by using a wave gauge located at x = 4.8 m (x is the location of the wavenumber at rest) from the location of the wave-maker. As shown in Figure 4, although there are some differences between the existing results and the experimental data, the general trend is consistent.

4. The Liquefaction Criterion

4.1. Calculation of Pore Pressure around the Pipeline

Supposing that the seabed is infinitely deep and the soil is homogeneous, the governing equation and boundary conditions for the pore water pressure of p can be written as follows [21]:

$$\left\{\begin{array}{l}{\nabla }^{2}p=\frac{{\gamma }_w}{k_s}\left(\frac{n_s}{K_w}+\frac{1}{2G}\frac{1-2\nu }{1-\nu }\right)\frac{\partial p}{\partial t}\\ p=p_w, z=d_p\\ \frac{\partial p}{\partial r}=0, r=R\\ p=0, z\to -\infty \end{array}\right.$$

(14)

where ∇² is the two-dimensional Laplace operator; p is pore water pressure; γ_w is the weight density of the seawater; $k_s$ is the coefficient of permeability; $n_s$ is porosity; K_w is the bulk modulus of water (K_w = 2.18 × 10⁹ Pa); G is the shear modulus of soils; ν is Poisson’s ratio of soils; and z is the depth into the soils, as defined in Figure 5. $d_p$ is the buried depth to the center line of the pipe. r = (x² + z²)^1/2 is defined in Figure 5. x is the horizontal coordinate axis, as defined in Figure 5. R is the radius of the pipe.

When calculating the pore water pressure of the seabed surface, the viscous and frictional forces of water are always ignored. Consequently, the vertical effective stress and shear stress on the surface of the seabed are not considered. Thus, it is approximately equal to the wave pressure at the bottom of the seabed caused by waves. That is,

$$p_w={\gamma }_{w}H \mathrm{c}{\mathrm{n}}^2\left(\sqrt{\frac{3H}{4k^2{d}^2\left(d+\frac{3}{4}H\right)}}\left(x-t\sqrt{g\left(d+H\right)}\right), k\right)+{\gamma }_{w}d$$

(15)

The fluid in the seabed will scatter when it reaches pipelines. Therefore, p can be divided into two parts: the pore water pressure caused by waves, denoted by p₁, and the perturbation pressure induced by the pipe, denoted by p₂.

$$p=p_1+p_2$$

(16)

When there is no pipeline in the seabed, the p₁ caused by waves meets the following governing equations and corresponding boundary conditions.

$$\left\{\begin{array}{l}\frac{{\partial }^2{p}_1}{\partial x^2}+\frac{{\partial }^2{p}_1}{\partial z^2}=C_s\frac{\partial p_1}{\partial t}\\ p_1=p_w, z=d_p\\ p_1=0, z\to -\infty \end{array}\right.$$

(17)

According to the boundary conditions, p₁ can be easily obtained.

$$p_1=\left\{{\gamma }_{w}H \mathrm{c}{\mathrm{n}}^2\left(\sqrt{\frac{3H}{4k^2{d}^2\left(d+\frac{3}{4}H\right)}}\left(x-t\sqrt{g\left(d+H\right)}\right), k\right)+{\gamma }_{w}d\right\}{e}^{C_s\left(z-d_p\right)}$$

(18)

where $C_s=\frac{{\gamma }_w}{k_s}\left(\frac{n_s}{K_w}+\frac{1}{2G}\frac{1 - 2\nu }{1 - \nu }\right)$.

Usually, each pipe is a circular tube; thus, the coordinates of the governing equations are transformed for convenience. Consequently, the governing equations and boundary conditions of p₂ are

$$\left\{\begin{array}{l}\frac{{\partial }^2{p}_2}{\partial r^2}+\frac{1}{r}\frac{\partial p_2}{\partial r}+\frac{1}{r^2}\frac{{\partial }^2{p}_2}{\partial {\theta }^2}=C_s\frac{\partial p_2}{\partial t}\\ p_2=0, r=d_p, \theta =\frac{\pi }{2}\\ \frac{\partial p_2}{\partial r}=-\frac{\partial p_1}{\partial r}, r=R\\ p_2=0, r\to -\infty \end{array}\right.$$

(19)

Using the separation of variables method, the perturbation pressure p₂ caused by pipelines can be expressed as follows.

$$p_2=-\left\{{\gamma }_{w}H \mathrm{c}{\mathrm{n}}^2\left(\sqrt{\frac{3H}{4k^2{d}^2\left(d+\frac{3}{4}H\right)}}\left(x-t\sqrt{g\left(d+H\right)}\right), k\right)+{\gamma }_{w}d\right\}{e}^{C_s\left(z-d_p\right)}\frac{J_n(\left(C_s\sin \theta +im\cos \theta \right)\left(d_p-r\right))}{J_n(\left(C_s\sin \theta +im\cos \theta \right)\left(d_p-R\right))}$$

(20)

where Jn(‧) is the first-order Bessel function.

The pore water pressure p at any point in the seabed can be approximately described as follows:

$$p=\left\{{\gamma }_{w}H \mathrm{c}{\mathrm{n}}^2\left(\sqrt{\frac{3H}{4k^2{d}^2\left(d+\frac{3}{4}H\right)}}\left(x-t\sqrt{g\left(d+H\right)}\right), k\right)+{\gamma }_{w}d\right\}{e}^{C_s\left(z-d_p\right)}\left(1-\frac{J_n(\left(C_s\sin \theta +im\cos \theta \right)\left(d_p-r\right))}{J_n(\left(C_s\sin \theta +im\cos \theta \right)\left(d_p-R\right))}\right)$$

(21)

Zen and Yamazaki [16] presented a further analysis of the determination standard of seabed liquefaction around pipelines in shallow water. In addition, this liquefaction criterion has been verified by the measured data of Luan et al., 2008 [22]; so, it is the most widely used method at present. Their criterion is written as Equation (3). According to Equations (3) and (21), the liquidation criterion for seabed around buried pipelines in shallow water can be described as follows:

$$-({\gamma }_s-{\gamma }_w)z+\left\{{\gamma }_{w}H \mathrm{c}{\mathrm{n}}^2\left(\sqrt{\frac{3H}{4k^2{d}^2\left(d+\frac{3}{4}H\right)}}\left(x-t\sqrt{g\left(d+H\right)}\right), k\right)+{\gamma }_{w}d\right\}{e}^{C_s\left(z-d_p\right)}\left(1-\frac{J_n(\left(C_s\sin \theta +im\cos \theta \right)\left(d_p-r\right))}{J_n(\left(C_s\sin \theta +im\cos \theta \right)\left(d_p-R\right))}\right)\le 0$$

(22)

4.2. Verification of the Present Model

Sudhan et al. [23] conducted a flume test during their investigation of the distribution situation of seepage pressure of the seabed around buried pipelines under wave force. The model material used for the seabed was sand (d₅₀ = 0.57 mm) and it was very uniform. The wave condition of the India East Sea coast was adopted in the test. The schematic of the test is shown in Figure 6. The values of the parameters used in the test are given in

Table 1. Experimental parameters and values.

Parameter	Value
The density of fluid ρw (kg/m3)	1025
Permeability ks (m/s)	8.1 × 10−4
Deformation modulus Es (N/m2)	4.8 × 107
Porosity ns	0.4
Poisson’s ratio υ	0.25
Pipeline buried depth dt (m)	0.18
The radius of pipe R (m)	0.1
The dry density of soil γd (kN/m3)	14.83

.

From Figure 7, it is clear that the pore water pressure around pipelines has a sinusoidal distribution; the largest dynamic pore pressure is at the top of the cylinder and the smallest is at the bottom. The theoretical values agree well with the test values.

The minor differences between the values are mainly due to the following: (a) because of the accuracy and sensitivity of the sensor in the test process, the measurement data also have errors; (b) the theory does not consider the influence of wave scattering, but wave scattering always exists in reality; (c) the theory does not consider the influence of the inertial effect of the acceleration of the soil and the pipeline.

Luan et al. [22] analyzed the wave-induced dynamic response of the seabed around the pipeline by using the finite element method. Their calculation boundary conditions are shown in Figure 8. The calculation parameters are given in

Table 2. Experimental parameters and values.

Parameter	Value
The density of fluid ρw (kg/m3)	1025
Permeability ks (m/s)	1 × 10−3
Deformation modulus Es (N/m2)	1 × 108
Porosity ns	0.4
Poisson’s ratio υ	0.33
Pipeline buried depth dt (m)	1.5
The radius of pipe R (m)	0.5
The density of soil ρs (kg/m3)	1600

.

It can be seen in Figure 9 that the theoretical calculation and the numerical results are slightly different because the finite element model of Luan et al. [22] considered the inertial effect of the acceleration of the soil and the pipeline. However, the results obtained by the two methods are very close and have the same fundamental trend. Thus, the analytical method proposed in this paper is feasible.

5. An Engineering Example

The Chengdao oilfield is located in southern Bohai. Based on the borehole sample in the research areas and the representative borehole data, the basic physical parameters of the seabed soil are shown in

Table 3. Properties of the seabed soil at Chengdao oilfield [24].

Parameter	Value
Density of fluid ρw (kg/m3)	1025
Elastic modulus E (MPa)	3.9
Density of seabed ρs (kg/m3)	1750
Poisson’s ratio υ	0.4
Porosity ns	1.052
Permeability ks (m/s)	10−8

.

The wave conditions of the different return period waves in the Chengdao oilfield are shown in

Table 4. Wave conditions of Chengdao oilfield [26].

d (m)	10 Year			50 Year
	H1/3 (m)	T (s)	L (m)	H1/3 (m)	T (s)	L (m)
3	1.8	8.1	42.6	1.8	8.6	45.4
4	2.3	8.1	48.6	2.3	8.6	51.9
5	3.0	8.1	53.8	3.0	8.6	57.5

. Due to Wang’s research [25], the wave condition is consistent with the actual condition. According to the research results of Le [10], the wave conditions satisfy the cnoidal wave theory.

The diameter of the Chengdao oilfield pipeline is 325 mm, and the buried depth is 0.66 m. The pore pressure at a point in the seabed under 10-year and 50-year return period waves is calculated using Equation (21). Further, the instantaneous liquefaction of the seabed around the pipelines is determined using Equation (22). The results are shown in Figure 10 and Figure 11.

The intersection point between the graph ${\left(p_w-p\right)}_{\max }~z$ and $\left({\gamma }_s-{\gamma }_w\right)z~z$ is the maximum instantaneous liquefaction depth calculation of the seabed under the action of waves in different water depths.

It can be seen from Figure 10 that the intersection point is between 1–2.0 m below the seabed. The effective stress in the soil corresponding to these intersections is approximately the same as the pore water pressure, and the soil is in the limit state. The soil is liquefied when the maximum instantaneous liquefaction depth of the soil in different depths of water is generally between 1 and 1.6 m under the action of 10-year return period effective waves. At a water depth of 5 m, the maximum instantaneous liquefaction depth of the soil is at its maximum, which is 1.54 m. At a water depth of 3 m, the maximum instantaneous liquefaction depth of the soil is at its minimum, which is 1.03 m. The maximum instantaneous liquefaction depth of the seabed soil increases with the water depth.

From Figure 11, the intersection point of the graph and under the action of the 50-year return period waves is between 1 and 2.0 m below the seabed. The maximum instantaneous liquefaction depth of the soil in different depths of water is generally in the range of 1–1.6 m under the action of 50-year return period effective waves. At a water depth of 5 m, the maximum instantaneous liquefaction depth of the soil is at its maximum, which is 1.56 m. At a water depth of 3 m, the maximum instantaneous liquefaction depth of the soil is at its minimum, which is 1.04 m.

Because the diameter of the pipeline is 325 mm, and the buried depth is 0.66 m, the pipeline is located in the seabed at depths of about 1 m. From Figure 10 and Figure 11, it can be seen that the submerged weight of the soil particle is less than the seepage pressure for different return period waves and water depths, and most of the soil around the pipeline would be momentarily liquefied. The pipeline loses stability more easily in the research area, and attention should be paid to the operation of the pipeline over time.

As can be seen from Figure 12, the maximum liquefaction depth under the 10-year return period wave is similar to that of the 50-year return period wave. This is because the elements of the wave tend to be consistent in different recurrence periods. From this figure, it is clear that the maximum liquefaction depth increased nonlinearly with the water depth.

6. Conclusions

In this paper, the approximate cnoidal wave theory was presented. Based on the Biot’s consolidation theory and the approximate cnoidal wave theory, analytical solutions for the pore water pressure around buried pipelines in the seabed with infinite depth were derived. Further, the criteria for soil liquefaction around pipelines under cnoidal waves are given, and the results of wave flume tests are compared with the theoretical results.

The results show that the pore water pressure in the seabed around a pipeline has a sinusoidal distribution under the action of nonlinear waves, and the largest value occurs at a position of 90 around the pipeline, whereas the minimum value exists at a position of 270. The results of the application of the theory to ascertain the stability of the pipeline in the Chengdao oilfield indicate that the pipeline will lose stability easily in the research area, and therefore attention should be paid to its operation over time.

It should be noted that the research on soil liquefaction in this paper ignores the influence of wave action time, and does not consider the residual liquefaction process caused by pore water pressure accumulation. However, for the highly permeable sandy seabed, the effect of pore pressure accumulation is not significant. Therefore, the model proposed in this paper applies to the stability judgment of buried pipes on a sandy seabed in shallow water.