An Analytical Solution by HAM for Nonlinear Simulation of Deepwater SCR Installation

Abstract

Steel catenary riser (SCR) is a cost-effective riser system that is widely used in deepwater offshore oilfields development. During SCR J-lay installation, the movement of pull-head must be carefully controlled to ensure riser safety. Since the SCR installation path calculation through numerical simulation software is usually time-consuming, this paper has established a mechanical model for SCR installation by making use of homotopy analysis method (HAM) to simplify its analytical solution, and dimensional analysis was considered in making initial guess solution. Based on this analytical solution, a program within the framework of MATLAB was developed to predict the two-dimensional riser behavior during installation, and a sensitivity analysis for different values of the control variables was carried out. Engineers may efficiently optimize the installation path by the application of this technique.

1. Introduction

In response to increasing global demand of energy from fossil fuels and the replacement of depleting oil and gas reserves in most matured fields in the world, operating companies in the oil industry are expanding their exploration and production operations into deepwater. In deepwater exploration, SCRs are widely used as a cost-effective riser system which is connecting offshore platforms and subsea production systems. During SCR installation as shown in Figure 1, the pull-head is transferred from installation vessel to platform by Abandon & Recovery (A&R) wire from installation vessel and pull-in wire from the platform. The transfer process is usually carried out by J-lay vessel, since the S-lay vessel cannot install SCR independently [1, 2]. The transfer process is divided into prelay and postlay, depending on whether the offshore platform is on site [3].

Figure 1 

SCR installation procedure.

The shape of SCR will be affected by the route of pull-head during installation controlled by both the A&R and the pull-in wire, and the maximum stresses to be encountered during the transfer process must be obtained before installation [4]. A large number of papers have been published on this issue. The catenary theory is a simple model for evaluating the tension and curvature of SCR [5], but it cannot simulate the rapid change of the inclination angle because it ignores the bending stiffness [6]. The nonlinear large deformation beam theory is more appropriate for simulating the riser near touchdown point (TDP) considering the large-angle deformation [7, 8]. Dixon and Rultledge [9] applied Plunkett’s expansions in analyzing J-lay method. Guarracino and Mallardo [10] developed the expansions to analyze the S-lay method. Dai et al. [11] used line integration technique to analyze the tensions of A&R wire while considering the movement of installation vessel. Xing et al. [12] established a nonlinear mechanical model to analyze the pipeline lifting process and applied a shooting method in solving the moving boundary problem. Lenci and Callegari [13] investigated some analytical models to analyze the J-lay method, but the solution is not easy to obtain because the equations are highly nonlinear. García-Palacios et al. [14] used two-dimensional Navier-Bernoulli beam to analyze pipeline laying process, and an updated Lagrangian formulation for the nonlinear analysis was obtained. Nowadays, finite element software such as OrcaFlex is commonly used to solve optimal installation path problem. But it might be time-consuming depending on the performance of computer. Therefore, a simple and effective SCR installation model is still important for engineers to understand the nature and relevance of the complex phenomena during installation. Thus, an analytical solution for SCR installation can substantially reduce design time and provide a quick evaluation of actual installation setting.

The structural model presented in this paper is a simple and practical method to obtain the static configuration and the mechanical load parameters for SCR transfer process. The bending stiffness and large deformation of the part suspended in water are taken into consideration. The governing equation system is derived and the analytical approximate solution is obtained by means of HAM. Compared with the available commercial software such as OrcaFlex, the calculation of the present model has the advantage of high stability and being time-saving. A series of parameters such as initial installation angle, maximum lower depth of pull-head, water depth, and distance between installation vessel and offshore platform are considered during the stress analysis of SCR installation.

2. Nonlinear Model of SCR Installation

The mechanical model used to simulate the behavior of transfer process during SCR installation is composed of two parts, as presented in Figure 2. In pull-head control part, the catenary theory is used to simulate the A&R wire and pull-in wire [15]. In SCR part, the nonlinear large deformation beam theory is used to simulate the suspended segment [16]. The hypotheses considered in the model are summarized as follows.(1)The dynamic movement of the installation vessel and platform are not considered.(2)The gravitational and hydrostatic forces are the only loads upon the riser during installation operations.(3)The SCR material is linear elastic, and the behavior of SCR is modeled as a two-dimensional beam subjected to axial and bending deformations. Torsional and shear deformation are not considered.(4)The seabed is rigid. Two parts of the model are solved in the local coordinate system X′O′Y′, respectively and transformed to the global coordinate system XOY, finally. The TDP of SCR is the origin of local coordinate system X′O′Y′.

Figure 2 

Mechanical model for transfer process of SCR installation.

2.1. Pull-Head Control Part

The location of pull-head is important for SCR shape control. It is controlled by the length of A&R wire from installation vessel and pull-in wire from platform. Based on catenary theory as shown in Figure 3, the following equations can be obtained: 𝑆=𝑦²𝑇+2^𝑠𝑤𝑦𝑇𝑥=^𝑆𝑤𝑤ln𝑇_𝑆𝑦+1+𝑤𝑇_𝑆𝑦+1²−1, (1)(2) where 𝑇 is axial tension at pull-head, 𝑇_𝑠 is axial tension at TDP, 𝑤 is the submerged weight per unit length of the riser, and 𝑆 is the length of SCR.

Figure 3 

Catenary axis and forces.

The transfer process is typically carried out by two steps. The first step is to lower the pull-head by increasing the length of A&R wire from installation vessel, and the second step is to pull-in the pull-head by decreasing the length of pull-in wire from platform.

The geometrical relationship in lowering step is shown in Figure 4. Assuming the initial position of pull-head before transfer process is at (𝑥₀, 𝑦₀), the increasing length of A&R wire Δ𝐿 will change the pull-head position to (𝑥₁, 𝑦₁), 𝑆₀ is the suspended segment length of the SCR at the initial position, 𝑆₁ is the suspended segment length of the SCR after lowering, Δ𝑥 is the horizontal position’s change of TDP, Δ𝑥₁ is the horizontal length of 𝑆₁, Δ𝑥₂ is the horizontal position change of pull-head, and Δ𝑦 is the vertical position change of pull-head. The following relations can be obtained: 𝑥₀=Δ𝑥+Δ𝑥₁+Δ𝑥₂𝑠₀=Δ𝑥+𝑠₁. (3)(4)

Figure 4 

Geometrical relationship for lowering step.

Based on (1)–(4), the pull-head control model for the lowering step can be obtained: 𝑥₁=𝑦₀√1+2𝐴−𝑦₁√1+2𝐵+𝑦₁1𝐵ln𝐵+1+1𝐵+1²−1+Δ𝐿cos𝜃₁, (5) where 𝐴=cos𝜃₀/(1−cos𝜃₀), 𝐵=(1−cos𝜃₁)/cos𝜃₁.

The geometrical relationship in pull-in step is shown in Figure 5. Assuming the initial position of the installation vessel is at (𝑥_𝑠,𝑦_𝑠), with the initial position of platform being at (𝑥_𝑣,𝑦_𝑣), while the initial position of pull-head is at (𝑥₀,𝑦₀), the increasing length of pull-in wire Δ𝐿 will change the pull-head position to (𝑥₁,𝑦₁). 𝑆₀ is the suspended segment length of the SCR at initial position, and 𝑆₁ is the suspended segment length of the SCR after pull-in, whereas Δ𝑥 is the horizontal position’s change of TDP, Δ𝑥₁ is horizontal length of 𝑆₁, and Δ𝑥₂ is horizontal position’s change of pull-head. Therefore, the following equations can be obtained: 𝐿_𝑆=𝑥_𝑆−𝑥₀²+𝑦_𝑆−𝑦₀²=𝑥_𝑆−𝑥₁²+𝑦_𝑆−𝑦₁²Δ𝐿=𝑥_𝑣−𝑥₀²+𝑦_𝑣−𝑦₀²−𝑥_𝑣−𝑥₁²+𝑦_𝑣−𝑦₁²𝑥₀=Δ𝑥+Δ𝑥₁+Δ𝑥₂. (6)(7)(8)

Figure 5 

Geometrical relationship for pull-in step.

By combining (1), (2), and (6) in (8), the pull-head control model for the pull-in step can be obtained: 𝑥₁=𝑦₀√1+2𝐴−𝑦₁√1+2𝐵+𝑦₁1𝐵ln𝐵+1+1𝐵+1²−1. (9)

2.2. SCR Part

An infinitesimal element with length 𝑑𝑠 based on the nonlinear large deformation beam theory is presented in Figure 6. The force equilibrium equation normal to the segment’s axial direction is established by (10), and the force equilibrium equation in the segment’s axial direction is established by (11). Consider 𝛿𝐹−𝑇𝛿𝜃+𝑤𝛿𝑠cos𝜃=0𝛿𝑇=𝑤𝛿𝑠sin𝜃, (10)(11) where 𝐹 is shear force (𝐹=𝑑𝑀/𝑑𝑠), 𝑇 is axial tension, and 𝑀 is bending moment. Then, (10) can be written as follows: 𝑑²𝑀𝑑𝑠²−𝑇𝑑𝜃𝑑𝑠+𝑤cos𝜃=0. (12)

Figure 6 

Forces on a large deformation beam segment 𝛿𝑠.

According to large deformation beam theory, the curvature 1/𝑅=𝑀/EI=𝑑𝜃/𝑑𝑠, where EI is the flexural rigidity. Therefore, (12) can be written in the following form: 𝑑²𝑑𝑠²EI𝑑𝜃𝑑𝑠−𝑇𝑑𝜃𝑑𝑠+𝑤cos𝜃=0. (13)

By simplifying (11) and (13), the mechanical model for large deformation beam can be obtained as follows: 𝑑EI³𝜃𝑑𝑠³−𝑇𝑑𝜃𝑑𝑠+𝑤cos𝜃=0𝑑𝑇𝑑𝑠=𝑤sin𝜃. (14)(15)

The boundary condition at top point of riser is 𝑑𝜃_𝑇/𝑑𝜀=0, and the boundary conditions at TDP are 𝜃_𝐷=𝜃^∗, 𝑑𝜃_𝐷/𝑑𝜀=sin𝜃^∗, and 𝑇(𝜀)=𝐹_𝑇cos𝜔_𝑇cos𝜃^∗, where 𝐹_𝑇 is the lifting load at the top of riser, 𝜔₀ is the angel between lifting load and the 𝑥-axis, 𝜃^∗ is the angel between riser and the 𝑥-axis at TDP.

Equation (14) is a third-order nonlinear differential equation with an unknown variable 𝜃. It can be solved using numerical method such as the finite element method [17] or the finite difference method [18]. However, we provide the analytical solution by HAM in this paper.

3. HAM Solution for SCR Installation Model

3.1. Basic Idea of HAM

Perturbation technique has been widely used for nonlinear problem [19] which is dependent on small physical parameters. Liao [20] proposed a general analytical method known as HAM for nonlinear problems by using the basic ideas of the homotopy in topology. HAM provides us with great freedom to select proper base functions for approximate solutions of nonlinear problems and a simple way to get enough accurate analytical approximations.

For the nonlinear differential equations with general form: 𝐴[𝑢(𝑥)]=0𝑥∈𝑅^𝑛. (16)

𝐴 is the nonlinear operator for all equations, 𝑢(𝑥) is an unknown solution for all equations, and 𝑥 is the independent variables. Based on the basic concept of HAM, the embedded parameter 𝑝∈[0,1] and the initial guess of the exact solution 𝑢₀(𝑥) are introduced. The homotopy which is the so-called zero-order deformation equation can be constructed as follows: 1−𝑝𝐿𝜙𝑥;𝑝−𝑢₀(𝑥)=𝑝⁻ℎ𝐻(𝑥)𝐴𝜙𝑥;𝑝𝑝∈[0,1], (17) where ₋ℎ is a nonzero auxiliary parameter, 𝐻(𝑥) is a nonzero auxiliary function, 𝐿 is an auxiliary linear operator that satisfies the property 𝐿[0]=0, and 𝜙(𝑥;𝑝) is an unknown function. It is obvious that, when 𝑝=0 and 𝑝=1, the following relations hold, respectively: 𝜙(𝑥;0)=𝑢₀(𝑥),𝜙(𝑥;1)=𝑢(𝑥). (18)

Thus, as 𝑝 increases from 0 to 1, the solution 𝜙(𝑥;𝑝) varies from the initial guess 𝑢₀(𝑥) to the accurate solution 𝑢(𝑥). Expand 𝜙(𝑥;𝑝) by Taylor’s theorem in a power series of 𝑝: 𝜙𝑥;𝑝=𝑢₀(𝑥)+^+∞_𝑘=1𝑢_𝑘(𝑥)𝑝^𝑘, (19) where 𝑢_𝑘(𝑥)=(1/𝑘!)(𝜕^𝑘𝜙(𝑥;𝑝)/𝜕𝑝^𝑘)|_𝑝=0.

Assuming that 𝐿, 𝑢₀(𝑥), ₋ℎ, and 𝐻(𝑥) are properly chosen; the power series (19) converges at 𝑝=1, and the solution series can be obtained as follows: 𝑢(𝑥)=𝑢₀(𝑥)+^+∞_𝑘=1𝑢_𝑘(𝑥). (20)

Differentiating the zero-order deformation equation (17) 𝑚 times with respect to 𝑝, dividing by 𝑚!, and setting 𝑝=0, 𝑚th-order deformation equation can be obtained: 𝐿𝑢_𝑘(𝑥)−𝜒_𝑘𝑢_𝑘−1(𝑥)₋=ℎ𝐻(𝑥)𝑅_𝑘(𝑥), (21) where 𝜒_𝑘=𝑅0,𝑘⩽11,𝑘>1,_𝑘(𝑥)=1()!𝜕𝑘−1^𝑘−1𝐴𝜙𝑥;𝑝𝜕𝑝^𝑘−1_𝑝=0. (22)

3.2. Solution of the Mechanical Model for SCR Part by HAM

A dimensionless parameter 𝜀=𝑠/𝑙 is introduced to (14) and (15), and HAM is used to obtain the analytical approximation of this mechanical model.

Assuming that 𝜑^𝜃(𝜀,𝑞) is the homotopy which is connected to the original equation, solution 𝜃(𝜀), and the initial guess solution 𝜃₀(𝜀), 𝜑^𝑇(𝜀,𝑞) is the homotopy which is connected to the original equation solution 𝑇(𝜀) and the initial guess solution 𝑇₀(𝜀). The embedded parameter 𝑞∈[0,1], the nonzero auxiliary parameters ₋ℎ_𝜃 and ₋ℎ_𝑇, the auxiliary linear operators 𝐿_𝜃 and 𝐿_𝑇, and the nonzero auxiliary functions 𝐻_𝜃(𝜀) and 𝐻_𝑇(𝜀) are introduced. The nonlinear operators for (14) and (15) are 𝑁_𝜃𝜑^𝜃𝜀,𝑞,𝜑^𝑇=𝜕𝜀,𝑞₃𝜑^𝜃𝜀,𝑞𝜕𝜀³−𝑙²𝜑EI^𝑇𝜀,𝑞𝜕𝜑^𝜃𝜀,𝑞+𝜕𝜀𝑤𝑙³𝜑EIcos^𝜃𝑁𝜀,𝑞_𝑇𝜑^𝜃𝜀,𝑞,𝜑^𝑇=𝜀,𝑞𝜕𝜑^𝑇𝜀,𝑞𝜑𝜕𝜀−𝑙𝑤sin^𝜃𝜀,𝑞. (23)

Thus, zero-order deformation equations can be obtained as follows: 1−𝑞𝐿_𝜃𝜑^𝜃𝜀,𝑞−𝜃₀(𝜀=₋)ℎ_𝜃𝐻_𝜃(𝜀)𝑞𝑁_𝜃𝜑^𝜃𝜀,𝑞,𝜑^𝑇𝐿𝜀,𝑞1−𝑞_𝑇𝜑^𝑇𝜀,𝑞−𝑇₀(𝜀=⁻)ℎ_𝑇𝐻_𝑇(𝜀)𝑞𝑁_𝑇𝜑^𝜃𝜀,𝑞,𝜑^𝑇𝜀,𝑞 (24) which satisfies the initial conditions 𝜑^𝜃(𝜀,0)=𝜃₀(𝜀) and 𝜑^𝑇(𝜀,0)=𝑇₀(𝜀).

Applying (20) to this case, we can write 𝜃(𝜀)=𝜑^𝜃(𝜀,1)=𝜑^𝜃(𝜀,0)+^+∞_𝑚=11𝜕𝑚!𝜕𝑞^𝑚𝑚𝜑^𝜃𝜀,𝑞_𝑞=0=𝜃₀(𝜀)+^+∞_𝑚=1𝜃_𝑚(𝜀)𝑇(𝜀)=𝜑^𝑇(𝜀,1)=𝜑^𝑇(𝜀,0)+^+∞_𝑚=11𝜕𝑚!𝜕𝑞^𝑚𝑚𝜑^𝑇𝜀,𝑞_𝑞=0=𝑇₀(𝜀)+^+∞_𝑚=1𝑇_𝑚(𝜀), (25) where 𝜃_𝑚(𝜀)=(1/𝑚!)(𝜕^𝑚/𝜕𝑞^𝑚)𝜑^𝜃(𝜀,𝑞)|_𝑞=0 and 𝑇_𝑚(𝜀)=(1/𝑚!)(𝜕^𝑚/𝜕𝑞^𝑚)𝜑^𝑇(𝜀,𝑞)|_𝑞=0.

The 𝑚th-order deformation equations (21) for this particular case are 𝐿_𝜃𝜃_𝑚(𝜀)−𝜒_𝑚𝜃_𝑚−1(𝜀=₋)ℎ_𝜃𝐻_𝜃(𝜀)𝑅^𝜃_𝑚𝜃_𝑚−1(𝜀),𝑇_𝑚−1(𝜀𝐿_𝑇)𝑇_𝑚(𝜀)−𝜒_𝑚𝑇_𝑚−1(𝜀=⁻)ℎ_𝑇𝐻_𝑇(𝜀)𝑅^𝑇_𝑚𝜃_𝑚−1(𝜀),𝑇_𝑚−1(𝜀), (26) where 𝜒_𝑘=𝑅0,𝑘⩽11,𝑘>1,_𝜃𝑚𝜃_𝑚−1(𝜀),𝑇_𝑚−1(𝜀)=𝜃^′′′_𝑚−1(𝜀)−𝑙²EI^𝑚−1_𝑘=0𝑇_𝑘(𝜀)𝜃^′_{𝑚−1−𝑘}(𝜀)+𝑤𝑙³()!𝜕EI𝑚−1𝜕𝑞^{𝑚−1𝑚−1}cos𝜑^𝜃𝜀,𝑞_𝑞=0,𝑅^𝑇_𝑚𝜃_𝑚−1(𝜀),𝑇_𝑚−1(𝜀)=𝑇^′_𝑚−1(𝜀)−𝑙𝑤()!𝜕𝑚−1𝜕𝑞^{𝑚−1𝑚−1}sin𝜑^𝜃𝜀,𝑞_𝑞=0. (27)

We select the nonzero auxiliary function as 𝐻_𝜃(𝜀)=𝑠²EI𝑇,𝐻_𝑇(𝜀)=1.EI𝑇 (28)

Referring to the catenary equation, we select the initial guess solution 𝜃₀(𝜀) and 𝑇₀(𝜀) as 𝜃₀(𝜀)=arctan𝑎𝑤𝑙𝜀^𝑘𝐺,𝑇₀(𝜀)=(𝑙𝑤𝜀)²+𝐺², (29) where 𝐺=𝐹₀cos𝜔₀ and 𝑎 and 𝑘 are adjustment parameters which is related to the length of riser 𝑙. Through dimensional analysis, the following relationship can be obtained: 𝑙𝑙₀=EI𝑤₀EI₀𝑤^1/3. (30)

If we select the length of riser 𝑙₀ under the condition EI₀=50 and 𝑤₀=350.59, the adjustment parameters 𝑎 and 𝑘 can be expressed with respect to 𝑙₀: 𝑎=1.02755−1.84502𝑒^{−𝑙0/1.43616}𝑙₀≤60.947+0.009𝑙₀𝑙₀≤101.035𝑙₀≤141.08931−0.00385𝑙₀𝑙₀≤230.98683+688.37803𝑒^{−𝑙0/2.11687}𝑙₀≤260.96059+0.00113𝑙₀𝑙₀≤401𝑙₀>40𝑘=0.00541𝑒^{𝑙0/0.48212}+1.35738𝑙₀≤21.28217𝑒^{−𝑙0/2.36014}−0.00781𝑙₀+1.19157𝑙₀≤251𝑙₀>25. (31)

Considering the boundary conditions at TDP (𝜀=0) and top point of riser (𝜀=1) in mechanical model for part I, the following boundary conditions can be obtained: (𝜀)(𝜀)𝜀=0:𝜃=0,𝑑𝜃(𝜀)𝑑𝜀=0,𝑇=𝐹₀cos𝜔₀(𝜀)𝜀=1:𝑑𝜃𝑑𝜀=0. (32)

When the nonzero auxiliary parameters are selected as ₋ℎ_𝜃=0.1 and ₋ℎ_𝑇=1, the analytical approximate solution of 𝜃(𝜀) and 𝑇(𝜀) can be written as 𝜃(𝜀)=arctan𝑎𝑤𝑙₀𝜀^𝑘𝐺₋+ℎ_𝜃𝜀²EI₀𝑇×𝐶𝐷−3𝐶−𝑘𝐷8𝑎²𝜀^2𝑘𝑙²₀𝑤²₀𝐶𝐺²𝐷²+2𝐶𝑘²𝐷+6𝑎²𝑠^3𝑘𝑙²₀𝑤²₀𝐶𝜀^𝑘𝑘𝐺²𝐷²+8𝑎⁴𝑠^5𝑘𝑙⁴₀𝑤⁴₀𝐶𝜀^𝑘𝐺⁴𝐷³+𝑙³₀𝑙₀𝑤₀𝑠²+𝐺^20.5𝜀²𝐶EI₀𝑘²𝑙₀𝐷+𝑙³₀𝑤₀EI₀𝐷^0.5𝑇(𝜀)=(𝑙𝑤𝜀)²+𝐺²+⁻ℎ_𝑇×𝑙EI𝑇₂𝑤²𝜀(𝑙𝑤𝜀)²+𝐺^20.5−𝑙𝑤sinarctan𝑤𝑙𝜀𝐺, (33) where 𝐶=𝑎𝜀^𝑘𝑘³𝑙₀𝑤₀/𝜀³𝐺, 𝐷=1+(𝑎²𝜀^2𝑘𝑙²₀𝑤²₀/𝐺²).

4. Results and Discussion

In order to implement the SCR installation model as described above, a computer program has been developed within the framework of MATLAB language program. The efficiency and accuracy of this model is verified by comparison with the numerical results of OrcaFlex. The parameters of SCR used for verification and analysis are detailed in Table 1.

Table 1 

Parameters of SCR.

The overall configuration and axial tension calculated by the proposed method and OrcaFlex are compared as shown in Figures 7 and 8, respectively. In different water depth conditions, the results are well coincided, which prove that the analytical method is accurate and can be used to analyze SCR installation problem. Some critical parameters are summarized in Table 2, where 𝑋 is horizontal distance from TDP to the top point of riser, 𝑌_top is vertical distance from seabed, and 𝑇_max is maximum axial tension at the top point of riser.

Table 2 

Critical results.

Figure 7 

Configuration results comparison between HAM and OrcaFlex in different water depth.

Figure 8 

Axial tension results comparison between HAM and OrcaFlex in different water depth.

Postlay method is selected for SCR installation parameters analysis. In this method, the offshore platform is on site. When SCR is laid near to the offshore platform by installation vessel, SCR pull-head is firstly connected to the A&R wire and the cable from offshore platform, which is lifted only by the A&R wire. Increase the length of A&R wire to lower pull-head to the maximum depth and then decrease the length of cable from offshore platform until pull-head is finally lifted to the hang-off position. Note that the dynamic positioning (DP) system always maintains the installation vessel and offshore platform position in the original place during SCR installation.

The safety of SCR during installation is affected by many factors. The most important one is that the variation of SCR maximum stress caused by the change of installation shape should be within the allowable stress. In order to investigate the influence of the initial installation angle, the maximum lower depth of pull-head, water depth, and the distance between installation vessel and offshore platform, the parameters shown in Table 1 are used for SCR, with following hypotheses.(1)Assume that the initial installation angle is 𝜃_𝐼, the maximum lower depth of pull-head is 𝐷_𝐿, water depth is 𝐷, and the distance between installation vessel and offshore platform is 𝐷_𝑉.(2)Assume that the horizontal position of pull-head is 𝑋_ph, which has zero value at the position of the installation vessel and has maximum value at the position of the offshore platform.(3)Assume that the vertical position of pull-head is 𝑌_ph, which has zero value at sea level and has maximum value at 𝐷_𝐿.

The maximum stress results under different conditions are summarized in Table 3.

Table 3 

Maximum stress results.

4.1. Influence of 𝜃_𝐼

Usually, deepwater SCR is installed by J-lay vessel, and the installation angle (the angle between the J-lay tower and the vertical direction) will affect the SCR installation shape. Under the conditions that 𝐷_𝐿 is 30 m, 𝐷_𝑉 is 35 m, 𝐷 is 1000 m, and 𝜃_𝐼 is 8°, 10° and 12°, respectively, the maximum stresses of SCR during installation are shown in Figure 9. It clearly shows that, with the small rise of 𝜃_𝐼, the maximum stresses on SCR increase rapidly. It can be observed that the J-lay method is more suitable for deepwater SCR installation than S-lay method, since it can control the 𝜃_𝐼 nearly to zero.

Figure 9 

Maximum stresses in different 𝜃𝐼.

4.2. Influence of 𝐷_𝐿

Under the condition that 𝜃_𝐼 is 10°, 𝐷_𝑉 is 35 m, 𝐷 is 1000 m, and 𝐷_𝐿 is 30 m, 33 m, and 36 m, respectively, the maximum stresses of SCR during installation are shown in Figure 10. The calculated results show that, with the rise of 𝐷_𝐿, the maximum stresses on SCR increase accordingly. It is necessary to control 𝐷_𝐿 at its minimum value in order to avoid the interference of auxiliary installation equipment to keep SCR under safe condition.

Figure 10 

Maximum stresses in different 𝐷𝐿.

4.3. Influence of 𝐷

Under the condition that 𝜃_𝐼 is 10°, 𝐷_𝑉 is 35 m, 𝐷_𝐿 is 35 m, and 𝐷 is 1000 m, 1500 m, and 2000 m, respectively, the maximum stresses on SCR during installation are shown in Figure 11. The calculated results show that, with the decrease of 𝐷, the maximum stresses on SCR increase rapidly. It is necessary to pay more attention to the control of installation path in shallow water. This is because the SCR length increases as water depth increases, and the relative change in installation shape is greater in shallow water than in deep water.

Figure 11 

Maximum stresses in different 𝐷.

4.4. Influence of 𝐷_𝑉

Under the condition that 𝜃_𝐼 is 10°, 𝐷 is 1000 m, 𝐷_𝐿 is 35 m, and 𝐷_𝑉 is 30 m, 35 m, and 40 m, respectively, the maximum stresses on SCR during installation are shown in Figure 12. The calculated results show that, with the rise of 𝐷_𝑉, the maximum stresses on SCR increase accordingly. It can be inferred that, for the safety of installation vessel and offshore platform, smaller values of 𝐷_𝑉 are preferred for SCR installation.

Figure 12 

Maximum stresses in different 𝐷𝑉.

5. Conclusion

A simple model for analyzing the static behavior of the deepwater SCR during installation is proposed, while the nonlinear large deformation beam theory is applied and HAM is used to obtain an analytical approximate solution for this model. This model has the main advantage of time-saving and its practicality. In comparison with the results calculated using the software OrcaFlex, a positive agreement is obtained, which demonstrates that the analytical approximate solution is reliable.

The presented model is applied to analyze the influence of different parameters. Some valuable conclusions can be drawn as follows.(1)Larger initial installation angle causes higher maximum stress during SCR installation, and the J-lay tower is preferred to be placed in an almost vertical position during SCR installation in order to reduce the initial installation angle.(2)The maximum stress during SCR installation increases with the lower depth of pull-head. The pull-head should be controlled at the minimum lower depth to keep the safety of SCR during installation.(3)As water becomes deeper, the maximum stress during SCR installation becomes smaller, which is beneficial to the safety of SCR. However, the increasing axial tension induced by its self-weight brings higher requirements on the capacity of the installation vessel.(4)A longer distance between the installation vessel and the offshore platform can cause a little increase in the maximum stress during SCR installation. To avoid the interference between the installation vessel and the offshore platform, a smaller distance between them is preferred for SCR installation.

This paper reports reasonable approach to the deepwater SCR installation analysis. However, as some assumptions are made for simplifying the investigation, further work needs to be carried out to integrate these assumptions, such as the effect of pipe-soil interaction.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper is financially supported by the National Natural Science Foundation of China (Grant no. 51349002), the China National Science and Technology Major Projects on Pipe Laying and Lifting Vessel (Grant no. 2011ZX05027-002), and the Higher Education Specialized Research Fund for the Doctoral Program (Grant no. 20130007120009). Sincerely, the authors’ thanks also go to colleagues in the COOEC Ltd. and Offshore Oil/Gas Research Centre of CUP, who are involved in this wide range of researches. The authors also highly appreciate the reviewers and the editors’ careful checking and valuable comments on this paper.