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

Authors: Yi Wang; Ji-jun Gu; Chen An; Meng-lan Duan; Ning He
Journal: Journal of Applied Mathematics
 (2014)

---

## 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.


---

## Body

## 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 Figure1, 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 Figure2. 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 Figure3, the following equations can be obtained:
(1)
S
=
y
2
+
2
T
s
w
y
(2)
x
=
T
S
w
ln
⁡
[
w
T
S
y
+
1
+
(
w
T
S
y
+
1
)
2
-
1
]
,
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 Figure4. Assuming the initial position of pull-head before transfer process is at (x
0, y
0), the increasing length of A&R wire Δ
L will change the pull-head position to (x
1, y
1), S
0 is the suspended segment length of the SCR at the initial position, S
1 is the suspended segment length of the SCR after lowering, Δ
x is the horizontal position’s change of TDP, Δ
x
1 is the horizontal length of S
1, Δ
x
2 is the horizontal position change of pull-head, and Δ
y is the vertical position change of pull-head. The following relations can be obtained:
(3)
x
0
=
Δ
x
+
Δ
x
1
+
Δ
x
2
(4)
s
0
=
Δ
x
+
s
1
.Figure 4
Geometrical relationship for lowering step.Based on (1)–(4), the pull-head control model for the lowering step can be obtained:
(5)
x
1
=
y
0
1
+
2
A
-
y
1
1
+
2
B
+
y
1
B
ln
⁡
[
1
B
+
1
+
(
1
B
+
1
)
2
-
1
]
+
Δ
L
cos
⁡
θ
1
,
where A
=
cos
⁡
θ
0
/
(
1
-
cos
⁡
θ
0
), B
=
(
1
-
cos
⁡
θ
1
)
/
cos
⁡
θ
1.The geometrical relationship in pull-in step is shown in Figure5. 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 (x
0
,
y
0), the increasing length of pull-in wire Δ
L will change the pull-head position to (x
1
,
y
1). S
0 is the suspended segment length of the SCR at initial position, and S
1 is the suspended segment length of the SCR after pull-in, whereas Δ
x is the horizontal position’s change of TDP, Δ
x
1 is horizontal length of S
1, and Δ
x
2 is horizontal position’s change of pull-head. Therefore, the following equations can be obtained:
(6)
L
S
=
(
x
S
-
x
0
)
2
+
(
y
S
-
y
0
)
2
=
(
x
S
-
x
1
)
2
+
(
y
S
-
y
1
)
2
(7)
Δ
L
=
(
x
v
-
x
0
)
2
+
(
y
v
-
y
0
)
2
-
(
x
v
-
x
1
)
2
+
(
y
v
-
y
1
)
2
(8)
x
0
=
Δ
x
+
Δ
x
1
+
Δ
x
2
.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:
(9)
x
1
=
y
0
1
+
2
A
-
y
1
1
+
2
B
+
y
1
B
ln
⁡
[
1
B
+
1
+
(
1
B
+
1
)
2
-
1
]
.


### 2.2. SCR Part

An infinitesimal element with lengthd
s 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
(10)
δ
F
-
T
δ
θ
+
w
δ
s
cos
⁡
θ
=
0
(11)
δ
T
=
w
δ
s
sin
θ
,
where F is shear force (
F
=
d
M
/
d
s
), T is axial tension, and M is bending moment. Then, (10) can be written as follows:
(12)
d
2
M
d
s
2
-
T
d
θ
d
s
+
w
cos
⁡
θ
=
0
.Figure 6
Forces on a large deformation beam segmentδ
s.According to large deformation beam theory, the curvature1
/
R
=
M
/
EI
=
d
θ
/
d
s, where EI is the flexural rigidity. Therefore, (12) can be written in the following form:
(13)
d
2
d
s
2
(
EI
d
θ
d
s
)
-
T
d
θ
d
s
+
w
cos
⁡
θ
=
0
.By simplifying (11) and (13), the mechanical model for large deformation beam can be obtained as follows:
(14)
EI
d
3
θ
d
s
3
-
T
d
θ
d
s
+
w
cos
⁡
θ
=
0
(15)
d
T
d
s
=
w
sin
θ
.The boundary condition at top point of riser isd
θ
T
/
d
ε
=
0, and the boundary conditions at TDP are θ
D
=
θ
*, d
θ
D
/
d
ε
=
sin
θ
*, and T
(
ε
)
=
F
T
cos
⁡
ω
T
cos
⁡
θ
*, where F
T is the lifting load at the top of riser, ω
0 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.


## 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 Figure3, the following equations can be obtained:
(1)
S
=
y
2
+
2
T
s
w
y
(2)
x
=
T
S
w
ln
⁡
[
w
T
S
y
+
1
+
(
w
T
S
y
+
1
)
2
-
1
]
,
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 Figure4. Assuming the initial position of pull-head before transfer process is at (x
0, y
0), the increasing length of A&R wire Δ
L will change the pull-head position to (x
1, y
1), S
0 is the suspended segment length of the SCR at the initial position, S
1 is the suspended segment length of the SCR after lowering, Δ
x is the horizontal position’s change of TDP, Δ
x
1 is the horizontal length of S
1, Δ
x
2 is the horizontal position change of pull-head, and Δ
y is the vertical position change of pull-head. The following relations can be obtained:
(3)
x
0
=
Δ
x
+
Δ
x
1
+
Δ
x
2
(4)
s
0
=
Δ
x
+
s
1
.Figure 4
Geometrical relationship for lowering step.Based on (1)–(4), the pull-head control model for the lowering step can be obtained:
(5)
x
1
=
y
0
1
+
2
A
-
y
1
1
+
2
B
+
y
1
B
ln
⁡
[
1
B
+
1
+
(
1
B
+
1
)
2
-
1
]
+
Δ
L
cos
⁡
θ
1
,
where A
=
cos
⁡
θ
0
/
(
1
-
cos
⁡
θ
0
), B
=
(
1
-
cos
⁡
θ
1
)
/
cos
⁡
θ
1.The geometrical relationship in pull-in step is shown in Figure5. 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 (x
0
,
y
0), the increasing length of pull-in wire Δ
L will change the pull-head position to (x
1
,
y
1). S
0 is the suspended segment length of the SCR at initial position, and S
1 is the suspended segment length of the SCR after pull-in, whereas Δ
x is the horizontal position’s change of TDP, Δ
x
1 is horizontal length of S
1, and Δ
x
2 is horizontal position’s change of pull-head. Therefore, the following equations can be obtained:
(6)
L
S
=
(
x
S
-
x
0
)
2
+
(
y
S
-
y
0
)
2
=
(
x
S
-
x
1
)
2
+
(
y
S
-
y
1
)
2
(7)
Δ
L
=
(
x
v
-
x
0
)
2
+
(
y
v
-
y
0
)
2
-
(
x
v
-
x
1
)
2
+
(
y
v
-
y
1
)
2
(8)
x
0
=
Δ
x
+
Δ
x
1
+
Δ
x
2
.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:
(9)
x
1
=
y
0
1
+
2
A
-
y
1
1
+
2
B
+
y
1
B
ln
⁡
[
1
B
+
1
+
(
1
B
+
1
)
2
-
1
]
.


## 2.2. SCR Part

An infinitesimal element with lengthd
s 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
(10)
δ
F
-
T
δ
θ
+
w
δ
s
cos
⁡
θ
=
0
(11)
δ
T
=
w
δ
s
sin
θ
,
where F is shear force (
F
=
d
M
/
d
s
), T is axial tension, and M is bending moment. Then, (10) can be written as follows:
(12)
d
2
M
d
s
2
-
T
d
θ
d
s
+
w
cos
⁡
θ
=
0
.Figure 6
Forces on a large deformation beam segmentδ
s.According to large deformation beam theory, the curvature1
/
R
=
M
/
EI
=
d
θ
/
d
s, where EI is the flexural rigidity. Therefore, (12) can be written in the following form:
(13)
d
2
d
s
2
(
EI
d
θ
d
s
)
-
T
d
θ
d
s
+
w
cos
⁡
θ
=
0
.By simplifying (11) and (13), the mechanical model for large deformation beam can be obtained as follows:
(14)
EI
d
3
θ
d
s
3
-
T
d
θ
d
s
+
w
cos
⁡
θ
=
0
(15)
d
T
d
s
=
w
sin
θ
.The boundary condition at top point of riser isd
θ
T
/
d
ε
=
0, and the boundary conditions at TDP are θ
D
=
θ
*, d
θ
D
/
d
ε
=
sin
θ
*, and T
(
ε
)
=
F
T
cos
⁡
ω
T
cos
⁡
θ
*, where F
T is the lifting load at the top of riser, ω
0 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:(16)
A
[
u
(
x
)
]
=
0
x
∈
R
n
.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 u
0
(
x
) are introduced. The homotopy which is the so-called zero-order deformation equation can be constructed as follows:
(17)
(
1
-
p
)
L
[
ϕ
(
x
;
p
)
-
u
0
(
x
)
]
=
p
ℏ
H
(
x
)
A
[
ϕ
(
x
;
p
)
]
p
∈
[
0,1
]
,
where ℏ 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:
(18)
ϕ
(
x
;
0
)
=
u
0
(
x
)
,
ϕ
(
x
;
1
)
=
u
(
x
)
.Thus, asp increases from 0 to 1, the solution ϕ
(
x
;
p
) varies from the initial guess u
0
(
x
) to the accurate solution u
(
x
). Expand ϕ
(
x
;
p
) by Taylor’s theorem in a power series of p:
(19)
ϕ
(
x
;
p
)
=
u
0
(
x
)
+
∑
k
=
1
+
∞
u
k
(
x
)
p
k
,
where u
k
(
x
)
=
(
1
/
k
!
)
(
∂
k
ϕ
(
x
;
p
)
/
∂
p
k
)
|
p
=
0.Assuming thatL, u
0
(
x
), ℏ, and H
(
x
) are properly chosen; the power series (19) converges at p
=
1, and the solution series can be obtained as follows:
(20)
u
(
x
)
=
u
0
(
x
)
+
∑
k
=
1
+
∞
u
k
(
x
)
.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:
(21)
L
[
u
k
(
x
)
-
χ
k
u
k
-
1
(
x
)
]
=
ℏ
H
(
x
)
R
k
(
x
)
,
where
(22)
χ
k
=
{
0
,
k
⩽
1
1
,
k
>
1
,
R
k
(
x
)
=
1
(
k
-
1
)
!
∂
k
-
1
A
[
ϕ
(
x
;
p
)
]
∂
p
k
-
1
|
p
=
0
.


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

A dimensionless parameterε
=
s
/
l is introduced to (14) and (15), and HAM is used to obtain the analytical approximation of this mechanical model.Assuming thatφ
θ
(
ε
,
q
) is the homotopy which is connected to the original equation, solution θ
(
ε
), and the initial guess solution θ
0
(
ε
), φ
T
(
ε
,
q
) is the homotopy which is connected to the original equation solution T
(
ε
) and the initial guess solution T
0
(
ε
). The embedded parameter q
∈
[
0,1
], the nonzero auxiliary parameters ℏ
θ and ℏ
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
(23)
N
θ
[
φ
θ
(
ε
,
q
)
,
φ
T
(
ε
,
q
)
]
=
∂
3
φ
θ
(
ε
,
q
)
∂
ε
3
-
l
2
EI
φ
T
(
ε
,
q
)
∂
φ
θ
(
ε
,
q
)
∂
ε
+
w
l
3
EI
cos
⁡
[
φ
θ
(
ε
,
q
)
]
N
T
[
φ
θ
(
ε
,
q
)
,
φ
T
(
ε
,
q
)
]
=
∂
φ
T
(
ε
,
q
)
∂
ε
-
l
w
sin
[
φ
θ
(
ε
,
q
)
]
.Thus, zero-order deformation equations can be obtained as follows:(24)
(
1
-
q
)
L
θ
[
φ
θ
(
ε
,
q
)
-
θ
0
(
ε
)
]
=
ℏ
θ
H
θ
(
ε
)
q
N
θ
[
φ
θ
(
ε
,
q
)
,
φ
T
(
ε
,
q
)
]
(
1
-
q
)
L
T
[
φ
T
(
ε
,
q
)
-
T
0
(
ε
)
]
=
ℏ
T
H
T
(
ε
)
q
N
T
[
φ
θ
(
ε
,
q
)
,
φ
T
(
ε
,
q
)
]
which satisfies the initial conditions φ
θ
(
ε
,
0
)
=
θ
0
(
ε
) and φ
T
(
ε
,
0
)
=
T
0
(
ε
).Applying (20) to this case, we can write
(25)
θ
(
ε
)
=
φ
θ
(
ε
,
1
)
=
φ
θ
(
ε
,
0
)
+
∑
m
=
1
+
∞
1
m
!
∂
m
∂
q
m
φ
θ
(
ε
,
q
)
|
q
=
0
=
θ
0
(
ε
)
+
∑
m
=
1
+
∞
θ
m
(
ε
)
T
(
ε
)
=
φ
T
(
ε
,
1
)
=
φ
T
(
ε
,
0
)
+
∑
m
=
1
+
∞
1
m
!
∂
m
∂
q
m
φ
T
(
ε
,
q
)
|
q
=
0
=
T
0
(
ε
)
+
∑
m
=
1
+
∞
T
m
(
ε
)
,
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.Themth-order deformation equations (21) for this particular case are
(26)
L
θ
[
θ
m
(
ε
)
-
χ
m
θ
m
-
1
(
ε
)
]
=
ℏ
θ
H
θ
(
ε
)
R
m
θ
[
θ
m
-
1
(
ε
)
,
T
m
-
1
(
ε
)
]
L
T
[
T
m
(
ε
)
-
χ
m
T
m
-
1
(
ε
)
]
=
ℏ
T
H
T
(
ε
)
R
m
T
[
θ
m
-
1
(
ε
)
,
T
m
-
1
(
ε
)
]
,
where
(27)
χ
k
=
{
0
,
k
⩽
1
1
,
k
>
1
,
R
m
θ
[
θ
m
-
1
(
ε
)
,
T
m
-
1
(
ε
)
]
=
θ
m
-
1
′′′
(
ε
)
-
l
2
EI
∑
k
=
0
m
-
1
T
k
(
ε
)
θ
m
-
1
-
k
′
(
ε
)
+
w
l
3
EI
(
m
-
1
)
!
∂
m
-
1
∂
q
m
-
1
cos
⁡
[
φ
θ
(
ε
,
q
)
]
|
q
=
0
,
R
m
T
[
θ
m
-
1
(
ε
)
,
T
m
-
1
(
ε
)
]
=
T
m
-
1
′
(
ε
)
-
l
w
(
m
-
1
)
!
∂
m
-
1
∂
q
m
-
1
sin
[
φ
θ
(
ε
,
q
)
]
|
q
=
0
.We select the nonzero auxiliary function as(28)
H
θ
(
ε
)
=
s
2
EI
T
,
H
T
(
ε
)
=
1
EI
T
.Referring to the catenary equation, we select the initial guess solutionθ
0
(
ε
) and T
0
(
ε
) as
(29)
θ
0
(
ε
)
=
arctan
(
a
w
l
ε
k
G
)
,
T
0
(
ε
)
=
(
l
w
ε
)
2
+
G
2
,
where G
=
F
0
cos
⁡
ω
0 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:
(30)
l
l
0
=
(
EI
w
0
E
I
0
w
)
1
/
3
.If we select the length of riserl
0 under the condition EI
0
=
50 and w
0
=
350.59, the adjustment parameters a and k can be expressed with respect to l
0:
(31)
a
=
{
1.02755
-
1.84502
e
-
l
0
/
1.43616
l
0
≤
6
0.947
+
0.009
l
0
l
0
≤
10
1.035
l
0
≤
14
1.08931
-
0.00385
l
0
l
0
≤
23
0.98683
+
688.37803
e
-
l
0
/
2.11687
l
0
≤
26
0.96059
+
0.00113
l
0
l
0
≤
40
1
l
0
>
40
k
=
{
0.00541
e
l
0
/
0.48212
+
1.35738
l
0
≤
2
1.28217
e
-
l
0
/
2.36014
-
0.00781
l
0
+
1.19157
l
0
≤
25
1
l
0
>
25
.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:
(32)
ε
=
0
:
θ
(
ε
)
=
0
,
d
θ
(
ε
)
d
ε
=
0
,
T
(
ε
)
=
F
0
cos
⁡
ω
0
ε
=
1
:
d
θ
(
ε
)
d
ε
=
0
.When the nonzero auxiliary parameters are selected asℏ
θ
=
0.1 and ℏ
T
=
1, the analytical approximate solution of θ
(
ε
) and T
(
ε
) can be written as
(33)
θ
(
ε
)
=
arctan
⁡
(
a
w
l
0
ε
k
G
)
+
ℏ
θ
ε
2
E
I
0
T
×
{
l
0
3
[
(
l
0
w
0
s
)
2
+
G
2
]
0.5
ε
2
C
E
I
0
k
2
l
0
D
C
D
-
3
C
k
D
-
8
a
2
ε
2
k
l
0
2
w
0
2
C
G
2
D
2
+
2
C
k
2
D
+
6
a
2
s
3
k
l
0
2
w
0
2
C
ε
k
k
G
2
D
2
+
8
a
4
s
5
k
l
0
4
w
0
4
C
ε
k
G
4
D
3
+
l
0
3
[
(
l
0
w
0
s
)
2
+
G
2
]
0.5
ε
2
C
E
I
0
k
2
l
0
D
+
l
0
3
w
0
E
I
0
D
0.5
}
T
(
ε
)
=
(
l
w
ε
)
2
+
G
2
+
ℏ
T
EI
T
×
{
l
2
w
2
ε
[
(
l
w
ε
)
2
+
G
2
]
0.5
-
l
w
sin
[
arctan
(
w
l
ε
G
)
]
}
,
where C
=
a
ε
k
k
3
l
0
w
0
/
ε
3
G, D
=
1
+
(
a
2
ε
2
k
l
0
2
w
0
2
/
G
2
).


## 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:(16)
A
[
u
(
x
)
]
=
0
x
∈
R
n
.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 u
0
(
x
) are introduced. The homotopy which is the so-called zero-order deformation equation can be constructed as follows:
(17)
(
1
-
p
)
L
[
ϕ
(
x
;
p
)
-
u
0
(
x
)
]
=
p
ℏ
H
(
x
)
A
[
ϕ
(
x
;
p
)
]
p
∈
[
0,1
]
,
where ℏ 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:
(18)
ϕ
(
x
;
0
)
=
u
0
(
x
)
,
ϕ
(
x
;
1
)
=
u
(
x
)
.Thus, asp increases from 0 to 1, the solution ϕ
(
x
;
p
) varies from the initial guess u
0
(
x
) to the accurate solution u
(
x
). Expand ϕ
(
x
;
p
) by Taylor’s theorem in a power series of p:
(19)
ϕ
(
x
;
p
)
=
u
0
(
x
)
+
∑
k
=
1
+
∞
u
k
(
x
)
p
k
,
where u
k
(
x
)
=
(
1
/
k
!
)
(
∂
k
ϕ
(
x
;
p
)
/
∂
p
k
)
|
p
=
0.Assuming thatL, u
0
(
x
), ℏ, and H
(
x
) are properly chosen; the power series (19) converges at p
=
1, and the solution series can be obtained as follows:
(20)
u
(
x
)
=
u
0
(
x
)
+
∑
k
=
1
+
∞
u
k
(
x
)
.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:
(21)
L
[
u
k
(
x
)
-
χ
k
u
k
-
1
(
x
)
]
=
ℏ
H
(
x
)
R
k
(
x
)
,
where
(22)
χ
k
=
{
0
,
k
⩽
1
1
,
k
>
1
,
R
k
(
x
)
=
1
(
k
-
1
)
!
∂
k
-
1
A
[
ϕ
(
x
;
p
)
]
∂
p
k
-
1
|
p
=
0
.


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

A dimensionless parameterε
=
s
/
l is introduced to (14) and (15), and HAM is used to obtain the analytical approximation of this mechanical model.Assuming thatφ
θ
(
ε
,
q
) is the homotopy which is connected to the original equation, solution θ
(
ε
), and the initial guess solution θ
0
(
ε
), φ
T
(
ε
,
q
) is the homotopy which is connected to the original equation solution T
(
ε
) and the initial guess solution T
0
(
ε
). The embedded parameter q
∈
[
0,1
], the nonzero auxiliary parameters ℏ
θ and ℏ
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
(23)
N
θ
[
φ
θ
(
ε
,
q
)
,
φ
T
(
ε
,
q
)
]
=
∂
3
φ
θ
(
ε
,
q
)
∂
ε
3
-
l
2
EI
φ
T
(
ε
,
q
)
∂
φ
θ
(
ε
,
q
)
∂
ε
+
w
l
3
EI
cos
⁡
[
φ
θ
(
ε
,
q
)
]
N
T
[
φ
θ
(
ε
,
q
)
,
φ
T
(
ε
,
q
)
]
=
∂
φ
T
(
ε
,
q
)
∂
ε
-
l
w
sin
[
φ
θ
(
ε
,
q
)
]
.Thus, zero-order deformation equations can be obtained as follows:(24)
(
1
-
q
)
L
θ
[
φ
θ
(
ε
,
q
)
-
θ
0
(
ε
)
]
=
ℏ
θ
H
θ
(
ε
)
q
N
θ
[
φ
θ
(
ε
,
q
)
,
φ
T
(
ε
,
q
)
]
(
1
-
q
)
L
T
[
φ
T
(
ε
,
q
)
-
T
0
(
ε
)
]
=
ℏ
T
H
T
(
ε
)
q
N
T
[
φ
θ
(
ε
,
q
)
,
φ
T
(
ε
,
q
)
]
which satisfies the initial conditions φ
θ
(
ε
,
0
)
=
θ
0
(
ε
) and φ
T
(
ε
,
0
)
=
T
0
(
ε
).Applying (20) to this case, we can write
(25)
θ
(
ε
)
=
φ
θ
(
ε
,
1
)
=
φ
θ
(
ε
,
0
)
+
∑
m
=
1
+
∞
1
m
!
∂
m
∂
q
m
φ
θ
(
ε
,
q
)
|
q
=
0
=
θ
0
(
ε
)
+
∑
m
=
1
+
∞
θ
m
(
ε
)
T
(
ε
)
=
φ
T
(
ε
,
1
)
=
φ
T
(
ε
,
0
)
+
∑
m
=
1
+
∞
1
m
!
∂
m
∂
q
m
φ
T
(
ε
,
q
)
|
q
=
0
=
T
0
(
ε
)
+
∑
m
=
1
+
∞
T
m
(
ε
)
,
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.Themth-order deformation equations (21) for this particular case are
(26)
L
θ
[
θ
m
(
ε
)
-
χ
m
θ
m
-
1
(
ε
)
]
=
ℏ
θ
H
θ
(
ε
)
R
m
θ
[
θ
m
-
1
(
ε
)
,
T
m
-
1
(
ε
)
]
L
T
[
T
m
(
ε
)
-
χ
m
T
m
-
1
(
ε
)
]
=
ℏ
T
H
T
(
ε
)
R
m
T
[
θ
m
-
1
(
ε
)
,
T
m
-
1
(
ε
)
]
,
where
(27)
χ
k
=
{
0
,
k
⩽
1
1
,
k
>
1
,
R
m
θ
[
θ
m
-
1
(
ε
)
,
T
m
-
1
(
ε
)
]
=
θ
m
-
1
′′′
(
ε
)
-
l
2
EI
∑
k
=
0
m
-
1
T
k
(
ε
)
θ
m
-
1
-
k
′
(
ε
)
+
w
l
3
EI
(
m
-
1
)
!
∂
m
-
1
∂
q
m
-
1
cos
⁡
[
φ
θ
(
ε
,
q
)
]
|
q
=
0
,
R
m
T
[
θ
m
-
1
(
ε
)
,
T
m
-
1
(
ε
)
]
=
T
m
-
1
′
(
ε
)
-
l
w
(
m
-
1
)
!
∂
m
-
1
∂
q
m
-
1
sin
[
φ
θ
(
ε
,
q
)
]
|
q
=
0
.We select the nonzero auxiliary function as(28)
H
θ
(
ε
)
=
s
2
EI
T
,
H
T
(
ε
)
=
1
EI
T
.Referring to the catenary equation, we select the initial guess solutionθ
0
(
ε
) and T
0
(
ε
) as
(29)
θ
0
(
ε
)
=
arctan
(
a
w
l
ε
k
G
)
,
T
0
(
ε
)
=
(
l
w
ε
)
2
+
G
2
,
where G
=
F
0
cos
⁡
ω
0 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:
(30)
l
l
0
=
(
EI
w
0
E
I
0
w
)
1
/
3
.If we select the length of riserl
0 under the condition EI
0
=
50 and w
0
=
350.59, the adjustment parameters a and k can be expressed with respect to l
0:
(31)
a
=
{
1.02755
-
1.84502
e
-
l
0
/
1.43616
l
0
≤
6
0.947
+
0.009
l
0
l
0
≤
10
1.035
l
0
≤
14
1.08931
-
0.00385
l
0
l
0
≤
23
0.98683
+
688.37803
e
-
l
0
/
2.11687
l
0
≤
26
0.96059
+
0.00113
l
0
l
0
≤
40
1
l
0
>
40
k
=
{
0.00541
e
l
0
/
0.48212
+
1.35738
l
0
≤
2
1.28217
e
-
l
0
/
2.36014
-
0.00781
l
0
+
1.19157
l
0
≤
25
1
l
0
>
25
.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:
(32)
ε
=
0
:
θ
(
ε
)
=
0
,
d
θ
(
ε
)
d
ε
=
0
,
T
(
ε
)
=
F
0
cos
⁡
ω
0
ε
=
1
:
d
θ
(
ε
)
d
ε
=
0
.When the nonzero auxiliary parameters are selected asℏ
θ
=
0.1 and ℏ
T
=
1, the analytical approximate solution of θ
(
ε
) and T
(
ε
) can be written as
(33)
θ
(
ε
)
=
arctan
⁡
(
a
w
l
0
ε
k
G
)
+
ℏ
θ
ε
2
E
I
0
T
×
{
l
0
3
[
(
l
0
w
0
s
)
2
+
G
2
]
0.5
ε
2
C
E
I
0
k
2
l
0
D
C
D
-
3
C
k
D
-
8
a
2
ε
2
k
l
0
2
w
0
2
C
G
2
D
2
+
2
C
k
2
D
+
6
a
2
s
3
k
l
0
2
w
0
2
C
ε
k
k
G
2
D
2
+
8
a
4
s
5
k
l
0
4
w
0
4
C
ε
k
G
4
D
3
+
l
0
3
[
(
l
0
w
0
s
)
2
+
G
2
]
0.5
ε
2
C
E
I
0
k
2
l
0
D
+
l
0
3
w
0
E
I
0
D
0.5
}
T
(
ε
)
=
(
l
w
ε
)
2
+
G
2
+
ℏ
T
EI
T
×
{
l
2
w
2
ε
[
(
l
w
ε
)
2
+
G
2
]
0.5
-
l
w
sin
[
arctan
(
w
l
ε
G
)
]
}
,
where C
=
a
ε
k
k
3
l
0
w
0
/
ε
3
G, D
=
1
+
(
a
2
ε
2
k
l
0
2
w
0
2
/
G
2
).


## 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 Table1.Table 1
Parameters of SCR.
Parameter
Value
Outer diameter (m)
0.3048
Inner diameter (m)
0.2743
Flexural rigidity (N·m2)
3.1340
E
07
Weight submerged (N/m)
350.59
Density (kg/m3)
7850The overall configuration and axial tension calculated by the proposed method and OrcaFlex are compared as shown in Figures7 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.
Item
D = 500 m
D = 1000 m
D = 2000 m
X (m)
Y
top (m)
T
max
⁡ (N)
X (m)
Y
top (m)
T
max
⁡ (N)
X (m)
Y
top (m)
T
max
⁡ (N)
HAM
297.97
499.41
217071
545.43
1004.09
429937
1050.67
2006.54
859997
OrcaFlex
299.89
499.91
214430
546.57
1002.77
428228
1056.20
2003.68
853240
Difference
0.64%
−0.1%
−1.23%
0.21%
−0.13%
−0.4%
0.53%
−0.14%
−0.79%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 Table1 are used for SCR, with following hypotheses.(1)
Assume that the initial installation angle isθ
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 isX
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 isY
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 Table3.Table 3
Maximum stress results.
θ
I (°)
D
L (m)
D (m)
D
V (m)
σ
max
⁡ (MPa)
8
30
1000
35
169.37
10
30
1000
35
215.15
12
30
1000
35
273.91
10
33
1000
35
218.09
10
36
1000
35
221.10
10
35
1000
35
219.98
10
35
1500
35
135.79
10
35
2000
35
100.23
10
35
1000
30
217.84
10
35
1000
40
222.21


### 4.1. Influence ofθ
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 thatD
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 differentθ
I.


### 4.2. Influence ofD
L

Under the condition thatθ
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 differentD
L.


### 4.3. Influence ofD

Under the condition thatθ
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 differentD.


### 4.4. Influence ofD
V

Under the condition thatθ
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 differentD
V.


## 4.1. Influence ofθ
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 thatD
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 differentθ
I.


## 4.2. Influence ofD
L

Under the condition thatθ
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 differentD
L.


## 4.3. Influence ofD

Under the condition thatθ
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 differentD.


## 4.4. Influence ofD
V

Under the condition thatθ
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 differentD
V.


## 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.



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*Source: 801781-2014-07-02.xml*