Experimental and numerical study of dynamic performance of CVAR subjected to regular wave and platform motion

Ocean Engineering 199 (2020) 106946

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Experimental and numerical study of dynamic performance of CVAR subjected to regular wave and platform motion Min Lou *, Weixing Liang, Run Li College of Petroleum Engineering, China University of Petroleum (East China), Ministry of Education, Qingdao, 266580, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Compliant vertical access riser Vibration equation Model experiments Dynamic performance

Compliant vertical access risers (CVAR) offer significant economic benefits and prospects for development because of their special configuration. In this paper, the dynamic performance of CVAR is studied by model experiments and numerical simulations. Based on virtual work and variational principles, a vibration equation considering the large deformation mechanical behaviors of CVAR with the influences of bending stiffness and non-uniform internal flow is proposed. To verify it, model experiments on the CVAR are firstly conducted and compared the results with those of a numerical simulation. Time-domain analyses of key nodes, regular wave analysis and dynamic response at different positions of the buoyancy module were then carried out. The vi­ bration near the platform was the largest due to the wave load, and decreased rapidly when in the transitional region. The minimum tension of the transition zone was even subzero at the far end, which means the CVAR is unstable when the platform subject to regular wave conditions. When the buoyancy block was above 1000 m, tension at the bottom was smaller than 672 kN, which will cause the lower region to become catenary shaped and the CVAR to lose its unique operational advantage.

1. Introduction In the past century, to meet the requirements of offshore oil explo­ ration and development, a number of risers have been developed that are suitable for different marine environments. As this exploration moves into deeper water, new forms of risers are being developed. Four types of risers are mainly used in deepwater platforms: top tension risers (TTRs), steel catenary risers (SCRs), flexible risers (FRs), and hybrid risers (HRs). TTRs have been popular in recent decades because of their high strength and good weldability, but also have the disadvantages of poor compliance and corrosion resistance (William Toh et al., 2018). The lower part of the SCR is in direct contact with subsea soil. Frequent interactions make it prone to fatigue fracture. Research has shown that weak points of the SCR are generally located at the top hanging and bottom touchdown zones (Lucile M. Queau et al., 2015; Rasoul Hejazi et al., 2016; Hodjat Shiri, 2014). SLWRs (Steel Lazy Wave Risers) derive from the SCR, and have a longer hanging part and larger buoyancy module that increases its outer diameter. According to Morrison’s equation, this causes the SLWR to be subjected to a greater fluid force, and the maximum offset can usually reach more than 10 m (Wang and

Duan, 2015). To solve these problems, it is usually necessary to install top bending stiffeners and subsea foundations. Owing to the special nature of the material, FR is unsuitable in high temperature and high pressure fields. For flexible risers with a large diameter, the cost of production is high so that the space for its development is minimal. Furthermore, the connection joints of HRs are complex, and failure due to fatigue from towing and installation is common (Ruilong Tan et al., 2018). A CVAR is a compliant rigid riser with a differentiated geometric configuration that allows for the exploitation of oil and gas in deepwater fields, and provides several operational advantages for offshore systems (Fig. 1). Compared with the SCR, the upper and lower parts of the CVAR are nearly vertical. Therefore, the fatigue lives of the top and bottom zones are longer. Compared with the SLWR, the CVAR offers greater economic benefits because the design of its weighted part does not require the top bending stiffener and tension system. Another major feature of CVAR systems is that they allow for direct intervention pro­ cedures in the well bore, thus enabling workover operations to be per­ formed directly from the production platform. This feature eliminates the need for hiring specific units for this purpose and makes this new riser economically attractive. Thus, CVAR systems have the potential to

* Corresponding author. E-mail address: [email protected] (M. Lou). https://doi.org/10.1016/j.oceaneng.2020.106946 Received 13 July 2019; Received in revised form 16 December 2019; Accepted 14 January 2020 Available online 1 February 2020 0029-8018/© 2020 Elsevier Ltd. All rights reserved.

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C et al., 2004). Kuroiwa and Nishigaki et al. proposed the design concept of the FPSO, called the CVAR-FPSO, to study the interaction between the CVAR and tubing. Martins optimized this design using the NSGA-II (the Elitist Nondominated Sorting Genetic Algorithm) method and DOE (Design of Experiments) method, and analyzed and compared several key parameters of the CVAR. He then designed the optimal solution of its dynamic response (Michele et al., 2012; Martins et al., 2013). Zhen and Yang et al. used the DOE (Design of Experiments) method to establish an approximate model, and performed a simple weight optimization for the CVAR (Zhen, W. Q et al., 2010). A structural analysis of CVAR can be conducted by software such as OrcaFlex, Riflex, Offpipe, Cable3D, etc. Though the above commercial software can lead to refined enough results, they are also relatively timeconsuming due to the cumbersome graphical user interface in the meantime, thus that software are usually adopted in the final stages of the design (Pavel A. Trapper, 2020). Consequently, engineers tend to use convenient methods to conduct numerous simulations at least for the intermediate stages of the design to analyze the initial riser configura­ tion (Lenci and Callegari, 2005). Nevertheless, it is required that these models need to be sufficiently accurate on an efficient basis. Many scholars has have done extensive studies on simplified models of risers (Ruan et al., 2014; Wang et al., 2015; Li et al., 2010). These work often consider the riser as several segments, i.e., upper segment, buoyancy segment, lower segment, which use continuity conditions at the segment boundaries to connect each segment as a whole segment and each segment are usually modeled to be a simple catenary without considering the effect of bending stiffness and non-uniform internal flow. At present, research on CVAR is in the stage of conceptual design, whilst no reported literature has been devoted to simplified model of CVAR. To solve this problem in a more relatively accurate way, it is necessary to derive the vibration equations, thus we propose a vibration equation of CVAR based on virtual work and variational principles. The model we develop takes large deformation mechanical behaviors of CVAR with the influences of bending stiffness and non-uniform internal flow into account and will make it possible to consider the whole riser as a single continuous segment. Moreover, most experiments on marine risers have focused on vortex-induced vibrations and the stability of other types of risers, and no model experiment on the dynamic perfor­ mance of the CVAR has been reported. To explore the dynamic perfor­ mance of CVARs using model experiments and numerical simulations, we firstly conducted model experiments on the CVAR and compared the results with those of a numerical simulation. Time-domain analyses of key nodes, regular wave analysis, and dynamic response at different positions of the buoyancy module were also carried out.

Fig. 1. Concept of the compliant vertical access riser.

2. Vibration equation of CVAR

Fig. 2. Three configurations of a 3D marine riser.

Owing to its large slenderness ratio (ratio of diameter to length), the deepwater CVAR can resist the movement and hydrodynamic load of the floating platform, and thus geometric nonlinearity is a prominent feature. In addition, when the CVAR operates in its service period, the internal fluid runs at a high temperature and pressure. Studies have shown that the internal fluid affects the nonlinear mechanical charac­ teristics of the structure and is thus an important factor. In this section, the vibration model of the CVAR is considered in light of its geometric nonlinear characteristics, including the internal fluid at high tempera­ ture and pressure. To consider geometric nonlinearity, the equation of static equilib­ rium of the CVAR needs to be established at the corresponding position, because of which it is necessary to use a special coordinate system. There are two methods available: the global Lagrange method and the updated Lagrange method. The vibration equation used in this paper is based on the updated Lagrange method. The elemental local coordinate system is not fixed but changes with the displacement of the structure.

reduce the costs of completion and well intervention (Chris et al., 2004). The two ends of the CVAR connect the wells and the platform with a stress joint or flexible joint. A horizontal distance is maintained between the wellhead and the platform to provide compliance for the CVAR. This compensates for the motion of the floating platform and allows it to operate in deeper sea. The CVAR is divided into three regions: an upper region, a transition region, and a lower region. In general, a buoyancy module with a large diameter is arranged at the top of the lower region to maintain rigidity. The lower region is similar to the TTR. The middle region is evenly arranged with a relatively low buoyancy module, and is called the transition region. The upper region is similar to the SCR and has an additional weight. This special configuration of the CVAR de­ termines its unique mechanical properties. Compared with work on the TTR, SCR, and FR, little research has been devoted to the CVAR. Mungall and Haverty proposed the concept of a semi-submersible platform combined with the CVAR in the Gulf of Mexico. They applied this setup below 8000 feet, and extreme response analysis and fatigue analysis of the CVAR proved its feasibility (Mungall, 2

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2.1. Vibration equation

In the above equation, the superscript (‘) represents the partial de­ rivative of the independent variable α. The geometric curvature of the 3D riser is expressed as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dθ 1 κ ¼ ¼ 0 3 ðx00 y0 x0 y00 Þ2 þ ðy00 z0 y0 z00 Þ2 þ ðx00 z0 x0 z00 Þ2 (10) ds s

The virtual work principle involves only the active force, and thus simplifies the establishment of the vibration equation of the CVAR with ideal constraints. The variational principle can directly deal with the entire elastic system by considering its energy relation, and the problem of elasticity is reduced to the variational problem of finding the func­ tional extremum under a given constraint. This model is based on the virtual work principle and variational principle (Chucheepsakul et al., 2003). To consider its generality, an independent variable α (Fig. 2) is introduced to the derivation process. The strain energy is mainly pro­ duced by axial deformation and bending deformation (Athisakul and Chucheepsakul, 2008; Bernitsas, 1982). The main virtual work of the external loads consists of effective gravity, buoyancy, wave, current, and vortex-induced lift (Bernitsas,M.M.,1985; Chucheepsakul,S. et al., 2002).

According to Equation (6), it is necessary to take the derivative of time t. Riser velocity Vr and acceleration ​ ar are expressed as:

vs ¼

vd ¼

(1)

0

(2)

0

s ¼

0

0

ss 0

ss

� 0 0 0 0 0 0 xs us þ ys vs þ zs ws 02

0

¼

s s0s

1¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2vs pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2vd

02

1 2

s0 s

02

�

02

0

0

0

0

0

02

us þvs þws 2

ss

0

xd ud þ yd vd þ zd wd

α0

0

(5)

(13b)

1

(13c)

02

� (14a) 02

ud þvd þwd 2

� (14b)

α0

α0

0 0 0 �x0 � 0 �y0 � 0 �z0 � 0 ∂s 0 ∂s 0 ∂s 0 δu þ 0 δv þ 0 δw 0 δu þ 0 δv þ 0 δw ¼ ∂u ∂v ∂w s0 s s

(16)

2.1.2.2. Bending strain energy. The variation in bending strain energy is: Z αt 0 0 � Mδ θ θ0 dα (17) δUb ¼

(6)

(9)

ss s

s0 ¼1 s0

¼1

1

δs ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 0 0 0 2 2 2 x0 þ y0 þ z0 ¼ ðx0 þ u0 Þ þ ðy0 þ v0 Þ þ ðz0 þ w0 Þ

0

s0 0

(13a)

The variation in s’ is:

(4)

(8)

0

ss

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2vs

2.1.2.1. Axial strain energy. Axial deformation in the riser is caused by two factors: tension in the riser, and pressure difference between the inner and outer fluids (Chucheepsakul et al., 2002). The variation in axial strain energy of the infinitesimal ds is: Z αt Z αt Z αt 0 0 δUa ¼ Te δðdsÞ ¼ Te δðs dαÞ ¼ Te δs dα (15)

(3)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffi 2 2 2 0 0 0 2 2 2 x0 þ u0s þ y0 þ v0s þ z0 þ w0s x0 s þ y0 s þ z0 s ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2vd

¼

ss

2.1.2. Virtual work equation For the CVAR in deepwater, shear deformation and torsional defor­ mation are small enough to be ignored. The axial deformation and bending deformation of the riser are thus mainly considered when building the virtual work principle (Chainarong Athisakul et al., 2011; Wenwu Yang et al., 2018; Mengmeng Zhang et al., 2015).

α0

The arc lengths of the undeformed state, static equilibrium state, and dynamic state can be expressed as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 2 2 s0 ¼ x0 0 þ y0 0 þ z0 0 (7) ss ¼

0

s0 0

The Green strain vs ; vd at each state can be represented by the displacement of the riser:

Its shape can be expressed as: rðα; tÞ ¼ rs ðαÞ þ rd ðα; tÞ ¼ ðxs þ ud Þi þ ðys þ vd Þj þ ðzs þ wd Þk

0

Dynamic ​ strain: εd ¼

The total displacement of the riser at any time is: uðα; tÞ ¼ us ðαÞ þ ud ðα; tÞ ¼ uðα; tÞi þ vðα; tÞj þ wðα; tÞk

s

0

Due to waves, the internal fluid, or movement of the floating plat­ form, the riser enters into its third state: the dynamic state. The dynamic displacement ​ ud is defined as the displacement of the riser from the equilibrium state to the dynamic state: ud ðα; tÞ ¼ ud ðα; tÞi þ vd ðα; tÞj þ wd ðα; tÞk

(12)

Static strain: εs ¼

Vector us is the static displacement at one point on the riser: us ðαÞ ¼ us ðαÞi þ vs ðαÞj þ ws ðαÞk

ar ¼ r€ðα; tÞ ¼ u€d ðα; tÞi þ v€d ðα; tÞj þ w€d ðα; tÞk

Total strain: εt ¼

The parameter α is scalar, and is used to define the shape of the curve of the riser. It is used to render the derived equation general, and can represent any component of any coordinate systems. The second state of the riser is the static equilibrium state. Co­ ordinates of points on curves of the riser are represented by the position vector rs : rs ðαÞ ¼ r0 ðαÞ þ us ðαÞ ¼ xs ðαÞi þ ys ðαÞj þ zs ðαÞk

(11)

where the superscript ð ⋅Þ represents the derivative with respect to time. According to the updated Lagrange formulation, the total, static, and dynamic strains are:

2.1.1. Motion and deformation Relative to its outside diameter, the length of the riser is longer, and the CVAR thus has a large slenderness ratio. Therefore, the riser is usually considered a three-dimensional (3D) rod element rather than a cylindrical shell element (Meng and Chen, 2012). The centerline of the riser can be described by a 3D orthogonal coordinate system (Fig. 2). The entire coordinate system uses the Cartesian coordinate system x; y; z and the corresponding unit base vectors are i; j; k. The local co­ ordinates are in the orthogonal coordinate system ​ x1 ;x2 ;x3 . Directions t; u; b are the tangential, normal, and binormal of the vertical riser, respectively, where ​ b ¼ t � n, and the corresponding base vectors are e1 ; ​ e2 ; e3 . Three states of motion of riser are defined in this paper. The first is the ideal state defined as an undeformed state. The position of the vertical riser is expressed by the position vector r0 : r0 ðαÞ ¼ x0 ðαÞi þ y0 ðαÞj þ z0 ðαÞk

_ α; tÞ ¼ u_d ðα; tÞi þ v_d ðα; tÞj þ w_ d ðα; tÞk V r ¼ rð

In the above equation, M is the moment of the dynamic state: M ¼

EIr κð1 þ εd Þ, θ0 ¼ s0 κ0 . The riser is linear in the undeformed state, and geometric curvature κ0 ¼ 0. Thus, Z αt 0 δUb ¼ Mδθ dα (18) 0

0

α0

Combined with (10), it yields

3

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Ocean Engineering 199 (2020) 106946

1 � 00 0 00 0 x ðy y þ z z00 Þ

f ¼

0

2

x y00 þ z00

05

2 ��

s κ 1 � 00 0 00 0 0 0 2 2 2 �� x ðx x þ y y00 þ z z00 Þ x x00 þ y00 þ z00 s κ 1 � 0 0 2 2 2 �� ¼ 0 5 x00 s s00 x x00 þ y00 þ z00 s κ 0 0 0 s00 �x00 s x s00 x s00 � 1 0 00 2 2 2� ¼ 03 þ 02 x x þ y00 þ z00 02 05 s κ s s s κ ¼

05

(26)

Substituting Equation (14) into Equation (26), f¼ Fig. 3. Loads on infinitesimal pipe.

0

0

0

0

(19)

In the equation, 02

0

0

∂s ∂u00 02

F ¼ s00

0

03

s κ2s

0

∂s ∂u00

(20)

F¼

2

0

0

0

0

2

2

x00 þ y00 þ z00

2 ��

(27)

2� 0

00

(28) 00

1 � 0 00 0 0 ðx x þ y y00 þ z z00 Þ2 02

¼

� (21)

0

0

00

2

00

2

x00 þ y00 þ z00

2�

02

02

x þy þz

02

0

0

1 � 00 0 ðx y 02 s

0

x y00 Þ2 þ ðy00 z

0

0

0

y z00 Þ2 þ ðx00 z

(30)

04

∂ �x � 0 s κ ∂α s s00

0

κx s0

0

0 0 0 ∂ �x � x00 s x s00 ¼ 02 ∂α s0 s

(23)

Substituting Equation (31) into Equation (25): � 00 0 0� ∂θ s ∂ �x � κx þ 0 ¼ 0 0 3 ∂u s s0 κ ∂α s

∂θ 1 ∂ �x � ¼ ∂u00 s0 2 κ ∂α s0

(24)

By simplify the expression, we get:

¼

1 � 00 0 00 0 x ðy y þ z z00 Þ

05

s κ

0

2

x y00 þ z00

2 ��

0

03

s κ2s

0

2κx ¼ 0 s

f

2κx 0 s

0

∂s 0 ∂u

(29)

�

κ2 s

f¼

0

0

x z00 Þ2

(22)

Examining the second item of (20) reveals � 0 0 0 0 0 ∂θ 1 2ðx00 y x y00 Þð y00 Þ þ 2ðx00 z x z00 Þð z’’ Þ 0 2 s 0 ¼ 04 03 ∂u s 2s κ

��

Substituting Equation (10) into Equation (18):

0

∂θ 1 x00 ðx þ y þ z Þy x ðx x00 þ y y00 þ z z00 Þz 1 x00 s x s s00 ¼ ¼ 02 ∂u00 s0 2 κ s κ s0 3 s0 3

0

02

Substituting Equation (28) into Equation (29): F¼

0

2

s

¼0 02

2

x00 þ y00 þ z00

Substituting s s ¼ x x þ y y þ z z into Equation (28),

0

∂θ ∂θ 0 ∂θ ∂θ 0 ∂θ ∂θ 0 δθ ¼ 00 δu00 þ 0 δu þ 00 δv00 þ 0 δv þ 00 δw00 þ 0 δw ∂u ∂u ∂v ∂v ∂w ∂w Examining the first item yields � 0 0 0 0 0 0 ∂θ 1 2ðx00 y x y00 Þy þ 2ðx00 z x z00 Þz 0 2 ¼ 04 s 03 00 ∂u s 2s κ

0

0

0

0

0

03

and

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0 0 θ ¼ s κ ¼ 0 2 ðx00 y0 x0 y00 Þ2 þ ðy00 z0 y0 z00 Þ2 þ ðx00 z0 x0 z00 Þ2 s 0

∂ �x � x � 2 þ 0 5 s00 0 s κ ∂α s s κ s00

03

(31)

0

(32)

Table 1 Parameters for the section design of CVAR.

�

0

(25)

and

Parameter

Value

Outer diameter Thickness Length Upper region (Bare) Upper region (Weighted) Transition region (bare) Transition region (Small buoyancy) Lower Region (Large buoyancy) Lower Region (Bare)

10 mm 1 mm 1.2m 62 cm 5 cm 5 cm 22 cm 6 cm 20 cm

Fig. 4. Buoyancy module and weighted section. 4

Ocean Engineering 199 (2020) 106946

M. Lou et al.

Fig. 5. Experimental equipment.

Similarly,

∂θ 1 ∂ �y � ​ ¼ ∂v00 s0 2 κ ∂α s0 0

0

�

(33)

∂ �y � κy þ 0 ∂α s0 s s κ

0

∂θ ¼ ∂v0

0

s00

0

α0

(34)

α0

∂θ 1 ∂ �z � ¼ ∂w00 s0 2 κ ∂α s0 0

�

(35)

∂ �z � κz þ 0 0 s s κ ∂α s

0

∂θ ¼ ∂w0

0

s00

α0

2.1.2.3.2. Virtual work of structural resistance Z αt 0 δWc ¼ ðcuδu _ þ cvδv _ þ cwδwÞs _ dα

�

03

0

2.1.2.3. Virtual work of external force 2.1.2.3.1. Virtual work of effective gravity Z αt Z αt 0 we dsδv ¼ we s δvdα δWe ¼

0

� (36)

03

Combining Equations (20), (24), and (32–36), the variation in the bending strain energy is expressed as: Zαt �� α0

Zαt ��

�

0 � B ∂ �z � δw00 0 02 s ∂α s

�

α0

Zαt �� α0

� 0 0� s00 ∂ �y � κ2 y 0 þ 0 δv dα 0 03 s s ∂α s

0 � B ∂ �y � 00 δv 0 02 s ∂α s

þ

þ

� 0 0� s00 ∂ �x � κ2 x 0 δu dα þ 0 0 03 s s ∂α s

�

0 � B ∂ �x � 00 δu 0 02 s ∂α s

δUb ¼

(37)

� 0 0� s00 ∂ �z � κ2 z 0 þ 0 δw dα 0 03 s s ∂α s

where B ¼ EIr ð1 þεd Þ is the bending rigidity and εd is dynamic strain.

Fig. 6. Position of model in the sink. 5

(38)

(39)

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Ocean Engineering 199 (2020) 106946

Fig. 7. Dynamic response of CVAR model.

Assuming that structural resistance is proportional to structural ve­ locity, where c is the coefficient of structural resistance, the dynamic analysis uses Rayleigh damping. 2.1.2.3.3. Virtual work of hydrodynamic force. According to the Morison equation, the riser is also subjected to a vortex lift force due to the current (S.Lei et al., 2017). Hydrodynamic virtual work done by the external fluid is expressed as: Z αt � � �0 δWH ¼ fHx δu þ fHy δv þ fHz þ FL δw s dα (40)

0

0

ρ s w€

FL ¼ fL

Cw_

(42)

ρe – Seawater density;

De – Riser diameter; CDt – Tangential drag coefficient; CDn – Normal drag coefficient; CDbn – Binormal drag coefficient; Ut – Tangential velocity of external fluid; Un – Normal velocity of external fluid; ut ; vn ; wbn – Tangent, binormal, and normal direction displacements; CM – Inertia force coefficient; CA – Added mass coefficient. CA ¼ 1; 0 ρ; – Mass of fluid added per unit length. ρ ¼ 1=4CA ρe πD2e ; 0 C; – Fluid damping. C ¼ γωs ρe D2e ; γ– Viscous force parameter. γ ¼ CD =4πSt ; ωs – Circular frequency of vortex shedding. ωs ¼ 2πSt Ue =D; fL – Lift force of vortex shedding; CL –Instantaneous lift coefficient. fL ¼ 1=2ρe U2e De CL . 2.1.2.3.4. Virtual work of inertial force

α0

fHx ; fHy ; fHz are components of the wave force and current force in the three directions x; y; z . For the inclined cylinder, the hydrodynamic force generated by the wave and current can be calculated by Morison’s equation: 9 8 9 9 8 8 _ < πCDt ðUHt ut ÞjUHt ut j = < fHt = 1 πD2e < U_ Ht = FH ¼ fHn ¼ ρe De CDn ðUHn vn ÞjUHn vn j þ ρe U ; 2 : : ; 4 : _ Hn ; CDbn ðUHbn vbn ÞjUHbn wbn j fHbn U Hbn 8 9 _ πD2e < U_ Ht u_t = þ ρe CA U v_n : _ Hn ; 4 U Hbn w_ bn (41)

δWI ¼ Z

The first term is fluid resistance, the second is the Froude–Krylov force, and the third term is the added mass force. The vortex lift force in the vertical direction is:

αt �

α0

6

� �0 ðmr arx þ mi aix Þδu þ mr ary þ mi aiy δv þ ðmr arz þ mi aiz Þδw s dα (43)

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Ocean Engineering 199 (2020) 106946

Fig. 7. (continued).

��

mr —Riser mass per unit length; mi –Internal fluid mass per unit length; arx ; ​ ary ; arz – Three components of riser acceleration; aix ; ​ aiy ; aiz – Three components of riser acceleration. 2.1.2.3.5. Internal fluid. In oil and gas exploration, simulating flow states inside the riser is a complex task. In general, the internal fluid is simplified as a slender piston model (Monprapussron et al., 2004). We assume that each point along the cross-section has the same velocity and there are six acceleration items of the internal fluid. The first is the ac­ celeration of the mass of the internal fluid, the second item is the Coriolis acceleration, the third is centripetal acceleration caused by the change in the direction of flow, the fourth item is local acceleration caused by unsteady flow, the fifth item is convective acceleration caused by non-uniform flow, and the final two items of the equation form the sixth item. They are the relative acceleration caused by rotation and displacement in the local coordinate system, respectively. � � 2 � � � �2 2 0 U_ i Ui Ui Ui s_ U 2i s00 ∂r ∂2 r Ui ∂r Ui ∂ r ai ¼ 2 þ 0 þ þ (44) þ 0 0 ∂t s ∂α∂t s ∂α2 s ∂α s0 2 s0 2 s0 2

0

x þ U_ i 0 s

�

� 0 2� �x0 y0 � 2 y y_ x_ þ 0 03 03 s s s 0 � y 0 0 0 y ðx x00 þ z z00 Þ þ U_ i 0 s

aiy ¼ y€ þ

�

� �y0 z0 � 2 y _ þ 3 s0 s0 0 � z 0 0 0 z ðx x00 þ y y00 Þ þ U_ i 0 s

aiz ¼ z€þ

0

0

�x0 z0 � x_ 3 s0

�y0 z0 � � U2 � 02 0 2� z_ Ui þ 0 4i y00 x þ z 03 s s

�� U2 � 02 0 2� z_ Ui þ 0 4i z00 x þ y 03 s s 02

z

(45)

According to the virtual work principle, the deformation energy generated by virtual work done by the internal force is equal to the sum of the external forces: δπ ¼ δUa þ δUb

In the above equation, r ¼ xi þ yj þ zk, s s00 ¼ x x00 þ y y00 þ z z00 and 0 0 0 _ Substituted into Equation (35), the three compo­ _ s s ¼ x x_ þ y y_ þ z z. nent expressions of fluid acceleration are expressed as: 0

0 2� �x0 y0 � �x0 z0 � � � 2 x U2 � 02 0 2� 0 0 0 x_ y_ x ðy y00 þz z00 Þ z_ Ui þ 0 4i x00 y þz 0 03 s s s0 3 s0 3 s

aix ¼ x€þ

0

δWc

δWe

δWH

δWI

(46)

Substituting Equations (15) and (37)~(40) into (46) yields the CVAR equation of 3D vibration:

0

7

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Fig. 7. (continued).

shear force, and bending moment (Fig. 3). The static equilibrium equation in the tangent direction is: � 0 � � 0 � θ dα θ dα 0 0 0 þ we s dαcos þ θ þ mr art s dα þ mi ait s dα Te cos 2 2 � 0 � � 0 � θ dα θ dα 0 0 0 0 ​ ¼ fHt s dα þ ðQ þ Q dαÞsin (48) þ Q sin þ Te þ T e dα 2 2

� � � 0 � 0 � 0 � B ∂ �x � 00 B ∂ �y � 00 B ∂ �z � δu þ 0 2 δv þ 0 2 δw00 dα 0 0 0 02 s ∂α s s ∂α s s ∂α s

Zαt �� δπ ¼ α0

þ

Zαt �� � 0 � x Te 0 s

α0

B

� 2 0 0 �� κ x s00 ∂ �x � 0 þ δu 0 0 03 s s ∂α s

� � 0� � 2 0 0 �� y κ y s00 ∂ �y � 0 B δv þ Te 0 þ 0 0 03 s s s ∂α s � � � 0� � 2 0 0 �� z κ z s00 ∂ �z � 0 B δw dα þ Te 0 þ 03 0 0 s s s ∂α s

By simplifying and omitting the higher-order term ðdαÞ2 , 0

and Q ¼ dM ds ¼

Zαt f½fHx α0

mr arx

mi aix

cu�δu _

Z Tα0 ¼ Tαt þ

� fHy

� fHz

0

0

0

0

0

we y dα þ mr art s dα þ mi ait s dα ¼ fHt s dα þ Qθ dα þ T e dα

� mr ary mi aiy cvw _ a δv � �0 mr arz mi aiz cw_ þ fL δw s dα

M s0

αt h

α0

0

(49)

, κ ¼ θs0 . Then, 0

� 0 0 M κ þ s fHt

we y s0

0

mr art

mi ait

�i dα

(50)

2.1.3.2. Top tension is unknown. We hypothesize that regardless of whether the riser is in static or dynamic equilibrium, it can elongate but its total length is almost constant. Thus, the constraint equation is: �2 � T 0 0 r r ¼ ð1 þ εÞ2 ¼ 1 þ (51) EA

(47) 2.1.3. Constraint equation Because of the unknown item Te , it is necessary to add a constraint equation. There are the following cases:

where EA is the axial tensile stiffness of the riser. Because the axial strain is small, the high-order item is omitted:

2.1.3.1. Top tension is known. Taking the infinitesimal ds for analysis, the micro-element is subjected to effective gravity, and the internal and external fluid forces. Both ends are affected by the effective tension, 8

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Table 2 Parameters for the section design of different modules. Section type

Position1

Position 2

Position 3

Position 4

Upper region (Bare) Upper region (Weighted) Transition region (bare) Transition region (Small buoyancy) Lower Region (Large buoyancy) Lower Region (Bare)

67 cm 5 cm 5 cm 22 cm 6 cm 15 cm

62 cm 5 cm 5 cm 22 cm 6 cm 20 cm

52 cm 5 cm 5 cm 22 cm 6 cm 30 cm

42 cm 5 cm 5 cm 22 cm 6 cm 40 cm

1 0 0 ðr r 2

1Þ ¼

Te

2νðρe Ae

ρi Ai Þ þ mi U 2i EA

s as independent variable. To consider geometric nonlinearity, the static equilibrium equation of the CVAR must be established at the corresponding position, and the 3D Euler equation (47) and constraint equation (50) or (52) constitute the basic governing equations. At the equilibrium position, δux ¼ δxs ; δvs ¼ δys . The internal fluid is assumed to be in steady flow, and its mechanical equation is as follows:

(52)

2.2. Mechanical equation of CVAR In this paper, the mechanical equation is universal and applied in different situations. For different forms of the riser, we flexibly choose independent variables to solve the problem more efficiently. The vari­ able s is suitable for any type of riser but it is necessary to know the riser’s length. Considering the special geometry of the CVAR, we choose

Fig. 8. Effect of the size of the buoyancy module on the maximum displacement. 9

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Fig. 8. (continued).

Zαt �� � 0 � x δπ ¼ Te 0s ss α0

� � 0� y þ Te 0s ss

Bs

Constraint equation of elongation condition:

� 2 0 � 0 ��� κ xs s00s ∂ xs 0 þ 03 δxs 0 0 ∂α ss s ss s

Z

α0

� 2 0 � ��� � κ ys s00s ∂ ys 0 þ 03 Bs δys dα 0 0 ss s s ∂α ss

α0

� 0 �� � 0 �� � � Bs ∂ xs Bs ∂ ys 00 δx δy00s dα þ 0 0 0 0 s ss2 ∂α ss ss2 ∂α ss Zαt �� α0

� 0 ys 0 fHt ss

0

0

0

xs ys 0 fHt þ 0 fHn ss ss 0

xs κxs U 2i 0 fHn þ mi 0 ss ss

1 02 0 x þ ys2 2 s

1

�

Te

2νðρe Ae

ρi Ai Þ þ mi U 2i EA

� dα ¼ 0

(54)

Equations (53) and (54) are the static mechanics equation of the CVAR.

Zαt �� þ

αt �

3. Model experiment on CVAR

(53)

The main purpose of the experiment was to conduct a preliminary exploration of the unique configuration of CVAR, verify the numerical model proposed above, and summarize the response analysis of the CVAR under various working conditions.

� 0 κy U 2 mi s 0 i δxs ss � � wa 0 δys ss dα 1 þ εs

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Fig. 8. (continued).

3.1. Design of experimental model

similarity between the flow systems under the action of gravity, the ratio of the inertial force of the fluid to gravity must remain unchanged, which is Froude’s law of similarity (Morooka and Tsukada, 2013). And the detailed parameters of model are as shown in Table 1.

3.1.1. Riser model The purpose of the experiment was to study the mechanical prop­ erties of the unique configuration of the CVAR. Therefore, the primary condition for the riser model was for it to naturally exhibit the proper configuration in a water tank and move with the water under different conditions. However, the high strength steel used in practice cannot be used to form a flexible tubular shape at a small scale, because of which selecting appropriate materials is a major difficulty. The pipe requires a certain rigidity in the local part and flexibility overall. By screening 20 materials, PVC (Polyvinyl chloride) was finally used as riser material. The buoyancy block was made of expandable polyethylene, EPE pearl cotton. It was non-absorbent, and had a very low density of only 0:022g=cm3 that can be neglected in weight-related experiments. It is the ideal experimental material of buoyancy module (Fig. 4a). And the weighted section is made of mirrored thick-walled seamless steel pipe, which is placed on the outer wall of the riser model to provide gravity (Fig. 4b). To use local similarity in the overall design while maintaining

3.1.2. Experimental equipment Model experiments were carried out in a wave and current flume, the flume can produce well controlled steady regular and irregular waves with constant velocity and wave height of 0.02m–0.2m according to different requirements and give a complete view of the model. The model support (Fig. 5a) is designed by using the aluminum alloy, it is fixed on the inner wall of the flume and a slot is installed in the middle of its bottom to make sure that the riser can be replaced quickly without water drainage. Therefore, the support can not only fix the riser model conveniently, but also improve the experimental efficiency. The mea­ surements show that the natural frequencies of the support and the riser model are far from each other, and the fluid action will not lead to the resonance. Besides, flexible joint (Fig. 5b) is adopted at both ends of the riser model to ensure its movement in any direction. Moreover, the 3DPDT Precision Displacement Meter is a non-contact 3D video 11

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results. While for the vibrational amplitude, there was a slight difference between the two sets of results because the numerical simulation rep­ resented an ideal state. Thus, the vibrational amplitude was very stable in it. However, in the model experiment, the measured amplitude fluc­ tuated due to the influence of wave reflection, cross-flow, and noise. By comparing displacements at three key points, it was evident the dis­ placements of the transition region and the weighted section were large, and decreased rapidly in the large buoyancy module. The numerical and experimental results showed the same trend. This model was thus verified.

Table 3 Parameters used in dynamic analysis of CVAR. Detailed parameters

Value

Horizontal distance between the wellhead and the platform xL (m)

610

Water depth yH (m)

2438

Total length of the riser S (m)

2601

Upper region length Lu (m)

1695

Transitional region length Lt (m)

416

Lower region length Ll (m)

490

Buoyancy factor of the buoyancy module in the lower region Cf

6

Length of the buoyancy module in the lower region Llb (m)

190

Buoyancy factor of the buoyancy module in the transitional region Cf

2

Location of buoyancy module (from wellhead) (m) Outer diameter of the riser De (m)

491 to 907 0.3

Thickness of the riser t(m)

0.03

Buoyancy factor of the buoyancy module in the upper region Cf

1.5

Elastic modulus E(Pa)

2.07e11

Poisson’s ratio Yield strength σr

0.3 720

Seawater density de (kg/m3)

1025

Riser material density dr (kg/m3)

7850

Internal fluid density di (kg/m3)

1000

Velocity of sea surface Ueh (m/s)

1.0

Normal drag force coefficient CDn

0.7

Gravitational acceleration g(m/s2)

9.807

Velocity of internal fluid Ui (m/s)

Tangential drag force coefficient CDt

3.2.2. Sensitivity analysis of position of buoyancy module In this section, we describe the use of the model experiment to analyze the influence of the position of the buoyancy module on dy­ namic performance. The length of the lower region (bare) was set to 15 cm, 20 cm, 30 cm, and 40 cm, and the other parameters are as shown in Table 2. The time history curve of the maximum displacement was measured. Fig. 8 shows that as the position of the buoyancy module moved upward, the displacement of the model increased in all directions. The upward displacement of the buoyancy module also led to an increase in the dynamic response of the riser and rendered it more unstable because the upward movement of the buoyancy blocks meant that the transition zone moved upward, and the influence of waves increased accordingly. At the same time, the upward movement of large buoyancy blocks caused the lower riser to become too long to provide sufficient tension, and the stiffness of the lower riser section decreases, which made it easier for vibrations to occur. In practice, thus, the buoyancy module should be placed as low in the riser as possible. But at the same time, the length of the lower region should have a certain feasible value.

30 0.03

4. Dynamic analysis of CVAR Dynamic analysis is a key step in the design of a marine riser. Based on the formula described in Section 2, we created a finite element pro­ gram in MATLAB to discuss the dynamic performance of the CVAR (Aamo and Fossen, 2000). In this part of the analysis, we defined the farthest drift from the wellhead to be the far end (i.e., Far, xL ¼ 730 m), the intermediate po­ sition was set as the equilibrium position (i.e., Cross, xL ¼ 610 m), and the near end was defined as the position where the offset was close to the wellhead (i.e., Near, xL ¼ 490 m). Thus, assuming that the offset of the floating body was 120 m in extreme marine conditions, we used a water depth of 2400 m as an example and performed a static mechanical CVAR analysis at three locations. Our design parameters are listed in Table 3. 4.1. Time domain analysis of key nodes For further analysis, we used four key nodes (Node 1–Node 4), as shown in Fig. 9, at 0 m (platform), 1710 m (top of small buoyancy modules), 1910 m (bottom end of small buoyancy modules), and 2200 m (bottom end of large buoyancy modules). The time history curves of each key node are shown below. Fig. 10 shows that the vibration near the platform was the largest due to the wave load, and decreased rapidly when in the transitional region. The amplitude of vibration of the lower region decreased to 1/10 due to its buffering effect. The maximum vertical and horizontal displacements were reduced to less than 1 m, which ensured the verticality of the lower region. Fig. 10g shows that stress in the transition zone of the riser was generally high, and the corresponding bending moment was large (Fig. 10h), indicating the stress was mainly caused by the bending moment of the riser.

Fig. 9. Four key nodes.

displacement sensor (Fig. 5c) used to measure spatial position, object displacement, and strain. This experiment used it to measure the displacement of the riser model. 3.2. Experimental results 3.2.1. Verification of results A typical case was used in the experiment involving a regular wave (height 0.04 m, period 2 s). The bare tube and the CVAR model are shown in Fig. 6. We compared the experimental results with the results of the nu­ merical simulation as shown in Fig. 7. As shown in Fig. 7, the results of the numerical simulation in terms of vibrational frequency were in good agreement with the experimental

4.2. Regular wave analysis CVARs are designed for using in ultra-deep water under complex 12

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Fig. 10. Dynamic characteristics of CVAR.

13

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Fig. 10. (continued).

conditions. Therefore, dynamic response analysis under environmental loads is important. Regular wave analysis with variation of period over a certain range is used for the extreme response studies. Maximum wave height and associated maximum wave period are used for different load cases, and the wave period is varied by þ= 1:5 seconds on either side of Tmax to estimate variations in load. The data for regular wave analysis were based on oceanic conditions in the Gulf of Mexico and the specific parameters of platform motions are identified in Table 4.

4.2.1. Regular wave analysis under hurricane conditions Fig. 11 shows that regardless of the position of the far, mid-, or near end, the vertical displacement of the riser was the same. The vertical displacement of the upper region reached 7.1 m and that of the lower region was limited to 0.1 m or smaller. Because the transition region effectively buffered the vertical displacement of the upper portion, this ensured the stability of the lower region. Fig. 12 shows that while that the top displacement was significantly affected by the platform, the transition region had the largest horizontal displacement in the transition and lower regions because the transition region had a smaller stiffness and was relatively flexible, which rendered it more susceptible to disturbance. At the near end, the horizontal displacement decreased but the range was very small, However, hori­ zontal displacement at the far end increased prominently. In the tran­ sition and lower regions, the maximum displacement even exceeded the top displacement with a value of 10.2 m, and the maximum horizontal displacement of the lower region reached 3.17 m (at 2295 m). Fig. 13 shows that the trend of change in the maximum Mises stress was the same when the platform was at the far, mid-, and near end. Both ends were low stress areas, and high stress was concentrated in the transition region. The maximum Mises stress was controlled to be below 200 MPa and the maximum stress was 422.4 MPa in the transition zone (1740 m in the riser). For the upper and lower regions, the stress increased to a greater extent when the riser was located at the far end, whereas for the transition area of the riser, the maximum stress

Table 4 Environmental loads and platform motions parameters. Type

Current

Hurricane

Height Hmax (m)

2.68

23.8

Period Tmax (s)

5.2

23.6

Surface velocity (m/s) Bottom velocity (m/s)

2.7 0

1.76 0

Platform displacement (m)

102.5

102.5

Platform motions

0.02sint 1E (-6)sint 0.02sint 1E (-6)sint 0.01sint 1E (-6)sint

10sin0.5t 1E (-3)sin0.5t 10sin0.5t 1E (-3)sin0.5t 5sin0.5t 1E (-3)sin0.5t

Wave Current

Surge (m) Sway (m) Heave (m) Roll (deg) Pitch (deg) Yaw (deg)

14

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Fig. 11. Maximum vertical displacement along length.

Fig. 12. Maximum horizontal displacement along length.

Fig. 13. Maximum Mises stress along length.

increased by about 50 MPa when the riser was located at the far and near ends. Therefore, the middle position was the safest, and the offset of the upper platform should be minimized. As shown in Fig. 14, the weighting module and large buoyancy module provided sufficient tension for the upper and lower sections of the riser. The minimum tension of the transition zone was even subzero at the far end, which is not empirically possible. Therefore, the CVAR was unstable when the platform was at the far end under hurricane conditions. Changes in the stress of the transition region should thus be

considered to avoid negative values. 4.2.2. Regular wave analysis under current conditions Fig. 15 shows that compared with the hurricane conditions, the response of CVAR under current conditions is less, and displacement was controlled below 0.1 m because the current as the main load was a static load, the impact on the CVAR was mostly reflected in the static configuration, and the response in the time history was minor. Fig. 16 shows that compared with the hurricane conditions, the 15

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Ocean Engineering 199 (2020) 106946

Fig. 14. Minimum effective tension along length.

Fig. 15. Maximum displacements along length.

Fig. 16. Maximum Mises stress along length.

16

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Fig. 17. Minimum tension along length.

the bottom as possible.

maximum Mises stress of the upper and the lower regions were nearly identical, and the difference was mainly concentrated in the transition area. The three cases differed by nearly 100 MPa because the current had a significant influence on the configuration of the riser, especially in the relatively flexible transition region. The maximum Mises stress was 313.9 MPa (1600 m) and 314.9 MPa (1750 m) when the platform was at the near end. Fig. 17 shows that tension in the riser was not considerably different when the platform was at the far, mid-, and near end. The riser also did not have a negative tension point, which was safer in the three cases of current conditions. And the three conditions were clearly sufficient to provide at least 672 kN of tension for the bottom end stress joint.

5. Conclusion To explore the dynamic performance of CVARs using model experi­ ments and numerical simulations, based on virtual work and variational principles, the vibration equation of the CVAR was established and validated by model experiments in this study. Time-domain analyses of key nodes, regular wave analysis, and dynamic response at different positions of the buoyancy module were conducted as well. The results showed the following: The vibration near the platform was the largest due to the wave load, and decreased rapidly when in the transitional region. The amplitude of vibration of the lower region decreased to 1/10 due to its buffering ef­ fect. The maximum vertical and horizontal displacements were reduced to less than 1 m, which ensured the verticality of the lower region. Regardless of the position of the platform, the vertical displacement of the riser was constant under hurricane conditions. The vertical displacement of the upper region reached 7.1 m and that of the lower region was limited to 0.1 m or smaller. The largest horizontal displacement was significantly affected by the platform, and horizontal displacement at the far end increased prominently. Both ends were lowstress areas, and high stress was concentrated in the transition region. The maximum Mises stress was controlled to below 200 MPa and the maximum stress was 422.4 MPa in the transition zone. The minimum tension in the transition zone was even subzero at the far end, which is not empirically possible. Therefore, the CVAR was unstable when the platform was at the far end under extreme wave conditions. The response of CVAR subjected to current conditions is less than that of hurricane conditions, and displacement was controlled to below 0.1 m. The maximum Mises stresses of the upper and the lower regions were nearly identical, and the difference was mainly concentrated in the transition area when the platform was at the far-, mid-, and near-ends. The three cases differed by nearly 100 MPa. The position of a higher buoyancy block not only increased the dy­ namic response of the riser, but also led to an increase in the bending moment at the bottom. As the large buoyancy module moved upward, the length of the lower region increased and tension at the wellhead decreased. When the buoyancy block was above 1000 m, the bottom tension was smaller than 672 kN, which did not satisfy the requirements of the bottom joint, and caused the lower region to become catenary shaped and thus losing its unique operational advantage.

4.3. Dynamic response analysis with different positions of buoyancy module The parameters for dynamic response analysis with different posi­ tions of buoyancy module are given in Table 5. Fig. 18a shows that as the buoyancy module moved upward, the maximum vertical displacement of the lower region increased, which corresponded to the results of the above model experiment. Fig. 18b shows that when the large buoyancy block was at 1200 m, a high stress point appeared at the bottom of the riser. This point also had a high bend moment point. Because the lower region was too long, it was catenary shaped, and thus lost its unique operational advantage. It is clear from Fig. 18d that as the large buoy­ ancy module moved upward, the length of the lower region increased and tension at the wellhead decreased. When the buoyancy block was above than 1000 m, the bottom tension was smaller than 672 kN, which did not satisfy the needs of the bottom joint. To further corroborate this, the riser profile is shown in Fig. 18 with the buoyancy module located at different positions. Fig. 19 shows that when the buoyancy module was too high, the upper portion was too curved to provide a certain vertical length for operation. At this time, the lower portion was too long to close to the catenary shape, and the advantage of CVAR could not be exploited with this type. Therefore, the buoyancy module should be placed as close to Table 5 Parameters for the section design of CVAR. Section Type

1

2

3

4

5

6

Upper region (Bare)/m Upper region (Weighted)/m Transition region (bare)/m Transition region (Small buoyancy)/m Lower Region (Large buoyancy)/m Lower Region (Bare)/m

1500 100 100 410

1300 100 100 410

1100 100 100 410

900 100 100 410

700 100 100 410

500 100 100 410

190

190

190

190

190

190

200

400

600

800

1000

1200

Author contributions section Min Lou contributed significantly to the conception and the prepa­ ration of the study; Weixing Liang performed the data analyses and revised the 17

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Ocean Engineering 199 (2020) 106946

Fig. 18. Influence of position of buoyancy module.

Fig. 19. Shapes of CVAR at different positions of the buoyancy module.

manuscript; Run Li carried out the model experiments.

manuscript is approved by all authors for publication. I would like to declare on behalf of my co-authors that the work described was original research that has not been published previously, and not under consideration for publication elsewhere, in whole or in part. All the authors listed have approved the manuscript that is enclosed.

Declaration of competing interest No conflict of interest exits in the submission of this manuscript, and 18

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Ocean Engineering 199 (2020) 106946

Acknowledgments

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