Hydrodynamic performance of a novel semi-submersible platform with nonsymmetrical pontoons

Ocean Engineering 110 (2015) 106–115

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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

Hydrodynamic performance of a novel semi-submersible platform with nonsymmetrical pontoons Shuqing Wang a,n, Yijun Cao a, Qiang Fu b, Huajun Li a a b

College of Engineering, Ocean University of China, Qingdao 266100, China CIMC Offshore Institute Co. Ltd., Yantan 264670, China

art ic l e i nf o

a b s t r a c t

Article history: Received 6 March 2015 Accepted 7 October 2015 Available online 26 October 2015

The hydrodynamic performance of a novel semi-submersible platform (SEMI) was investigated in the present study. This new type of SEMI, which features two nonsymmetrical pontoons and no horizontal connection braces, can serve as a semi-submersible crane vessel. For a comparative study, two conventional SEMI models, i.e., one SEMI having both twin symmetric pontoons and horizontal braces and the other SEMI having twin symmetric pontoons with no horizontal braces, were established. The hydrodynamic characteristics were first investigated and compared with a special focus on the heave, roll and pitch motions. Furthermore, the motion response of the novel SEMI was studied under normal operation in eight different wave directions. The numerical results demonstrate that there exists significant heave-roll/heave-pitch coupling effects in the novel SEMI due to the nonsymmetrical pontoon shapes. These coupling effects require that more attention be paid to long waves whose periods are close to the heave resonance period. The motion responses of the novel SEMI are generally satisfactory for a typical normal sea state. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Semi-submersible platform Nonsymmetrical pontoons Hydrodynamic performance Motion response

1. Introduction With the rapid development of hydrocarbon exploration and exploitation in deepwater ocean, floating crane vessels play an increasingly important role in the operation of offshore installations. Basically, there are three different types of floating crane vessels around the world (Clauss and Vannahme, 1999). One type is the semisubmersible crane vessel (SSCV) that has a semisubmersible hull shape and revolving cranes. SSCVs have favorable sea-keeping behavior and an immense deckload capacity as compared to other crane ships. These vessels require a sophisticated ballast system for counter ballast in order to operate cranes in different slewing angles. Other types of crane vessels comprise monohull vessels and catamaran-shaped crane vessels, outfitted with revolving cranes. Advantages of monohulls are higher transit speed, associated with lower mobilizing costs. However, they are more weather sensitive compared to SSCVs due to larger waterplane area, and need also strong support and sophisticated ballast system. Shear leg cranes are another type of floating cranes, having a barge-shaped hull with large water-plane area, operating in sheltered waters of harbors where waves are relatively low and in offshore regions at moderate weather conditions. n

Corresponding author. Tel.: þ 86 532 66781672; fax: þ86 532 66781550. E-mail address: [email protected] (S. Wang).

http://dx.doi.org/10.1016/j.oceaneng.2015.10.012 0029-8018/& 2015 Elsevier Ltd. All rights reserved.

Among those floating crane vessels, SSCVs are widely employed in all types of heavy lifting at offshore projects around the world. Whereas the monohull crane vessels have excessive response to the wind-driven waves and swell, this limits the workability of the vessel. A semi-submersible obtains its buoyancy from ballasted, watertight pontoons that are fully submerged during operations, supporting four to eight columns that extend through the water plane and in turn support the deck. The vessel is therefore subject to minimum wave loading, and corresponding motions. The semisubmersible concept was first developed for offshore exploratory drilling but the advantages of the semi-submersible vessel stability were soon recognized for offshore construction. Crane vessels and barges operating in the North Sea were quite sensitive to wave action, and this made operations during the winter months virtually impossible. To increase heavy-lift capacity and operability in the North Sea, Heerema Marine Contractors constructed the world's first SSCVs – sister vessels Balder and Hermod In 1978. The SSCVs were much less sensitive to sea swell, therefore it was possible to operate on the North Sea during the winter months. The high stability also allowed for heavier lifts than was possible with a monohull. The larger capacity of the cranes reduced the installation time of a platform from a whole season to a few weeks. So far, there are several SSCVs in use. The representative of them are DCV Thialf, DCV Balder and HLV Hermod of Heerema Marine Contractors, Saipem7000 of Saipem, DB-101 of J. Ray

S. Wang et al. / Ocean Engineering 110 (2015) 106–115

McDermott, etc. (Clauss and Riekert, 1990). Generally, these SSCVs share common structural features such as identical twin pontoons, four/six/eight columns, horizontal braces and large upper deck. And the twin cranes are installed at the left and right parts of the hull deck on the ship bow, respectively. At present, this kind of structure has been regarded as the classic SSCV type. A major consideration of any floating production vessel is the dynamics and performance of these vessels, which is of great importance in overall field development (Söylemez, 1998). Early studies concerned with the stability of the semi-submersibles under intact and damaged conditions (Numata et al., 1976; Dahle, 1981). Later, extensive studies have been investigated on the hydrodynamic performance and motion responses (Wu, 1986; Yilmaz and Incecik, 1996; Wu et al., 1997; Söylemez, 1998; Feng et al., 2009; Ng et al., 2010; Li et al., 2011; Zhu and Ou, 2011; Yang, et al., 2012). As for the SSCV, the nonlinear dynamic responses of moored crane vessels to regular waves are investigated experimentally and theoretically (Ellermann et al., 2002). Both the developed models and the analytical tools can be used to identify the limits of the operating range of floating cranes. Jacobsen and Clauss (2005) investigated the sea-keeping behavior of SSCV Thialf in detail in time and frequency domain deriving definitions of operational limits and the coupling effects with a barge floating nearby. In order to examine the effects of the water depth on heave and pitch motion, model tests accompanied by numerical calculations were conducted on SSCV Thialf (Clauss et al., 2009). Motion behavior of the crane vessel is investigated focusing on the effect of the water depth on the hydrodynamic coefficients, i.e. potential damping, added mass and exciting forces. From the literature review, one can see that for both the semi drilling rigs and SSCV, the main hull structure of semi-submersible shares common structural features of identical twin pontoons and several columns, connected with braces. An innovative semisubmersible platform, named Explorer Lifter, was designed and constructed by Yantai CIMC Raffles Offshore Ltd., China. This new type of SEMI, which served as a semi-submersible crane & accommodation vessel, integrated the functions of heavy-lifting, storage of cargo, and accommodation. It is the first asymmetric semi-submersible unit without bracing in the world. As shown in Fig. 1, the novel SSCV has several significant innovations and notable features. One major innovation is the different sizes of the two pontoons. Classic SEMIs commonly have two identical pontoons. However, the external shapes of the two pontoons of this novel SSCV have different dimensions, i.e., one is large and one is small. Accordingly, the two groups of columns also have different dimensions. Another characteristic is that there are no horizontal connection braces between the columns of the SEMI. In addition, both of the lifting cranes are installed on the same side of the upper deck, above the large pontoon. This new SEMI

107

configuration has several advantages. Asymmetric pontoon outline with pneumatic de-ballast system is very useful for quick ballast adjustment to suit heavy lifting operation, and the time for ballast adjustment is reduced from normally 1 h to 15 min. Because the two cranes are placed on the same side as the large pontoon, the new SEMI is able to get closer to the lifting object. Furthermore, the lack of horizontal connection braces results in a reduced towing resistance and dynamic positioning load, which improved the transit speed from about 8–9 knots to more than 11.3 knots. As a new type of deep-water floating structure, the first asymmetric semi-submersible unit without bracing in the world, the hydrodynamic performance of this novel SEMI is worthy of attention and investigation. To present the results of the investigation, the paper is organized as follows. In Section 2, the theoretical background related to the study is briefly described. The three different SEMI models used for the comparative investigation, including the principal parameters and mooring systems, are introduced in Section 3. In Section 4, the hydrodynamic performances of these SEMIs are investigated and compared, and the conclusions are summarized in Section 5.

2. Theoretical background Floating structures positioned with mooring systems generally consist of large- and small-scale structures and are subjected to different types of wave forces. In this section, the theoretical basis related to the hydrodynamic analysis is briefly summarized. 2.1. Potential theory For large-scale structures compared with the wavelength, such as the columns and the pontoons, wave diffraction and radiation force are considered the main wave loads on the structures. The 3D diffraction-radiation theory can be used to calculate the hydrodynamic loads on the structures. When a floating structure is subjected to wave action, the wave incident upon the floating structure is diffracted to produce a scattered wave field and sets the structure in motion to produce a radiated field. Through linear superposition, the velocity potential can be decomposed into three parts as follows (Faltinsen, 1993):

Φðx; y; z; tÞ ¼ ΦI ðx; y; z; tÞ þ ΦD ðx; y; z; tÞ þ ΦR ðx; y; z; tÞ

ð1Þ

where potential ΦI represents the incident wave potential, ΦD the scattered wave potential, and ΦR the radiation potential. Each type of potential must satisfy Laplace's equation with the associated boundary conditions throughout the domain of the fluid. After solving the resulting boundary-value problem, the potentials ΦI ,

Fig. 1. Geometrical model and stereogram of the novel SEMI. (a) Geometrical model, (b) Photo of Explorer Lifter.

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S. Wang et al. / Ocean Engineering 110 (2015) 106–115

ΦD , and ΦR can be readily computed, and the hydrodynamic loads can be calculated directly using Bernoulli's equation. For the first-order computation, the six components of the force and moment vectors are obtained through the direct integration of the dynamic pressure p (from the linearized Bernoulli's equation) over the body average wet surface (sH ): " # F k ¼ ∬ pnk ds ¼ ρRe iωae  iωt ∬ ðϕI þ ϕD Þnk ds sH

þ ρRe

"

sH

6 X j¼1

#

iωυj e  iωt ∬ sH jϕj nk ds

ð2Þ

where ρ is the water density; a is the incident wave amplitude; ϕI and ϕD are the spatial parts of the incident wave potential and the diffracted wave potential, respectively; ϕj ðj ¼ 1; 2; …; 6Þ is the radiation potential; υj is the complex amplitude component in the jth mode when the floating body undergoes small motions from a regular wave; and nk ðk ¼ 1; 2; 3; …; 6Þ is the outward unit normal vector at the body wet surface sH . The first term on the right-hand side of Eq. (2) is related to the linear wave exciting forces and moments and are proportional to the incident wave amplitude: " #    iωt F wk ¼ ρRe iωae ∬ ϕI þ ϕD nk ds ð3Þ sH

The second term on the right-hand side of Eq. (2) corresponds to the added mass and damping coefficient of the floating body with the restriction that body motions are small and sinusoidal in time. It can be rewritten as ' j F Rk ¼  μkj ξj  λkj ξ

ð4Þ

where F Rk is the radiation force (moment); μkj and λkj represent ̇

}

the added mass and potential damping, and ξj and ξj are the linear motion velocity vector and acceleration vector of the floating structure. 2.2. The Morison equation For small-scale structures compared to the wave length, such as braces, the diffraction effect is usually negligible, and the viscous effect becomes significant. For these members, the Morison equation is commonly used for evaluating the wave force. Floating structures undergo large motions, and it is necessary to consider the relative motions between the structure and the fluid particles during a dynamic analysis. Therefore, the general equation for fluid forces acting on a cylinder, considering the relative motion of the body in the fluid, can be given as   } ̇  ̇  ̇ 1 F ¼ C M ρVu  C m ρVξ þ C D ρAðu  ξÞu  ξ ð5Þ 2 where C M is the inertial coefficient; C m is the add mass coefficient (C m ¼ C M  1); C D is the drag coefficient; V is the volume matrix for the body; A is the area matrix for the body; u is the water ̇

particle velocity; u is the water particle acceleration; and }  V ¼ kT ðln ϵ1 = ln ϵ2 Þ=ψ ðσ 1 σ 2 Þ and ξ are the velocity and acceleration of the small-scale structure, respectively. The last term in the right hand side of Eq. (5) represents the drag force in the relative velocity form, which indicates that the drag force contributes to both exciting force and damping to the motion of the structure. In the following numerical study, the values of C M and C D are set to 1.8 and 1.0, respectively (DNV, 2010). It should be noted that the drag forces are nonlinear with respect to the wave particle velocity. In order to perform a

frequency domain analysis using the transfer function concept, it is necessary to approximate the nonlinear problem as a linear one. This involves replacing the nonlinear function with an equivalent linear function by minimizing the errors involved in some way. In other words, the drag force is approximated as (WADAM, 2005)     ̇ ̇ 1 8 F D ¼ C D ρA V max u  ξ ¼ B u  ξ ð6Þ 2 3π where B is the linearized viscous damping matrix expressed as B ¼ 12C D ρA38π V max . The term 38π V max is a standard result obtained by assuming equal work done at resonance by the non-linearised and the equivalent linear damping term. V max is a linearising velocity amplitude specified as input to Wadam (WADAM, 2005). 2.3. Frequency domain analysis By applying Newton's law and including the added mass, damping and exciting force contributions acting on the panel and   Morison parts of a floating model, the motion vector X ω; β can be found from the equation of motion (Ran, 2000; WADAM, 2005)          ω2 ðM þMa Þ iω λðωÞ þ Bv þ K þ Ke X ω; β ¼ F ω; β ð7Þ where M is the mass matrix of the floating structure; Ma , the added mass matrix, includes either the frequency dependent added mass or added mass from Morison equation, or both of them; λðωÞ is the frequency dependent potential damping matrix; Bv is the linearized viscous damping matrix; K and Ke are the hydrostatic restoring and external restoring matrices, respectively;   F ω; β is the exciting force vector for frequency ω and incident wave heading angle β . 2.4. Coupled dynamic analysis in the time domain As floating production systems extend to deeper waters, the effects of mooring and risers become increasingly significant when predicting the response of the floater. In the time-domain simulation, the motion of the semi-submersible platform and the mooring system are fully coupled. The floating vessel is treated as a nodal component, assuming that the vessel acts as a rigid body. The forces on the vessel, which are represented by a large-volume body, are computed separately at each time step. The equation of motion for the coupled system in the time domain is given as follows (Ran, 2000): Z t ̇  } M þ μð1Þ ξ ðt Þ þ Rðt  τÞξðτÞdτ þ Kξðt Þ ¼ F w ð8Þ 1

where μð1Þ is the added mass at the infinite frequency; F w is the wave's exciting force, including forces from the Morison equation and from the mooring system (and other structures connected to the floating structure, such as the risers). Wind and current forces are not considered in the hydrodynamic analysis performed in this study. RðtÞ is the retardation function and is related to the frequency domain solution of the radiation problem as follows: Z 2 1 Rðt Þ ¼ λðωÞcos ωtdω ð9Þ

π

0

3. Numerical models of the floating systems 3.1. SEMI models As described in Section 1, the novel SEMI features two pontoons of different sizes, and accordingly, it has two different groups of columns. The large main pontoon has a length of 137.75 m, a breadth of 19.5 m and a depth of 12 m, and the size of

S. Wang et al. / Ocean Engineering 110 (2015) 106–115

the small one is 113 m  16.5 m  12 m. Similarly, the main dimensions of the large columns are 20.25 m  19.5 m  18 m, and the small columns have a length of 16.5 m, a breadth of 13.5 m and a height of 18 m. The operational draft is 20 m, and the total displacement of the novel SEMI is 55088 t in this case. The other main particulars are listed in Table 1. To investigate the hydrodynamic performance of this novel SEMI and to compare it with the performances of classic SEMIs, two traditional SEMI models were created, each having a hull displacement identical to that of the novel SEMI. The first traditional SEMI has both twin identical pontoons and horizontal connection braces, and the second one has twin identical pontoons but no horizontal connection braces. For brevity and convenience, in the present paper, the novel SEMI is referred to as model A. The conventional SEMI with both symmetrical pontoons and connection braces is referred to as model B, and the conventional SEMI with symmetrical pontoons and without connection braces is referred to as model C. The particulars of models B and C are listed in the last two columns of Table 1 for comparison purposes. As observed, the dimensions of the pontoons and the columns are identical for models B and C. The length, width and height of the pontoons are 129.28 m, 16.5 m and 12 m, respectively, and the corresponding dimensions of the columns are 20.25 m, 16.5 m and Table 1 Structural particulars of the SEMI models. Description

Unit

Deck size Pontoon length

m m

Pontoon width

m

Pontoon height Number of columns Column length

m \ m

Column width

m

Column height Brace length Brace diameter Operation draft Total displacement COG position

m m m m T

COG above the baseline COB above the baseline Ixx Iyy Izz Local fillet radius Water plane area

m

Model A

Large Small Large Small

Large Small Large Small

m

m 2

Kg*m Kg*m2 Kg*m2 m m2

Model B

Model C

81  81 137.75 113 19.5 13.5 12 4

81  81 129.28

81  81 129.28

16.5

16.5

12 4

12 4

20.25 16.5 19.5 13.5 18 \ \ 20 55088

20.25

20.25

16.5

16.5

18 48 3 18.59 55088

18 \ \ 19.16 55088

(0,  7.4, 8.02) 28.02

(0.53, 0, 9.08) (0.53, 0, 8.57) 27.67

27.73

8.02

7.67

7.73

6.21e þ 10 6.64e þ 10 1.19e þ11 2.25 1211.53

6.45e þ10 6.96e þ10 1.22e þ11 2.25 1193.70

6.39e þ 10 6.90e þ 10 1.22e þ11 2.25 1193.70

109

18 m, respectively. The horizontal connection braces of model B are constructed of cylindrical steel and have a length and external diameter of 48 m and 3 m, respectively. Because the hull displacement is the most important factor for the hydrodynamic performance of an offshore floating structure, the premise of the comparison study is to ensure an identical displacement of these three models under operational conditions. However, due to their different external shapes, the drafts of models A, B and C are slightly different, namely 20 m, 18.59 m and 19.16 m, respectively. In addition, the center of gravity (COG) of model A is not in the middle longitudinal section but approximately 7.4 m away from the longitudinal y-axis. However, the COGs for both models B and C are located on the longitudinal y-axis due to their structural symmetry. Moreover, the moments of inertia about the x-axis (Ixx) and the y-axis (Iyy) of model A are smaller than those of models B and C. The global coordinate system is a right-handed system with the x-axis forward, the y-axis toward the port side and the z-axis upward. The origin is placed in the middle of the unit in the transverse direction, in the middle of the unit in the longitudinal direction and 12 m above the bottom of the unit in the vertical direction. The hydrodynamic models were created using SESAM GeniE, as shown in Fig. 2. As observed, there are three features worthy of attention. First, only model A is not symmetrical about the middle longitudinal section. Second, none of these models is symmetrical about the middle transverse section because the external shape of the bow and stern of the pontoons is different. Third, only model B has horizontal connection braces. Furthermore, models A and C consist of only panel elements, whereas model B includes both panel elements and Morison elements due to the presence of connection braces. 3.2. Mooring systems The purpose of a mooring system is to limit the movement of a floating structure such that the platform remains in its operational region. In this study, these SEMI models are positioned by a taut mooring system. The layout of the mooring system is illustrated in Fig. 3. The mooring system consists of 12 mooring lines in four groups, each of which has three mooring lines. The four groups of mooring lines are each separated by an angle of 90 degrees, and the separation angle between the mooring lines in each group is 15°. The mooring line is 700 m in length. Each mooring line is divided into three segments, forming a chain-polyester-chain structure. More detailed parameters regarding the mooring lines are presented in Table 2.

4. Results and discussion Numerical investigations of the hydrodynamic characteristics of the three SEMI models were first performed in the frequency domain, as conducted in Section 4.1. For the motion response

Fig. 2. Hydrodynamic models of the SEMI models A, B and C.

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S. Wang et al. / Ocean Engineering 110 (2015) 106–115

simulation in Section 4.2, a coupled dynamic analysis was then conducted in the time domain due to the possible presence of nonlinearities from mooring systems. The comparison results were analyzed in detail in each subsection. 4.1. Hydrodynamic characteristics As described above, the bow and stern of the pontoons of all of these SEMI models are asymmetrical about the middle transverse section. Furthermore, the pontoons of the novel SEMI model are also asymmetrical about the middle longitudinal section. To fully investigate the hydrodynamic performance, the angle range of the incident wave must be sufficiently wide. Wave directions with headings from 0° to 360° with an interval of 15° were chosen, and the wave periods were chosen to be 1–40 s. A linear regular wave was used in the analysis. The hydrodynamic parameters were investigated in detail using the SESAM HydroD module. Because the DOFs of the heave, roll and pitch motions are of concern, only the hydrodynamic coefficients in the vertical DOFs are investigated in the following section. 4.1.1. Added mass and potential damping When floating objects undergo forced harmonic motion in still water, even though there is no incident wave, the motion will generate outgoing radiation waves, which result in radiation loads on the floating object. Added mass and potential damping are important hydrodynamic parameters for a deep-water floating structure. In this subsection, the added masses in the heave, roll and pitch motions of the three SEMI models are first investigated. Fig. 4 shows the added mass in the heave motion. One can see that the added masses of each of the three SEMI models show similar characteristics as the wave frequency increases. When the

frequency is less than 0.6 rad/s, the added mass of model A is nearly the same as that of model C and slightly lower than that of model B. In the frequency range from 0.63 rad/s to 0.9 rad/s, the value of the added mass of model A was the lowest. When the frequency exceeded 0.9 rad/s, the added mass of model A was always larger than that of the other models. Similar variation trends can be observed for added masses in the roll and pitch motions, which are not shown here due to space limitations. The potential damping in the heave, roll and pitch motions was then investigated; Fig. 5 shows the potential damping in the heave motion. Damping plays an important role in decreasing the corresponding motion amplitude, which means that a large magnitude of potential damping is good for a platform's motion performance. One can observe that the three potential damping curves associated with models A–C show similar characteristics as the frequency increases, and these are generally close to each other over most of the frequency range. In a narrow-band frequency range from 0.8 rad/s to 1.16 rad/s, these curves deviate somewhat from each other, and the potential damping of the novel model A was the lowest. In other words, the potential damping characteristic is as good as the conventional one, except in the local narrow-band frequency range. As expected, similar variation trends can be found for potential damping in the roll and pitch motions; the results are not shown here due to space limitations. 4.1.2. Motion RAOs The motion characteristics for a floating object are normally described by response amplitude operators (RAOs). In this subsection, the RAOs of the novel SEMI are be investigated in detail. The heave RAO is shown in Fig. 6, in which the regular wave period is used as the abscissa and the response amplitude per wave amplitude is used as the ordinate. As observed, the three curves show similar characteristics. There exist sharp peaks around the wave period of 20 s, which corresponds to the resonance periods of models A, B and C, respectively. To the right of the natural period in Fig. 6, the water plane stiffness mostly governs the motion, and the RAO asymptotically approaches 1.0 for long waves. To the left of the natural period, the total mass and hydrodynamic excitation forces dominate the motions. At the natural period, heave damping dominates the motions. As observed, the heave response of the novel SEMI model (model A) is generally slightly smaller than that of models B and C in the period range from 4 to 16 s. In addition, the heave natural periods of models A, B and C are slightly different, with resonance periods of 19.62 s, 19.94 s and 19.79 s, respectively. The deviations of the resonance periods and peak values can be explained theoretically as follows. It is well known that the uncoupled and undamped natural period of a floating structure in heave motion is a function of the total mass and the water plane stiffness, and the equation is as follows (Faltinsen, 1993): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M þ μ33 T 33 ¼ 2π ð10Þ ρgAw where T 33 is the heave resonance period of the floating structure, M the floater mass, μ33 the heave added mass, and Aw the water

Fig. 3. Mooring system layout. Table 2 Mooring line characteristics. Mooring line

Mooring material

Length (m)

Diameter (m)

Wet density (kg/m)

Axial stiffness (kN/m)

Mini-breaking loads (kN)

Upper chain Middle polyester Bottom chain

chain polyester chain

60 550 90

0.095 0.095 0.095

171 43.68 171

678705 79629 678705

10512 9987 10512

S. Wang et al. / Ocean Engineering 110 (2015) 106–115

Added mass matrix μ

7

9

x 10

111

33

model A model B model C

8

Added mass(kg)

7 6 5 4 3 2 1 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Wave Angular Frequency(rad/s) Fig. 4. Added mass in heave motion (μ33 ).

Potential damping matrix λ 33

7

6

x 10

model A model B model C

Potential damping(kg/s)

5 4 3 2 1

0 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Wave Angular Frequency(rad/s) Fig. 5. Heave potential damping (λ33 ).

Heave Amplitude (m/m)

2.5

model A model B model C

2

1.5

1

0.5

0 5

10

15

20

25

30

35

40

Wave Period(second) Fig. 6. Heave RAO at 0°.

plane area of the floater. In this study, the masses of models A–C are equivalent due to the identical displacements of the three models. Therefore, the heave period is directly proportional to the heave added mass μ33 and inversely proportional to the water plane area Aw . The water plane areas Aw of models A, B, and C are 1211.53 m2, 1193.70 m2 and 1193.70 m2, respectively. One can note that a larger value of μ33 results in a larger heave resonance period. As shown in Fig. 4, the heave added mass μ33 of model A at 0.32 rad/s, which corresponds to the heave resonance period, is slightly smaller than that of model B and almost the same as that of model C. As a result, the heave resonance period of model A is slightly smaller than those of the other two models. At the same

time, the resonance period of model B is slightly larger than that of model C. Noting that the water plane areas of models B and C are identical, this resonance period difference possibly comes from a slightly larger added mass of Model B due to a additional contribution of the connection braces. The RAO magnitudes at the resonance periods for models A, B and C are very close to each other, and they are 2.131 m, 1.980 m and 2.073 m, respectively. As mentioned above, the heave amplitude is dominated by damping in the resonance frequency range. One can note that the potential damping values in the resonance frequency range of the three models are very close to each other, which results in their similar RAO magnitudes. It should also be

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This heave-roll coupling effect of model A may be a result of the distinctive pontoon shapes of the novel SEMI. As described above, the novel SEMI has two pontoons of different sizes, one large and one small, which are not symmetrical about the middle longitudinal section. When the SEMI moves harmonically up and down, the wave loads acting on the two pontoons in the z-direction are not equivalent, which definitely results in a roll motion of the novel SEMI. However, because both models B and C have two identical pontoons, there are no heave-roll coupling effects for these two SEMIs. The RAOs of the pitch motion for models A–C are shown in Fig. 9. Compared with Fig. 7, the pitch RAO and the roll RAO have similar features; in particular, they share common sharp peaks at approximately 20 s. It is interesting to find that for all three RAOs, peaks exist, and the peaks, which have different amplitudes, are associated with different resonance periods. This phenomenon is similar to the heave-pitch coupling effect called Mathieu instability that often occurs with single column floaters, such as the Spar platform. The premise of Mathieu instability is that the heave period is exactly half of the pitch natural period (Shen and Tang, 2009). However, the calculations show that the pitch natural period of the novel SEMI is approximately 65 s, which is three-fold greater than the heave natural period. Therefore, it is unlikely that this reflects the Mathieu instability problem. The resonance periods associated with models A, B and C are approximately 19.62 s, 19.94 s and 19.79 s, respectively. The comparison of Fig. 9 with Fig. 6 reveals that these pitch resonance periods are exactly equal to the heave resonance periods of models A–C. Therefore, one can speculate that this is a heave-pitch coupling phenomenon.

noted the heave RAO magnitude of model B is the smallest, due to additional viscous damping of connection braces in model B. Roll motion is another important motion mode that requires attention for a semi-submersible lift unit. The RAOs of the roll motion for models A–C are illustrated in Fig. 7. As observed, the three curves show similar characteristics, and the roll motion performance of model A is better than that of models B and C in most of the wave frequency range with the exception of the existence of a sharp peak in model A; the period associated with this peak is approximately 19.62 s. This is a unique phenomenon for the novel SEMI model. The comparison of Fig. 7 with Fig. 6 shows that this roll peak period is exactly equal to the heave resonance period of model A. Therefore, one can conclude that the heave resonance will affect the roll motion for model A, which can be regarded as the coupling effect between the heave and roll motions. However, there are no coupling effects between the heave and roll motions for models B and C. It should be mentioned that a 2:22  106 N U s=m heave damping value, estimated from a model test, has been added in the above analysis for considering the viscous effect. To further validate this heave-roll coupling phenomenon, the heave damping of model A was artificially varied. Five heave damping values were chosen for the damping sensitivity analysis and these values were determined artificially from 1:90  106 N U s=m to 2:22  106 N Us= m by an interval 0:3  106 N U s=m. The roll RAOs associated with these five heave damping values are illustrated in Fig. 8. One can note that the amplitude of the coupling peak increases as the heave damping decreases, which means that the RAO peak at approximately 20 s is indeed caused by the heave resonance motion. 0.8

model A model B model C

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0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 5

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Wave Period(second) Fig. 7. Roll RAO at 90°.

Roll amplitude(degree/m)

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Heave damping 1.00E+6N*s/m Heave damping 1.30E+6N*s/m Heave damping 1.60E+6N*s/m Heave damping 1.90E+6N*s/m Heave damping 2.22E+6N*s/m

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0.4

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0 5

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Wave Angular Frequency(rad/s) Fig. 8. Heave-roll coupling peak values of model A under different heave damping values.

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0.7

model A model B model C

Roll Amplitude (degree/m)

0.6 0.5 0.4 0.3 0.2 0.1 0 5

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Wave Period(second) Fig. 9. Pitch RAO at 0°.

To further confirm this heave-pitch coupling, a similar procedure to that used to investigate the heave-roll coupling was conducted. The heave damping of model A was artificially adjusted to different values to investigate the heave and roll motions. As expected, the peak amplitude in the roll DOF will increase with the heave resonance amplitude as the heave damping is reduced. Therefore, this heave-pitch coupling effect is further validated. The results are not shown here due to space limitations. The distinctive pontoon shapes of the SEMI models may also be the main reason for the heave-pitch coupling effect. When the external shapes of the stern and bow of the pontoons are identical, that is, if the stern and bow of the pontoons are symmetrical about the middle transverse section, there are no such heave-pitch coupling effects (Li et al., 2011). As described in Section 3.1, the bow and stern of the pontoons of these three SEMI models are not symmetrical about the middle transverse section. When the SEMI moves harmonically up and down, the wave loads impacting the stern and bow of each pontoon in the z-direction are different, which would affect the pitch motion. Based on this comprehensive analysis, it can be concluded that the hydrodynamic characteristics of the novel SEMI model are generally satisfactory, as good as the other two models, when the wave period is less than the heave resonance period. It is worth noting that the innovative asymmetric pontoon outline with pneumatic de-ballast system is very useful for quick ballast adjustment to suit heavy lifting operation, and no horizontal connection braces reduce the towing resistance and dynamic positioning load. However, more attention should be paid to the heave-roll/pitch coupling effect for long waves whose period is close to the heave resonance period.

Fig. 10. Vertical view of the novel SEMI.

and in a real operation, these forces would be compensated for by the DP system and/or mooring systems that restrain the ship in place and are therefore not included in the analysis. In this study, the mooring system was simulated by a taut mooring system, which is described in Section 3.2. The operational areas of novel SSCV are mainly in the Gulf of Mexico, the UK North Sea and the west of Africa. The researched sea states are expected to occur at a water depth of approximately 500 m. The wave load is represented by the JONSWAP spectrum (Goda, 1999) in this study: h i

2 2 5 ωm 4 exp  ðω2σ2ωωm2 Þ  2 ωm m Sη ðωÞ ¼ α H s 5 exp  γ ð11Þ 4 ω ω

4.2. Motion response analysis The novel SEMI, Explorer Lifter, which is a lifting installation and accommodation facility, has very strict requirements for motion response. Because it works as a crane vessel, the prevention of capsizes is crucially important. In addition, because the SEMI is also used as an accommodation platform, the horizontal offset range cannot exceed the length of the gangway, which is a connecting structure for both an accommodation platform and an operational platform. Therefore, more attention should be paid to the excursions of surge, sway and roll motions, particularly the roll motion response. To address this, a study of the coupled dynamic motion response is discussed in this section. The primary exciting forces result from the waves, and the purpose of the motion analysis is to investigate the dynamic behaviors of the novel SSCV under the effects of the waves. The current, wind and drift forces will have effects on large motions and/or slowly varying motions,

α ¼

0:0624 0:230 þ 0:0336γ  0:185ð1:9 þ γ Þ  1

ð12Þ

where H s is the significant wave height, ωm is the wave spectral peak frequency, γ is the wave spectral peak factor, and σ is the wave peak shape factor. ( ω r ωm ; σ ¼ 0:07 ð13Þ ω 4 ωm ; σ ¼ 0:09 As a lifting facility, the operational case of the novel SEMI should usually be a benign sea state. Therefore, a once-in-a-year sea state from the UK North Sea was selected. The significant wave height is 7.13 m. The peak spectral period is 12 s, and the spectral peak factor is 3.3. Because the configuration of the novel SEMI model is not symmetrical about either the x or y axis, as shown in Fig. 10, studies of the motion response in different wave directions

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270°, which shows a delay of 90° about the occurrence location of the maximum response, and on the other hand, a scrupulous reader will observe that the values of these two peaks are different in the variation period. The larger one, which represents the maximum motion response, occurs at 90°, which indicates that the wave hit perpendicular to the large pontoon. The smaller one expresses the maximum motion response that occurred at 270°, which indicates that the wave hit perpendicular to the small pontoon. Therefore, a possible reason for the different peaks is that the novel SEMI is not symmetrical about the middle longitudinal section. As described above, the novel SEMI has two pontoons of different sizes, one large and one small, which are not symmetrical about the middle longitudinal section. When the wave acted on each pontoon, the wave loads would be not equivalent, and the loads on the large pontoon side would be greater than that on the small pontoon, which means that if the wave hit perpendicular to the large pontoon, the maximum motion response of sway and roll would occur. However, this phenomenon is not representative of actual operations. Therefore, having the wave hit perpendicular to the large pontoon should be avoided. In addition, the yaw motion response curve, whose variation period is 180° and where the maximum occurs at approximately 75° and 255°, is very special. A possible reason for this phenomenon also exists in the distinctive pontoon shapes of the novel

were conducted. The wave directions head from 0° to 360°, and the step was 15°. The wave direction is distributed counterclockwise, as shown in Fig. 10. The water density was set to 1025 kg/m3. The coupled dynamic analysis of the novel SEMI under the selected sea state was conducted after considering the taut mooring system. Fig. 11 illustrates the roll motion response for a 0° incident wave. The statistical results in all six-DOF motions for different wave incident directions are shown in Fig. 12. As shown in Fig. 12, one can note that all of the motion responses are very sensitive to the wave incident angle with the exception of the heave motion, and these responses show similar patterns of change. Among these figures, the surge and pitch response curves show similar characteristic variations associated with the incident wave direction. The variation periods of surge and pitch are 180°, and the maximum motion responses in both surge and pitch DOFs occur at 0°, 180° and 360°, which means that when the incident wave is from the bow or the stern direction, the surge and pitch motions will each reach their maximum offset. The response curves in the sway and the roll directions demonstrate similar characteristic variations with a 180° variation period. The comparison of the motions of sway and roll with that of surge and pitch reveals that they differ in two aspects. On the one hand, the two peaks in the variation period occur at 90° and 1.5

Roll rotation (deg)

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7000

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10000 10800

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5 4 3 2 1 0 -1 -2 -3 -4 -5

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5 4 3 2 1 0 -1 -2 -3 -4

Sway displacement (m)

Surge displacement (m)

Fig. 11. Typical dynamic responses in roll motion.

0

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45 90 135 180 225 270 315 360 Incident angle(deg)

0 -2 -4 -6

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4 2 0 -2 -4 -6

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45 90 135 180 225 270 315 360 Incident angle(deg)

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45 90 135 180 225 270 315 360 Incident angle(deg)

Fig. 12. Motion response offset.

mean min max 45 90 135 180 225 270 315 360 Incident angle(deg)

6 4

4 3 2 1 0 -1 -2 -3 -4 0

4 3 2 1 0 -1 -2 -3 -4 0

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SEMI. When the wave incident angle is approximately 75° or 255°, the moment that leads to yaw motion is at its maximum. Additionally, as expected, the heave motion response is very stable and not sensitive to the wave incident angle as in a conventional symmetrical SEMI. Moreover, as shown in Fig. 12, the following maximum excursions exist in all six DOFs: 4.03 m, 4.77 m, 2.9 m, 5.60°, 4.42° and 2.38°. Although all of these are at a satisfactory level, the maximum excursion of roll is close to the specified critical value. Therefore, measures should be taken to reduce the values of the maximum motion responses, particularly roll motion. Based the analysis presented above, the motion responses of the novel SEMI model are generally satisfactory in a normal sea state, and most of them are very sensitive to the wave incident angle. However, in an actual project, reducing the maximum motion response is always crucial. Therefore, based on comprehensive considerations, adjusting the incident wave direction to an appropriate value, such as approximately 45°, 135°, 225° or 315°, would be a very effective measure.

5. Conclusion The semi-submersible unit is widely used for deep-water hydrocarbon exploration. The hydrodynamic performance of a novel SEMI was investigated in this study. This new type of SEMI, which has two nonsymmetrical pontoons and no horizontal connection braces, is used as an accommodation/crane unit. Its hydrodynamic characteristics were first investigated, paying special attention to the heave, roll and pitch motions. Furthermore, the motion response of the novel SEMI was studied under normal operation in eight different wave directions. For a comparative study of the hydrodynamic characteristics, three SEMI models, including the novel SEMI model and two conventional SEMI models, were established. Their hydrodynamic performances are first investigated and compared focusing on the heave, roll and pitch motions. The numerical results demonstrate the existence of significant heave-roll/heave-pitch coupling effects for the novel SEMI due to its nonsymmetrical pontoon shapes, which indicate that more attention should be paid to long waves that are close to the heave resonance period. A coupled dynamic motion analysis was conducted in the time domain by considering the mooring system. The motion response of the novel SEMI was investigated under normal operation in eight different wave directions. All of the motion responses with the exception of the heave motion are very sensitive to the wave incident angle. The motion responses of the novel SEMI are generally satisfactory for a typical normal sea state, but adjusting the SSCV to an appropriate incident wave direction, such as approximately 45°, 135°, 225° or 315°, may provide better operational performance.

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Acknowledgments The authors acknowledge the financial support by the National Basic Research Program of China (2011CB013704), the Major Program of the National Natural Science Foundation of China (51490675), the Taishan Scholar Program of Shandong Province, and the Shandong Provincial Science & Technology Development Project (2013GHY11503).

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