Motion and dynamic responses of a semisubmersible in freak waves

The present research aims at clarifying the effects of freak wave on the motion and dynamic responses of a semisubmersible. To reveal the effects of mooring stiffness, two mooring systems were employed in the model tests and time-domain simulations. The 6-DOF motion responses and mooring tensions have been measured and the 3-DOF motions of fairleads were calculated as well. From the time series, trajectories and statistics information, the interactions between the freak wave and the semisubmersible have been demonstrated and the effects of mooring stiffness have been identified. The shortage of numerical simulations based on 3D potential flow theory is presented. Results show that the freak wave is likely to cause large horizontal motions for soft mooring system and to result in extremely large mooring tensions for tight mooring system. Therefore, the freak wave is a real threat for the marine structure, which needs to be carefully considered at design stage.


Introduction
Extreme waves are generally believed to pose a threat to ships and offshore structures, which probably result in extreme wave loads and large motion responses. Freak waves are special appearances of such extreme waves, which usually refer to waves with the maximum wave height twice larger than the significant wave height for specified sea states (Kharif et al., 2009). They are normally regarded as 'the monsters of the deep', which appear surprisingly and disappear without a trace (Kharif and Pelinovsky, 2003). Freak waves were long thought to exist only in the imaginative minds of seamen. However, the observations have now proved that such waves could occur in water of arbitrary depth, as a single giant wave or as a group of waves could occur in both moderate and severe sea states, unfortunately more frequently than expected (Clauss, 2002).
Freak waves are considered as the culprits of a great number of severe damages to offshore structures and ships. For example, more than 22 supercarriers were assumed to be lost after collisions with freak waves between 1969 and 1994 (Kharif and Pelinovsky, 2003). Moreover, offshore platforms are also vulnerable to freak waves. In 1982, a drilling rig located at the Grand Banks of Newfoundland capsized and sank shortly after encountering a giant wave, and 84 people on board were killed (Kharif et al., 2009). On December 20, 2015, the COSL Innovator drilling rig was struck by a freak wave and one person was killed and four suffered injuries in this accident. Besides, most of freak wave records come from oil-platform measurements. Photographic evidences and the simulations show that freak waves are of large volume water and enormous energy, they are less like waves than like mountains on the move (Rudman and Cleary, 2013). Unfortunately, the marine structures cannot escape, and meet giant waves "as they are" in most cases. It becomes more obvious that fully considerations of freak wave impacts at the early design state of marine structures are indispensable.
The interactions between freak waves and structures are highly non-linear processes. Moreover, the responses due to transient freak waves rather than that during a long period are more concerned. For the impacts of freak waves, available works predominantly focus on the vertical bending moments and heave, pitch responses of vessels, such as bulk carrier (Clauss et al., 2010), container (Clauss et al., 2010) and FPSO (Fonseca et al., 2005;Soares et al., 2006). As for the offshore platforms, El Moctar et al. (2009) studied the freak wave impacts on a mobile jack-up drilling platform with Computational Fluid Dynamic (CFD) and Finite Ele-ment Method (FEM) techniques and suggested special consideration of the strong nonlinearity of the wave loads. Chandrasekaran and Yuvraj (2013) focused on the dynamic responses of a Tension Leg Platform (TLP) under freak waves with different wave approaching angles. Rudman and Cleary (2013) adopted the Smoothed Particle Hydrodynamics (SPH) to simulate the fully non-linear interactions between a TLP and freak waves. It was concluded that the maximum surge, heave and pitch vary slightly with wave angle, and the duration of heave and pitch excursions decreases and peak mooring tension slightly increases with increasing pre-tensions. In additions, Clauss et al. (2003) investigated the heave and pitch responses and splitting forces of a semisubmersible GVA 4000 with both the time-domain simulation and model tests. Deng et al. (2014) investigated the effects of wave group characteristics on a semi-submersible in freak waves and found that the horizontal motion response becomes much larger due to the freak waves.
To satisfy the increasing demand of oil resource, a large amount of deepwater semisubmersibles have been constructed and come into operation in recent decades. These structures are constantly exposed to rough environment, and probably meet with freak waves during their life time. However, the researches on the freak wave impacts on semisubmersibles are by no means sufficient. It still remains unclear about how and to which degree the freak waves affect the horizontal motion responses and the maximum mooring tensions.
In this study, experimental and numerical investigations on the motion and dynamic responses of a semisubmersible were carried out to highlight how freak wave significantly influences the mooring tensions and the motion responses. Two different mooring configurations were considered to make a more comprehensive demonstration of the freak wave impacts on a semisubmersible. Besides, the responses under both freak wave and the combination action of freak wave, wind and current were investigated as well.

Descriptions of the reference platform
Model tests of the moored semisubmersible were carried out in the deepwater offshore basin of State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University. The basin is 50 m in length, 40 m in width and 10 m in the maximum effective depth. A large-area movable bottom allows the flexible modeling of the water depth from 0.0 m to 10.0 m. Based on the test capacities, a scale ratio of 1:64 was adopted in the model tests. The reliability and accuracy of measurements in State Key Laboratory of Ocean Engineering have been proven in many published works, such as Zhao et al. (2017). The sampling frequency during the whole tests is 40 Hz in model scale.
A deepwater drilling semisubmersible with mooring positioning systems is selected as the reference in this study. The semisubmersible consists of four columns, double pontoons, four horizontal braces and box-shaped deck. The maximum operation depth is 1500 m and the maximum drilling depth is about 10000 m. Fig. 1a shows the platform model for model tests. The main dimensions and the mass parameters of the semisubmersible in prototype and model scale are listed in Table 1 and Table 2, respectively. Only the survival condition was investigated for the freak wave environment.
Numerical analyses on the motion and dynamic responses of the moored semisubmersible are performed using the hydroD and deepC modules within SESAM. The calculation is based on linear 3D potential flow theory. Newman approximation (Newman, 1974) was adopted to calculate the difference-frequency forces. The numerical models are built in full scale. Fig. 1b presents the panel model for the numerical analyses. The element size is about 2 m×2 m.
Two catenary mooring systems were employed in the model tests, corresponding to 300 m and 1500 m water depths, respectively. Each mooring system consists of 12  Table 3. It is noted that for the 1500 m fulldepth mooring system exceeds the water depth of the water basin with the scale ratio 1:64, therefore, a truncated mooring system has been designed with the horizontal stiffness remaining unchanged. The horizontal stiffness tests were also conducted to validate the accuracy modelling of mooring systems.

Wave environments
Freak waves were modeled and calibrated prior to the formal tests. Various formation mechanisms were proposed with respect to freak waves and various modeling methods had been successfully applied to generate freak wave trains in physical or numerical wave tanks (Deng et al., 2015;Dysthe et al., 2008;Kharif and Pelinovsky, 2003). In this study, the dispersive focusing model (Kriebel and Alsina, 2000) and phase iteration method (Chaplin, 1996), as successfully employed in previous works (Deng et al., 2016a(Deng et al., , 2016b, were used to generated the freak wave train. The JONSWAP spectrum was adopted and the parameters of the selected spectrum are H s =13.0 m, T p =15.4 s and γ=2.5, corresponding to a severe sea state. This is to comply with the fact that freak waves more probably occur during harsh environment (Guedes Soares et al., 2004). Fig. 3 shows the time series of the calibrated freak waves. The time axis in this figure and in the figures hereafter is translated to make the freak wave occur at 0 s. The maximum wave height is H max =30.2 m and the crest height is ζ c =23.2 m. The calibrated maximum wave height is close to the maximum wave height (29.8 m) detected by Europe satellite ERS-2 during a three-week registration (Kharif et al., 2009). The comparisons of wave parameters between the famous New Year wave record (Hayer and Anderson, 2000) and the calibrated wave are given in Table 4. T tt and L tt in Table 4 denote the trough-to-trough wave period and the corresponding wave length. It is observed that the calibrated H max /H s , ζ c /H max and H max /L tt values are close to those of New Year wave record. The power spectrum density of the measured freak wave train and the target spectrum are also presented in Fig.  4. It shows that the freak wave spectrum agrees well with the target wave spectrum, indicating that the model of Kriebel and Alsina retains the realistic properties of a given sea state. All of the above suggest that the calibrated freak wave train is a representative freak wave, with extremely high wave and highly asymmetric features.

Static offset tests and decay tests
Static offset tests were conducted prior to the formal tests, aiming at verifying the modelling of the mooring systems. Figs. 5 and 6 present the offset static stiffness curves for 300 m and 1500 m full-depth mooring systems, respect-   ively. Good agreements were achieved between the measured results and target values. Moreover, it is obvious that the horizontal stiffness of 300 m full-depth mooring system is much larger than that of 1500 m full-depth mooring system. This difference results in different characteristics of responses under freak waves. Decay tests with and without mooring systems were performed to obtain the natural periods and damping coefficients and to verify the modelling of semisubmersible model and mooring systems as well. The natural period results of both experimental and numerical data are provided in Table 5. Good agreements were observed between the experimental and numerical results. More specifically, the relative errors between the measured and calculated results are below 1% in this study. The results of static offset tests and decay tests fully demonstrate the accuracy for modeling of both the semisubmersible model and the mooring systems.

Response amplitude operators (RAOs)
Frequency-domain analysis is an efficient method to inspect the hydrodynamic performance of marine structures. The experimental and numerical models verify each other with the RAO results. The motion RAOs from both white noise tests and regular wave tests in head sea (β=180°) are compared with the calculation data in Fig. 7. Good agreements were achieved between the experimental results and calculations, both on the magnitudes and the trends. A continual increase in surge RAOs is observed with large wave periods, suggesting that the surge motion probably has significant responses in low-frequency area. Moreover, the heave and pitch RAOs at either the peak period 15.4 s or the trough-to-trough period of freak wave 12.9 s are significant. It means that the heave and pitch motions probably have large transient responses due to the freak wave.

Motion behaviors in freak wave
Freak wave impacts on a semisubmersible with mooring system are very complex. To investigate the capability of the time-domain coupled analysis in such a nonlinear wave, both model tests and numerical analyses were conducted for comparisons. Fig. 8 shows the snapshot of the semisubmersible in freak wave in the basin.
In order to comprehensively investigate the impacts of freak wave on the motion responses, a total of 4 cases were carried out. Specifically, the 300 m full-depth mooring system was employed in Case 1 and Case 2, and 1500 m fulldepth mooring system was for Case 3 and Case 4. Besides,     Fig. 9 presents the time series of the measured motion responses. On the whole, the surge responses are dominant by the low-frequency motions. The heave responses take the form of wave-frequency motions. For the pitch responses, in addition to the primary wave-frequency responses, the lowfrequency responses are visible as well. With regard to the responses due to freak wave, it appears that the freak wave has caused significant transient heave and pitch responses. However, the maximum heave and pitch amplitudes do not occur immediately after the passing of freak wave, but occur during the next wave period. It seems that the following wave somewhat enhances the heave and pitch responses. For different mooring systems, the heave and pitch responses are quite similar to each other, indicating that the mooring system has little effect on the heave and pitch responses.
To investigate the capability of the time-domain analysis based on frequency-dependent hydrodynamic coefficients in simulating the motion behavior of the moored semisubmersible, the time-domain analyses corresponding to Case 1 and Case 3 were conducted. No extra viscous damping was specified on the semi-submersible except for the potential damping. In the calculations, the semi-submersible is treated as a nodal component in the finite element model, and mooring lines are modeled by the finite elements representing the slender structures. The equations accounting for the semi-submersible and slender elements are solved simult-    aneously using a nonlinear approach. Fig. 10 and Fig. 11 respectively present time series of the wave-frequency and low-frequency motion responses. The division frequency adopted to separate the wave-frequency components and the low-frequency components from each motion response is 0.05 rad/s. For the surge responses, it is observed that the freak wave results in significant wave-frequency oscillations and amplifies the low-frequency responses as well. The main diffe-rence of surge responses for different mooring systems lies in the low-frequency surge responses. It shows that the freak wave impacts on the low-frequency surge are more noticeable for the 1500 m full-depth mooring system. Through comparisons of the motion responses between Case 1 and Case 3, the wave-frequency motion responses are close to each other while the low-frequency responses significantly differ from each other due to the mooring systems.
In Fig. 10, the simulated wave-frequency responses as a whole agree with the measured results, indicating the competence of 3D potential flow theory in dealing with wavefrequency motions. The relatively large disagreement in the wave-frequency heave responses might be resulted from the variations of the waterline area. As shown in Fig. 1, the four horizontal braces are located close to the waterline under the survival condition. When the braces pass through the free water surface, they contribute significantly to the waterline area and thus result in noticeable differences between the measurement and simulation.
In Fig. 11, there are noticeable differences between the simulated and measured low-frequency responses, particularly for the heave and pitch responses. The inaccuracy of  LI Xin et al. China Ocean Eng., 2017, Vol. 31, No. 6, P. 754-763 759 simulated heave and pitch responses are caused by ups and downs of the four horizontal braces. Though the effects on the low-frequency surge responses of the horizontal braces are limited, the simulated maximum surge response could not reach the measured value. Therefore, it is unlikely to acquire accurate predictions on the motion responses of a semisubmersible in harsh environment with linear potential flow theory. Despite the strong nonlinearity of the wave loads, the complexity of the drift damping and mooring damping, the variation of the waterline area due to the large vertical motions might bring remarkable errors.
To provide an intuitive view of the motion responses under freak wave, the measured trajectories of the center of gravity (COG) are presented in Fig. 12. The green up-triangles in the figure denote the occurrences of freak waves. It shows that the platform generally follows the elliptical motion of the water particles. For the tight mooring system (Case 1), the platform appears to move in circles around a nearly constant point. However, for the soft mooring system (Case 3), the platform moves forward in a spiral until being pulled back by the mooring system. The effect of freak wave on the low-frequency surge motion is that the impulse provided by freak wave allows the semisubmersible to move further before being pulled back by the restoring forces, and apparently, this effect is more noticeable for the soft mooring system. Fig. 13 summarizes the maximum motion responses under the freak wave. It is shown that the maximum and average surge responses of the soft mooring system are much larger than those of the tight mooring system. Besides, the actions of wind and current loadings would significantly increase the average offsets, which lead to much larger surge responses. It can be seen that the maximum surge response in Case 4 has reached 8.6% of the full water depth, indicating possible damage to risers, etc.

Mooring dynamics responses in freak wave
The mooring tension response at each fairlead was measured during the model tests. In this study, the head sea condition was considered and the maximum mooring tensions occur at Line 1 and Line 12. Fig. 14 shows the mooring tension responses of Line 1. Compared with the average level of the mooring tensions, the tension force becomes ex-  tremely large under the action of freak wave for the tight mooring system. However, the tension forces gain limited increase as the freak wave passes by for the soft mooring system. It is observed from Fig. 14a that the minimum tension force is close to zero, indicating a possible taut-slack process. This is the possible reason for the extremely large tension responses. Similar to the heave responses, the maximum tension force does not occur during the freak wave period but at the next wave period following the freak wave. The reason is that the following wave enhances the motion responses, which results in larger tension responses. Fig. 15 presents the maximum and average mooring tensions of Line 1. It shows that the maximum mooring tension is large when the wind and current are playing a role, particularly for the tight mooring system. It is noted that the maximum tension force for Case 2 is unbelievably getting close to 1400 MT. The extremely large tension force is very likely to result in the breaking of mooring lines. It could be imagined that if Line 1 is broken, Line 2 and Line 3 may be also in danger.
The 3-DOF motions of Fairlead 1 were calculated based on the 6-DOF motions of the semisubmersible to reveal the relation between the fairlead motion and the mooring tension responses. The trajectories of Fairlead 1 are presented in Fig. 16. The red and yellow diamonds in the figure represent the moments of the maximum and the second maximum mooring tensions, respectively. The trajectories of Fairlead 1 are similar to COG. It is observed that the maximum tension forces usually occur when the fairlead reaches the highest position and is moving backward with the largest horizontal velocity in a circle. It is at this moment that the mooring tension is the largest, leading to the maximum downward acceleration of the fairlead.
To investigate the capability of deepC module in dealing with taut-slack process, a single line model of Line 1 was established and the above 3-DOF motions of Fairlead 1 of Case 1 were adopted in below simulation. It is observed in Fig. 17 that the predicted mooring tension agrees perfectly with the measured tensions beyond the taut-slack process. However, during the taut-slack process, the simulation provides much larger mooring tensions and the minimum of the simulated results is below zero. This might partly contribute to the underestimation of the above horizontal response. Therefore, the Riflex module with FEM model is not capable of precisely simulating the taut-slack process of mooring lines.

Conclusions
In this study, the motion and dynamic responses of a semisubmersible in freak wave have been investigated   LI Xin et al. China Ocean Eng., 2017, Vol. 31, No. 6, P. 754-763 761 through model tests and numerical simulations. Prior to formal freak tests, a representative freak wave was calibrated, and the decay tests and static offset tests were performed to verify the accuracy modelling of the reference semisubmersible and the mooring systems. Two mooring systems corresponding to 300 m and 1500 m full water depths were employed to reveal the effects of mooring stiffness on the motion behaviors and mooring tension responses. Moreover, the numerical simulations were performed and compared with experiments. Main conclusions are drawn as follows.
(1) The freak wave might result in significant transient heave and pitch responses, and the actions of wind and current and the mooring configurations have limited effects on the maximum responses of the heave and pitch motions.
(2) For the soft mooring configuration, the low-fre-quency surge responses could be significantly enlarged with the freak wave action. For the tight mooring configuration, the freak wave probably results in taut-slack processes, leading to extremely large mooring tensions.
(3) The maximum tension forces usually occur when the fairlead reaches the highest position and moves backward with the largest horizontal velocity in a circle.
(4) Predictions on the motion responses based on linear potential flow theory are unlikely to acquire accurate results in harsh environment and the Riflex is incompetent in dealing with the taut-slack processes.
Above all, the freak wave is likely to cause damage to the risers and mooring lines due to large horizontal responses and extremely mooring tensions, respectively. The interactions between the freak wave and marine structures are complex, which call for a more comprehensive study on these topics.