Parametric Dimensional Analysis on the Structural Response of An Innovative Subsurface Tension Leg Platform in Ultra-Deep Water

The innovative Subsurface Tension Leg Platform (STLP), which is designed to be located below Mean Water Level (M.W.L) to minimize direct wave loading and mitigate the effect of strong surface currents, is considered as a competitive alternative system to support shallow-water rated well completion equipment and rigid risers for large ultra-deep water oil field development. A detailed description of the design philosophy of STLP has been published in the series of papers and patents. Nonetheless, design uncertainties arise as limited understanding of various parameters effects on the structural response of STLP, pertaining to the environmental loading, structural properties and hydrodynamic characteristics. This paper focuses on providing quantitative methodology on how each parameter affects the structural response of STLP, which will facilitate establishing the unique design criteria as regards to STLP. Firstly, the entire list of dimensionless groups of input and output parameters is proposed based on Vaschy- Buckingham theory. Then, numerical models are built and a series of numerical tests are carried out for validating the obtained dimensionless groups. On this basis, the calculation results of a great quantity of parametric studies on the structural response of STLP are presented and discussed in detail. Further, empirical formulae for predicting STLP response are derived through nonlinear regression analysis. Finally, conclusions and discussions are made. It has been demonstrated that the study provides a methodology for better control of key parameters and lays the foundation for optimal design of STLP. The obtained conclusions also have wide ranging applicability in reference to the engineering design and design analysis aspects of deepwater buoy supporting installations, such as Grouped SLOR or TLR system.


Introduction
Petroleum exploration and production of offshore fields continue to increase and proceed into ever deeper water. In this scenario, an innovative STLP concept (Huang et al., 2013) is proposed as a competitive alternative system to support shallow-water rated well completion equipment and rigid risers for large ultra-deep water oil field development. It primarily consists of three parts: a tethered subsurface Sea-star Pontoon (SSP), rigid risers and well completion equipment, as shown in Fig. 1. Flexible jumpers then link the STLP to the Floating Production Unit (FPU). The overall view of the STLP with flexible jumpers is shown in Fig.  2. Further detailed description of the design philosophy as well as the working principle of the STLP has been given in the series of papers Zhen et al., 2014;Zhen and Huang, 2017). It can be seen that some of the STLP components share the design with well proven deepwater buoy supporting installations (Hatton et al., 2002;Tellier and Thethi, 2009;Zimmermann et al., 2002), more particularly, such as Grouped SLOR (Franciss, 2005;Dale et al., 2007) and TLR system (Zimmermann et al., 2001;de Araújo et al., 2011).
Nonetheless, it should be noted that substantial design uncertainties arise as there is still limited understanding of various parameters effects on the structural response of the STLP, pertaining to the environmental loading, structural properties and hydrodynamic characteristics. Furthermore, the structural response of the STLP is critical criterion for identifying its security state, which will directly affect the stress of rigid risers and flexible jumpers as well as the well-head stability. Parametric studies to investigate the influence of parameters, particularly where the value is uncertain are recommended, in order to improve the understanding of the overall behavior and provide more accurate modelling and increase confidence in design (Det Norske Veritas, 2010;Quéau et al., 2013Quéau et al., , 2014. As regards to complex multi-parameter problems to which the parametric studies are applied, it is necessary to identify the dimensionless groups of the parameters that influence the key response variables (Palmer, 2008). A limited set of dimensionless groups have been used in previous parametric studies for the subsurface buoy stability (Sundaravadivelu and Varaprasad, 1991;Sundaravadivelu et al., 2011). However, it is essential to identify the complete set of dimensional groups correlated to the problem, aiming to clarification of the relationships among all the parameters. This paper focuses on providing the quantitative methodology on how each parameter affects the structural response of the STLP, which will facilitate establishing the unique design criteria as regards to the STLP (American Petroleum Institute, 2005). Firstly, the entire list of dimensionless groups of input and output parameters is proposed based on the Vaschy-Buckingham theory. Then, the numerical models are built and a series of numerical tests are carried out to validate the obtained dimensionless groups. On this basis, the calculation results of a great quantity of the parametric studies on the structural response of the STLP are presented and discussed in detail. Further, the empirical formulae for predicting the STLP response are derived. Finally, the conclusions and discussions are made.

Dimensional analysis of the STLP response
In order to clarify the dependencies among the magnitudes of quantities pertaining to the structural response of the STLP, their dimensions are considered by the dimensional analysis. The first step of the method is to select appropriate initial parameters including input and output parameters for the STLP response. The second step is to group the initial parameters into dimensionless groups and form the new relationship among various parameters. It should be noted that the accurate selection of the initial parameters is of the primary challenge and concern as it is necessary that the unique relationship exists among the selected parameters (Palmer, 2008;Quéau et al., 2013).

Determination of dimensionless groups
Thirteen initial parameters are selected for identifying the non-dimensional groups of the parameters that can quantify the structural response of the STLP, as shown in Table 1. On this basis, the Vaschy-Buckingham theorem predicts that ten corresponding dimensionless groups can be obtained from the selected initial parameters, as shown in    Table 2. Further, the output groups of the SSP offsets can be expressed as the function of eight groups, as shown in Eq.
(2) 2.2 Numerical models for verification of the dimensionless groups A set of the numerical analyses are performed to validate the obtained dimensionless groups. Validation is achieved through comparing the SSP horizontal offset X/h and vertical offset Y/h of different numerical models that have identical values of the dimensionless groups, but different values of initial parameters.
Two equivalent groups including a base case (BCi) and three model tests (MTi) are defined. Initial parameters of BC1 come from the designed STLP (Zhen and Huang, 2017) while the nominal diameter of the tether in BC2 is smaller than that in BC1. Model tests are established from the corresponding base cases by scaling some of the para-meters while ensuring dimensionless groups constant. The distance from the SSP to the seabed (h), tether diameter (D) and Young's modulus (E) are scaled by α, SSP drag area (A) and tether mass (m) by α 2 , tether pretension (T) by α 3 , current velocity (V) by α 0.5 whereas SSP drag coefficient (C d1 ) and tether drag coefficient (C d2 ) keep unchanged. Characteristics of the numerical models are shown in Table 3. Table 4 presents the dimensionless group values of each equivalent group. Three load cases are selected for the validations in this study. Table 5 presents the values of current velocity (V) and π 3 (V 2 /(gh)) for each model.

Validation of the dimensionless groups
As all these dimensionless groups remain unchanged in the models which come from the same equivalent group, the same X/h and Y/h can be expected. Table 6 summarizes X/h and Y/h values for all models in each load case. In addition, the relative differences between MTi and corresponding BCi as well as the maximum differences among the same equivalent group are presented. It can be seen from Table 6 that the maximum difference is 0.022%, which indicates high accuracy of the established dimensionless groups.
Young's modulus (E) of the tether is modified to keep the value of π 10 (ED 2 /T) constant in the above numerical ED 2 /T π 10 1.2 1.2 π 10 (×10 2 ) 0.425 0.318 analyses. However, the change of Young's modulus means the material type alteration of the tether and thus the applicability of the dimensionless groups would be diminished. Another dimensionless analysis without Young's modulus is conducted for the purpose of enhancing the applicability as well as simplifying calculation. Finally, twelve selected parameters can be grouped into nine dimensionless groups, and the only difference compared with previous results lies in the absence of π 10 , as shown in Eq.
(3) and Eq. (4). Similar numerical analyses are performed for validating the newly obtained results. Related results are presented in Table 7. It can be seen from Table 7 that the maximum difference is within 1%. It suggests that the newly established dimensionless groups are accurate enough to be applied in further numerical analyses.
3 Parametric study on the structural response of the STLP Based on the initial parameters of BC1 that are presented in Table 3, a great quantity of the parametric studies are carried out by using ORCAFLEX calculation program to study the effects of various parameters on the structural response of the STLP. Table 8 presents the selected values for the initial parameters of the models, such as the current velocities ranging from 0.50 m/s to 1.50 m/s, in the step of 0.25 m/s. Fig. 3 and Fig. 4 present the effects of the parameters of the drag area and drag coefficient on the behavior of the STLP. In accordance with the hydrodynamic loads calculation theory, it can be found that increasing drag area results in the linear increase of the horizontal and vertical offsets of the SSP, due to the increasing drag forces on the SSP and tethers. Similarly, the higher drag coefficient is, the larger SSP offsets are. Besides, the horizontal and vertical offsets of the SSP increase more gently under minor drag coefficients than that under high values. For instance, a 60% increase in the drag area increases the horizontal offset by 11.8% and vertical offset by 20.4% at the drag coefficient of 0.5, whereas a 60% increase in the drag area increases the horizontal offset by 35.5% and vertical offset by 78.2% at the drag coefficient of 3.0. Fig. 5 and Fig. 6 illustrate the effects of the parameters of initial pretension and current velocity on the behavior of the STLP. It shows that the increase in the pretension is to reduce the SSP offsets while an increase in the current velocity increases the SSP offset. At the current velocity of 0.5 m/s, a 100% increase of the pretension leads to reduction of the horizontal offset by 58.2% and vertical offset by 83.6%.
In the actual design process, the tethers could be designed with various nominal diameters and may have differ-  . 3. Effect of the drag area (π 4 ) on the horizontal offset for various SSP drag coefficients (π 5 ).   Fig. 6. Effect of the pretension (π 7 ) on the vertical offset for various current velocities (π 3 ). ent drag coefficients. Fig. 7 and Fig. 8 present the effects of the parameters of the tether diameter and drag coefficient on the behavior of the STLP. It shows that the increase in the tether diameter as well as the drag coefficient is to increase the SSP offset. By the same token, the reason can be revealed by the hydrodynamic loads calculation theory. A 64% increase in the tether diameter causes 45.6% increase in the horizontal offset and 130.6% increase in the vertical offset at the tether drag coefficient of 0.5, whereas a 64% increase in the tether diameter increases the horizontal offset by 77.1% and the vertical offset by 240.4% at the tether drag coefficient of 3.0. Fig. 9 and Fig. 10 present the effects of the parameters of Young's modulus and nominal weight of the tether. Interestingly, the results indicate that the variation of Young's modulus has little influence on the SSP offset when its value exceeds 300. On this basis, it can be concluded that the selection on the material of the tether has little influence on the structural response of the STLP once its stiffness reaches some thresholds. Besides, it shows that the increase in the tether mass is to increase the SSP offset as the tether radian increases.

Contributions of various dimensionless groups
The contributions of each dimensionless group on the variation of the structural response of the STLP were deduced on the basis of above parametric studies.
In Fig. 11 and Fig. 12, the dimensionless groups are ranked according to their contribution on the variation of the SSP offsets (X/h and Y/h). The length of the column indicates the contribution level of the proposed dimensionless groups. It can be seen that for the selected ranges of input data, the current velocity (π 3 ) has the largest contribution on the structural response of the STLP, followed by the pretension (π 7 ) while Young's modulus (π 10 ) of the tether has the least contribution which could be negligible. Therefore, it can be concluded that the current velocity and the pretension of the tether are the two most important parameters controlling the structural response of the STLP. Further- Fig. 7. Effect of the tether diameter (π 8 ) on the horizontal offset for various tether drag coefficients (π 9 ). Fig. 8. Effect of the tether diameter (π 8 ) on the vertical offset for various tether drag coefficients (π 9 ).  9. Effect of Young's modulus (π 10 ) on the horizontal offset for various tether masses (π 6 ). Fig. 10. Effect of Young's modulus (π 10 ) on the vertical offset for various tether masses (π 6 ). Fig. 11. Contributions of each dimensionless group on the variation of X/h. more, it should be noted that the pretension is the design parameter while the current velocity is the environmental parameter, which is dependent on specific site.