Dynamic Response Study of Steel Catenary Riser Based on Slender Rod Model

A numerical model of the steel catenary riser (SCR) is built based on the slender rod model. The slender rod model, which describes the behavior of the slender riser in terms of the center line position, can solve the geometrical nonlinearity effectively. In a marine environment, the SCR is under the combined internal flow and external loads, such as wave and current. A general analysis considers only the inertial force and the drag force caused by the wave and current. However, the internal flow has an effect on the SCR; it is essential to explore the dynamic response of the SCR with the internal flow. The SCR also suffers the lift force and the fluctuating drag force because of the current. Finite element method is utilized to solve the motion equations. The effects of the internal flow, wave and current on the dynamic response of the SCR are considered. The results indicate that the increase of the internal flow density leads to the decrease of the displacement of the SCR, while the internal flow velocity has little effect on the SCR. The displacement of the SCR increases with the increase of the wave height and period. And the increasing wave period results in an increase in the vibration period of the SCR. The current velocity changes the displacements of the SCR in x- and z-directions. The vibration frequency of the SCR in y-direction increases with the increase of the current velocity.


Introduction
As principal energy sources for the daily life and industrial production, oil and gas resources play a highly significant role in promoting the growth of the world economy and the development of human civilization. The marine oil and gas reserves are approximately 100 billion tons, of which approximately 44% lie below 2000 m in deepwater. The exploration and development of oil and gas resources gradually shift from the land to the sea, specifically to deepwater. Marine risers have become the frontier of the scientific and technological innovation in the petroleum industry. Marine risers are used for production, transportation, and drilling, which are typical 'lifelines' in the advance of marine oil and gas resources. It is imperative to provide intellectual support for the design of marine risers.
Numerous studies have been done on marine risers. Lou et al. (2017) carried out a test model to investigate the role of a stationary fairing by varying the caudal horn angle to suppress riser VIV considering the effect of the wake inter-ference. Zhang and Tang (2015) presented the parametric instability analysis of top tensioned riser (TTR) considering the linearly varying tension along the length. Yang and Xiao (2014) investigated the non-linear dynamic response of the TTR under the combined parametric and external vortex excitations. Li et al. (2010) analyzed the nonlinear coupled dynamics of the in-line and cross-flow vortex-induced vibrations of the TTR in the time domain.
The steel catenary riser (SCR) is generally considered as a preferable solution in deepwater, as it has significant advantages over other risers, including a low cost, strong adaptability to the harsh environment. Wang et al. (2017) carried out a nonlinear dynamic analysis of the SCR by Abaqus/Aqua, with the SCR-seabed interaction taken into consideration. Dong and Sun (2015) studied the fatigue damage of the SCR in the touchdown zone based on a plasticity model. Wang et al. (2015) investigated the response of a truncated SCR under top vessel motion by performing a large-scale model test. Kim and Kim (2015) compared the SCR and the lazy wave riser under the same condition. Bai et al. (2015) calculated the dynamic response of the SCR by numerical calculation considering the interaction between riser and soil. Huang and Li (2006) described the SCR in detail and prospected the development of the SCR. Orcina Ltd (2015), a commercial software developed by Orcina, can analyze the dynamic behavior of the SCR by the lumped mass method. The SCR is subject to large-scale deflection and large-angle movement under various loads in the marine environment (Meng and Chen, 2012).
Although some commercial software related to marine riser is easily accessible, developing an in-house program has greater advantages in cost savings, custom designing and analytical understanding. In this paper, a calculation process for SCR is programmed based on the slender rod model (Garrett, 1982), which can solve the large deformation of the SCR effectively. The goal of this study is to examine the significance of the internal flow related to the design of the SCR and report the analyses of the SCR under several marine loads.
The research carried out in this paper provides a new perspective model of the SCR considering the internal flow. The internal flow is approximated as a plug flow. The effect of internal flow on the SCR includes the inertial force, the Coriolis force and the centrifugal forces.
Present studies are to estimate the effect of the current taking only the drag force. In addition to the drag force, the effect of the current on the riser includes the lift force and the fluctuating drag force. This study considers the effect of the lift force and the fluctuating drag force when simulating the SCR with the slender rod model.
Applying the internal flow and the lift force that incorporate the SCR's motion using the slender rod model can provide a realistic technology for predicting the response of the SCR under marine loads.

Slender rod model
The slender rod model is established in the global coordinate system. The configuration of the slender rod is described in terms of the position of the center line, as shown by the space curve in Fig. 1. The slender rod is the function of arc length and time (Chen et al., 2011).

F M
The internal stress at each point along the riser is described by the resultant force and bending moment . The motion equations of the SCR with the internal flow can be derived from the conservation of linear and angular momentum. (1) where is the applied force per unit length; and are the mass of the riser and the internal flow per unit length; is the applied moment per unit length. For slender structures, the torque and distributed torsional moment are usually neglected and , .
is the bending stiffness.
The deformation condition for elastic and extensible rod can be written as follows: , is the local curvature. is the local effective tension, where is the wall tension, and are the external and internal hydrostatic pressure respectively, and are the outer and inner cross-sectional areas respectively.
is the axial tensile stiffness. is the cross-sectional area.
The motion equation of the SCR can be written as: (4)

Load analysis q
The SCR suffers from various loads in the marine environment. The applied force per unit length can be described as follows.
where , , and are the gravity, the hydrostatic force and the hydrodynamic force per unit length.
The hydrostatic force contains the buoyancy force and the pressure.
where, is the buoyancy force per unit length; is the hydrostatic pressure, induced by the pressure difference between outer and inner pressure. The hydrodynamic force includes the inertial force and the drag force, resulted from the wave and current, and the lift force and the fluctuating drag force, caused by the current. The inertial force and the drag force can be calculated by the Morison equation (7) where is the outer diameter; is the added mass coefficient; is the drag coefficient; is the lift coefficient; is the fluctuating drag coefficient; and are the normal velocity and acceleration of the wave and current; and are the normal velocity and acceleration of the riser; is the normal velocity of the current; is the added mass per unit length; , is the Strouhal number, which is set to 0.2 in this study; is the velocity of the current. v The effect of internal flow on the SCR is highly complex. To simplify the modeling, the internal flow is approximated as a plug flow that is similar to an infinitely flexible rod where all points of the flow have the same velocity . The force induced by the internal flow on the riser can be obtained according to Païdoussis (1998).
where the first term is the inertial force, the second term is associated with the Coriolis force, and the third term is associated with the centrifugal forces as the flow has to keep the same curvature as the riser.

Finite element model
The motion equation and inextensibility condition of the SCR are: w where =w+B is the effective gravity per unit length.
The SCR is discretized into several elements along the length in finite element method.
where and are the shape functions; are the vectors of position and tangent; are the vectors of effective tension.
The final equations for the SCR are presented as follows: where, and are the mass matrices; and are the stiffness matrices induced by the material, the tension, and the curvature; is the damping matrix induced by the internal flow; is the load vector; ; ; ; .

Static and dynamic analysis
In static analysis, the inertial force term is excluded.
The above strongly nonlinear algebraic equations are solved by the Newton-Raphson method. This method assumes the unknown values in iterative step n are and . The values are extended by Taylor's formula with the higher order terms ignored. The unknown values are updated constantly and the true values can be obtained by loop iteration.
The equation for modal analysis can be obtained without damping and load.
ω {ϕ} where is the natural frequency; is the corresponding mode.
For the dynamic response, the equilibrium shape of the riser is taken as the initial condition, and the load is applied from the zero point with the slope function. The dynamic response of the SCR is obtained by a predictor-corrector scheme based on the Newmark-β integration algorithm.
The calculation process of the SCR is programmed into DRSSCR (Dynamic Response Study of Steel Catenary Riser) based on the Matlab platform.

Model validation
The detailed physical properties of the SCR are listed in Table 1. The SCR is discretized into 200 elements with its two ends set as pinned. The configuration of the SCR is shown in Fig. 2. The instantaneous shape of the SCR in a wave period under wave is illustrated in Fig. 3a. In order to describe the deformation of the SCR more obviously, the displacement of the riser is enlarged by 1500 times. The instantaneous displacement, velocity, and acceleration of the SCR in a wave period under wave are illustrated in Fig. 3b. The 1st to 6th order frequencies of the SCR are calcu-lated by OrcaFlex to verify the accuracy of the DRSSCR. It can be observed clearly from Table 2 that the DRSSCR is in good agreement with OrcaFlex.

Dynamic response study of SCR
3.1 Effect of internal flow density on SCR The effect of internal flow density on the nonlinear dynamic response of the SCR is studied. The SCR is subjected to a linear wave with the height of 6 m and period of 10 s. Three different internal flow densities are considered in this section, ρ 1 =800 kg/m 3 , ρ 2 =900 kg/m 3 , and ρ 3 =1000 kg/m 3 . Table 3 gives the maximum dimensionless displacement under different internal flow densities and the number of the node where the maximum displacement occurs. The envelope of the maximum dimensionless displacement along the riser and the time history at the extreme node are shown in Figs. 4 and 5. Where A represents the displacement and D represents the outer diameter of the SCR.
When the internal flow density is 800 kg/m 3 , the maximum dimensionless displacements in the x-and z-directions are 0.5297 and 0.4384, which appear at No. 193 and No. 12. When the internal flow density is 900 kg/m 3 , the maximum dimensionless displacements in the x-and z-directions are 0.4809 and 0.4343, which appear at No. 194 and No. 12. When the internal flow density is 1000 kg/m 3 , the maximum dimensionless displacements in the x-and z-directions are 0.4138 and 0.4276, which appear at No. 195 and No. 12. The results indicate that the maximum displacements in the x-and z-directions decrease with the increasing internal flow density. And the internal flow density has a greater effect on the x-direction than that on the z-direction. As can be observed in Fig. 5, the riser always vibrates near the equilibrium position as there is no current. The vibration period is equivalent to the wave period, no matter how much the internal flow density is.

Effect of internal flow velocity on SCR
The effect of internal flow velocity on the dynamic characteristics of the SCR is studied. The internal flow density is   Table 4. It can be seen from Table 4 that the natural frequency of the SCR decreases with the increase of the internal flow velocity, but the decrease is very small. The increasing internal flow velocity reduces the stiffness of the SCR, reducing the natural frequency of the SCR. However, internal flow velocity in engineering is not very large, and the effect of the internal flow velocity on the stiffness is too small compared with the riser's stiffness. As a result, the effect of internal flow velocity on the dynamic characteristic of the SCR is very small.

Effect of wave height on SCR
The effect of wave height on the nonlinear dynamic response of the SCR is studied in this section. The internal flow density is 900 kg/m 3 , and the internal flow velocity is 2 m/s. The wave period is 10 s. Three different wave heights are selected: H 1 =6 m, H 2 =8 m and H 3 =10 m. Table 5 gives the maximum dimensionless displacement under different wave heights and the number of the node where the maximum displacement occurs. Figs. 6 and 7 show the envelope of the maximum dimensionless displacement along the riser and the time history at the extreme node.
The results demonstrate that the wave height affects the nonlinear dynamic response of SCR. When the wave height is 6 m, the maximum dimensionless displacements in the xand z-directions are 0.4809 and 0.4343, which appear at No. 194 and No. 12. When the wave height is 8 m, the maximum dimensionless displacements in the x-and z-directions are 0.7490 and 0.5303, which appear at No. 195 and No. 12.
When the wave height is 10 m, the maximum dimensionless displacements in the x-and z-directions are 1.0829 and 0.6013, which appear at No. 195 and No. 12. The displacement of the SCR increases obviously along the riser with the increase of the wave height. While the position where the maximum displacement appears has little change.

Effect of wave period on SCR
The effect of wave period on the nonlinear dynamic response of the SCR is studied. Based on the same internal flow density (900 kg/m 3 ) and velocity (2 m/s), and the same wave height (6 m), three different wave periods are considered, T 1 =10 s, T 2 =14 s, and T 3 =18 s. Table 6 gives the maximum dimensionless displacement under different wave heights and the number of the node where the maximum displacement occurs. Figs. 8 and 9 show the envelope of the maximum dimensionless displacement along the riser and the time history at the extreme node.
When the wave period is 10 s, the maximum dimension-    LIU Zhen, GUO Hai-yan China Ocean Eng., 2019, Vol. 33, No. 1, P. 57-64 less displacements in the x-and z-directions are 0. 4809 and 0.4343,which appear at No. 194 and No. 12. When the wave period is 14 s, the maximum dimensionless displacements in the x-and z-directions are 0. 8534 and 0.6240,which appear at No. 190 and No. 15. When the wave period is 18 s, the maximum dimensionless displacements in the xand z-directions are 1. 3081 and 0.7743,which appear at No. 185 and No. 17. With the increase of the wave period, the displacements in the x-and z-directions of the SCR increase obviously. Meanwhile, the positions in which the maximum displacements appear also change distinctly. As the vibration period of the SCR is equivalent to the wave period, the increasing wave period leads to the increase of the vibration period of the SCR.
3.5 Effect of current velocity on SCR In this part, the effect of current velocity on SCR is studied, considering the lift force and fluctuating drag force. The lift coefficient is 0.4 and the fluctuating drag is 0.05. Fig. 10a shows the instantaneous shape of the SCR in a wave period under the wave (height 6 m, period 10 s) and current (velocity 0.3 m/s). Fig. 10b shows the instantaneous displacement, velocity, and acceleration of the SCR in a wave period under wave and current. In order to describe the deformation of the SCR more clearly, the displacement of the riser is enlarged by 100 times.
By comparing Fig. 10 and Fig. 3, it can be seen that the nonlinear dynamic response of the SCR under wave and current has a notable difference from the nonlinear dynamic response of the SCR under wave. Obviously, the current greatly affects the dynamic response of the SCR. The current changes the vibration modes of the SCR in the x-, y-, and z-directions. The SCR vibrates in the y-direction under current because of the lift force.
The effect of the current velocity on the nonlinear dy-   namic response of the SCR is studied. The SCR is subjected to linear wave whose height is 6 m and period is 10 s. The internal flow density and velocity are 900 kg/m 3 and 2 m/s. The current velocity used in the numerical simulation are v 1 =0.3 m/s, v 2 =0.4 m/s, and v 3 =0.5 m/s. Table 7 gives the maximum dimensionless displacement under different current velocities and the number of the node where the maximum displacement occurs. Figs. 11 and 12 show the envelope of the maximum dimensionless displacement along the riser and the time history at the extreme node. The vibration frequency of the SCR is illustrated in Fig. 13. The current velocity changes the maximum displacements in the x-and z-directions of the SCR. The displacements increase with the increasing current velocity. While the extreme nodes and vibration frequencies of the SCR in the x-and z-directions have little change. The effect of current velocity on the y-direction is different from that on the x-and z-directions. The current velocity has a significant effect on the vibration frequency and mode in the y-direction. As the frequency of the lift force is linked to the current ve-locity, the higher the current velocity is, the higher the frequency will be. As a result, the vibration frequency in the ydirection increases with the increase of the current velocity.

Conclusions
A numerical model of the SCR is developed based on the slender rod model. The internal flow is approximated as a plug flow. The effect of internal flow on the SCR includes the inertial force, the Coriolis force and the centrifugal forces. In addition to the inertial force and the drag force caused by the wave and current, the lift force and the fluctuating drag force caused by the current are also considered in this paper.
The effect of the internal flow density and velocity, wave height and period, and current velocity on the dynamic response of the SCR is calculated. The increase of the internal flow density leads to the decrease of the displacement of the SCR. Practical attention should be paid to the internal flow density in the design phase. The internal flow velocity has little effect on the SCR. Wave height and period have a great effect on the dynamic response of the SCR. The displacement of the SCR increases with the increase of the wave height and period. And the increasing wave period leads to an increase in the vibration period of the SCR. The SCR suffers the lift force because of the current. The current velocity changes the displacements of the SCR in the x-and z-directions. The vibration frequency of the SCR in the y-direction increases with the increase of the current velocity.