Effect of Internal Flow on the Dynamic Behavior of Top Tensioned Riser

LI Xiao-min,GUO Hai-yan,MENG Fan-shun

(1 College of Engineering,Ocean University of China,Qingdao 266100,China;

2 College of Marine Geo-science,Ocean University of China,Qingdao 266100,China)

1 Introduction

Top tensioned riser(TTR)which can be used for drilling,production and intervention has become a hotspot for research with the trend towards oil and gas exploitation in deep waters in the past two decades.It serves the offshore structure system as the link between the platform on the water surface and the well head on the seabed.Tension is applied at the top end of the riser to resist lateral loads such as vortex shedding,current and wave forces and the riser is usually kept almost straight.TTR can be modeled as an extremely long tensioned tubular structure in the analysis.In addition to its own weight,internal and external hydrostatic pressures,and vessel offsets,the riser also encounters severe current and wave forces and therefore the dynamic response of the riser is very complicated.Generally,the dynamic response of the riser induced by wave and current forces will not be coupled with the vibration caused by vortex shedding and they can be analyzed individually.

Though transporting fluid is the main function of the marine risers,it is only at the end of the 1980s the effect of the internal flow on the dynamic behavior began to draw more and more attention.Before this time,its effect is omitted or misconceived.When internal fluid travels along the curved path inside the deflected risers,it experiences centrifugal and coriolis accelerations because of the curvature of the risers and the relative motion of fluid to the time dependent risers motion,respectively.Those accelerations exert against the risers and,in turn,affect the dynamic behavior of the risers and cause additional vibrations.TTR has been successfully used in more than 6 000 feet deep waters recently.As the water depth increasing,the influence of the internal flow may become increasingly important and the drag force caused by ocean currents which may lead to considerable static and dynamic displacement should be given more attention in the analysis.

There is a vast literature on the investigations of the riser dynamics in the last few decades.Recent publications are briefly discussed herein.Moe and Chucheepsakul(1988)[1]investigated the effect of internal flow on vertical riser’s linear natural frequencies while in the analysis flexural rigidity was neglected.Patel and Seyed(1989)[2]examined the contributions of internal flow to both the governing equations and the dynamic excitation forces applied to a flexible riser.Wu and Lou(1991)[3]derived the governing equation of the riser considering the effects of internal flow and bending rigidity and found that the internal flow acts to reduce the effect of the top tension.Chucheepsakul et al(1999)[4]investigated the influence of fluid transported inside the riser and static offset on dynamic characteristic behaviors.The results showed that the natural frequencies of the riser decrease while both the internal flow speed and the static displacements increase;and the transported fluid reveals more significance on the high extensible risers than the low extensible ones.Chucheepsakul et al(2003)[5]launched the theoretical investigation of the large strain analysis and developed mathematical formulations of extensible flexible marine pipes transporting fluid.Recently,Kaewunruen et al(2005)[6]analyzed the influence of marine riser’s parameters such as flexural rigidity,top tensions,internal flow velocities,and static offsets on the nonlinear free vibrational behaviors by reformed the governing formulation to the eigenvalue problem.Morkookaza et al(2006)[7]analyzed the dynamic behavior of a top tensioned riser under different wave and riser conditions in frequency and time domain.Also active control of marine riser’s vibrations in order to reduce vibration amplitudes and angles and prevent the collisions between adjacent risers has become an issue of considerable concern and more details can be reviewed in Ioki et al(2006)[8],Rustad et al(2008)[9],How et al(2009)[10].Reviews on the specifics of riser modeling and analysis techniques are given in Patel and Seyed(1995)[11]and Pa?doussis(1998)[12].However,most of the publications mentioned above are either mainly on the analysis of the riser’s parameters on the effect of the riser mode shapes and frequencies or the internal flow was not considered in the dynamic response analysis.The dynamic response of the riser under the combined excitation of internal flow and external marine forces has not been scrutinized yet.

In this paper,the effect of the internal flow on the dynamic characteristic and dynamic behavior of the riser under the excitation of current and wave forces is analyzed in detail.The governing equation of the riser is derived based on virtual work-energy principles and the ki-netic energy of internal flow is considered in the equation.Then the Galerkin finite element approximation is implemented to derive the nonlinear matrix equation and the corresponding numerical program which solves the equation in time domain is compiled.Finally,the dynamic characteristic and dynamic behavior of the riser are analyzed in detail and the effect of internal flow on the riser behavior is given special attention in the analysis.

2 Model description

2.1 Governing equation

Considering an element has length dz at the equilibration state,then the length of the riser element after deformation ds can be expressed as:

in which a superscript prime denotes a differentiation with respect to z.

One can write the dynamic strain of the riser element as:

in which ε0is the initial strain of the riser element.

The curvature of the riser element can be expressed as:

The strain energy due to axial deformation can be expressed as:

where E is the Young’s modulus of elasticity,A is the cross section area of riser.Performing the variation of Eq.(4),one obtains:

where Te=EAε,it is the effective tension in the riser.

The strain energy due to bending can be expressed as:

Performing the variation of Eq.(6),one obtains:

where mris the mass of unit length of riser,miis the internal flow mass of unit length of riser,aris the acceleration of riser,ai=+x″V2+2V′,aiis the acceleration of internal flow,V is the internal flow velocity,（·）represents the derivation with respect to time t.The work done by damping of the riser system can be expressed as:The virtual work done by inertial force can be expressed as:

where c is the structure damping ratio of riser system.

Linear Airy Waves are adopted in the present study.The wave particle velocity u and its acceleration u˙are expressed by the following Equations.

where,H is wave height,T is wave period,ω is wave frequency given by the dispersion relationship ω2=gktanh k（d）,g is the gravity acceleration,k is the wave number,d is the water depth.

The virtual work-energy of the riser system can be written as:

Substituting Eqs.(5),(7-10)into Eq.(14)and integrating of Eq.(14)is performed,one obtain Euler’s equations associated with virtual displacements δx as follows:

2.2 Finite element discretization

Galerkin finite element approximation is implemented to obtain the mass,damping and stiffness matrices of the Eq.(15).

Eq.(16)is solved directly in time domain with Newmark-β time history analysis method and the corresponding MATLAB numerical programs are compiled.

The dynamic character equation can be expressed as:

The natural frequencies and their associated modes can be determined by solving Eq.(17).

3 Results and discussions

The parameters employed in the calculations are given as follows(Kaewunruen et al,2005)[6]:water depth=300m,riser length=300m,riser pipe outside diameter=0.26m,inside diameter=0.2m,top tension=476 198N,modulus of elasticity=2.07×1011N/m2,specific weight of the fluid of the sea water=1 025kg/m3,specific weight of the fluid in the riser bore=998kg/m3,specific weight of the riser wall material=7 700kg/m3,the inertia coefficient=2,drag coefficient=0.7.

3.1 Dynamic characteristic analysis

Fig.1 shows the comparisons of the fundamental frequencies under various internal flow velocities.Apparently,the results obtained in this paper are in very good agreement with Kaewunruen.Fig.2 gives the comparisons of the fundamental frequencies of the same riser with different top tensions,under various internal flow velocities.It can be seen that the fundamental frequency increases greatly with the increase of top tensions.From Figs.1 and 2,one can see that the fundamental frequency is slightly reduced at low internal flow velocities but significantly reduced at high velocities.This is also true with the analysis results of Moe and Chucheepsakul(1988)[1].Fig.3 gives the absolute value of differences between the frequencies considering the effect of internal flow and the frequencies without considering internal flow.It can be observed that the internal flow has more influence on the lower frequency orders of the riser.The influence decreases rapidly with the increase of frequency orders.The riser natural modal shape up to fifth mode obtained by FEM is shown in Fig.4 and the corresponding circular frequencies are 0.298 9,0.628 8,0.994 2,1.402 4 and 1.858 6rad/s.It can be seen that the envelope is damped from the sea bottom to the riser top and the reason is that the effective tension of the riser decreases gradually from the riser top to the sea bottom.

3.2 Dynamic response analysis

First,some results of a marine riser are presented for comparison with Yamamoto et al(2004)[13].The riser has the following characteristics:water depth=100m,riser length=120m,riser pipe outside diameter=0.25m,inside diameter=0.211 6m,top tension=200kN,modulus of elasticity=2.1×1011N/m2,specific weight of the fluid of the sea water=1 025kg/m3,specific weight of the fluid in the riser bore=800kg/m3,specific weight of the riser wall material=7 700kg/m3.Two shear flows considered for our comparisons vary linearly are illustrated in Fig.5.The computation results are illustrated in Fig.6.It can be observed that there are no big differences between the two results.

Hereinafter,the 300m long riser is used in calculation.Figs.7 and 8 present the maximum amplitude envelopes of the riser at the excitation of uniform current velocity 0.5m/s and 0.8m/s with different internal flow velocities.It can be observed that the amplitude envelopes increase with the increase of the internal velocities especially in high internal velocities this trend becomes more evident.The effect of current on the maximum amplitude of the riser is more evident and it can be observed that the simple increase of current speed of 0.3m/s results in great increase of amplitudes.The maximum amplitude envelopes of the riser subjected to a wave with period 10s and wave height 3m at various internal flow velocities are depicted in Fig.9.No significant difference can be seen at low internal flow velocities while significant difference appears at high velocities.Resembling phenomena can be observed in Fig.10,but in this case,the riser is subjected to a wave with period 8s and wave height 3m.It can also be seen from Figs.9 and 10 that the vibration amplitude of the riser at wave period 10s is much larger than at wave period 8s.This is because wave period 10s is just in the vicinity of the riser’s second natural frequency which may result in resonance of the riser.With the increase of internal fluid velocities,the riser’s second natural frequency deviates from the resonance region and the amplitude decreases.The riser’s envelopes at the conjunct excitation of wave and current forces are depicted in Fig.11.In this case,the current velocity,wave period and wave height are 0.3m/s,10s and 3m,respectively.The effect of internal flow velocity on the response of the riser is also very evident in this figure.

Fig.12 presents the envelopes of the riser subjected to a wave with period 10s and wave height 3m with different top tensions.It can be seen evidently that top tension has a strong influence on the dynamic behavior of riser.A simple modification on the riser top tension may cause a transition of the mode of vibration though the riser is excited for the same hypothetic load.One can also find that a simple increase of the riser top tension does not mean the amplitude of the riser may decrease.

Figs.13 and 14 present the result of the riser without incident wave and current forces.Only sinusoidal oscillations with amplitudes of 1m and 2m and period 10s are forced at the riser top end,different internal flow velocities are also considered in the analysis.It can be seen that the effect of the internal flow velocity on the forced oscillation of the riser is mainly on the middle amplitudes of the riser.

4 Conclusions

The governing equation of the top tensioned riser is derived based on work-energy princi-ples and the dynamic behavior of the riser is investigated in this paper.The influences of internal flow velocities on the dynamic characteristic and dynamic behavior of the riser are given special attention in the analysis.Based on the performed analyses the following conclusions can be drawn.

Internal flow tends to reduce the natural frequencies of the riser and the response amplitudes and vibration modes may have great differences at different internal flow velocities though the riser is excited for the same exterior load,especially the internal flow velocity are remained at a relatively high level.Hence the influence of internal flow on the dynamic response of riser should not be neglected and more attention should be given in the analysis.

Top tension has remarkable influence on the dynamic characteristic and dynamic behavior.Through changing the top tension of the riser,the resonance can be avoided and also the vibration amplitude maybe reduced to an appropriate degree.However the vibration mode of the riser may also transit from one to another when the top tension changes.

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