Dynamic responses of a riser under combined excitation of internal waves and background currents

International Journal of Naval Architecture and Ocean Engineering.
2014.
Sep,
6(3):
685-699

This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

- Published : September 30, 2014

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Dynamic response
;
Internal waves
;
Background currents
;
Taylor-goldstein equation
;
Newmark-β method.

INTRODUCTION

Marine risers are important equipment for the exploration of oil/gas in deep water, which usually serve as the bridge between an offshore platform and the well head on the seabed (
Kaewunruen et al., 2005
;
Ju et al., 2012
). As exploration activities move into deepwater, the long slender risers tend to undergo large-amplitude motions (
Xu et al., 2013
). Internal waves are believed to be responsible for a great deal of damage. This is because they can create enormous local loads and bending moments on offshore structures. The internal waves have been reported to induce an additional displacement of 200
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THEORY OF INTERNAL WAVES

- Governing differential equation of internal waves

With the Boussinesq and linear approximation, the second-order differential equations govern the vertical structure function
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- The relative velocity of internal waves

In the two-dimensional Cartesian x, z coordinate system, the water pulsation can be assumed as wave solution and displayed in the form (
Ye and Shen, 2005
):
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VARIATIONAL MODEL OF RISER AND SOLUTION METHOD

The following assumptions are stipulated in the present mathematical model: the material property of the marine riser is homogeneous and linearly elastic and the cross-section are uniform and remain plane perpendicular to the axis at all state; longitudinal strain is large, but the shear strain, torsional deformation and rotational inertia are neglected; the internal fluid is inviscid, incompressible, irrotational, flowing upward and the density is uniform along the riser.
The nonlinear free vibration is determined through the virtual oscillations along the marine riser. The riser will move nonlinearly from the static equilibrium configuration due to small perturbation, as shown in
Fig. 1
. In the Cartesian coordinates, the normal displacement
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- Virtual work formulation

- Total strain energy

The total strain energy of a marine riser, which consists of the strain energy due to axial deformation and bending, is given by (
Chucheepsakul et al., 2003
):
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- Virtual work done by external forces

Needed for dynamic response analysis, the virtual work done due to external forces included inertial force, structural resistance and the hydrodynamic force are taken into account for accurate reckons, i.e.,
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- Differential motion equation

Based on the principle of virtual work, the total virtual work of the apparent system is zero:
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- Finite element method

Discretize Eq. (21) by Hermite interpolation function, obtaining the general form of the finite element method:
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RESULTS AND DISCUSSION

- Program verification

Programs Fre.m and resp.m in MATLAB are developed to solve the natural vibration frequency and the dynamic response of riser in consideration of background currents only. The results of ANSYS analysis which are taking the pipe59 element to simulate and solved by the prestress method are chosen to verify the accuracy of the programs. The parameters of riser and fluid, which are applied in this work, are given in
Tables 1
and
2
respectively. The background currents are chosen from the one-year return currents profile in the Liuhua oilfield of South China Sea, as shown in
Fig. 2
.
Model properties of marine riser.

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Model properties of seawater.

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Comparison of Riser’s natural vibration frequencies.

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- Dynamic response of riser under internal waves

The parameters of Holmboe density distribution in ocean are given in
Table 4
(
Ye and Shen, 2005
). Then, the density distribution profile and the computed buoyancy frequency N are obtained and as shown in
Fig. 5
. According to vast field observations and investigations of the internal waves in the South China Sea in recent decades (
Bole et al., 1994
;
Qiu et al., 1996
;
Fang et al., 2000
;
Li et al., 2003
), the internal waves of high frequency (about 3~5
Parameters of Holmboe density distribution.

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- Dynamic response of a riser under combined excitation of internal waves and background currents

To study the dynamic response of riser under combined excitation of internal waves and background currents, the following five cases are considered as the follows. It should be noted that the main data of internal wave are kept the same as the previous section and the main parameters of uniform currents are taken form
Fig. 2
(1) Mean currents, i.e., the velocity
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The horizontal wave numbers of internal waves in different cases.

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CONCLUSION

In this paper, the dynamic performance of a marine riser under combined excitation of internal waves and background currents are presented. The dispersion relations and relative velocities of internal waves in horizontal plane are obtained, by solving the geometrically nonlinear equation for riser motion. The numerical solutions about displacement and transfiguration of riser with elapsing time are analyzed. The following conclusions have been drawn:
(1) Without background currents, the internal waves in the first two modes play a dominated role in dynamic response of a marine riser. The displacement of the riser diminishes rapidly in higher modes.
(2) The background currents will make waves that go along them get stretched and waves against them shrunk, which is in agreement with other investigations (
Fang et al., 2000
). Moreover, the displacements of the riser caused by internal waves in the first mode have an enormous increase when background currents exist. In this case, the distribution of both the currents speed
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Acknowledgements

This research is funded by the Fundamental Research Funds for the Central Universities (No. 11CX04018A) and the National Natural Science Foundation of China (No. 51309241). The authors are grateful to acknowledge the financial support.

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Citing 'Dynamic responses of a riser under combined excitation of internal waves and background currents
'

@article{ E1JSE6_2014_v6n3_685}
,title={Dynamic responses of a riser under combined excitation of internal waves and background currents}
,volume={3}
, number= {3}
, journal={International Journal of Naval Architecture and Ocean Engineering}
, publisher={The Society of Naval Architects of Korea}
, author={Lou, Min
and
Yu, Chenglong}
, year={2014}
, month={Sep}