Pipelines from offshore petroleum fields must frequentlypass over areas with uneven seafloor. In such cases the pipeline may have free spans when crossing depressions. Hence, if dynamic loads occur, the free span may oscillate and time varying stresses may give unacceptable fatigue damage. A major source for dynamic stresses in deep water free span pipelines is vortex induced vibrations (VIV) caused by current.
A substantial research effort related to various aspects of VIV and free span pipelines has been seen during the last years. The hydrodynamics of a cylinder close to a wall was investigated by Bearman and Zdravkovich (1978), while Bryndum et.al. (1989) and Marchesani et.al. (1995) have reported results from laboratory tests on long slender beams. Full scale measurements were carried out by Bruschi et.al. (1989). Such research activities gave the background for a new set of guidelines issued by Det Norske Veritas (1998), also referred to as DNV-G14. Important implications of these guidelines were published by Mørk et.al. (1998). Some work has also been published on direct calculation of VIV on free span pipelines, cf. Tura and Vitali (1990). Halse and Larsen (1998) used the combination of two-dimensional (2D) solutions of Navier-Stokes in combination with a 3D beam model.
It is interesting to observe that the offshore industry has applied guidelines such as DNV G-14 for fatigue analysis of free span pipelines, while direct calculations by use of empirical models have been carried out for vertical risers. The main differences are probably that the current is uniform for the pipeline while sheared profiles must be considered for risers, and that the pipeline will respond at low modes only, which is in contrast to a riser where higher order modes often are seen. However, the boundary conditions are more complex for a pipeline than a riser. One purpose of the present work is therefore to illustrate the significance of these boundary conditions and describe how they can be taken into account in a direct analysis.
Most empirical models are based on frequency domain dynamic solutions and linear structural models (Larsen 2000). A free span pipeline has, however, important non-linearities that should be taken into consideration. Both tension variation and pipe-seafloor interaction will contribute to nonlinear behaviour, which means that most empirical models will have significant limitations when dealing with the free span case. The need for time domain methods is therefore obvious.
The purpose of the present paper is to discuss nonlinear effects related to VIV of free span pipelines and describe how a best possible linear model can be established. An improved strategy for analysis will also be outlined. This is based on combined use of a traditional linear VIV analysis and a subsequent nonlinear simulation where information from the linear case is utilized. All models will be described and the influences from non-linearities illustrated by case studies.
An ideal model for analysis of VIV for free span pipelines should be able to take into account the following effects:
- Start with a correct static condition found from a non-linear static analysis that gives a correct tension, seafloor contact and 3D shape of the pipe. Such an analysis must know the tension during installation and consider later changes in pressure, density of contents and temperature. This condition will define the in-line and cross-flow eigenfrequencies for the pipe.
- Describe the necessary hydrodynamic coefficients (added mass, lift, drag and damping) for a pipe that oscillates in current close to a wall.
- Describe the local current profile including boundary layer effects close to the seafloor.
- Handle any current direction relative to the pipeline and also the combination of current and wave induced forces.
- Predict the correct dominating modes for in-line and crossflow response for a given current condition.
- Describe the interaction between in-line and cross-flow VIV
- Describe the interaction between the pipe and seafloor in terms of non-linear stiffness and damping including friction for in-line oscillations of a pipe with seafloor contact.
- Take into account the influence from tension variation on stiffness and hence also on the actual eigenfrequency.
- Be able to analyse adjacent spans and the dynamic interaction between them.
Most empirical models for VIV analysis have been developed for the analysis of marine risers, Larsen 2000). They are therefore not able to meet all demands related to the free span pipeline case. Some aspects related to this problem will be discussed in the following section.
OUTLINE OF STANDARD ANALYSIS PROCEDURE
The computer programs VIVANA, Larsen et.al. (2000) and RIFLEX, Fylling et.al. (1998) have been used in the present study. The VIV analysis program VIVANA applies RIFLEX modules for system modeling and static analysis, and RIFLEX has been used for all time domain simulations.A complete VIV analysis of a free span pipeline must
include steps as follows:
- Static analysis to define tension and shape of the span. Geometry of the seafloor and a model for pipe/seafloor interaction are essential features of this analysis.
- Eigenvalue analysis for the actual static condition.
- VIV analysis to identify response frequency and amplitudes for a given current condition.
Eigenvalue analysis
An eigenvalue analysis can be carried out on the basis of the static solution. This is straightforward for the still water case since the added mass coefficient is well known. The effect of seafloor proximity may be taken into account, cfr. Jensen et.al. (1993), but this effect has been neglected in the present study. Eigenvalue analyses are also applied in order to identify the response frequency for a given current condition. The response frequency will in general appear as a compromise between the still water eigenfrequency and the vortex shedding frequency for the fixed cylinder, Larsen et.al. (2001). This frequency is found from an iteration where the added mass coefficient is given as a function of the non-dimensional frequency, see Nomenclature. Convergence is obtained when consistency is obtained between the added mass and eigenfrequency.
Method for VIV response calculation
A free span pipeline is assumed to respond at one discrete frequency identified as an eigenfrequency with added mass valid for the given flow condition. The response may be calculated by using finite elements and the frequency response method. The equation of dynamic equilibrium may be written:
M&r& +Cr& +Kr = R (1)
The external loads will in this case be harmonic, but loads at all degrees of freedom are not necessarily in phase. It is convenient to describe this type of load pattern by a complex load vector with harmonic time variation:
R = X eiωt (2)
The response vector will also be given by a complex vector and a harmonic time variation. Hence we have:
r = x eiωt (3)
By introducing the hydrodynamic mass and damping matrices dynamic equilibrium can now be expressed as:
(4)
The damping matrix CS represents structural damping and will normally be proportional to the stiffness matrix. CB contains damping terms from pipe/seafloor interaction. A simple matrix with elements on the main diagonal for vertical displacement has been applied in the present study.
Elements in the excitation vector X are always in phase with the local response velocity, but a negative lift coefficient will imply a 180 degrees phase shift and hence turn excitation into damping. Since the magnitude of the lift coefficient depends on the response amplitude (cfr. Figure 5), an iteration is needed to solve the equation. Note that the response frequency is fixed during this iteration.
The iteration will identify a response shape and amplitude that gives consistency between the response level, lift coefficients and the local flow condition. The mode shape corresponding to the selected response frequency is used as an initial estimate for the response vector only.
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