Long-term stochastic heave-induced dynamic buckling of a top-tensioned riser and its influence on the ultimate limit state reliability
Introduction
A top-tensioned riser (hereafter referred to solely as a riser throughout this paper) consists of a vertical pipe, which is used in the offshore industry to convey drilling fluids, oil, gas, water or chemicals from its respective floater to a subsea system or vice versa. This type of structure is mostly used in deep water, where the relative motion between the floater and the subsea system is minor.
One of the technical challenges in the application of risers is the occurrence of heave-induced parametric excitation which may lead to dynamic buckling. A riser is held at its top end by a tensioning system, which keeps the riser's body under tension in order to avoid compressive loads; nevertheless, due to the floater's heave (vertical motion), the tension fluctuates with time, and lateral vibrations (dynamic buckling) can be excited. The unwanted consequence us such phenomenon is the riser's damage due to excessive stress which could lead to oil or gas spills with consequent pollution and economic loses (Yang et al., 2013). Moreover, it is known that the heave motions of floaters are responsible for bending and buckling conditions that can lead to fatigue damage of risers (Katifeoglou and Chatjigeorgiou, 2016).
From the cost perspective, it would be desirable to reduce the tensioning capacity and to permit the riser to operate in low tension (Patel and Vaz, 1996), nonetheless large heave motions can cause serious damage to the riser and thus it is necessary to balance the capital with the risk expenditures in order to optimize the riser's design and operations. To give idea of the consequences, the World Offshore Accident Database (DNV GL, 2016) has reported at least 2 accidents related to the partition of marine drilling risers during stormy conditions with large heave motions where the largest consequence is spill of drilling mud to sea, 1 accident where the drilling string broke after the maximum deflection of the heave compensator was reached, and 4 accidents where one riser tensioner wire failed. Unfortunate events as the above mentioned, can be avoided if inadequacies are solved at design stage, being the most common ones (Patel and Witz, 1991, Chapter 11): failure to predict multiple curvature, failure to predict high curvature, inadequate top-tension availability, inadequate tensioner rate, excessive bending in free-hanging condition and failure of buoyancy modules.
Regarding the description of dynamic buckling, the amplitude of the response is often larger near the bottom owing to the spatial variation of tension with depth. It can be excited via three main mechanisms (Kuiper et al., 2008). First, classic parametric resonance may develop when the frequency and amplitude of the floater's heave excite a specific riser mode or combination of modes. In this situation, the response frequency is about twice the excited eigen-frequency (Park and Jung, 2002), also as confirmed by experiments (Franzini et al., 2015). Second, sub-critical buckling can arise when the floater heaves with low frequency and large amplitude, and thus a single wave grows near the riser's bottom and then propagates along its length. Third, when frequency and amplitude are high enough, buckling waves are generated periodically near the bottom, travel and decompose in a combination of riser modes.
Many studies have been devoted to the issue of parametric excitation of offshore structures in which a coefficient appears as function of time in the governing differential equation. Said studies have included the research about risers and cables (Chatjigeorgiou, 2004, Chatjigeorgiou and Mavrakos, 2005, Chatjigeorgiou and Mavrakos, 2002, Franzini et al., 2015, Franzini and Mazzilli, 2016, Hsu, 1975, Kuiper et al., 2008, Lei et al., 2014, Mazzilli et al., 2016, Park and Jung, 2002, Prado et al., 2014, Wang et al., 2015, Wu et al., 2016, Xiao and Yang, 2014, Yang et al., 2013, Yang and Xiao, 2014, Zhang and Tang, 2015), tethers for tension-leg platforms (Patel and Park, 1995, Patel and Park, 1991), submerged floating pipelines (Yang et al., 2017) and parametric rolling of ships (Pipchenko, 2009, Thomas et al., 2010).
The general approach to investigate the parametric excitation of risers and tethers is as follows: (1) First, the nonlinear governing equation of motion with a time-dependent coefficient is derived. (2.a) Then the stability of the linear Mathieu's equation (for single-frequency excitation) (Chatjigeorgiou and Mavrakos, 2002, Hsu, 1975, Park and Jung, 2002, Patel and Park, 1995, Patel and Park, 1991, Prado et al., 2014, Wang et al., 2015) or Hill's equation (for multi-frequency excitation) (Xiao and Yang, 2014, Yang et al., 2013) is analyzed via the Strutt's diagram, where the stability is estimated analytically. (2.b) Another alternative is to analyze the linearized system by means of the Floquet theory (Kuiper et al., 2008, Lei et al., 2014, Zhang and Tang, 2015). (3) Finally, the nonlinear equation is solved in the time-domain to examine the effect of nonlinear terms (Chatjigeorgiou and Mavrakos, 2002) and the map of the steady-state amplitude can be plotted (Franzini and Mazzilli, 2016, Kuiper et al., 2008, Mazzilli et al., 2016, Prado et al., 2014).
Some variations of the analysis consist of adding forced vibrations, such as surge (horizontal) motions at the top end of the riser. Scholars have found that the response period of combined parametric and forcing excitation is dependent on the relative strengths of each type of excitation (Patel and Park, 1995). The finite element method has been used in order to address axial and torsional effects, and results have shown that the responses of combined excitations can be in general equal or larger than for surge forcing excitation alone (Park and Jung, 2002). Another study that used small deformation theory found that the bending stresses on top-tensioned risers increase as the amplitude of floater's drift motion does so, especially at the bottom end of the riser (Li et al., 2010). On the other hand, the frequency of said motion has dominant effect on the stresses at the upper part. Other researchers who investigated parametric excitation of risers have included other phenomena such as sea wave forces (Lei et al., 2014, Wu et al., 2016), vortex-induced vibrations (Wang et al., 2015, Yang and Xiao, 2014) and earthquake excitation (Wu et al., 2016).
A couple of experiments have been reported in the literature. The first one (Franzini et al., 2015) used spectral analysis of the experiments and Strutt diagrams to investigate the response at different frequency ratios of top excitation to eigen frequency. The second (Mazzilli et al., 2016) used the Galerkin method with Bessel functions to solve the riser's motion numerically and compared the results against experimental data.
The present paper focuses on the investigation of whether the random heave-induced dynamic buckling can lead to structural failure of a riser in the long-term. While useful methods and findings are available in the literature, there are some differences that might be highlighted for this study. (1) The riser's response due to the random floater's motion is investigated, while previous studies have focused either on the floater's harmonic motion or have assumed that the riser's top end follows the random sea surface. (2) Realistic probability density functions (PDFs) are employed to describe the long-term statistics of sea waves which are input for the reliability analysis. (3) Moreover, Monte Carlo simulation (MCS) is performed to assess the probability of dynamic buckling and the probability of failure after dynamic buckling, which up to the authors' knowledge have not been evaluated in the past.
The description of the paper is presented comprising six sections. After Section 1 for introduction, Section 2 presents the governing equation of a riser and its discretization by means of the finite difference into a system of nonlinear ordinary differential equations. Section 3 identifies combinations of significant wave height and wave spectral peak circular frequency that lead to dynamic buckling, where the dynamic stability of the riser is first analyzed and then a map of maximum amplitude response is used. In Section 4, the MCS method is introduced and then applied for the structural reliability analysis of a riser in Section 5. Finally, insights and findings from the present study are addressed in Section 6.
Section snippets
Governing equation
The straight vertical riser submerged in a fluid medium is considered as shown in Fig. 1. The tensioning system is approximated as a soft spring , and the flexible joints as rotational springs and .
Under the small-deformation theory of columns (Paik and Thayamballi, 2003, Chapter 9), and including the fluid's drag damping, the in-plane governing equation for a riser surrounded by quiescent water can be expressed assubjected to
Properties of a riser
Stability analysis is carried out and then the maximum absolute response map is computed for a marine drilling riser made of X-80-grade steel 21 in (533.4 mm) main conductor with properties taken from the literature (Permana, 2012) (see Table 1). Linear regression is applied to approximate , and as functions of the internal fluid's density . In this section, = = 1 and = 1600 kg m−3 are taken, unless otherwise indicated.
As a target floater, a semi-submersible with properties used
Probability of dynamic buckling
Based on the results of Section 4, it is realized that dynamic buckling is likely to happen in sea states of low wave frequency and high amplitude; nevertheless, one may be still interested in how often that event would happen and in whether such event can make the riser fail.
Concerning the probability of dynamic buckling, (i) the deformation imposed by the initial conditions will disappear in time if the riser finds the straight shape stability or (ii) it will grow until the quadratic damping
Applied example and discussion
Stochastic analysis is performed for a base case and then the influence of the minimum top-tension factor in the probability of dynamic buckling and failure is investigated. The riser and floater properties are the same as in Section 3.1. The procedure described in Fig. 6 is followed.
Concluding remarks
The aim of the present work has been to investigate the stochastic heave-induced dynamic buckling in a top-tensioned riser and its effect on the probability of failure. The governing differential equation of motion for the riser with nonlinear damping was derived and solved by means of the finite difference method together with a Runge-Kutta method for time integration. The evolution of the riser's motion showed that the dynamic buckling response is bounded by the presence of quadratic damping.
Acknowledgements
This study was undertaken at the Korea Ship and Offshore Research Institute at Pusan National University which has been a Lloyd's Register Foundation Research Centre of Excellence since 2008. This work was supported by a 2-Year Research Grant of Pusan National University. The first author would like to acknowledge the scholarship from the Mexican National Council for Science and Technology (CONACYT) (scholarship number 409711).
References (58)
- et al.
On the probabilistic distribution of loads on a marine riser
Ocean. Eng.
(2017) On the parametric excitation of vertical elastic slender structures and the effect of damping in marine applications
Appl. Ocean. Res.
(2004)- et al.
Nonlinear resonances of parametrically excited risers-numerical and analytic investigation for Ω = 2ω1
Comput. Struct.
(2005) - et al.
Bounded and unbounded coupled transverse response of parametrically excited vertical marine risers and tensioned cable legs for marine applications
Appl. Ocean. Res.
(2002) - et al.
Blind Kriging: implementation and performance analysis
Adv. Eng. Softw.
(2012) - et al.
Non-linear reduced-order model for parametric excitation analysis of an immersed vertical slender rod
Int. J. Non. Linear. Mech.
(2016) - et al.
Marine riser emergency disconnection analysis using scalar elements for tensioner modelling
Appl. Ocean. Res.
(2016) The response of a parametrically excited hanging string in fluid
J. Sound. Vib.
(1975)Regression and Kriging metamodels with their experimental designs in simulation: a review
Eur. J. Oper. Res.
(2017)- et al.
Destabilization of deep-water risers by a heaving platform
J. Sound. Vib.
(2008)
Frequency domain response of a parametrically excited riser under random wave forces
J. Sound. Vib.
Computing Lyapunov exponents of continuous dynamical systems: method of Lyapunov vectors
Chaos Solit. Fractals
Diseño óptimo robusto utilizando modelos Kriging: aplicación al diseño óptimo robusto de estructuras articuladas
Rev. Int. Métodos Numéricos para Cálculo y Diseño en Ing
Heave-imposed motion in vertical risers: a reduced-order modelling based on Bessel-like modes
Procedia IUTAM
A finite element method for dynamic analysis of long slender marine structures under combined parametric and forcing excitations
Ocean. Eng.
Combined axial and lateral responses of tensioned buoyant platform tethers
Eng. Struct.
Dynamics of tension leg platform tethers at low tension. Part I - Mathieu stability at large parameters
Mar. Struct.
Optimization — the basis of code-making and reliability verification
Struct. Saf.
A Matlab toolbox for Kriging metamodelling
Procedia Comput. Sci.
Coupled dynamic analysis of deepwater drilling riser under combined forcing and parametric excitation
J. Nat. Gas. Sci. Eng.
Instability analyses of a top-tensioned riser under combined vortex and multi-frequency parametric excitations
Ocean. Eng.
Parametric instability prediction in a top-tensioned riser in irregular waves
Ocean. Eng.
Parametric instability analysis of deepwater top-tensioned risers considering variable tension along the length
J. Ocean. Univ. China
Design, Selection, Operation, and Maintenance of Marine Drilling Riser Systems API Recommended Practice 16Q
Specification for Line Pipe API Specification 5L
Lyapunov Characteristic Exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them
Part 1 Theory. Mecc.
Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: numerical application
Meccanica
Offshore Structures: Volume I: Conceptual Design and Hydromechanics
Surrogate-based infill optimization applied to electromagnetic problems
Int. J. RF Microw. Comput. Eng.
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