Abstract:
A system and method is provided for determining curvature for subsea riser system, including but not limited to drilling risers, steel catenary risers, lazy-wave catenary risers and riser jumpers, comprising the steps of: periodically measuring acceleration in a first lateral direction at said vertical position to obtain a first acceleration timetrace processing said first acceleration timetrace to obtain a first acceleration spectra; applying a transfer function to said first acceleration spectra to obtain a first curvature spectra; and processing said first curvature spectra to obtain a first curvature timetrace. Preferably the transfer function is determined by a method comprising the step of modelling the riser as a Tensioned Timoshenko Beam. 
     The curvature may be used to determine stress and fatigue damage in a structure from motions measured at a single location or a combination of motions measured at a single location with or without tension measurement. The method can be used to determine curvature and hence stress and fatigue damage from any source of excitation, for example the excitation at the tension ring by the top tensioner system, and the vortex induced vibration locked in at any water depth.

Description:
[0001]    This application claims the benefit of U.S. Provisional Patent Application Ser. No. 61/920,145, filed on Dec. 23, 2013 and entitled “Riser Fatigue Monitoring”, which is incorporated herein by reference in its entirety for all purposes. 
     
    
     BACKGROUND 
       [0002]    a. Field of the Invention 
         [0003]    The present invention relates to field monitoring of riser systems used in the oil and gas industry. However, this method can be used for any type of similar structure such as cables, umbilicals, mooring lines and pipeline spans. In particular, this invention relates to the use of motions measured at a single location, in lieu of strain sensors, for curvature and fatigue damage monitoring under excitation from any source. 
         [0004]    b. Description of the Related Art 
         [0005]    Risers are used in the oil and gas industry as the pipe conduit between the ocean seabed and the surface. Risers and similar structures such as cables, umbilicals and mooring lines are exposed to the environment and are susceptible to excitation from various sources which can cause fatigue accumulation in the structures. Fatigue monitoring can be used to determine the riser fatigue accumulation, confirm riser integrity and assist in operational decisions. 
         [0006]    Fatigue monitoring can be achieved through strain and curvature measurement, motion measurements, or a combination of these. The benefit of strain or curvature measurement is that it is a direct measurement. However, depending on the application, subsea strain measurement may not be feasible. In addition, subsea strain measurement may not have the required track record, reliability and cost effectiveness of motion measurement. Hence, it is preferable to have a means of determining curvature from motion measurements. 
         [0007]    A number of methods currently exist to determine fatigue damage from motion measurements. Typically these take the form of reconstruction methods using mode shapes and/or mode superposition. As an example US Patent 2012/0303293 A1 presents a method based on the use of time synchronous measurements and mode superposition. Another approach is to use finite element analysis (FEA) to determine acceleration to stress transfer functions. 
         [0008]    The objective of the present invention is to provide an improved monitoring means allowing continuous measurement of riser fatigue from accelerations without the necessity for strain or curvature measurement devices, time synchronous measurements, theoretical mode shapes and/or finite element analysis (FEA). 
       SUMMARY OF THE INVENTION 
       [0009]    It is an object of the present invention to address the problems cited above, and provide a system and method for monitoring fatigue in risers owing to motion-induced stresses within the riser. 
         [0010]    The invention provides a method for determining curvature at a vertical position in a subsea riser system comprising the steps of: periodically measuring acceleration in a first lateral direction at said vertical position to obtain a first acceleration timetrace processing said first acceleration time trace to obtain a first acceleration spectra; applying a transfer function to said first acceleration spectra to obtain a first curvature spectra; and processing said first curvature spectra to obtain a first curvature timetrace. 
         [0011]    Preferably the transfer function is determined by a method comprising the step of modelling the riser as a Tensioned Timoshenko Beam. 
         [0012]    The invention has motion modules comprising motion sensors at points along the riser length, each sensor includes at least one accelerometer, and preferably two or three accelerometers for measuring acceleration in up to three orthogonal directions, which measure acceleration owing to excitation from vessels, waves and currents. Mean tension and mean curvature are determined at the module location. The riser dynamic acceleration is then measured in three dimensions. A transfer function based on the mean tension and mean curvature may be used together with measured riser accelerations to determine instant riser curvature. 
         [0013]    The transfer function is derived using an analytical method based on tensioned Timoshenko beam theory and wave propagation theory and thus no FEA is required. 
         [0014]    The measured riser accelerations may be translated into stress time traces for fatigue assessment using the derived transfer function. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0015]    The invention will now be further described, by way of example only, and with reference to the accompanying drawings in which: 
           [0016]      FIG. 1  is a schematic diagram of a riser fatigue monitoring system according to a preferred embodiment of the invention, having a plurality of motion modules comprising motion sensors spaced apart along the length of the riser each of which measure one or more accelerations at a single location; 
           [0017]      FIG. 2  is an illustration to explain the theory of converting riser motions to riser curvature along length; and 
           [0018]      FIG. 3  is a flowchart of the steps in a method of determining riser fatigue according to a preferred embodiment of the invention. 
       
    
    
     DETAILED DESCRIPTION 
       [0019]    As shown in  FIG. 1 , the present invention provides a flexible installation scheme for continuous determination of riser fatigue at any location. Motion modules are attached to the outside of a riser structure to monitor the motions of the riser at the attached locations. The motion modules can be placed at any location of concern on the riser, usually at expected hot spots in fatigue damage. The motion modules can be installed on any type of risers or similar structures, including but not limited to, drilling risers, production risers and hybrid risers. The riser configuration can be vertical riser, steel catenary riser, lazy-wave catenary riser and so forth. Similar structures include cables, umbilicals, mooring lines and pipeline spans. 
         [0020]      FIG. 1  shows a preferred embodiment of a riser fatigue monitoring system applied to a drilling riser  2 . The drilling riser  2  is typically used in deep water drilling operations. In this example, the drilling riser  2  comprises a riser string  4  including a plurality of riser pipe sections or joints. The riser string  4  will typically comprise both slick joints and buoyant joints. In the preferred embodiment illustrated in  FIG. 1  a central section  6  of the riser string  4  comprises buoyant joints and upper and lower sections  8  of the riser string  4 , above and below the central section  6 , comprise alternating slick and buoyant joints. 
         [0021]    The riser string  4  is connected at its lower end to a wellhead  10  and a conductor  12  located proximate the seabed  14 . Above the wellhead  10  a blow out preventer (BOP)  16  and a lower marine riser package (LMRP)  18  connect to a lower flex-joint  20  of the riser string  4 . The lower flex-joint  20  is used to absorb relative rotations between the LMRP  18  and the riser string  4 . The riser string  4  extends upwards from the lower flex-joint  20  to an upper flex-joint  22  at the upper end of the riser string  4  above the water&#39;s surface  24 . To maintain the stability of the riser string, a riser tensioner  26  is used to provide a tensile force to the riser string  4 . The tensioner system  26  comprises a tension ring  28  that is attached to the riser string  4  and tensioners or tensioning cables  30  which are connected between the tension ring  28  and a drill floor  32 . 
         [0022]    In this example, motion modules  34 , forming part of the fatigue monitoring system, are located on the outside of the riser string  4  in the upper and lower sections  8 . The motion modules  34  comprises at least one motion sensor and may also comprise a motion logger. The motion sensor preferably comprises one or more accelerometers, and is preferably arranged to sense motion in three orthogonal directions. The motion logger is preferably arranged to store the detected motions in a memory. It will be appreciated that, in other embodiments, the motion modules may be provided at any suitable locations along the length of the riser string  4 . Furthermore, any number of motion modules may be provided at any desired spatial intervals along the riser string  4 . 
         [0023]    In the present invention, the motion modules can be either standalone, battery powered or hard wired. Communication with a remote processor can be either through cables, wireless including but not limited to acoustic transmission, or data may be retrieved using Remotely Operated underwater Vehicles (ROVs). 
         [0024]    The remote processor is arranged to receive data from one or more motion modules and to process a received acceleration time trace to obtain as acceleration spectra. A transfer function is applied to the acceleration spectra to obtain a first curvature spectra; and to process said first curvature spectra to obtain a curvature timetrace as described below. 
         [0025]    In alternative embodiments data processing may be performed by a processor within the measurement module before sending to a remote system for further analysis. 
         [0026]    An advantage of the invention is that the motion modules are independent from each other with no requirement for common wire to connect all modules for signal synchronization. This provides improved efficiency in riser joint installation, maintenance and/or retrieval in terms of saved time and reduced interference. 
         [0027]    Referring now to  FIG. 2 , the input motion can be the vessel induced motion, or wave, current and/or vortex induced motion at any locations along the riser length. The boundary condition as an example herein assumes the riser is attached to a floater at the top hang-off location and latched to the seabed or subsea structure at the bottom. For different tension providers and floating units, the boundary condition at point B should be assessed accordingly, and this invention still applies. 
         [0028]      FIG. 2  is a schematic illustration of the curvature of a riser string  4  due to forces acting on the string. In this example, with no curvature of the riser string  4 , the string extends between a lower end at A and an upper end at B along an axial direction x. Also illustrated in  FIG. 2  are lateral directions y and z which are perpendicular to the axial direction x and to each other. The axial direction x refers to pipe axial direction at that location, and the lateral directions y and z are perpendicular to the local pipe axial direction. 
         [0029]    In this example three motion modules  34  are shown. A first, lowermost module  34   a  is located proximate the lower end A of the riser string  4 . A second, uppermost module  34   c  is located proximate the upper end of the riser string, but below the surface  24  of the water. A third module  34   b  is located approximately midway between the first and second module  34   a ,  34   c , in the central section  6  of the string  4 . 
         [0030]    Any movement of the top end (point B) of the riser string  4 , together with any lateral movement or deflections of the riser string  4  along its length, will cause a change in profile of the riser string  4  and, in particular, will induce curvature along the length of the string  4 . 
         [0031]    The motion measured by a motion module is attenuated in amplitude under various damping and inertia effects and frequency dispersion occurs. 
         [0032]    The motion sensors are able to measure accelerations in X, Y and Z directions. As an example in the near vertical application in  FIG. 2 , the lateral acceleration 
         [0000]    
       
         
           
             
               
                 
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         [0000]    is measured by the motion sensors. The axial motion 
         [0000]    
       
         
           
             
               
                 
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                   2 
                 
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                     ( 
                     
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                   t 
                   2 
                 
               
             
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         [0000]    is also measured but is not required in the present invention. 
         [0033]    This invention also applies to steel catenary riser (SCR) or lazy-wave SCR, which are not illustrated in  FIG. 2 . The motions in the lateral directions (Y and Z) are perpendicular to the pipe axis (X). 
         [0034]    In this specification the symbol y(L,t) refers to displacement of the riser in the y direction at a location with an arc length (or curvilinear length) of L along the riser and at time instant t. Similarly, z(L,t) denotes the riser displacement in the z direction at curvilinear length L and time t. For the near vertical application in  FIG. 2 , the difference between the curvilinear length L and coordinate x is negligible, and y(x,t) is used instead of y(L,t) merely for notation convenience. Likewise, z(x,t) is adopted in lieu of z(L,t). Furthermore, the y(x,t) or y and z(x,t) or z are interchangeable in all following equations. 
         [0035]    In this invention, the motion modules can be used for both as-built risers and new risers. To monitor the fatigue life of as-built risers, the motion modules can be clipped on the outer surface of the riser with help of remotely operated underwater vehicle (ROV) for deep water risers or divers for shallow water risers. 
         [0036]    The invention distinguishes itself from those methods that utilize Finite Element Analysis (FEA) models. It is not necessary to create a riser model in the invention, nor assume any damping coefficients, as opposed to those models where prior riser configurations should be modelled for FEA software, and waves, current profiles, damping coefficients and energy attenuation along the riser have to be predetermined subjectively based on Metocean data and other sources. The method of the present invention only assesses the measured data and true riser responses with unknown damping and attenuation effects, as well as travelling history of the stress signals under real-world environmental conditions. 
         [0037]    This invention employs an analytical method without any FEA model that has to be meshed for elements. The analytical method considers the riser string as a beam and anything attached to the beam, such as buoyant modules, is modelled as added mass or external loads. An unknown variable in the method is the mass due to “added mass”. Added mass represents the mass of the surrounding water that must be moved under lateral motion. The added mass coefficients to use are determined from field tests or lab experiments. Alternatively, it can be calibrated in the field if direct strain measurements are also available. 
         [0038]    For an ordinary Euler-Bernoulli beam, the bending stiffness El (where E is the Young&#39;s modulus and l is the second moment of area of the beam&#39;s cross section) and mass are key parameters for beam responses, while the beam in this invention refers to a modified beam with additional considerations for axial tension, shear stiffness, rotational inertia and cross-section shape, which is termed as tensioned Timoshenko beam (TTB). The TTB method includes the features of an ordinary beam and a vibrating cable (or string) and their coupled effect. When the shear modulus, G, becomes infinite and the tension and rotational inertia are neglected, the TTB is degraded to an Euler-Bernoulli beam. Likewise, if the bending stiffness El is negligible, the TTB is degenerated to a cable or string. With all parameters of TTB in play and non-trivial, TTB method simulates the coupled effects of bending, shear, rotational and tensile responses. 
         [0039]    In addition to Euler-Bernoulli beam, the TTB method applies to short beams and composite beams, especially for beams subject to high-frequency excitations. It differs from the Euler-Bernoulli beam by taking into account shear deformation, rotational inertia effects and the tension of a riser. 
         [0040]    The TTB method is a theory for converting accelerations to curvature. For prior art methods relying on a detailed riser configuration and FEA software, a riser has to be modelled again even if only the water depth is changed. For the TTB method, only the riser properties (tension, mass and sectional properties) at the measurement locations are required as defined below. The riser can be any length and the TTB method still applies. 
         [0041]    The TTB method is expressed in the following equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
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                   Eq 
                   . 
                   
                       
                   
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                   1 
                 
               
             
           
         
       
       
         where: 
         t is time, measured in sec 
         y is lateral displacement, measured in m 
         m is mass per unit length of riser, including added mass and internal fluid, if any measured in kg/m 
         E is Young&#39;s modulus of riser material measured in Pa 
         G is shear modulus of the riser cross section measured in Pa 
         A is cross-section area of the riser measured in m 2    
         l is second moment of the riser cross section area measured in m 4    
         k is shear reduction factor of riser cross section, a function of Poisson&#39;s ratio 
       
     
         [0000]    
       
         
           
             J 
             = 
             
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         [0000]    is rotational inertia
   q(x,t) is lateral force per unit riser length   S is axial tension   damping force f(η) in the form of viscous damping can be expressed as   
 
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         [0000]    where η(x) is the proportional constant determined from riser configuration. The TTB equation Eq. 1 can be solved in the frequency domain or by using differential methods in the time domain. 
         [0054]    When an explicit analytical form is unavailable as in circumstance of complicated riser profiles and boundary conditions, a differential method is preferred. 
         [0055]    In real-world environmental conditions, analytical expressions for local current profiles q and for damping force f are usually unavailable. Validation with measured data shows that the hydrodynamic effect of current profiles and damping forces are equivalent to that of extra added mass. Therefore, in the following analytical expressions and solutions, the damping force f is excluded. 
         [0056]    When the axial tension S=0, the TTB simplifies to a conventional Timoshenko beam ie 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
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         [0057]    If axial tension S=0, rotational inertia J=0 and the shear modulus of the riser pipe material G becomes very large tending towards infinity, the TTB is further degenerated to an Euler-Bernoulli beam as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
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                   4 
                 
               
             
           
         
       
     
         [0058]    When axial tension S≠0, bending stiffness El=0 and rotational inertia J=0 and shear modulus of the riser pipe material G becomes very large tending towards infinity, the TTB describes the behaviour of a vibrating string or cable: 
         [0000]    
       
         
           
             
               
                 
                   
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         [0059]    For near vertical risers such as drilling risers, the tension S can be expressed as the weight of the riser plus the base tension S 0 , or 
         [0000]        S ( x,t )= S   0 ( x,t )+∫ 0   x   mgdx   Eq.6
 
         [0060]    For risers with large static or dynamic curvature, such as catenary risers or jumpers, the tension S can be modified as 
         [0000]    
       
         
           
             
               
                 
                   
                     S 
                      
                     
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                         , 
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                   . 
                   
                       
                   
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                   7 
                 
               
             
           
         
       
     
         [0061]    Where N(x,t) is the axial tension and x is the riser length in concern. The equation above leads to a nonlinear Timoshenko beam equation due to the term 
         [0000]    
       
         
           
             
               
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                     y 
                   
                   
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                 ) 
               
               2 
             
             . 
           
         
       
     
         [0000]    It is found that the use of time-averaged curvature can simplify the nonlinear equation to a linear equation with adequate accuracy. 
         [0062]    Without loss of generality and to avoid enumerating all the solutions for all the boundary conditions and initial conditions, a solution in frequency domain for risers with negligible variations in geometric properties (ie Young&#39;s modulus E, the second moment of the riser cross section area l, cross-section area of the riser pipes A and the shear modulus of the riser pipe material G remain substantially constant) is demonstrated as an example for drilling risers, then the Eq. 1 can be simplified as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       EI 
                        
                       
                         
                           
                             ∂ 
                             4 
                           
                            
                           y 
                         
                         
                           ∂ 
                           
                             x 
                             4 
                           
                         
                       
                     
                     + 
                     
                       m 
                        
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           y 
                         
                         
                           ∂ 
                           
                             t 
                             2 
                           
                         
                       
                     
                     - 
                     
                       
                         ( 
                         
                           J 
                           + 
                           
                             EIm 
                             kAG 
                           
                         
                         ) 
                       
                        
                       
                         
                           
                             ∂ 
                             4 
                           
                            
                           y 
                         
                         
                           
                             ∂ 
                             
                               x 
                               2 
                             
                           
                            
                           
                             ∂ 
                             
                               t 
                               2 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         Jm 
                         kAG 
                       
                        
                       
                         
                           
                             ∂ 
                             4 
                           
                            
                           y 
                         
                         
                           ∂ 
                           
                             t 
                             4 
                           
                         
                       
                     
                     - 
                     
                       S 
                        
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           y 
                         
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       q 
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                     + 
                     
                       
                         J 
                         kAG 
                       
                        
                       
                         
                           
                             ∂ 
                             4 
                           
                            
                           q 
                         
                         
                           ∂ 
                           
                             t 
                             4 
                           
                         
                       
                     
                     - 
                     
                       
                         EI 
                         kAG 
                       
                        
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           q 
                         
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   8 
                 
               
             
           
         
       
     
         [0063]    As an alternative to numerical solutions, Eq. 1 can be solved by the superposition of a general solution with lateral force per riser unit length q=0 and specific solutions for non-trivial q. The general solution is the one for the equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       EI 
                        
                       
                         
                           
                             ∂ 
                             4 
                           
                            
                           y 
                         
                         
                           ∂ 
                           
                             x 
                             4 
                           
                         
                       
                     
                     + 
                     
                       m 
                        
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           y 
                         
                         
                           ∂ 
                           
                             t 
                             2 
                           
                         
                       
                     
                     - 
                     
                       
                         ( 
                         
                           J 
                           + 
                           
                             EIm 
                             kAG 
                           
                         
                         ) 
                       
                        
                       
                         
                           
                             ∂ 
                             4 
                           
                            
                           y 
                         
                         
                           
                             ∂ 
                             
                               x 
                               2 
                             
                           
                            
                           
                             ∂ 
                             
                               t 
                               2 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         Jm 
                         kAG 
                       
                        
                       
                         
                           
                             ∂ 
                             4 
                           
                            
                           y 
                         
                         
                           ∂ 
                           
                             t 
                             4 
                           
                         
                       
                     
                     - 
                     
                       S 
                        
                       
                         
                           
                             ∂ 
                             2 
                           
                            
                           y 
                         
                         
                           ∂ 
                           
                             x 
                             2 
                           
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   9 
                 
               
             
           
         
       
     
         [0064]    It can be solved by separation of variables: 
         [0000]        y=Y ( x ) T ( t ) where  Y ( x )= e   iλx   ,T ( t )= e   −iωt   Eq. 10
 
         [0065]    The corresponding Eigen equation is 
         [0000]      λ 4 +βλ 2 +γ=0  Eq. 11
 
         [0000]    where 
         [0000]    
       
         
           
             β 
             = 
             
               
                 
                   S 
                   
                     2 
                      
                     EI 
                      
                     
                         
                     
                      
                     
                       ω 
                       2 
                     
                   
                 
                 - 
                 
                   
                     m 
                     
                       2 
                        
                       EA 
                     
                   
                    
                   
                     ( 
                     
                       1 
                       + 
                       
                         E 
                         kG 
                       
                     
                     ) 
                   
                    
                   
                       
                   
                    
                   and 
                    
                   
                       
                   
                    
                   γ 
                 
               
               = 
               
                 
                   m 
                   
                     4 
                      
                     EI 
                   
                 
                  
                 
                   ( 
                   
                     
                       mI 
                       
                         
                           kA 
                           2 
                         
                          
                         G 
                       
                     
                     - 
                     
                       1 
                       
                         ω 
                         2 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0066]    The four roots of A are determined by substituting 
         [0000]    
       
         
           
             γ 
             = 
             
               
                 m 
                 
                   4 
                    
                   EI 
                 
               
                
               
                 ( 
                 
                   
                     mI 
                     
                       
                         kA 
                         2 
                       
                        
                       G 
                     
                   
                   - 
                   
                     1 
                     
                       ω 
                       2 
                     
                   
                 
                 ) 
               
             
           
         
       
     
         [0000]    and β into Eq. 11. 
         [0000]    
       
         
           
             
               
                 
                   
                     λ 
                     2 
                   
                   = 
                   
                     
                       ω 
                       2 
                     
                     [ 
                     
                       
                         - 
                         β 
                       
                       + 
                       
                         
                           
                             β 
                             2 
                           
                           - 
                           
                             
                               m 
                               EI 
                             
                              
                             
                               ( 
                               
                                 
                                   mI 
                                   
                                     
                                       kA 
                                       2 
                                     
                                      
                                     G 
                                   
                                 
                                 - 
                                 
                                   1 
                                   
                                     ω 
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   12 
                 
               
             
           
         
       
     
         [0067]    For angular frequency ω in rad/s and remembering that 
         [0000]    
       
         
           
             
               J 
               = 
               
                 mI 
                 A 
               
             
             , 
           
         
       
     
         [0068]    If 
         [0000]    
       
         
           
             
               ω 
               ≤ 
               
                 
                   GkA 
                   J 
                 
               
             
             , 
           
         
       
     
         [0000]    the solution is 
         [0000]        y ( x,t )=( P   1  cos  hλ   1   x+P   2  sin  hλ   2   x+P   3  cos λ 3   x+P   4  sin λ 4   x ) e   −iωt   Eq. 13
 
         [0069]    For 
         [0000]    
       
         
           
             
               ω 
               &gt; 
               
                 
                   GkA 
                   J 
                 
               
             
             , 
           
         
       
     
         [0000]    the solution becomes 
         [0000]        y ( x,t )=( P   1  cos λ 1   x+P   2  sin λ 2   x+P   3  cos λ 3   x+P   4  sin λ 4   x ) e   −iωt   Eq. 14
 
         [0070]    The 4 constants of P (P 1  to P 4 ) are determined by initial and boundary conditions. For instance, for drilling risers with an upper and lower flexjoint, the corresponding general solution between these two points in the riser system is 
         [0000]        y=p  sin λ xe   −iωt   Eq.15
 
         [0000]      Denoting 
         [0000]        Y (ω)= p  sin λ x   Eq. 16
 
         [0000]    the sum of the response from all frequencies is 
         [0000]    
       
         
           
             
               
                 
                   
                     y 
                      
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       ω 
                     
                      
                     
                         
                     
                      
                     
                       
                         Y 
                          
                         
                           ( 
                           ω 
                           ) 
                         
                       
                        
                       
                          
                         
                           
                             - 
                             ω 
                           
                            
                           
                               
                           
                            
                           t 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   17 
                 
               
             
           
         
       
     
         [0000]    or 
         [0000]        y ( x,t )=IFFT[ Y (ω)]  Eq. 18
 
         [0000]      and hence 
         [0000]        Y (ω))=FFT( y ( x,t ))  Eq. 19
 
         [0071]    The acceleration is 
         [0000]    
       
         
           
             
               
                 
                   
                     a 
                      
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           ∂ 
                           2 
                         
                          
                         
                           y 
                            
                           
                             ( 
                             
                               x 
                               , 
                               t 
                             
                             ) 
                           
                         
                       
                       
                         ∂ 
                         
                           t 
                           2 
                         
                       
                     
                     = 
                     
                       
                         - 
                         
                           ω 
                           2 
                         
                       
                        
                       
                         y 
                          
                         
                           ( 
                           
                             x 
                             , 
                             t 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   20 
                 
               
             
           
         
       
     
         [0000]    or in frequency domain 
         [0000]    
       
         
           
             
               
                 
                   
                     a 
                      
                     
                       ( 
                       ω 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           ∂ 
                           2 
                         
                          
                         
                           y 
                            
                           
                             ( 
                             
                               x 
                               , 
                               t 
                             
                             ) 
                           
                         
                       
                       
                         ∂ 
                         
                           t 
                           2 
                         
                       
                     
                     = 
                     
                       
                         - 
                         
                           ω 
                           2 
                         
                       
                        
                       
                         
                           Y 
                            
                           
                             ( 
                             ω 
                             ) 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   21 
                 
               
             
           
         
       
     
         [0072]    The curvature χ in the time domain is obtained from Eq. 16: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       χ 
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                     ≈ 
                     
                       
                         
                           ∂ 
                           2 
                         
                          
                         
                           y 
                            
                           
                             ( 
                             
                               x 
                               , 
                               t 
                             
                             ) 
                           
                         
                       
                       
                         ∂ 
                         
                           x 
                           2 
                         
                       
                     
                   
                   = 
                   
                     
                       - 
                       
                         λ 
                         2 
                       
                     
                      
                     
                       y 
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   22 
                 
               
             
           
         
       
     
         [0000]    or in frequency domain 
         [0000]      χ(ω)=λ 2   Y (ω)  Eq. 23
 
         [0073]    The relation between acceleration and curvature becomes 
         [0000]    
       
         
           
             
               
                 
                   
                     χ 
                      
                     
                       ( 
                       ω 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         λ 
                         2 
                       
                       
                         ω 
                         2 
                       
                     
                      
                     
                       a 
                        
                       
                         ( 
                         ω 
                         ) 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   24 
                 
               
             
           
         
       
     
         [0074]    The transfer function from acceleration to curvature in frequency domain is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       c 
                        
                       
                         ( 
                         ω 
                         ) 
                       
                     
                     ≡ 
                     
                       
                         λ 
                         2 
                       
                       
                         ω 
                         2 
                       
                     
                   
                   = 
                   
                     
                       - 
                       β 
                     
                     + 
                     
                       
                         
                           β 
                           2 
                         
                         - 
                         
                           
                             m 
                             
                               E 
                                
                               
                                   
                               
                                
                               I 
                             
                           
                            
                           
                             ( 
                             
                               
                                 mI 
                                 
                                   k 
                                    
                                   
                                       
                                   
                                    
                                   
                                     A 
                                     2 
                                   
                                    
                                   G 
                                 
                               
                               - 
                               
                                 1 
                                 
                                   ω 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   25 
                 
               
             
           
         
       
     
         [0075]    Compared with Eq. 9, the plus sign is adopted for the square root in the transfer function of the aforementioned drilling riser example whose boundary conditions give the general solution of Eq.12. For other riser types or boundary conditions, the transfer function can be derived accordingly. The general solution remains in the form of Eq.10 and Eq.11, The 4 roots of A are determined by Eq.9. Different boundary conditions entail different  4  constants of P (P 1  to P 4 ) for corresponding roots of A, 
         [0076]    With the use of Eq. 21, the curvature in the frequency domain in Eq. 20 becomes: 
         [0000]      χ(ω)= c (ω) a (ω)  Eq. 26
 
         [0000]    The curvature time trace is calculated using an Inverse Fast Fourier Transform (IFFT) as follows: 
         [0000]      χ( x,t )=IFFT( c (ω) a (ω))  Eq. 27
 
         [0077]    Substituting Eq. 26 into Eq. 27, one obtains: 
         [0000]      χ( x,t )=IFFT( c (ω) a (ω))  Eq. 28
 
         [0000]    which is 
         [0000]    
       
         
           
             
               
                 
                   
                     χ 
                      
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                   
                   = 
                   
                     IFFT 
                      
                     
                       { 
                       
                         
                           
                             [ 
                             
                               
                                 - 
                                 β 
                               
                               + 
                               
                                 
                                   
                                     β 
                                     2 
                                   
                                   - 
                                   
                                     
                                       m 
                                       EI 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           mI 
                                           
                                             k 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               A 
                                               2 
                                             
                                              
                                             G 
                                           
                                         
                                         - 
                                         
                                           1 
                                           
                                             ω 
                                             2 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           · 
                         
                         * 
                         
                           FFT 
                            
                           
                             [ 
                             
                               a 
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             ] 
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   29 
                 
               
             
           
         
       
     
         [0078]    The “·*” symbol in the equation denotes multiplication of the two series in the frequency domain item by item for each frequency. 
         [0079]    The specific solutions for non-trivial q depend on applications. Most FEA software assumes the current profile remains constant with time t, and the specific solutions can be obtained analytically. However, field measurements indicate that, current speed varies with time, depth and orientation. In practice, current profiles in real time are not available, thus analytical specific solutions are groundless. Fortunately, validation with field measurements from strain sensor data shows that the uncertainty in fluid dynamics due to varying current and other lateral loading and damping is in the same magnitude of the uncertainty in added mass, which is determined by an added mass coefficient and an inertia outer diameter. Therefore, the effect of unknown current q can be accounted for in terms of added mass in the general solution, and the specific solutions become dispensable. 
         [0080]    With the curvature calculated from the accelerations, the stress time trace is 
         [0000]    
       
         
           
             
               
                 
                   
                     σ 
                      
                     
                       ( 
                       
                         x 
                         , 
                         t 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       χ 
                        
                       
                         ( 
                         
                           x 
                           , 
                           t 
                         
                         ) 
                       
                     
                      
                     E 
                      
                     
                       D 
                       2 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   30 
                 
               
             
           
         
       
     
         [0000]    where D is the outer diameter of the riser at the motion module location, and E is the Young&#39;s modulus of the steel pipe, 
         [0081]    Fatigue life can be calculated in many ways. For example, the Rain Flow stress cycle counting approach may be used to obtain number of stress cycles n and stress ranges Δσ, then use an S-N curve to obtain damage. A typical S-N curve gives 
         [0000]        N=a (SCF*Δσ) −k   Eq. 31
 
         [0000]    where N is the number of cycles to failure, a and k are S-N curve intercept and slope parameters, SCF is stress concentration factor. There might be other factors, for example, thickness correction factor for thick-walled pipe. The fatigue damage for each stress range is n/N. The total damage can be calculated using Miner&#39;s rule. Fatigue life is reciprocal of the total damage factored by a year. 
         [0082]      FIG. 3  is a flow chart that illustrates the primary steps in a method of monitoring riser fatigue according to a preferred embodiment of the invention. 
         [0083]    At a first step  50  acceleration data, preferably in the form of acceleration time traces, is acquired by the motion modules attached to the riser string. Depending on the sampling frequency and monitoring equipment hardware, the measured acceleration time traces may include data at frequencies that are not of interest for use in the method of the present invention. Accordingly, at a second step  52 , the measurements are filtered to remove any unwanted frequency ranges. The measurements may be filtered using a suitable band pass filter. 
         [0084]    The filter acceleration data is then converted into the frequency domain using a Fast Fourier Transform (FFT) to obtain an acceleration spectrum, at a third step  54 . To obtain a curvature spectrum, the acceleration spectrum is then multiplied by a transfer function as shown in Eq. 25 
         [0085]    It will be appreciated by those skilled in the art that filtering step  52  may be performed either before or after conversion step  54 . 
         [0086]    At a fourth step  56  the curvature spectrum is converted into a curvature time trace using an inverse FFT (IFFT) as shown in Eq. 29. 
         [0087]    At a fifth step  58  the curvature time trace is used to calculate a stress time trace, Eq. 30. 
         [0088]    Finally, at a sixth step  60 , the stress time trace is post-processed to determine fatigue damage and life. 
         [0089]    A common approach to determine fatigue damage is rainflow stress cycle counting followed by fatigue calculation using the stress/number of cycles approach as described above.