EXTRACTION OF MODAL PARAMETERS OF RISERS IN DEEP WATER OFFSHORE PLATFORMS UTILIZING THE FOURIER SPECTRAL ANALYSIS AND THE FREQUENCY DOMAIN DECOMPOSITION METHODS Rodríguez-Rocha. Ramsés1*, Rivero-Angeles. Francisco J2 and Vázquez-Hernández. Alberto O2 1

Instituto Politécnico Nacional Escuela Superior de Ingeniería y Arquitectura, SEPI-Estructuras Av. Juan de Dios Batiz s/n, edificio 10, CP. 07738, Mexico City, Mexico E-mail: [email protected] 2

Instituto Mexicano del Petróleo Programa de Explotación de Campos en Aguas Profundas Av. Eje Central Lázaro Cárdenas 152, Edificio 32, Col. San Bartolo Atepehuacan, CP. 07730, Mexico City, Mexico E-mail: [email protected], [email protected] Keywords: Risers, modal parameters, acceleration records. ABSTRACT Steel catenary risers (SCR) are widely used for transportation of hydrocarbons in deep and ultra-deep water fields. The design of these structures is demanding and, thus, full scale testing, instrumentation and monitoring are essential to improve our understanding of their dynamic behaviour. The present paper shows a procedure for the identification of modal parameters, such as periods and mode shapes of a SCR, based solely on output data. A mathematical model of a riser was developed considering different sea states and soil stiffness. Accelerations records were computed at a few discrete locations along the riser; Fourier Spectral Analysis and Frequency Domain Decomposition methods were used. The identified modal parameters are based on a realistic movement of the riser, instead of the classical assumption of a riser connected at the seabed and the floating unit, with only movement in between both ends. Nonetheless, the identified and the theoretical parameters show good agreement. The proposed procedures prove to be an efficient tool for structural health monitoring purposes, especially for future damage detection, fatigue analysis and risk-based management.

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1.

INTRODUCTION

With the increasing demand for oil and gas products in the world and the new offshore discoveries, the offshore industry has moved into deep and ultra deep waters. This has resulted in many challenges in analysis and design of offshore platforms and risers [1]. Risers are vertical tubular elements used for the transportation of oil produced by wells on the seabed to the floating production unit, or vice versa, transporting oil or gas already separated in the production unit to marine pipelines. At first sight, the riser appears to be a simple structure, but in reality, its response is very complicated. The global response is time dependant and geometrically nonlinear. Riser designers take into account the depth in which it will be installed, the severity of environmental loads which it is exposed, as well as its possible dynamic behavior. Risers may have failures during their lifetime of operation, due to the kind of dynamic behavior that may be subject, mainly when subject to severe storm condition like the hurricane [2]. Therefore, with the increasing discoveries of new oil fields in deep waters, it is needed to study in further detail their structural dynamic behavior. The risers’ analysis must consider the present non-linearities, mainly due to the intermittent contact with the seabed at the touch down zone, vortex-induced vibrations (VIV), large displacements of the floating systems and possible structural hysteretic behavior [3]. Due to the complexity, not all possible effects are considered simultaneously, so separate analysis are carried out, and are focused on each type of study. One way of understanding the dynamic behavior of structures is through their modal parameters [4]. Even in the case of existing structures, this information is used to identify critical resonant periods and be employed by other methodologies for identification, prevention of possible critical areas, or identification of damages [5, 6, 7, 8, 9]. In the case of the Steel Catenary Risers (SCR), existing software, developed by the industry, considers some simplifications, for example only the initial tension and configuration of the SCR, without regard of the changes in time is considered; mainly the point of contact at the seabed, due to the movements of the floating system caused by environmental acting loads, as well as other types of coupled attached effects as VIV. Since the environmental loads act in different directions and magnitudes, the floating system response will vary during its lifetime. In the same way, the dynamic behavior of the SCR is different under changing environmental conditions, so the main natural periods of the SCR are also variable. However, in spite of the large variation of natural periods that an SCR can have, numerical techniques are needed to obtain the main modal parameters from the behavior that the SCR exhibits [10, 11, 12]. In the case of an already installed SCR, the response can be obtained by means of instrumentation and acquisition systems [13]. In this case the signal of the riser response will be the real effects of its dynamic behavior under other conditions such as VIV, movement of the floating system and variation of contact of the riser and the seabed, among others, by which this methodology will have the ability to get the realistic vibration modes. Also, for the placement of sensors, it can be based on obtaining major modes of vibration identified by a series of simulations, which consider the variability of environmental loading conditions or even VIV response [14]. The present research focuses on the estimation of modal parameters, to be precise, periods and mode shapes which represent the vibration characteristics of a structure. This is achieved by post-processing the dynamic response of the riser through appropriate methods. This process is discussed in further sections. Thus, a methodology is proposed that considers the realistic behavior of the SCR to obtain the modal parameters. In this work we are considering some numerical simulation of the SCR installed in 1800m water depth.

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2.

FOURIER SPECTRAL ANALYSIS

This method is used to identify the modal parameters of a structure using its dynamic response. According to Bendat and Piersol [10], the autospectral density function of a signal y for positive frequencies is defined as: gy(f)=2 sy(f)

(1)

where sy(f) is the autospectral density function of a signal y defined as: sy(f)=y*(f) y(f)

(2)

where y(f) is the Fourier transform of a signal y in terms of the frequency f . The asterisk indicates the operation of conjugated complex. On the other hand, when a structure is subject to an excitation, eq. (1) will present some maximum value, either due to the excitation frequency or to the frequencies of the system. In order to determine the mode shapes corresponding to each identified vibration frequency of the structure it is needed to calculate the transfer functions from eq. (1) and the signal from the base as: tfyo(f)= [gy(f)/ go(f)]1/2

(3)

where tfyo(f) is the transfer function of the signal y at a floor and the signal o at the base. The values of this function are adimensional. Theoretically, each mode shape of the structure can be determined if dynamic measurements at every location along the riser are available. Otherwise, mode shapes would be incomplete and interpolation is an option to complete them. 3.

FREQUENCY DOMAIN DECOMPOSITION METHOD

According to Brincker et al. [15] the power spectral density matrix of the response must be obtained. One way to determine this matrix is: Gy(f)=y*(f)T y(f)

(4)

This matrix operates at discrete frequencies f = fp where p is a discrete series for each frequency in the domain. Eq. (5) expresses eq. (4) as a Singular Value Decomposition, SVD: Gy(f)=Up Sp UpT

(5)

Where Up is a matrix containing singular vectors up. Sp is a diagonal matrix containing singular values sp. These singular values can be plotted in the frequency domain and peak values may be observed which correspond to the natural frequencies of the system. A mode shape associated to each extracted frequency can be determined as well through SVD. 4.

NUMERICAL EXAMPLE

The finite element mathematical model was developed with commercial software [16] and analyzed nonlinearly in time domain. It is a 0.2286m inside diameter SCR in 1800m depth with 2720 elements of 1m long each. A flexible joint at the top side was considered, and three

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different soil characteristics were modeled to account for the interaction by an elastic seabed, which defines the resistance of the soil to lateral and vertical movements over a range of depths below the mud line [17]. The stiffness of the soil (Table 1) is considered at each node of the finite element model along the supported length on the sea floor. Currents and offsets were discarded since only soil nonlinearity under wave loading was to be analyzed. RAOs on a 135° heading and five different sea states were used that depend on significant wave hight (Hs) and peak period (Tp) (Table 2). Property (SI Units) Seabed stiffness Longitudinal coefficient of friction Transverse coefficient of friction Lateral seabed stiffness Suction stiffness Suction zone extent Maximum longitudinal characteristic length Maximum transverse characteristic length

Soil type 1 143,400.0 0.2 0.4 0.0 0.0 0.0 2.0 2.0

Soil type 2 71,700.0 0.0 0.0 7,170.0 717.0 1.0 2.0 2.0

Soil type 3 35,850.0 0.0 0.0 3,585.0 358.5 1.0 2.0 2.0

Table 1. Soil properties selected for the mathematical models. Sea state No. Sea state 1 Sea state 2 Sea state 3 Sea state 4 Sea state 5

Hs (m) 11.2 12.1 7.5 8.0 4.4

Tp (s) 14.0 14.0 10.0 10.0 9.1

Year return period 100 250 100 250 25

Table 2. Pierson-Moskowitz sea spectra data. A time domain solution was performed for 3600 seconds with a time step of 0.2 seconds. 15 nodes were selected along the riser, shown in Figure 1, to calculate acceleration and displacement data for in and out of plane responses of each of the 15 analyses performed (3 types of soils and 5 sea spectra). Theoretical eigenvalue analysis was performed assuming pinned boundary conditions at both ends of the riser. These assumptions are an approximation of the real behaviour since the touchdown point is not unique, due to the fact that at the seabed there is intermittent contact, and at the top, the floating unit is in constant motion. This section compares the eigenvalue results versus the identified ones using the Fourier Spectral Analysis (FSA) of the calculated acceleration records. Table 3 contains 30 eigenvalues and the associated direction (in or out of plane, as given directly by the software).

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Figure 1. Nodes along the riser where motion was calculated. Eigenpair 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Eigenvalue 0.0084 0.0177 0.0339 0.056 0.0761 0.1144 0.1351 0.1886 0.2112 0.282 0.3046 0.3911 0.4154 0.5197 0.5441 0.6643 0.6908 0.829 0.8558 1.0104 1.0396 1.2126 1.2425 1.4322 1.4648 1.6734 1.707 1.9328 1.9695 2.2148

Period (s) 68.4064 47.2023 34.1065 26.5478 22.7771 18.5781 17.0935 14.4669 13.672 11.8317 11.3852 10.0465 9.7482 8.7161 8.5182 7.7088 7.5598 6.9008 6.7918 6.2507 6.1623 5.7058 5.6369 5.2503 5.1915 4.8571 4.8091 4.5195 4.4772 4.222

Direction Out of plane In plane Out of plane In plane Out of plane In plane Out of plane In plane Out of plane In plane Out of plane In plane Out of plane In plane Out of plane In plane Out of plane In plane Out of plane In plane Out of plane In plane Out of plane In plane Out of plane Unknown Out of plane In plane Out of plane Unknown

Table 3. Theoretical eigenpairs and associated directions of the riser.

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For identification purposes, it is important to mention that the most demanding sea state (sea state 1) has a peak period Tp of 14 seconds, thus, the excited periods of the riser will be close to this period, and so, the periods far from this one will not be observable in the Fourier spectra. Figures 2 and 3 plot the Fourier spectra of the signals in sea state 1, soil type 1, in and out of plane of the SCR, respectively. From these plots, amplification peaks of the spectra could be observed and selected quite easily. These plots are shown in amplitude vs. period in order to compare directly to Table 3. It was interesting to see that the computed spectra were very similar for all three types of soil. Table 4 contains the identified periods from the spectra and their associated theoretical eigenpair in Table 3. It is evident that the energy concentration is found between eigepairs 6 and 14 with a range of periods that go from 9 to 18 seconds. This is, of course, related to the energy of the sea state, around a period of 14 seconds.

It is common to use natural periods in the modal analysis of offshore structures in order to relate them to sea states that excite these structures. One of the most commonly used parameters of sea states in the analysis of offshore platforms is the spectral period of the wave (seconds) being the wave loading the main dynamic source of excitation.

Figure 2. Fourier spectra of all the calculated motions in sea state 1, soil type 1, in plane of the SCR.

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Figure 3. Fourier spectra of all the calculated motions in sea state 1, soil type 1, out of plane of the SCR.

Identified period (s) 19.355 16.941 16.000 15.303 14.724 14.187 13.701 13.247 12.800 12.371 11.921 11.483 11.018 10.534 10.014 9.4426 8.7966 8.0044 6.9164 5.7994 4.8847

Associated period (s) 18.5781 17.0935 --14.4669 -13.672 ---11.8317 11.3852 --10.0465 9.7482 8.7161 -6.9008 5.7058 4.8571

Associated eigenpair 6 7 --8 -9 ---10 11 --12 13 14 -18 22 26

Table 4. Identified periods for sea state 1. If a sea state with lesser demand is used, the identified periods shift to a lower range, as it could be observed in Figure 4 and Table 5.

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Figure 4. Fourier spectra of all the calculated motions in sea state 5, soil type 1, in plane of the SCR. Identified period (s) 12.58 11.02 10.4 9.95 9.57 9.23 8.92 8.62 8.33 8.04 7.76 7.47 7.17 6.86 6.52 6.15 5.73 5.23 4.54

Associated period (s) --10.0465 9.7482 --8.7161 8.5182 --7.7088 7.5598 -6.9008 6.7918 6.1623 5.7058 5.2503 4.5195

Associated eigenpair --12 13 --14 15 --16 17 -18 19 21 22 24 28

Table 5. Identified periods for sea state 5. The previous tables are interesting because they show that the periods of the riser could be identified directly from a few acceleration signals. But it is even more interesting to see the identified mode shapes using the proposed technique. The following plots show the theoretical and identified mode shapes for each sea state, using the FSA over the calculated accelerations. Only modes 6 to 12 are shown for soil type 1. Notice that the theoretical mode shape (eigenvector) is calculated in the commercial software assuming that the riser is not allowed to move at the ends, while the identified ones do not hold that restriction, thus, large motions could be observed at the top. Of course, the comparison between mode shapes is just

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to clarify the issue that commercial software assume that the riser, at the top, does not move; and Fourier spectra of the signals do not, because the signals consider that the semisubmersible vessel is moving throughout the simulation time. However, notice the resemblance of the identified shapes to the theoretical ones. Also notice that with more demanding sea states, the larger the top motions will be. These motions dissipate through the bottom part of the riser. With lesser sea states, the top motions are not that significant and larger motions are observed in the middle and lower parts of the riser. It was also found that the higher the mode, the better the agreement between the theoretical and the identified mode shapes. As an example Figure 5 is presented. Identified periods and mode shapes were also obtained applying the Frequency Domain Decomposition Method, FDD. Figure 6 shows the singular values of the acceleration response for soil 3 sea state 5 in plane direction. 19 peaks were observed, each corresponding to a natural period of vibration of the riser. A similar plot was obtained for the out of plane direction.

Mode 8

Mode 14 Figure 5. Theoretical and identified mode shapes 8 and 14.

Singular values 105 10-0 10-5 10-10 10-15

10-20

T (s)

9 36 12 18 0 4 4.5 6 7.2 spectral Figure 6. Singular values of the5.1response power density matrix utilizing in-plane accelerations. Sea state 5, soil type 3.

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Table 6 shows the identified periods using both FSA and FDD and relative error values with respect to the theoretical ones. Note that both signal processing techniques provided error values less than 5 %, which is acceptable in engineering. Also, the absolute values of the errors using the FDD were smaller than the ones from FSA for most of the modes (seven out of twelve), which means that the FDD was more precise to determine periods of vibration for the studied riser. Figure 7 shows two modes. In general, a good agreement between theoretical and identified shapes can be observed. T (s) Mode 12 13 14 15 16 17 18 19 21 22 24 28

Theoretica l 10.05 9.75 8.72 8.52 7.71 7.56 6.90 6.79 6.16 5.71 5.25 4.52

Direction In -plane Out of plane In-plane Out of plane In-plane Out of plane In-plane Out of plane Out of plane In-plane In-plane In-plane

FSA 10.40 9.95 8.92 8.62 7.76 7.47 6.86 6.52 6.15 5.73 5.23 4.54

FDD 10.37 9.92 8.89 8.59 7.74 7.45 6.84 6.51 6.14 5.72 5.22 4.53

error(%) FSA -3.52 -2.07 -2.34 -1.20 -0.66 1.19 0.59 4.00 0.20 -0.42 0.39 -0.45

FDD -3.27 -1.74 -1.98 -0.87 -0.43 1.41 0.82 4.15 0.31 -0.31 0.63 -0.19

Table 6. Theoretical and identified periods.

Theoretical FDD FSA

Mode 15

Mode 16

Figure 7. Theoretical and identified mode shapes 15 and 16. 5.

SUMMARY

This paper presents the use of Fourier Spectral Analysis (FSA) and Frequency Domain Decomposition Method (FDD) to compute periods and mode shapes of a steel catenary riser, which represent its vibration characteristics, based solely on output acceleration records. Simulations with 5 sea states and 3 soil types were performed to calculate data at 15 discrete locations along the riser. Comparisons between theoretical eigenvectors and identified mode

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shapes show that the proposed method could be a valuable tool for in-service analysis of risers. One interesting advantage of the methods is that does not assume boundary conditions at the ends of the riser, thus, leading to more realistic mode shapes. Risers dynamic analysis is very demanding so new methodologies are needed to understand the behaviour that can be present during its lifespan. The methodologies purposed herein can be used also in conjunction with other simulation processes to identify possible damage or critical areas for integrity and monitoring purposes. The authors are currently working on this issue. ACKNOWLEDGMENTS The authors specially acknowledge Instituto Mexicano del Petróleo and the Instituto Politécnico Nacional for supporting this research. REFERENCES [1] J. P. Kenny, Deepwater Riser Design, Fatigue Life and Standards Study Report, Minerals Management Service, October 22, 2007. [2] K. M. Lund, P. Jensen, D. Karunakaran, K. H. Halse, A Steel Catenary Riser Concept for Statfjord C, 17th International Conference on Offshore Mechanics and Arctic Engineering, OMAE 981361, Lisbon, Portugal, July 5-9, 1998. [3] N. Dale, C. D. Bridge, Measured VIV Response of a Deepwater SCR, Proceedings of the ISOPE Conference, Paper No. ISOPE-2007-JSC-473, 2007. [4] R. Garrido, F. J. Rivero, Hysteresis and Parameter Estimation of MDOF Systems by a Continuous-Time Least Squares Method, Journal of Earthquake Engineering, 10(2), pp. 237264, 2006. [5] M. Podskarbi, R. Theti, H. Howells, Fatigue Monitoring of Deep Water Drilling Risers, IADC Conference, 2005. [6] M. Podskarbi, D. Walters, Review and Evaluation of Riser Integrity Monitoring Systems and Data Processing, Deep Offshore Technology Conference, 2006. [7] R. Rodriguez, F. J. Rivero, E. Gomez, Limited Modal Information and Noise Effect on Damage Detection without Baseline Modal Parameters, Proceedings of the IMAC-XXVII, Society for Experimental Mechanics, Orlando, Florida, 2009. [8] R. Rodriguez, F. J. Rivero, E. Gomez, Damage Detection of a 7-Storey Concrete Building Utilizing Frequency Domain Decomposition and Independent Component Analysis for Modal Parameter Extraction, 3rd International Operational Modal Analysis Conference, IOMAC, Ancona, Portonovo, 2009, pp. 773-780, 2009. [9] R. Theti, F. Botros, Performance Monitoring of Deepwater SCRs, Subsea Tieback Forum, 2004. [10] J.S. Bendat, A.G. Piersol, Random data. Analysis and measurement procedures, John Wiley & Sons, 1986. [11] J. N. Juang, R. S. Pappa, An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction, AIAA Journal of Guidance, Control and Dynamics, 8(4) , pp. 620-627, 1985

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[12] N. Kalouptsidis, S. Theodoridis, Adaptive System Identification and Signal Processing Algorithms, Prentice Hall International Series in Acoustics, Speech and Signal Processing, 1993. [13] M. Chezhian, K. Mørk, T. S. Meling, C. Makrygiannis, P. Lespinasse, NDP Review of State-of-the-Art in Riser Monitoring: Lessons Learned and Experiences Gained, Offshore Technology Conference, Houston, May 1-4, 2006. [14] S. Natarajan, H. Howells, D. Deka, D. Walters, Optimization of Sensor Placement to Capture VIV Response, Proceedings of the OMAE-2006, Hamburg, 2006. [15] R. Brincker, L. Zhang, P. Andersen, Modal identification from ambient responses using frequency domain decomposition, 18th International Modal Analysis Conference, IMAC XVIII, San Antonio, Texas, 2000. [16] Marine Computation Services Ltd., Flexcom, Galway Technology Park, Galway, Ireland, 2008. [17] 2H Offshore Engineering, STRIDE JIP – Seabed Stiffness Parametric Study, Woking, UK, 1094-RPT-023, 1997.

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