an investigation on an alternative approach to compute the
TRANSCRIPT
AN INVESTIGATION ON AN ALTERNATIVE APPROACH TO COMPUTE THE
100-YEAR LOAD RESPONSES FOR TURRET MOORED SYSTEMS
Guilherme Viana Rosa e Silva Valsa
Projeto de Graduacao apresentado ao
Curso de Engenharia Naval e Oceanica da
Escola Politecnica, Universidade Federal do
Rio de Janeiro, como parte dos requisitos
necessarios a obtencao do tıtulo de Engen-
heiro.
Orientador: Paulo de Tarso Themistocles Es-
peranca
Rio de Janeiro
Novembro de 2019
AN INVESTIGATION ON AN ALTERNATIVE APPROACH TO
COMPUTE THE 100-YEAR LOAD RESPONSES FOR TURRET
MOORED SYSTEMS
Guilherme Viana Rosa e Silva Valsa
Orientador:
Prof. Paulo de Tarso Themistocles Esperanca, D.Sc
Examinador:
Prof.Claudio Alexis Rodrıguez Castillo, D. Sc
Examinador:
Prof. Luis Volnei Sudati Sagrilo, D.Sc
2019
Viana Rosa e Silva Valsa, Guilherme
An Investigation on an Alternative Approach to Compute the 100-
Year Load Responses for Turret Moored Systems/Guilherme Viana
Rosa e Silva Valsa. - Rio de Janeiro: UFRJ/Escola Politecnica,
2019.
XV, 44p.:il.;29,7cm
Orientador: Paulo de Tarso Themistocles Esperanca
Projeto de Graduacao - UFRJ/Escola Politecnica/Curso de Engen-
haria Naval e Oceanica, 2019.
Referencias Bibliograficas: p.43-44.
1.Hidrodynamics. 2.Seakeeping. 3.Reliability. 4.Extreme Analy-
sis. 5.Mooring. I. Themistocles Esperanca, Paulo de Tarso. II.
Universidade Federal do Rio de Janeiro, Escola Politecnica, Curso
de Engenharia Naval e Oceanica. III.An Alternative Approach to
Compute the 100-Year Load Responses for Turret Moored Systems.
iii
ACKNOWLEDGEMENTS
Primeiramente gostaria de agradecer a minha famılia por todo apoio fornecido nao so
durante o perıodo de graduacao, mas por todo o suporte antes do inıcio da faculdade e
por todo apoio que, sem duvida, sera prestado apos a graduacao. Teria sido impossıvel
sem voces.
Gostaria de agradecer aos meus colegas de curso e amigos. Obrigado pelos mo-
mentos, risadas e ajuda nas materias. A amizade de voces foi fundamental durante os
anos de curso e espero reve-los sempre que possıvel. Os levarei para sempre comigo.
Gostaria de agradecer a UFRJ, que me forneceu nao somente a possibilidade de
aprender e me tornar um profissional de engenharia como tambem a inesquecıvel ex-
periencia de intercambio. A dupla titulacao foi um patamar alcancado que certamente
moldou e que ainda molda minha vida. Aproveito a oportunidade para agradecer a todos
que conheci no exterior, amigos, professores e colegas.
Agradeco ao Prof. Paulo de Tarso por aceitar ser meu orientador e pela ajuda
ao longo do trabalho e a Prof. Marta Tapia por ter possibilitado minha participacao no
programa de intercambio da UFRJ.
iv
Resumo do Projeto de Graduacao apresentado a Escola Politecnica/UFRJ como parte
dos requisitos necessarios para obtencao do grau de Engenheiro Naval e Oceanico.
An Investigation on an Alternative Approach to Compute the 100-Year Load Responses
for Turret Moored Systems
Guilherme Viana Rosa e Silva Valsa
Novembro/2019
Orientador: Paulo de Tarso Themistocles Esperanca
Curso: Engenharia Naval e Oceanica
O metodo de analise de valores extremos para calcular a resposta de perıodo de retorno
de 100 anos ja e bem estabelecido. Para se contabilizar a variacao de longo prazo das res-
postas, usam-se condicoes ambientais equivalentes a um perıodo de retorno de 100 anos.
Enquanto para a variabilidade de curto prazo, usam-se percentis equivalentes a 37% ou
50% nas simulacoes. Uma investigacao de um metodo alternativo usando condicoes ambi-
entais equivalentes de um perıdo de retorno de 40 anos com um percentil de 90% para as
simulacoes de curto prazo e um fator de correcao [1] e usado para o calculo de tensoes em
linhas de ancoragem em um sistema de torre. Os valores obtidos pelo metodo proposto
obtiveram resultados cerca de 30% maiores que o padrao. Por motivos de limitacao tem-
poral, o calculo da resposta exata que forneceria uma ferramenta efetiva de comparacao
entre os metodos nao foi realizado.
Palavras-Chave: Hidrodinamica, Seakeeping, Confiabilidade, Analise Extrema, Ancor-
agem
v
Abstract of Undergraduate Project presented to POLI/UFRJ as a partial fulfillment of
the requirements for the degree of Engineer.
An Investigation on an Alternative Approach to Compute the 100-Year Load Responses
for Turret Moored Systems
Guilherme Viana Rosa e Silva Valsa
November/2019
Advisor: Paulo de Tarso Themistocles Esperanca
Course: Naval and Oceanic Engineering
The reliability investigations that compute the 100-year responses follow a standard
practice. The 100-year environmental conditions are used to account for the long-term
variability of the responses while the MPM or 50% quantile values are used for the
short-term variability. A new method using a 40-year return period with the use of 90%
quantile values for the short-term simulations and a correction factor of 1.04 [1] is tested
to verify its applicability to the turret loads. The values obtained from the alterantive
method are over 30% higher than those of the standard method. The computation of
the exact response would give a comparison tool between the responses. Due to time
and computational issues this response is not computed, leaving opportunities for future
research.
Key-words: Hidrodynamics, Seakeeping, Reliability, extreme analysis, mooring
vi
Nomenclature
Fdiff Diffraction Forces
FK Froude-Krylov Forces
Frad Radiation Forces
CoG Center of Gravity
DOF Degree of Freedom
Ln Natural Logarithm
LT Long-term
MBL Minimum Breaking Load
MPM Most Probable Maximum
PFE ”Projet des fins d’Etudes”
Q Quantile
QTF Quadratic Transfer Function
RAO Response Amplitude Operator
RMS Root Mean Square
RP Return Period
RV Response Value
ST Short-term
TLP Tension Leg Platforms
vii
TMS Turret Moored System
α Wave Phase
β Gumbel Scale Parameter
βLT Variable in Gaussian Space Related to the Long-term Variable in the Physical
Space
βST Variable in Gaussian Space Related to the Short-term Variable in the Physical
Space
χ Wave Direction
ε Phase Angle
κ Wave Number
λ Gumbel Location Parameter
µ Response Parameter Evaluated
ω Wave Frequency
x Data Average Value
Φ Standard Normal Distribution
φD Diffraction Potential Flow
φI Incident Potential Flow
φR Radiation Potential Flow
φt Total Potential Flow
ρ Density [ tonsm3 ]
ξ Shape Parameter for the Long-term Probabilistic Distribution
A∞ Fluid Added Mass at Infinite Frequency
Aj Wave Amplitude
viii
Fh Total Horizontal Force applied onto the Chain Table
FLT Long-term Probabilistic Distribution of the Responses
Frad Total Horizontal Force applied onto the Turret
fST Short-term Probabilistic Distribution of the Responses
FXCT Mooring and Riser Forces Applied along the Turret x-axis
FXrad Mooring, Riser, Inertia and Entrapped Water Forces Applied along the Turret
x-axis
FY CT Mooring and Riser Forces Applied along the Turret y-axis
FY rad Mooring, Riser, Inertia and Entrapped Water Forces Applied along the Turret
y-axis
TMax Maximum Tension of the Mooring Lines
Vi Fluid Velocity in the i Direction
C Damping Matrix
K Stiffness Matrix
M Added Mass Matrix
Fx Cumulative Probabilistic Function for a Specific Response Parameter
Cs Current Speed
h Acceleration Impulse Function Matrix
Hs Significant Wave height
I Index Number used in the Gumbel Fit
M Mass
N Total Number of seeds
P Probability
ix
PI Probability of a Specific Index I in a Data Distribution
Tp Modal Peak Period
Ws Wind Speed
g Gravity[ms2
]
x
Contents
List of Figures xii
List of Tables xiii
1 Introduction 1
1.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Problem Statement and Objectives . . . . . . . . . . . . . . . . . . . . . 2
1.3 Structure of the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Project Presentation 4
2.1 Reference Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Mooring Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Risers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Turret . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 Environmental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Theoretical Formulation 9
3.1 Diffraction/Radiation Analysis . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.1 First-order Approach . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.2 Second-Order Approach . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.3 Radiation forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Line Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Turret Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.4 Extreme Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4.1 Random/Stochastic Process . . . . . . . . . . . . . . . . . . . . . 16
3.4.2 Seed Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4.3 Short-term Variability . . . . . . . . . . . . . . . . . . . . . . . . 18
xi
3.4.4 Gumbel Probabilistic Model . . . . . . . . . . . . . . . . . . . . . 19
3.4.5 Return Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4.6 Long-term Variability . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4.7 Methodology for the 100-year RP Responses . . . . . . . . . . . . 24
3.4.8 Exact Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4.9 IFORM Contour . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4.10 Reference Article/New Method summary . . . . . . . . . . . . . . 29
4 Numerical Studies 30
4.1 Software Versions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2 Hydrostar-ARIANE-OrcaFlex Methodology . . . . . . . . . . . . . . . . 30
4.3 100-year Return Period Configuration . . . . . . . . . . . . . . . . . . . . 32
4.4 40-year Return Period Configuration . . . . . . . . . . . . . . . . . . . . 33
4.4.1 Design Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 100-year RP Turret Loads Investigations 35
5.1 Hydrostar-ARIANE-OrcaFlex Investigations . . . . . . . . . . . . . . . . 35
5.1.1 Standard Method Investigations . . . . . . . . . . . . . . . . . . . 36
5.1.2 Prosed Method Investigations . . . . . . . . . . . . . . . . . . . . 36
5.1.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6 Conclusion 41
6.1 Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
xii
List of Figures
2.1 Bird’s eye view of the mooring and riser systems. . . . . . . . . . . . . . 4
2.2 Vessel reference system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.1 Example of a Gumbel fit. . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Example of an environmental contour [6]. . . . . . . . . . . . . . . . . . . 22
3.3 Standard practice flow chart . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Example of a circle in gaussian space defined by two independent variables
[6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.5 IFORM Contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.1 Summary chart describing the work flow of the Hydrostar-ARIANE-OrcaFlex
investigations. The square boxes represent the software used. The rounded-
edged boxes represent the type of computation performed. The ellipses
represent the output from the computations. . . . . . . . . . . . . . . . . 32
5.1 OrcaFlex model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
xiii
List of Tables
2.1 Summary of the reference systems . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Segment Lengths Summary . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Environmental Characteristics Main Parameters Summary . . . . . . . . 8
3.1 Description of turret loads . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Example of table that relates the response values with their indexes and
cumulative probability values. This table exemplifies one condition ran
with 50 different seeds. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Summary of the RP values and their probabilities considering 2920 simu-
lations of 3h in one year interval . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Joint frequency of significant wave height and spectral peak period [2] . . 23
3.5 IFORM Summary table values . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1 Table that summarizes the loads taken into account when computing the
motions in ARIANE [11]. This is a general table that indicates ARIANE’s
computational capacities. It does not represent all loads computed in this
study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 100-year return period turret loads reference environmental conditions . . 33
4.3 Summary of Metocean Return Periods. . . . . . . . . . . . . . . . . . . . 33
4.4 40-year return period design environmental conditions . . . . . . . . . . . 34
5.1 Fh and Frad values obtained with the standard approach. . . . . . . . . 36
5.2 40-year return period turret load responses . . . . . . . . . . . . . . . . . 36
5.3 Summary of environmental directions evaluated for design points 1 and 2. 37
5.4 Summary of turret load values with all environmental directions evaluated
for design points 1 and 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.5 Results comparison with the two proposed methods. . . . . . . . . . . . . 39
xiv
Chapter 1
Introduction
FPSOs are largely used for oil and gas production. The environmental conditions
for this kind of activity can be very aggressive. Waves, wind and current play an impor-
tant role in defining the vessels’ motions and the loads that are applied to them. When
severe sea conditions are added into the equation, the problem becomes even more com-
plex. It is therefore vital for oil companies to accurately predict the vessel’s behavior in
open waters.
The seakeeping performance of offshore units is a crucial part when studying the
development of an oil and gas exploration field. A ship’s seakeeping performance can be
tied to its capacity to withstand external environmental loads and its motions response.
Comfort and damage to the ship and its cargo being the most used parameters for quality
standards. The investigations presented in this report relate to both seakeeping and to
the ability of the mooring system to keep the FPSO on station i.e station-keeping. This
is highly important when studying the chaintable loads.
The offshore unit analyzed in the investigations here presented is an internally
turret-moored FPSO vessel. This means that it is a floating system and that it requires
mooring lines to keep it in place, or within an imposed range of offsets. In addition to
these underwater cables, there are also risers that connect the seabed to the vessel. These
pipes are responsible for transferring fluids and information. More notably, the crude oil
from the wells.
The use of the Return Period (RP) responses is a standard practice to determine
which sea conditions the offshore unit might have to withstand during its lifetime. The
Return Period characterizes the probability of exceedance of an event during a previously
determined time frame. The most common RP analyzed in the industry nowadays are the
1
1, 10, 100 and 10.000 years. The 1-year RP being the least severe, while the 10.000-year
RP is the most extreme case.
Usually, the 100-year RP environment is the main criteria to assess the vessel’s
seakeeping and mooring system station-keeping performaces. Since the environment is
described as a probabilistic model, the actual sea conditions vary from one study to
the other. Therefore, a wide number of 3h time-domain simulations is done, each of
them giving a different response. For every 3-hour environmental conditions, the Most
Probable Maximum (MPM) or the median reponse (50% Quantile) value of the response
is computed. The highest MPM or Q50% of all simulations performed is considered to
be the 100-year RP response.
Despite being a consolidated practice in the industry, an article [1] from BV pro-
poses a new method to better model the 100-year RP response. They use a 40-year RP
with a 90% quantile for the short-term variability, instead of the current 100-year RP
with a 50% quantile. They also propose a 1.04 correction factor. This new method is
investigated. The main variables analyzed in the investigations are the loads applied to a
turret mooring system. These efforts are the ones applied to the turret and are primarily
due to the mooring lines restoring forces.
1.1 Context
The study presented in this report is part of the internship activities done in
the context of a double degree program in France. This internship, called ”Projet des
Fins d’Etudes”or ”PFE” in French, lasted for 5 months and a half. This is a vital and
mandatory step for students at ENSTA-Bretagne to finish their studies and obtain their
diploma.
The nature of the work is mainly in the field of Hydrodynamics, with focus on
seakeeping. The offshore unit in question is an internally moored FPSO.
1.2 Problem Statement and Objectives
Even though the current practice for calculating the 100-year return period re-
sponse is well established in the offshore industry, there are other ways of computing this
response. A recent BV paper[1] tackled this issue. Their new proposed method indicates
2
an approach to better estimate the 100-year RP response. However, their analyses focus
on the mooring lines maximum tensions (TMax). It is unknown if this approach can be
applied to other parameters (such as offset, mooring forces etc.).
If such approach is proven to be more precise than the one largely used in the
industry, it may (depending on the outcome of future research) indicate a need to update
the current long-term analysis methodology. The precision of the answer is indicated by
how far it is from the exact 100-year response (see Section 3.4.8 Exact Response).
The main activities consist of the computation of the 100-year RP response of
the tensions on the mooring turret using dedicated software. Several simulations are
performed to compute the reponses using the standard and the proposed methods.
1.3 Structure of the report
• Chapter 2 summarizes the characteristics of the Turret Moored Systems (TMS)
and its components.
• Chapter 3 details the theoretical formulations and hypotheses used to compute the
desired response values.
• Chapter 4 introduces the software used to do the computations and focuses on the
methodology used to obtain the 100-year RP responses. In addition, it describes
initial investigations important for the setup of the project.
• Chapter 5 details the investigations made and results obtained using the method-
ology described in Chapter 4.
• Chapter 6 is dedicated to the final conclusions of the project.
3
Chapter 2
Project Presentation
As previously mentioned, the vessel being studied is a FPSO. Even though the
seabed studied has a slight slope, the seabed is considered as a flat surface. The mean
water depth is over 250m . The vessel has an internal turret system that allows it to
weathervane. The turret is connected to all mooring lines and risers. There is a total of
3 bundles composed of 5 mooring lines each, which makes a total of 15 mooring lines. In
addition, there are 21 risers. In total, there are 36 lines in the model. Figure 2.1 shows
the complete system.
Figure 2.1: Bird’s eye view of the mooring and riser systems.
4
2.1 Reference Systems
There are three main reference systems to be considered: the global earth-fixed
reference system, the vessel axis system and the turret axis system. The origin (O) of the
earth-fixed system corresponds to the turret reference position at the mean water line. A
summary of the reference systems can be seen in Table 2.1 and Figure 2.2. The software
used during design have their own definition of the reference systems. As a result the
turret system has an orientation towards west, while the earth-fixed system is towards
east.
Table 2.1: Summary of the reference systems
Earth-Fixed Reference System
Axis Origin Positive Direction
X Turret CoG Towards North
Y Turret CoG Towards East
Z Mean Water Line Downwards
Vessel Axis System
Axis Origin Positive Direction
X FPSO Aft Perpendicular (AP) From AP to bow
Y FPSO Center Line (CL) From CL to Port Side
Z FPSO Base Line (BL) From BL upwards
Turret Axis System
Axis Origin Positive Direction
X Turret Center Line (CL) Towards North
Y Turret Center Line (CL) Towards West
Z FPSO Base Line (BL) From BL upwards
5
Figure 2.2: Vessel reference system.
2.2 Mooring Lines
Each mooring line is composed of different segments, each of them with different
properties. They all possess a bottom chain, which is the part in contact with the seabed,
thus subjected to friction forces. In addition, they have a chain-steel wire component
(SSSW) and a top chain. Finally, they also have a small connector component that is
considerably heavier than the others and is used as the connection segment with the
fairleads. The mooring lines composition can be seen in Table 2.2.
Table 2.2: Segment Lengths Summary
Segment Unit Bundle 1 Bundle 2 Bundle 3
Bottom Chain m 520 420 720
SSSW m 355 355 355
Top Chain m 50 50 50
Connector m 3.5 3.5 3.5
Total Length m 1125 1025 1325
6
2.3 Risers
The system operates with 21 risers when in full capacity, this condition is the one
considered for the computations. All risers have a buoyancy section dedicated to the
creation of underwater arches. This pliant-wave configuration seeks to lower the overall
tensions on the pipes. The risers are implemented on the computation model only for the
line dynamics computation. They do not take part on the computation of the Response
Amplitude Operators (RAOs).
2.4 Turret
The turret is a crucial component for this system. Besides grouping all risers and
mooring lines, it also allows the FPSO to weathervane. This ability is very important
when the system is subject to severe environmental conditions that can also come from
multiple directions. The main focus of the investigations presented is related to the turret
loads.
2.5 Environmental Data
The environmental data is obtained by a hindcast database. The metocean data
allows for the creation of long-term statistics that describe wave, current and wind char-
acteristics. The main aspects to be considered are the intensities (wave height and natural
period, wind and current speed) and their governing directions. Table 2.3 shows the main
characteristics of the environmental conditions studied.
7
Table 2.3: Environmental Characteristics Main Parameters Summary
Parameter Symbol Unit
Waves Wave Height Hs [m]
Waves Spectral Peak Period Tp [s]
Waves Mean Wave Heading (-) [deg]
Wind Wind Speed Ws [m/s]
Wind Mean Wind Heading (-) [deg]
Current Current Speed at surface Cs [m/s]
Current Mean Current Heading (-) [deg]
Current Current Profile (-) [m/s] and [m]
8
Chapter 3
Theoretical Formulation
3.1 Diffraction/Radiation Analysis
The radiation and diffraction analysis is a method that seeks to predict the re-
sponse of offshore structure forces/motions due to wave excitation. The first-order or
linear approach limits the analysis of the excitation efforts with the same frequencies as
the incoming waves, while the second-order approach allows the evaluation of loads with
frequencies besides those of the waves.
3.1.1 First-order Approach
The wave theory is important to predict sea induced loads on ships and offshore
structures as well as wave-induced motions. This approach considers that these efforts
and motions oscillate with the same frequency as the waves that excite the structures.The
linear theory allows the modelling of irregular waves by the simple addition of waves with
different amplitudes, lengths and directions. The wave elevation can be described as seen
in Equation 3.1 [2].
ζ =∞∑j=1
Aj sin(ωjt− κjx+ εj) (3.1)
Where Aj is the wave amplitude, ω the wave frequency, κ the wave number and ε the
phase angle.
Due to the linearity of the problem, the potential flow can be divided in three
parts, as shown in Equation 3.2. This assumption is not valid for large waves.
9
φt = φI + φD + φR (3.2)
Where φI is the incident potential flow, φD the diffraction potential flow and φR the
radiated potential flow.
Considering the linear behaviour of the problem, the incident wave can be modelled
as the Airy wave. One main assumption is that the body does not affect the incoming
wave, more precisely its pressure field. It discards the diffraction contribution, therefore
only the incoming wave component is considered. From the incoming wave it is possible
to determine the loads that are applied on the body. These efforts are known as the
Froude-Krylov forces, as shown in Equation 3.3.
FK = −ρ∫∫
∂φI∂t
n dS (3.3)
Where ρ is the fluids’ density.
The diffraction phenomenon consists of the reflection and scattering of waves once
they encounter the body. The structure does not move, it is considered as a fixed body.
Diffraction originates excitation forces. The radiation problem takes into consideration
the waves created by the body without the contribution of the incoming wave. The body
oscillates and originates waves independently of any external excitation. The resulting
forces from this phenomenon are the added-mass and radiation damping contributions.
The contributions from the diffraction and radiation loads are shown in Equations 3.4
and 3.5.
Fdiff = −ρ∫∫
∂φD∂t
n dS (3.4)
Fradj = −ρ∫∫
∂φradj∂t
n dS j = 1, ... 6 (3.5)
Where ρ is the fluids’ density.
For the linear approach to be used, the fluid’s viscosity is neglected. In the linear
approach the dynamic responses of the structures (motions and loads) are excited with
the same frequency as those of the external waves.
10
3.1.2 Second-Order Approach
The second-order approach allows the evaluation of loads that cannot be analyzed
with the linear approach since the latter works at a limited frequency range. Some of the
forces applied at the offshore structures are not at the same frequency of the incoming
waves. Therefore, the excitation loads can be close to the natural frequency of the
horizontal planar motions of the body, which may cause resonance issues.
The frequencies of these efforts can be smaller or higher than the wave frequency.
The most common effect analyzed at second-order is the wave drift load, an horizontal
effort linked to the structure’s capacity of creating waves. There is also a resonating
phenomenon known as springing that is observed at high frequencies in moored structures
such as TLPs.
The external forces can be developed as seen in Equation 3.6. [3]
F = F (0) + εF (1) + ε2F (2) + ε3F (3)... (3.6)
Where ε is a small linear parameter related to the wave amplitude. The ε2F (2) component
is the second-order term.
As shown in [2] the non-linear effects can be obtained by analyzing the Bernoulli’s
equation for the fluid pressure, as shown in Equation 3.7.
−ρ2
(V 21 + V 2
2 + V 23 ) =
−ρ2|∇φ|2 (3.7)
Where Vi is the fluid velocity at the i direction. An approximation of the velocity vector
for an idealized sea state composed of two distinct wave components ω1 and ω2 is proposed
[2], as shown in Equation 3.8.
V1 = A1 cos (ω1t+ ε1) + A2 cos (ω2t+ ε2) (3.8)
By analyzing only the V1 component, Equation 3.7 can be expanded as shown in Equation
3.9.
−ρ2
(V 21 ) =
−ρ2
(A2
1
2+A2
2
2+A2
1
2cos (2ω1t+ 2ε1)+
A22
2cos (2ω2t+ 2ε2) + A2
1A22 cos ((ω1 − ω2)t+ ε1 − ε2)+ (3.9)
11
A21A
22 cos ((ω1 + ω2)t+ ε1ε2))
Through this analysis a constant term appears : −ρ2
(A2
1
2+
A22
2). This effort is the
mean drift load and its contribution is only noticeable by doing this second-order analysis,
this component is called wave drift load. Another term that appears is the (ω1−ω2) which
represents the difference wave frequencies. This factor represents a pressure contribution
that oscillates with a frequency equal to the difference of both wave frequencies. When
modelling more complex and realistic sea conditions, the velocity component can be
described as the sum of N different waves. Therefore, there will be several combinations
of difference wave frequencies. As a result, the difference frequency force components
oscillate at other frequencies than those of the waves. Thus, the combination of incoming
waves might introduce resonating forces and moments on the body [2].
This example demonstrates that the first-order analysis simplifies the problem by
not taking into consideration terms of higher order. By doing a second order analysis
other components such as the wave drift load and the difference frequencies come into
play. These efforts can impact greatly the movement and integrity of the structures since
the efforts being applied might be in the same range as that of their natural frequencies.
The higher-order approaches are not limited to the second-order analysis, they can
be a third-order, fourth-order approach or even higher. However, the more the order is
increased the more the solution becomes complex. Therefore, the study can become very
expensive since it takes larger amounts of time to solve a high-order problem. Besides,
approaches of order higher than two introduce terms that can be neglected because their
impact on the structure is small in comparison to first and second-order components.
Equation 3.9 also introduces a sum of frequencies. However, they are more specific to
TLP and taut moored structures that have oscillating periods of no more than 4 seconds.
3.1.3 Radiation forces
The radiation problem solution consists of forcing the ship to oscillate for the six
degrees of freedom in calm water. The system is excited with the same frequency as
the excitation waves. The equation of motion of the system can be written as shown in
Equation 3.10.
12
MX(t) + CX(t) + KX(t) = F(t) (3.10)
Where M is the added mass matrix, C the damping matrix, K the stiffness matrix and
F(t) the external forces.
The added mass matrix and the hydrodynamic damping in the damping matrix are
frequency dependant. Since this method considers that external forces (such as mooring
and riser loads) do not affect motions in a significant manner at wave frequency, it does
not evaluate the full problem. In some cases this assumption may be correctly applied,
in others not. This approach computes the radiation forces as shown in Equation 3.11
[4].
[F (t)]rad =∑
Re{am(ωi)
[ω2i (A(ωi)− A0) + iωiB(ωi)
][u(ωi)]e
i[−ωit+ki(x cosχ+y sinχ)+αi]}
(3.11)
Where A(ωi) and B(ωi) are the added mass and damping matrix at the i-th wavelet
frequency, [u(ωi)] is the translation or rotation motion RAO. A0 is the added mass at
drift frequency. ωi is the encounter wave frequency, ki the wave number, χ the wave
direction and αi the wave phase of the i-th wavelet.
3.1.3.1 Convolution
The equation of motion is solved in the frequency domain. This is not compatible if
the external forces (wind, waves . . . ) are in time domain. The convolution options allows
for a more precise result since it computes the radiation forces by taking into account
all frequency values. The convolution integration method is an alternative approach to
compute the radiation forces.
The convolution method uses the system’s added mass and damping information
to compute a time domain response. The equation of motion of the system (Equation
3.10) is slightly modified. This approach is shown in Equation 3.12 [4].
(M + A∞)X(t) + CX(t) + KX(t) +
∫ t
0
h(t− τ)X(τ) dτ = F (t) (3.12)
Where M is the structural mass matrix, A∞ is the fluid added mass matrix at infinite fre-
quency, B is the damping matrix, K is the total stiffness matrix, and h is the acceleration
impulse function matrix, as shown in Equation 3.13 [4].
13
h(t) =2
π
∫ ∞0
B(ω)sin(ωt)
ωdω =
2
π
∫ ∞0
{A(ω)−A(∞)} cos(ωt)dω (3.13)
Equation 3.12 is the equation of motion expressed in convolution integral form.
In the computations that do not consider convolution, the radiation forces are restricted
to the excitation wave frequencies. However, the convolution method is capable of cal-
culating the forces and their impacts on the structure motions to all frequencies by the
usage of the integral form of the radiation forces. It passes the parameters from frequency
domain to time domain.
The convolution considers then, that the radiation forces are analyzed as a separate
force in time domain. The convolution method basically means that frequency dependant
parameters such as the added mass and damping are treated in time domain instead of
the frequency domain by the means of a numerical computation, which is represented by
the integral in Equation 3.12.
3.2 Line Dynamics
Offshore units are either fixed or moored structures. Moored vessels and platforms
are subjected to forces originated from the interaction of the mooring lines and the
environment: friction force with the seabed, wave and current loads, inertia etc. This is
true for all underwater cables, therefore also for risers as well.
The vessels possess a large mass in comparison to the risers and moorings. Never-
theless, the dynamic response of the cables to external loads affect the vessels’ motions,
specially the lines drag forces. There are different approaches to consider the effects of
the mooring lines on a numeric model [5].
One approach is the use of the quasi-static method. This method solves the
mooring lines contributions by solving the catenary equations at each time step. With this
approach, the low frequency motions due to environmental excitation and the dynamic
effects of the lines are difficult to be solved.
In order to properly take into account the impact of the lines into the vessels’
motions, a full dynamic solution is to be used. This method allows for the computation
of the full effects of the moorings in the analysis by solving the motion equation at each
time step. This approach considers:
14
• Seabed effects - The contact between the lines and the seabed create friction
forces. These loads are nonlinear. Besides, the portion of the lines that is in
contact with the floor changes as the vessel moves. As a result, there is also a
nonlinear geometrical factor to be considered.
• Drag forces - The Morison equation is commonly used to take into account the
fluid contributions on the underwater cables. The drag force is of nonlinear be-
haviour, it is proportional to the square of the velocity (relative velocity between
line and fluid).
3.3 Turret Loads
The objective of this study is to investigate the loads applied onto the turret. There
are two parameters investigated: the chain table loads and the turret loads. The former
takes into account the mooring lines and risers contributions, while the latter considers
also the entrapped water and the inertia contributions of the turret. The design load at
the chain table level is called Horizontal Force (Fh). See Equation 3.14.
Fh =√
(FXCT )2 + (FY CT )2 (3.14)
Where FXCT represents the mooring and riser forces along the turret x-axis and FY CT
represents the mooring and riser forces along the turret y-axis. Fh is then, the total
horizontal force applied onto the chain table. The turret axes are in accordance with the
previously defined turret reference system.
The turret load contributions have the same components as Fh. Moreover, it also
considers the turret inertia and entrapped water for its computations. The entrapped
water is the amount of water in the turret cylinder. This entrapped water does not
impact the static analysis. However, when the vessel moves, the turret accelerations also
impose accelerations on the entrapped water. Therefore, the water generates loads onto
the turret due to its inertia. The total horizontal forces applied onto the turret are called
turret loads (Frad). See Equation 3.15.
Frad =√
(FXrad)2 + (FY rad)2 (3.15)
15
Where FXrad is the mooring and riser forces along the turret x-axis plus the turret inertia
and entrapped water loads along this axis. FY rad is the mooring and riser forces along the
turret y-axis plus the turret inertia and entrapped water loads along this axis. Frad is
then, the total horizontal force applied onto the turret. The turret axes are in accordance
with the previously defined turret reference system.
Table 3.1 summarizes the loads that are taken into consideration for each variable
investigated. These loads depend on the motions of the vessel and therefore are time-
dependent.
Table 3.1: Description of turret loads
Load Types Chain Table Loads Turret Loads
Dynamic Mooring Loads√ √
Dynamic Risers Loads√ √
Turret Mass Inertia Load ×√
Entrapped Water Inertia Load ×√
3.4 Extreme Analysis
The extreme analysis study is used to estimate the response values of the vari-
ables being studied. In this case, this analysis is focused on the 100-year Return Period
responses of the turret loads.
This analysis is made by computing several 3h numerical simulations using ded-
icated software. Each simulation gives a response value. These values can be used to
create a probabilistic distribution that gives in return the response desired according
to the Return Period being studied. The metodology and concepts of this study are
explained in this section.
3.4.1 Random/Stochastic Process
The random process field of study evaluates the random changes in numerical
values that compose the system being analyzed. Unlike the deterministic processes, in
the stochastic processes the behaviour of the variables being studied cannot be determined
16
by the use of equations and so forth. A stochastic process is characterized by an ensemble
of empirical data. In this case, the empirical data is obtained through the simulations
ran.
The procedure for the simulations in this report consists of the definition of wave
and wind spectra and the realization of time domain evaluations. The waves are defined
by a significant wave height, spectral peak period and direction. The wind is characterized
by wind speed and direction. When transformed into the physical space of time domain
simulations, these spectra are responsible for the definition of the wave elevations and
wind behaviour with time in a simulation. They define how the environment behaves and
therefore the structure in it.
These random processes are then classified as stationary since the statistical pa-
rameters do not change with time. Another important concept is the ergodic theory.
An ergodic process is a class of random process as well as a sub-type of the stationary
process. In this type of analysis, the mean values for the time domain responses for a
specific sample are the same as the mean statistic values of the data ensemble.
3.4.2 Seed Number
The seed is a random number used to give the initial conditions for a pseudo-
random generator. The waves and wind are characterized as spectra, which are in fre-
quency domain. The wind or waves will have different time-domain behaviour depending
on the seed first used to ”populate” their spectra. In other words, every seed number
used to characterize a spectrum will give a different time-domain behaviour for the waves
and the wind. In this study, every 3h short-term simulation has a different seed number.
The seed number is computationally represented by the εj in Equation 3.1. The
spectrum used to represent the waves is the Jonswap spectrum. Due to the different wave
elevations and wind behaviour, the offshore systems being analyzed will have different
response values (ex. motions, turret loads) for each seed used. This means that even
though the spectra used are statistically similar (same significant wave height and spectral
peak period), the physical response encountered in each simulation is different from the
others.
Since the environmental characteristics are described by a probabilistic model, a
simulation emulating a specific environment does not necessarily output the same re-
17
sponses every time it is run. This happens because the waves and wind are defined as
spectra. By using a different seed for each simulation, a different time-domain response is
obtained. The variation in results for the same sea state is called short-term variability.
3.4.3 Short-term Variability
The short-term variability is the term used to indicate that environmental condi-
tions that possess constant statistical parameters give different response values in accor-
dance to the seed that populates the spectra.
For instance, the statistical parameters that characterize a wave spectra are usually
the spectral peak period, the main direction and the wave height. For each simulation
run with a different seed number, the offshore unit will also respond differently in time
domain.
In this study several 3h simulations with different seeds for the same sea state are
run. By doing this procedure, it is possible to establish a probabilistic function that gives
the likelihood of obtaining a certain response value.
For each 3h simulation, the maximum response value is taken to create the curve.
The used method for building the probabilistic function out of empirical data is a Gumbel
fit that describes the probability of obtaining a response value for a specific sea state.
The number of 3h simulations to be run indicates the precision obtained to fit
the probabilistic curve. In other words, the larger the number of simulations ran, the
larger the number of points that were evaluated to create the curve. This is useful to get
a better understanding of the extreme values that define the probabilistic distribution
since its tail will be better discretized.
Convergence is an important aspect when doing a probabilistic function fit with
empirical data. Response values related to large quantiles, require a larger number of
simulations to be precise. For example, response values for the mean response may require
about twenty simulations, whereas values related to high quantile values i.e Q80% may
require more than fifty runs. In this study, the convergence criterium used is of a 1%
difference in value of the turret loads for Q90%. This translates into about 100 seeds.
This leads to the conclusion that investigations using this methodology can become
very expensive (CPU time) due to the number of simulation runs they require.
18
3.4.4 Gumbel Probabilistic Model
The Gumbel distribution is a common probabilistic model used in engineering to
predict extreme values based on empirical data. After running the required number of
seeds for the environmental conditions to ensure convergence of the results, the Gumbel
fit can be made. Equation 3.16 shows the probabilistic density function of the Gumbel
distribution.
F (x;λ, β) = e−e−(x−λ)/β
(3.16)
Where λ is the location parameter and β is the scale parameter.
In the approach used for the calculations, the parameters are obtained from the
empirical data. In other words, they vary according to the data being analyzed. By
ordering the empirical values in increasing order, it is possible to create a distribution
of data. An index number is given to each response value as shown in Table 3.2. The
cumulative probability value in respect to the index is defined in Equation 3.17.
Table 3.2: Example of table that relates the response values with their indexes and
cumulative probability values. This table exemplifies one condition ran with 50 different
seeds.
Index Sorted response values Cumulative probability
1 2.11 E+4 0.010
2 2.28 E +4 0.030
. . . . . . . . .
50 5.56 E+4 0.990
PI =I − 0.5
N(3.17)
Where I is the index number and N the total number of seeds used.
After the variable values and their probabilities are correlated, a linear regression
is performed. The objective of the linear regression is to predict a response value from
an empirical data set. In this study, this is done by the fit of the values obtained through
the simulations and their corresponding probability as exemplified in Table 3.2. In order
19
to properly do a linear regression, it is necessary that the variables behave linearly to the
the probabilities. That is why the regression is actually made by analyzing the logarithm
of the probability. This is obtained by the linearization of Equation 3.16. See Equation
3.18.
X = λ− β · ln(ln(P )) (3.18)
Where X is one response value and P the probability attached to this value.
If well behaved, the distribution takes the shape of a line - y = ax + b. The
shape and scale parameters can be obtained once the linear regression is fit. The shape
parameter is the slope of the line (a) and the scale parameter the intercept in the y-axis
(b). Figure 3.1 shows an example of distributed data in increasing values.
Finally, the response values can be obtained by using Equation 3.18. The cumu-
lative probability shown in Table 3.2 represents the quantile. For instance, the value
5.56E+4 represented in the table is linked to the cumulative probability of 0.990 or Q
99%.
Therefore, the MPM would correspond to the value of the Q 37%, the median to
the Q 50%. The proposed method is then related to Q 90% of the probabilistic fit done
with this method.
Figure 3.1: Example of a Gumbel fit.
The x-axis corresponds to the ln(ln(P)), where P is the probability of the sorted
max value in the Gumbel distribution and the y-axis corresponds to the sorted values
themselves.
20
3.4.5 Return Period
The Return Period is a theoretical concept that can be understood as the inverse of
the average frequency of occurence of a given variable. For instance, a 100-year response
value represents a 1/100 chance of this value being exceeded in a year time.
In the analysis here made, the 100-year responses are evaluated based on an anual
event. In other words, an 100-year return period response has 1% chance of exceedence
in a year time.
The simulations made in this study are 3h-long. By considering an one year’s
time, there are 2920 ”pieces” of 3h. See Equation 3.19.
1 Year = 365 days = 365 days · 24hours
days= 8760 hours −→ 8760 hours
3 hours= 2920 (3.19)
Therefore, by considering the 100-year response (or 1% chance of exceendence) of
2920 groups of 3h, the likelihood for a specific RP event to happen can be described as
seen in Equation 3.20. Table 3.3 shows examples of probabilities of different RP values
for 2920 groups.
PRP =1
RP · 2920(3.20)
Table 3.3: Summary of the RP values and their probabilities considering 2920 simulations
of 3h in one year interval
Return Period Probability Probability
1 year 1/(1 · 2920) 3.42 E−4
10 years 1/(10 · 2920) 3.42 E−5
100 years 1/(100 · 2920) 3.42 E−6
10.000 years 1/(10.000 · 2920) 3.42 E−8
3.4.6 Long-term Variability
The short-term variability describes the variation in response within the same
environmental conditions. However, the sea states are in constant change. The long-
term variability is the term used to take into account such changes.
21
The sea states are considered to be constant for a 3h period. After such time has
passed, the environmental conditions are changed, thus the mean response values are also
changed. By grouping all the sea states in 3h constant conditions, there is a total of 2920
conditions for a one-year period.
The long-term variability can be better exemplified by the use of environmental
countours. Figure 3.2 [6] is an example of a significant wave height and spectral period
contour used to characterize the different combinations possible for the design
Figure 3.2: Example of an environmental contour [6].
Each curve of this contour represents a specific return period environmental condi-
tion. The contour in the middle (red line) is representative of the 100-year return period
wave characteristics. Each point on this contour represents a possible (Significant wave
heigt, Spectral peak period) combination to characterize the waves. As a result, every
possible combination on this line has the same probability of occurence. In this case,
3.42E−6 as seen in Table 3.3.
The computation of all possible points of the contour prove to be a very demanding
procedure since there are several possible combinations to be evaluated. This is further
22
emphasized by the need of several runs of the same 3h conditions to ensure that the
short-term variability is properly taken into consideration.
However, it is evident that some environmental conditions are more severe than
others. These not necessarily mean the waves with the largest significant heights but
also those that have peak periods close to the structure’s natural frequencies. Therefore,
only simulations for specific design points from the wave contour are actually considered
in the simulations. This ensures that the worst conditions are taken into account while
trying to minimize global computational time.
The environmental contours are created by the collection and analysis of metocean
data. This data is collected on the desired offshore installation sites. These empirical
statistical data are obtained throughout the years. Companies specialized in these type of
data collection and analysis are usually hired to give information about the environmental
conditions of the installation sites.
Table 3.4 correlates significant wave heights and spectral peak periods and indi-
cates the frequency each combination happens (or wave-period joint frequency). By using
a probabilistic analysis it is possible to obtain the 100-year RP Hs-Tp contours from the
metocean data. The procedure for computing such results is explained in [2].
Table 3.4: Joint frequency of significant wave height and spectral peak period [2]
23
3.4.7 Methodology for the 100-year RP Responses
The standard practice in the industry nowadays to compute the 100-year response
uses the 100-year environmental contours to pick the design points. All of these points
characterize the 100-year condition and are likely to yield the maximum response. A 3h
simulation is run for each design point in time-domain and the MPM or 37% value of
non-exceedance or Q 37% is computed. The highest MPM over all desing points is taken
as the 100-year return period. In some cases the Q 50% value is used.
Even though this method considers the long-term variability of the environmental
conditions, it may not fully take into account the sensibility of the short-term variability
for the 3h simulations. Which means that the response values obtained to describe the
100-year RP response may not be properly estimated. This does not mean that the
systems will necessarily fail due to the big safety factors involved in offshore projects.
Figure 3.3 summarizes the standard practice procedure. This procedure is applied for
each design point being evaluated on the contour. See Figure 3.2.
Figure 3.3: Standard practice flow chart
The convergence criterium varies with each project. For this project the conver-
gence criterium is explained in Section 3.4.3 Short-Term Variability. The metodology
24
used for the proposed method follows the same philosophy of the flow chart with the
exception that instead of the use of the MPM values for the computation of the tur-
ret loads, the Q 90% is used. In addition, the input environmental conditions are not
those characterizing the 100-year conditions but those of the 40-year RP environmental
conditions.
3.4.8 Exact Response
The exact response is a value used for the comparison with the results found
with the methodology explained in Figure 3.3 for all the design points evaluated on the
environmental contours (Figure 3.2).
The computation of the exact response can be done by using Equation 3.21 [1].
F (x) =
∫fLT (µ)FST (x|µ)dµ (3.21)
Where FST is the conditional short-term distribution of the environmental conditions and
fLT the long-term distribution that characterizes the environmental conditions.
This approach requires the computation of the long-term distribution fLT = fTp|Hs
of all possible combinations of the metocean data. This means running simulations for
all values in Table 3.4. In addition, it is necessary to compute the conditional short-term
responses to each environmental condition. This can be represented as FST = FX|TpHs.
With X being the response desired (ex. Turret loads).
Besides, it is necessary to run each condition several times to ensure the correct
accountability of the short-term variability. It is therfore a very demanding and time
consuming approach. This response is not computed in this report. Only the comparison
between the standard and proposed method is done.
It is important to mention that even with the name of exact response, the value
obtained by this method is not necessarily what the offshore unit will encouter during its
service life. This happens because the tools used to obtain this value are based on em-
pirical observations (Table 3.4) and may not correspond to the real values. Nevertheless,
it is a very robust approach that serves to give a anchor point for research studies.
25
3.4.9 IFORM Contour
An alternative approach to compute the exact response is the FORM (First-Order
Reliability Method) [6]. With this approach the integral (Equation 3.21) is transformed
to the Gaussian space where three independant variables are defined: U1, U2 and U3. See
Equation 3.22[6].
FHs(h) = φ(U1)
FTp|Hs(t|h) = φ(U2) (3.22)
FX|TpHs(t|h) = φ(U3)
Where φ() is the standard Gaussian distribution function.
These variables form a sphere in the Gaussian space. The radius of the sphere β
defines points which the probability density is constant. See Equation 3.23 [6].
β =
√∑i
U2i (3.23)
Where β is the radius of the sphere in the gaussian space and Ui are the three independent
variables that define the sphere.
With this method it is possible to obtain the extreme values for a specific X
variable. The specific extreme response for a given probability of exceedence or return
period is obtained through an iterative procedure. See [6] for a full explanation of the
method.
The IFORM (Inverse First-Order Reliability Method)[7] is an approach that uses a
similar concept from the FORM method but instead, it seeks to find the extreme values
from an already defined probability of failure. For instance, for the 100-year RP the
probability of exceedence of 2920 periods of 3h in a year is 3.42 · 10−6 as shown in Table
3.3.
The IFORM contour is defined by a circle in the Gaussian space when considering
two variables. See Figure 3.4. Every point within a given radius distance of the circle
represents a combination between two independent variables. Every point within the
same radial distance defines points of the same probability of exceedence or return period
(RP1, RP10, RP100 etc.). By transforming the Gaussian space circle into the physical
space, it is possible to obtain a curve that defines the responses for a given RP. Each
26
response (motion, tensions etc.) has its own curve, therefore the IFORM contour is result
dependant.
Figure 3.4: Example of a circle in gaussian space defined by two independent variables
[6]
An alternative application of the IFORM contour [1] relates the short-term sim-
ulations maximum responses with their respective return periods. All the points in the
contour correspond to an equivalent (Q,RP) combination that simulate the 100-year re-
sponse. This approach relates then, the RP directly with the short-term simulations’
quantiles. The objective of this study is to give an indication of which (Q,RP) combina-
tion to use. By doing this analysis, it is possible to arrive at the proposed combination
by [1] of using (Q,RP) = (90%,40-year RP). This approach [1] relates the Q and RP with
the Gaussian Variables U1 and U2 as shown in Equations 3.24 and 3.25.
RP =1
2920(1− Φ(U1))(3.24)
Q = Φ(U2) (3.25)
Where Φ is the standard normal distribution. The terms U1 and U2 correspond to the
variables in the Gaussian space. These variables are independent from one another.
27
As stated before, β (Equation 3.23) represents the radius of the sphere in which
all points possess the same probability of exceedence. By computing the inverse of of
the normal distribution using the probability of exceedence as the input, it is possible to
find the value of the radius in the gaussian space that represents the 100-year RP in the
physical space.
In the Gaussian space, the combinations between Φ(U1) and Φ(U2) have the shape
of a circle. A circle with the radius 4.50 indicates then, that all combinations of U1 and
U2 possible in this radius give the 100-year response period. See Equation 3.26[1] and
Table 3.5.
β =√U21 + U2
2 = Φ−1(
1− 1
292000
)≈ 4.5 (3.26)
Table 3.5: IFORM Summary table values
Gaussian Space Physical Space
Combination # Φ(U1) Φ(U2) RP Q
1 3.396 2.952 1 99.842%
2 3.581 2.725 2 99.678%
. . . . . . . . . . . . . . .
100 4.500 0.000 100 50%
All U1 and U2 possible combinations are then calculated. Φ(U1) gives information
about the RP, while Φ(U2) the associated Q for a specific RP. By reverting these com-
binations from the Gaussian space (U1, U2) to the physical space (RP, Q) it is possible
to create an IFORM contour that relates the short-term variability with the RPs. See
Figure 3.5.
This approach is useful as the contour is valid for all responses. Otherwise, there
would be a different plot for every response being evaluated. Every point on the contour
express a different combination of RP and Q that gives the 100-year response. Any
possible combination of (Q,RP) is a valid point for computing the 100-year response.The
bigger the quantile, the more severe is the response value for the system. Likewise, the
bigger the return period, the more severe are the environmental conditions.
28
Figure 3.5: IFORM Contour
Figure 3.5 is coherent with the choice of using a Q 90% with 40-year return period
for environmental conditions [1]. As previously stated, simulating all possible combina-
tions is time consuming both due to the large number of points but also because for higher
Q values, it is necessary to run several 3h simulations to guarantee results convergence.
3.4.10 Reference Article/New Method summary
An alternative (Q,RP) combination the 100-year return period response compu-
tation is proposed by BV [1]. This method proposes the use of the 40-year return period
environmental condition. In addition, it uses a Q 90% for the short-term simulations,
which is in accordance with the IFORM contour, see Figure 3.5. Finally, they also propose
a 1.04 correction factor on the response.
29
Chapter 4
Numerical Studies
4.1 Software Versions
The software versions used during the project are stated below:
• HydroStar For Experts V7.22 [8]
• ARIANE 7.0.2 [9]
• OrcaFlex 10.2 [10]
4.2 Hydrostar-ARIANE-OrcaFlex Methodology
The geometry file is used as an input in Hydrostar. This software performs radia-
tion/diffraction computations and creates the vessel’s hydrodynamics database, including
RAOs and QTFs. The output is used as input in ARIANE.
After importing the database into ARIANE, the mooring lines are set. Besides, the
user inputs the parameters that describe the environmental parameters (waves, wind and
current). After the correct configuration of these parameters, the equilibrium position
and time domain simulations can be run. The outputs (first-order motions) are used in
OrcaFlex.
ARIANE solves the time-domain motions for the FPSO’s six degrees of freedom.
However, some assumptions are made: line-dynamics are not considered, lines stay in the
vertical plane and environmental conditions (waves, current and wind) are not applied
to the lines. In ARIANE, it is possible to compute the motions based on the low-
frequency excitation values. However, it also offers the possibility of using a unified
30
low-high frequency resolution [11]. This approach considers the coupling of both the
low frequency and the wave frequency motions, which gives a more accurate response.
This is the chosen approach. Table 4.1 shows the parameters taken into account in its
computations when considering only the low-frequency motions or the total motion.
Table 4.1: Table that summarizes the loads taken into account when computing the
motions in ARIANE [11]. This is a general table that indicates ARIANE’s computational
capacities. It does not represent all loads computed in this study.
It is necessary to use the OrcaFlex software because ARIANE does not use line
dynamics in its computations. Therefore, line loads are not accurate enough. By imposing
the time domain motions obtained in ARIANE into the OrcaFlex model, the software is
capable of calculating the turret loads while taking into account the line dynamics. The
entrapped water contribution is also implemented in the OrcaFlex model.Figure 4.1 shows
the numerical work flow procedures of the Hydrostar-ARIANE-OrcaFlex investigations.
The simulations performed in ARIANE and OrcaFlex are time domain simulations
that last for three hours. A transient of 30 minutes is used to take into account a steady
sea-state. A total of 100 seeds with the same environmental conditions are used for every
return period. This allows for Fh and Frad results to converge.
31
Figure 4.1: Summary chart describing the work flow of the Hydrostar-ARIANE-OrcaFlex
investigations. The square boxes represent the software used. The rounded-edged boxes
represent the type of computation performed. The ellipses represent the output from the
computations.
4.3 100-year Return Period Configuration
The 100-year environmental contours are ready-to-use at the beginning of the
project. As a result, there are several design points that can be picked from this contour.
Since the investigations are centered on the turret loads, the environmental conditions
that give the biggest responses for Fh and Frad are the conditions that will be chosen as
the 100-year design point.
Table 4.2 indicates the governing environmental conditions that are chosen as the
100-year RP environmental desing points.
32
Table 4.2: 100-year return period turret loads reference environmental conditions
Wave Heading [deg] 240
Hs [m] 14.0
Tp [s] 14.3
Wind Heading [deg] 210
Ws [m/s] 32.0
Current Heading [deg] 195.0
Cs [m/s] 1.23
4.4 40-year Return Period Configuration
The usual way of defining the environmental contours is by using the hindcast
data. However, due to the limited time available it is chosen to interpolate between the
1, 10, 100 and 10.000 year contours are already defined to create the 40-year RP wind
and wave contour. This approach is not standard and a more criterious evaluation of the
40-year RP contours should be done in future works.
The environmental conditions calculated consider mainly a wave/wind governing
condition. As a result, the current RP of 10 years is maintained for the alternative
approach as well. Table 4.3 summarizes the RP used for each environmental parameter.
Table 4.3: Summary of Metocean Return Periods.
Parameter Standard Method Proposed Method
Waves 100-year RP 40-year RP
Wind 100-year RP 40-year RP
Current 10-year RP 10-year RP
33
4.4.1 Design Points
Once the waves, wind and current design points for the 40-year RP environment
are defined, it is possible to compute the turret load responses and compare with the
standard approach. Table 4.4 summarizes the environmental design point for the 40-year
RP environmental conditions. The environmental headings correspond to the same as
those of the 100-year RP as seen in Table 4.2.
Table 4.4: 40-year return period design environmental conditions
Parameter Design Point 1 Design Point 2 Design Point 3
Wave Heading [deg] 240 240 240
Hs [m] 11.9 12.7 11.0
Tp [s] 13.0 14.0 12.0
Wind Heading [deg] 210
Ws [m/s] 31.3
Current Heading [deg] 195
Cs [m/s] 1.23
Since the 1-year and the 100-year RP conditions have the same wave, wind and
current directions that give the maximum values for the turret loads, it is chosen to
conserve these environmental headings. As previously stated, it is also chosen to keep
the 10-year RP condition for the current parameter.
34
Chapter 5
100-year RP Turret Loads
Investigations
5.1 Hydrostar-ARIANE-OrcaFlex Investigations
The main goal of this section is the computation of the 100-year turret load re-
sponses with the standard and proposed methods. Figure 5.1 shows the OrcaFlex model
comprising the turret and underwater cables.
Figure 5.1: OrcaFlex model.
35
5.1.1 Standard Method Investigations
The design point used to characterize the 100-year RP environmental conditions is
described in Table 4.2. This combination of environmental conditions simulate the most
critical environmental characteristics that give the maximum response for the turret loads.
It is chosen to run a batch with 100 different seeds in ARIANE to ensure con-
vergence (1% difference in value) for Frad. Then, the motions from every simulations
are inputted in OrcaFlex. After running all 3h simulations taking the line dynamics into
account, a Gumbel fit is made. The statistical analysis allows the computation of the
MPM value for the Fh parameter and the Q90% for Frad. Table 5.1 gives the values
found with the 100 seeds computed.
Table 5.1: Fh and Frad values obtained with the standard approach.
Parameter Quantile Investigations
Fh [kN] 37% 3.07 E+04
Frad [kN] 37% 2.99 E+04
5.1.2 Prosed Method Investigations
The design points used to characterize the 40-year RP environmental conditions
are shown in Table 4.4. A total of 100 seeds are used to obtain the Fh and Frad responses.
Table 5.2 summarizes the findings from the OrcaFlex simulations.
Table 5.2: 40-year return period turret load responses
Parameter Q Design Point 1 Design Point 2 Design Point 3
Fh [kN] 37% 2.48 E+4 2.58 E+4 1.95 E+4
Fh [kN] 90% 3.86 E+4 3.77 E+4 2.69 E+4
Frad [kN] 37% 2.53 E+4 2.49 E+4 2.18 E+4
Frad [kN] 90% 3.65 E+4 3.49 E+4 2.64 E+4
It can be seen that the design point 1 gives the most extreme responses with the
exception of Fh Q 37%. Therefore, design point 2 is chosen for Fh, otherwise design point
36
1 is used. Design point 3 is not used.
Another investigation that is done is the variation of the mean environmental
heading direction. This check looks for the wind, wave and current directions that give
the maximum turret loads responses. Two additional investigations are made for design
points 1 and 2. For each point, the environmental parameters are shifted ±15 degrees.
Table 5.3 summarizes the environmental directions analyzed. These give a package of
four simulations : ±15 deg for design point 1 and ±15 deg for design point 2.
Table 5.3: Summary of environmental directions evaluated for design points 1 and 2.
Environmental parameter Initial directions + 15 degrees - 15 degrees
Waves 240 255 225
Wind 210 235 195
Current 195 210 180
A total of 100 seeds are used for each investigation with design points 1 and 2.
Results can be seen in Table 5.4. This Table contains all turret load values obtained from
both design points 1 and 2 and it also considers the variation in environmental direction
shown in Table 5.3.
The values in bold represent the maximum values obtained for each parameter.
All maximum values are obtained with the initial direction case, except for Frad - Q
90%. However, the difference in value between the +15 degrees and the initial value is
of 0.97%. So as to simplify the procedure and due to this small discrepancy, the initial
direction is considered to give the maximum response.
As a result, all maximum values are obtained when the environment has the same
directions as in the reference. Likewise, all maximum values are obtained with design
point 1, except for Fh - Q37%.
The Fh Q37% and Frad Q90% give values smaller than the reference since they
consider the same quantile value for the short-term simulations but a less severe sea state
condition. However, by comparing the Fh Q90% for 40 RP with Fh Q37% for 100 RP
the new found values are indeed larger.
37
Table 5.4: Summary of turret load values with all environmental directions evaluated for
design points 1 and 2.
Initial direction
Load parameter Q Design point 1 Design point 2
Fh [kN] 37% 2.48 E+4 2.58 E+4
Fh [kN] 90% 3.86 E+4 3.77 E+4
Frad [kN] 37% 2.53 E+4 2.49 E+4
Frad [kN] 90% 3.65 E+4 3.49 E+4
-15 degrees
Load parameter Q Design point 1 Design point 2
Fh [kN] 37% 2.29 E+4 2.33 E+4
Fh [kN] 90% 3.35 E+4 3.54 E+4
Frad [kN] 37% 2.35 E+4 2.33 E+4
Frad [kN] 90% 3.19 E+4 3.32 E+4
+15 degrees
Load parameter Q Design point 1 Design point 2
Fh [kN] 37% 2.47 E+4 2.47 E+4
Fh [kN] 90% 3.83 E+4 3.60 E+4
Frad [kN] 37% 2.56 E+4 2.43 E+4
Frad [kN] 90% 3.54 E+4 3.47 E+4
To fully apply the method proposed by [1], there is still a correction factor of 1.04
to be applied to the answers. Table 5.5 compares the reference values with the results
from the OrcaFlex simulations with the correction factor. The discrepancy is calculated
according to Equation 5.1. Where the reference values used are the ones obtained with
the standard method for comparison reasons.
Discrepancy =Result− ReferenceValue
ReferenceValue(5.1)
38
Table 5.5: Results comparison with the two proposed methods.
Standard Method Proposed method
Variable Q Values Q Results Discrepancy
Fh [kN] 37% 3.07 E+4 90% 4.01 E+4 +31%
Frad [kN] 37% 2.99 E+4 90% 3.79 E+4 +27%
For Fh, the new method evaluates the 100-year response as 31% larger than the
reference value. Likewise, new method predicts a 27% larger response value for Frad if
compared with the 100 RP MPM method.
For Frad, the 40-year RP using Q90% and the 1.04 correction factor give bigger re-
sponse values than the 100-year RP Q37%. This could indicate that the current practices
underestimate the 100-year RP responses as claimed in [1]. However, their conclusion is
only drawn and supported for the maximum line tensions.
5.1.3 Conclusion
In the BV article, they used Equation 3.21 with all environmental conditions
representing the 100-year responses and several short-term simulations to compute the
exact response. The values found with the new approach were compared with this exact
response. Furthermore, their approach only evaluates the TMax for the mooring lines, not
the turret loads.
When computing the maximum tensions for the mooring line, the BV article found
errors up to +30% in their computations using the standard design point method (RP
100, Q50%) and the exact response values. Similarly, the error found between the BV
method and the standard practice for our case is +31% for Fh. The fact that both [1] and
our simulations found similar values for the discrepancies between the different calculation
methods could indicate that the 40-year RP responses computed for the turret loads are
also closer to the exact response values. However, this consistency in the discrepancy
value may as well be a coincidence.
To properly verify if the new found 100-year response value is a more suited ap-
proach, the exact response is to be computed. If the reference value is closer to the exact
response, then the standard practice is the better option. Otherwise, the new approach is
39
the most suited method among the two. It does not mean however, that the new method
is the best method for the 100-year response prediction since that there may be other
points on the IFORM contour that better estimate the searched answer.
It is also possible that the design point may change with the evaluated offshore
system and the response parameters being searched. The BV article found a good design
point for the maximum tension of the mooring lines. The exact answer is to be calculated
as shown in Equation 3.21. This answer requires the computation of 3h short-term
simulations for all possible environmental conditions. Besides, there is also the need
to compute several seeds for every condition to ensure convergence. This approach is
extremely time consuming and can not be completed within the limited time frame of
the internship. Therefore, assuming that the approach proposed in the BV article can be
applied to the turret loads, the new method offers indeed a better design point. However,
this is still to be verified in the future by the comparison of the standard practice and
new method answers with the exact response.
Besides, the 40-year environmental contours are interpolated between known con-
tours (RP 1, 10, 100, 10.000). To simulate the exact 40-year RP conditions it is necessary
to derive the contour from a hindcast database.
Even though simplifications were made during the process and the response values
found can not be considered as the final 100-year response values, the investigations
created the opportunity for the subject to be further developed. Should this study be
picked up in the future and the more time consuming tasks performed, it will be possible
to verify or not the new approach for turret loads responses. Should the method be
confirmed as more precise than the standard practice, response values would be the
closest to the exact computation values, thus creating a more precise method of the
100-year RP responses computations.
40
Chapter 6
Conclusion
The objective of this study is to investigate the use of an alternative approach to
compute the 100-year Return Period response for turret loads. A design point approach
based on the IFORM contour is tested. This method proposed by BV [1] advises the
use of a 40-year RP environment for long-term conditions and the use of Q90% for the
short-term simulations, as well as a correction factor of 1.04.
The procedures are done with a combination of software: Hydrostar, ARIANE
and OrcaFlex.Three design points are picked based on the environmental contours avail-
able.Also, the environmental directions are varied for each design point to ensure that
the maximum response is found. These simulations use a 90% quantile value for the
short-term simulations, thus the decision to use 100 seeds for each case to ensure the
results converge.
On one hand, the investigations lead to a response that is coherent with the BV
claim of the new method, in other words, bigger than the reference values. On the other
hand, it is not possible to claim that this response gives the closest response value to the
100-year response.
To do a full evaluation of the values found, it is necessary to compute the exact
100-year response [6] (Equation 3.21). This computation requires the study of all envi-
ronmental conditions to take into account the long-term variability. Furthermore, it is
necessary to run each condition several times to account for the short-term variability.
This procedure is extensive and requires time. The initial goal was to perform this inves-
tigation but the time reserved for these calculations were not enough to pursue the inital
plan.
Extreme analysis studies are very comprehensive and even if the initially set goals
41
were not all fulfilled, progress was made. A preliminary study of the new method is
already made. The 40-year RP responses gave larger values than the standard 100-year
practice. However, in the current state of affairs it is an assumption to claim that the
new method gives indeed a more precise answer. To properly evaluate results, the exact
response must be computed.
6.1 Perspective
• The first procedure to be done is the computation of the exact 100-year RP response
based on the hindcast database. This would solve the question whether the new
approach is applicable to the turret loads and if they do indeed give a closer response
value to the exact computation.
• Computation of the 40-year RP environmental contour using the hindcast database.
• In the long-term, if the work done during the internship proves itself to be a better
method to compute the responses, it could maybe question the current standard
practices in these type of studies.
42
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