Empirical Estimation of Probability Distribution
of Extreme Responses of Turret Moored FPSOs
Presentation By:
Amir Izadparast
Authors:
Amir Izadparast
Arun Duggal
Background
Design Value
Extreme Statistics
Limited Data
Non-linearity
Extreme Value Analysis
Distribution
Model
Sample Data
Numerical
Experimental
Complexity of
data structure
Full Scale
Mixed Effects
Table of Content
•
•
•
•
•
•
•
Non-linear Responses of Turret Moored FPSOs
Probability Distributions
Distribution Parameters
Extreme Statistics
Case Studies
Results
Concluding Remarks
Non-linear Responses of Turret Moored FPSOs
Mooring Leg Tension
•
Windward
•
•
•
•
Low-Frequency
Wave Frequency
Total Tension
Leeward
•
•
•
Low-Frequency
Wave Frequency
Total Tension
Vessel Offset
•
Along the vessel length (X)
•
Low-Frequency
Sources of Non-linearity
•
•
•
•
Mooring System Stiffness
Loading Nature (low-drift forces, drag, etc.)
Environmental Condition (steep waves)
Damping
Probability Distributions
Normalized Random Variable
 
Model
Transformation

Linear Random Variable
(narrow-banded)
n 
2
2
 2

 n  A  B 
 2

Non-Linear Random Variable
n 

a  

2


1
2
 n      2  
Distribution
F  x   1  exp   x 2 2 
Rayleigh
F n  x   1  exp   x 
Exponential
 x

F n  x   1  exp     B  

 A
Stansberg
   x     
F n  x   1  exp   
  Weibull
    


     2 
 3-Par Rayleigh
F n  x   1  exp  
2


8


   2  4   x    
12
Distribution Parameters
Model
Linear Random Variable
(narrow-banded)
Distribution
Parameters
Rayleigh
,
Exponential
,
Stansberg
,
, ,
Weibull
,
, , ,
3-Par Rayleigh
,
, , ,γ
Non-Linear Random Variable
Extreme Statistics
Ordered Value Statistics Theory
(N independent cycles):
F max  x    F n  x  
N

Expected Maximum:
E  max    x dF max  x 
Asymptotic Distribution of Large N (Gumbel)
F max  x   exp  exp    x  a N  bN 



E  max   a N  bN  EM
Number of Cycles (N)
Wave frequency
Narrow-banded process
N  Tstorm Tz
Low frequency
Non-narrow-banded– Correlation
time - Stansberg’s formula
  1 2
Combined Process
Difficult to estimate
 bandwidth of the spectrum
Number of
observed cycles
Case Studies: General Info
Deepwater System
Shallow-water System
Water depth (m)
~2000m
~45m
Area
West Africa
South East Asia
100Yr Condition
Hs = 4.5m, Tp = 17sec,
Ws = 6.3m/sec
Hs = 10m, Tp = 16sec,
Ws = 32m/sec
Mooring System
3G*4L Taut mooring legs
4G*3L Catenary mooring legs
Mooring Legs
Chain-Polyester-Chain
Chain-Heavy Chain-Chain
Case Studies: Response Characteristics
10
4
10
4
Line T (Leeward)
Line T (Windward)
10
1
10
0
1
Deepwater System
0
-2
10
10
10
0
10
4
10
-1
f ( Hz )
10
4
Line T (Leeward)
Line T (Windward)
Shallow-water System
10
10
3
X Motion
2
10
1
10
0
10 -3
10
10
10
-2
-1
10
f ( Hz )
10
0
10
3
2
1
0
2
10
2
10 -3
10
2
2
2
3
S X ( f ) /  X (1/sec)
10
10
SX ( f ) /  X (1/sec)
10
X Motion
S T ( f ) /  T (1/sec)
2
ST ( f ) /  T (1/sec)
10
3
Case Studies: Response Characteristics
14
Shallow Water
Deepwater
12
FX /  FX
10
8
6
4
2
0
0
1
2
3
X/X
4
5
Results: Wave-Frequency
7
 n-wave ( P )
6
5
7
SAMPLE
RAYLEIGH
6
EXPONENT IAL
3PAR-WEIBULL
5
3PAR-RAYLEIGH
4
4
3
3
2
2
1
0 0
10
10
-2
10
10
-3
RAYLEIGH
EXPONENT IAL
3PAR-WEIBULL
3PAR-RAYLEIGH
1
Deepwater windward
-1
SAMPLE
-4
10
0 0
10
Deepwater leeward
10
-1
10
P
7
 n-wave ( P )
6
5
7
SAMPLE
RAYLEIGH
6
EXPONENT IAL
3PAR-WEIBULL
5
3PAR-RAYLEIGH
4
4
3
3
2
2
1
0 0
10
-1
10
10
-2
P=( 1 -u )
-3
10
(1
-4
10
0 0
10
10
-3
10
-4
)
SAMPLE
RAYLEIGH
EXPONENT IAL
3PAR-WEIBULL
3PAR-RAYLEIGH
1
Shallow-water windward
-2
Shallow-water leeward
-1
10
-2
10
P=( 1 -u)
-3
10
-4
10
Results: Low-Frequency
7
6
 n-low ( P )
5
4
7
SAMPLE
RAYLEIGH
6
EXPONENT IAL
EXP-ST ANSBERG
5
3PAR-WEIBULL
3PAR-RAYLEIGH
4
3
3
2
2
1
0 0
10
7
6
 n-low ( P )
5
4
-1
-2
-3
10
10
-4
10
0 0
10
7
SAMPLE
RAYLEIGH
6
EXPONENT IAL
EXP-ST ANSBERG
5
3PAR-WEIBULL
3PAR-RAYLEIGH
4
3
3
2
2
1
0 0
10
10
-2
10
P=( 1 -u )
10
-3
EXP-ST ANSBERG
3PAR-WEIBULL
3PAR-RAYLEIGH
Deepwater leeward
-1
10
-2
10
10
-4
0 0
10
-3
10
-4
10
SAMPLE
RAYLEIGH
EXPONENT IAL
EXP-ST ANSBERG
3PAR-WEIBULL
3PAR-RAYLEIGH
1
Shallow-water windward
-1
EXPONENT IAL
1
Deepwater windward
10
SAMPLE
RAYLEIGH
Shallow-water leeward
-1
10
-2
10
P=( 1-u)
-3
10
-4
10
Results: Total
7
6
n (P )
5
7
SAMPLE
RAYLEIGH
6
EXPONENT IAL
3PAR-WEIBULL
5
3PAR-RAYLEIGH
4
4
3
3
2
2
1
0 0
10
7
6
-3
10
10
EXPONENT IAL
3PAR-WEIBULL
3PAR-RAYLEIGH
-4
10
0 0
10
7
SAMPLE
RAYLEIGH
6
EXPONENT IAL
3PAR-WEIBULL
5
3PAR-RAYLEIGH
Deepwater leeward
-1
10
-2
10
-3
10
-4
10
SAMPLE
RAYLEIGH
EXPONENT IAL
3PAR-WEIBULL
3PAR-RAYLEIGH
4
4
n
n (P )
5
10
-2
RAYLEIGH
1
Deepwater windward
-1
SAMPLE
3
3
2
2
1
0 0
10
1
Shallow-water windward
-1
10
-2
10
P=( 1-u)
-3
10
-4
10
0 0
10
Shallow-water leeward
10
-1
10
-2
P=( 1 -u )
10
-3
10
-4
Results: Extreme Statistics
Model
Deep
water
Sample
Rayleigh
Exponential
3-Par. Rayleigh
3-Par. Weibull
Stansberg Exp.
Model
Sample
Shallow
water
Rayleigh
Exponential
3-Par. Rayleigh
3-Par. Weibull
Stansberg Exp.
Wave
Freq.
4.2
(5.4 - 3.3)
3.8
7.1
3.8
3.9
--
Windward
Low
Freq.
3.2
(3.5 - 2.8)
2.7
3.8
3.3
3.1
3.4
Wave
Freq.
5.8
(7.3 - 4.9)
3.8
7.3
6.3
5.6
--
Windward
Low
Freq.
4.1
(4.5 - 3.6)
3.2
5.2
4.7
4.4
4.5
Total
4.2
(5.2 - 3.6)
3.7
7.0
4.3
4.1
--
Total
5.8
(6.9 - 5.1)
3.7
6.8
6.4
5.7
--
Wave
Freq.
4.1
(5.3 - 3.4)
3.8
7.1
4.0
4.0
--
Leeward
Low
Freq.
3.1
(4.1 - 2.6)
2.8
4.0
3.1
3.1
3.5
Wave
Freq.
5.6
(5.8 - 5.4)
3.8
7.2
5.4
5.0
--
Leeward
Low
Freq.
5.0
(5.2 - 4.6)
3.2
5.2
4.8
4.5
4.5
Total
3.9
(4.9 - 3.3 )
3.7
7.1
3.7
3.8
--
Total
5.4
(6.4 - 4.4)
3.7
7.0
6.1
5.3
--
Concluding Remarks
•
The probability distribution of mooring leg tension and vessel offset in extreme
environmental condition were studied.
•
Two case studies of shallow water and deepwater turret moored FPSOs are
considered.
•
The characteristics of probability distribution of wave-frequency, low-frequency,
and the combined tension are studied.
•
The probability distributions of tension in the windward and leeward lines are
studied.
•
The performance of widely used distribution models as well as the three-parameter
Rayleigh distribution model is evaluated over the experimental data.
•
The effect of distribution model on the predicted extreme values is discussed.
Thank You!