wave theory offshore structures

Offshore Engineering and Technology

Day1AM 1

THREE DAY SHORT COURSE

Course Presentation

An introduction on


INSTRUCTOR: DR. SUBRATA K. CHAKRABARTI

University of Illinois at Chicago Offshore Structure Analysis Inc.

Plainfield, ILLINOIS


Day1AM 2


Overview of the 3-Day Course

COURSE OUTLINE

---------- DAY 1 ----------

Morning Afternoon

Introduction to Design Environment &Construction: • History and current state of the art • Offshore environment • Design cycle of offshore structures • Construction and launching • Preliminary concept design

---------- DAY 2 ----------

Morning Afternoon

Design Tools and Evaluation Design Tools and Evaluation

• static and dynamic stability • structure response and sea-keeping • environmental forces

---------- DAY 3 ----------

Morning Afternoon

Design Tools and Evaluation Design validation and model testing

• probabilistic design • riser system • extreme load & strength & fatigue • Scaling laws & Model testing • anchoring and mooring system • Challenges in deepwater testing • Assessment of validation methods


Day1AM 3

History and current state of the art of offshore structures Definition of Offshore Structures

No fixed access to dry land May be required to stay in position in all weather conditions

Functions of Offshore Structures

♦ Exploratory Drilling Structures: A Mobile Offshore Drilling Unit [MODU] configuration is largely determined by the variable deck payload and transit speed requirements.

♦ Production Structures: A production unit can have several functions, e.g. processing, drilling, workover, accommodation, oil storage and riser support.

♦ Storage Structures: Stores the crude oil temporarily at the offshore site before its transportation to the shore for processing.


Day1AM 4

Types of Offshore structures

Fulmar jacket platform

Jack-up Drilling Unit with Spud Cans

DERRICKHELIDECK

SPUD CAN

LEG

CANTILEVERBEAM

QUARTERS

JACKINGUNIT

HULL


Day1AM 5

Historical Development of Fixed Offshore Structures

Gravity Base Structures placed on the seafloor and held in place by their weight

Troll A gas platform, world's tallest concrete structure in North Sea


Day1AM 6

Structures piled to the seabed

500,000 barrel oil storage gravity & piled structure in Persian Gulf

Compliant Tower

Deck

Fw (t)

Tower Frame

Ungrouted pilesliding inside thetower leg or pileguides

Pile top weldconnected tothe tower at top


Day1AM 7

Floating Offshore Structures moored to the seabed

Articulated Towers Ready for Tow in the 1970s

Floating structure types Tension Leg Platform: vertically (taut) moored (by tendons) compliant platform


Day1AM 8

Semisubmersible Platform: Semisubmersibles are multi-legged floating structures

Marine 7000 4th generation semi-submersible

Floating Production Storage and Offloading (FPSO) ship shaped with a turret

Balder: first "permanent" FPSO in the North Sea

Internal bow turret Cantilevered bow turret


Day1AM 9

Deepwater floater types

Ultra deepwater (>1500m) wells drilled in the Gulf of Mexico

1 1 1 37

12 11

37 36

56

0

10

20

30

40

50

60

No.

of W

ells

Spu

dded

1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

Year

Compliant TLP Semi SPAR FPSO

Tension leg platforms installed as of 2002


Day1AM 10

Progression of Spars

5th generation drilling semi-submersible “Deepwater Nautilus”

SeaStar MiniTLP for Typhoon Field


Day1AM 11

Failure of Offshore Structures Global failure modes of concern are

- capsizing/sinking - structural failure - positioning system failure

Examples of accidents which resulted in a total loss.

Alexander Kielland platform --fatigue failure of one brace, overload in 5 others, loss of column, flooding into deck, and capsizing Ocean Ranger accident sequence -- flooding through broken window in ballast control room, closed electrical circuit, disabled ballast pumps, erroneous ballast operation, flooding through chain lockers and capsizing. Piper Alpha suffered total loss after: a sequence of accidental release of hydrocarbons, as well as escalating explosion and fire events.


Day1AM 12

Structural damage due to environmental loads

Accident of P-36 semi-submersible

P-36 sank after 6 days after: an accidental release of explosive gas, burst of emergency tank, accidental explosion in a column, progressive flooding, capsizing and.

Gulf Oil Platform -- ThunderHorse Saturday, July 30, 2005

(a) Before hurricane hit (b) After hit The BP Thunderhorse semi took a hit from Hurricane Dennis listing 20-30 degrees with the topsides almost in the water. The platform costs over a billion dollars to build.


Day1AM 13

Design cycle of offshore structures

Design Spiral (API RP2T)


Day1AM 14

Preliminary concept design

Factors Affecting Field Development Concepts and Components


Day1AM 15

Offshore environment


Day1AM 16

Construction process, launching and operations

Construction of Caisson


Day1AM 17

Running Pile from Derrick Barge


Day1AM 18

Construction of the Spar (Technip Offshore)

Seastar® fabrication, Louisiana (SBM Atlantia)

Offloading of Spar Hull for Transport (Dockwise)


Day1AM 19

Truss Spar in Dry Tow

Holstein Spar being towed out of site

Brutus TLP


Day1AM 20

Spar Upending Sequence (Kocaman et al, 1997)


Day1AM 21

Spar Mooring Line Hookup (Kocaman et al, 1997)

Derrick Barge Setting Deck on Spar Platform (Technip Offshore)


Day1AM 22

Holstein Spar in place being outfitted

ThunderHorse being launched

ThunderHorse being Towed to place


Day1AM 23

Installation of the lower turret on FPSO Balder

APL Buoy Turret System


Day1AM 24

Rendition of the new bridge left of the existing bridge

Final Drafts of Caisson Showing the Mooring Lines Attached


Day1AM 25

Current Flow past New Caisson behind the Roughened Pier

Location of the Immersed Tunnel of the Busan-Geoje Fixed Link


Day1AM 26

Immersion of TE into trench

Physical Testing of Rigs Moored above the Trench


Day1AM 27

OCEAN ENVIRONMENT

Standing Wave vs. Progressive Wave

Linear Theory (also called Stokes First Order, Airy)

Stokes Second Order Theory

Stokes Fifth Order Theory

Stream Function Theory

Measured Wave Properties in Wave Tank

Offshore Applications of Wave Theory

Current -- Uniform and Shear

Wave-current Interaction

Wind and Wind Spectrum

1NOTE: Chapter 3, Hydrodynamics of Offshore Structures


Day1AM 28

Idealization of Waves

Actual Ocean Wave

Simple Idealizations to Regular Waves

Sine Wave

-1

0

1

0 90 180 270 360

Angle, deg.

Am

plitu

de

Nonlinear Regular Wave

-1.25

0.00

1.25

0 90 180 270 360

Angle, deg.

Am

plitu

de


Day1AM 29

STANDING WAVE VS PROGRESSIVE WAVE

Standing wave: Oscillation of waves generated in confined boundaries with no forward speed. Variation with space and time may be separated in a mathematical description. Examples include wave tank, liquid storage tanks, harbors, lakes etc. Space and time are uncoupled: sin kx sin ωt – some points do not see any waves

-1

0

1

0 90 180 270 360

Angle, deg.

Am

plitu

de

Standing wave at times t1 and t1 + T/2

Progressive wave: Waves have a forward speed in addition to oscillation. Variation with space and time is combined in a mathematical description. Waves generated in the open oceans are progressive waves. We discuss progressive waves here. Space and time are dependent: sin (kx -ωt) – all points see maximum waves

-1

0

1

0 90 180 270 360

Angle, deg.

Ampl

itude

Progressive wave at two times t1 and t2


Day1AM 30

WAVE THEORIES

Three Independent Parameters describe all waves so that they must

be specified for a wave. They are:

water depth, d

wave height, H

wave period, T

d: uniform water depth based on flat bottom

T: fixed interval of time at which a succession of wave crests

passes a given point

L: wave length i.e., length between two consecutive crests

c: celerity or velocity of wave propagation

c = L/T


Day1AM 31

WAVE THEORY DEVELOPMENT

The theory is described in terms of

A Boundary Value Problem -- BVP consisting of

A differential equation

certain boundary conditions

it will be shown that

A complete solution of BVP is not possible without approximation.

Approximate solution is obtained through simplified assumptions

Two general types of approximate theory:

(1) wave height as perturbation parameter (deep water)

(2) water depth as perturbation parameter (shallow water)

The theory is formulated in terms of

either Potential Function, Φ

or Stream Function, Ψ


Day1AM 32

Basic Assumptions

• Fluid is incompressible

Fluid density = ρ. Fluid is incompressible if ρ is constant. Then the variation of density with time, t is zero:

0=∂∂

tρ

since we are dealing with water it is considered incompressible.

• Flow is continuous

• Flow is irrotational

Starting points for the BVP are the following

• Equation of Continuity

• Bernoulli's Equation

Definitions of following important variables to follow:

• Stream Function

• Potential Function


Day1AM 33

TWO CLASSES OF WAVE THEORIES

A. Closed Form Linear wave theory

Stokes higher-order wave theory B. Numerical Solution

Stream Function theory -- Regular -- Irregular

Properties sought: Fluid particle velocity/acceleration components

Dynamic pressure of fluid particle

x,u,i~z,w,k~

y,v,j~

u, v, w = velocity components i, j, k = unit vectors along x, y, z

Equation of Continuity 2D 3D

0=∂∂

+∂∂

+∂∂

zw

yv

xu

u1

v1

u2

v2

∆y

∆x

u1

v1

u2

v2

w1

If the above equation is satisfied, then the Continuity of flow is maintained


Day1AM 34

Stream Function

existence of Ψ ⇔ continuity

Ψ = constant defines a streamline


Day1AM 35

Potential Function -- Irrotational Flow The concept of irrotationality is somewhat abstract/mathematical, but is one of the most important in hydrodynamics. Flow is called rotational, if each fluid particle undergoes a rotation, in addition to translation and deformation.

existence of Φ ⇔

irrotationality

in terms of Φ

Continuity Equation becomes ⇒

02

2

2

2

2

22 =

∂Φ∂

+∂

Φ∂+

∂Φ∂

=Φ∇zyx

This is the governing differential equation known as LaPlace’s equation

Similarly, irrotationality ⇒

02 =Ψ∇


Day1AM 36

The definition of potential function yields the wave particle kinematics

We still need a description for the wave particle dynamic pressure.

It comes from the Bernoulli's Equation which is derived from the Navier Stokes

equation under simplified assumptions .

Assumptions are made of

• Irrotational flow, and

• Perfect fluid – no change in viscosity

Then the wave particle pressure has the expression

∂Φ∂

+

∂Φ∂

+

∂Φ∂

−∂Φ∂

−−=222

zyx21

tgyp ρρρ

hydrostatic linear dynamic nonlinear dynamic


Day1AM 37

Example: Consider ( ) exp( )y A kyΦ = with amplitude A = 10 ft, and wave number k = 0.01,. Compute the three pressure components for this wave.

-100

-80

-60

-40

-20

0

0 2000 4000 6000 8000

Hydrostatic Pressure, psf

Dep

th, f

t

p1

Hydrostatic

-100

-80

-60

-40

-20

0

-8.5 -6.5 -4.5 -2.5 -0.5Linear Pressure, psf

Dep

th, f

t

p2

Linear hydrodynamic

pressures

-100

-80

-60

-40

-20

0

-0.015 -0.01 -0.005 0Nonlinear Pressure, psf

Dep

th, f

t

p3

Nonlinear hydrodynamic

pressures


Day1AM 38

Two-dimensional wave motion

Note that the wave theory is developed over a flat bottom

y is measured up from the SWL

thus y is negative below the SWL to the ocean floor

s is measured up from the floor so that

s = y + d


Day1AM 39

Boundary Value Problem

In terms of velocity potential Φ (x,y,z,t), the following equations are obtained in the presence of a body in fluid: Differential Equation

02 =∇ Φ -- Laplace's equation

Bottom boundary condition

0=∂∂

yΦ

at y = -d

Fixed body surface condition

0=∂∂

nΦ

on the fixed surface

Moving body surface condition

nVn

=∂∂Φ

on the moving surface


Day1AM 40

Linear Wave Theory Development

The following assumption is made:

Waves are two-dimensional Problem is posed as determining:

either, the velocity potential or, the stream function

Perturbation technique

Express Φ as a power series in wave slope (i.e., wave height/ wave length)

Perturbation parameter LHkH πε ==

2

Then, the velocity potential is expressed as a series

nn

nΦεΦ ∑

∞

=

=1 −− Φn is the nth order potential

similarly, the free surface profile

nn

nηεη ∑

∞

=

=1 −− ηn is the nth order profile


Day1AM 41

Linear Wave Theory (solution is obtained in a closed form) Dispersion relationship (relates the wave length or wave number to the wave frequency, wave period or wave speed)

kdkgc tanh2 =

or kdgk tanh2 =ω or kdTgL tanh2

2π=

Horizontal particle velocity

)](cos[coshcosh

2ctxk

kdksgkHu −=

ω

Vertical particle velocity

)](sin[coshsinh

2ctxk

kdksgkHv −=

ω

Horizontal particle acceleration

)](sin[coshcosh

2ctxk

kdksgkHu −=&

Vertical particle acceleration

)](cos[coshsinh

2ctxk

kdksgkHv −−=&

Dynamic pressure

)](cos[coshcosh

2ctxk

kdksHgp −= ρ


Day1AM 42

Deepwater Criterion and Wave Length

Certain simplifications in the mathematical expressions describing properties of linear waves may be made if the water depth is very deep or very shallow. The following table describes these criteria:

Depth of Water Criterion )tanh(kd Deep water Deep water d/L = > 1/2 1 π2/2

0 gTL =

Shallow water d/L < = 1/20 kd gdTL =

Intermediate water depth

1/20 < d/L < 1/2 [ ] 2/100 )/2tanh( LdLL π=

For example, in deep water Horizontal particle velocity becomes

)](cos[)exp( ctxkkyTHu −=

π

Note the exponential decay term with respect to the vertical coordinate y. Similar expressions follow for the other quantities.


Day1AM 43

Wave Particle Orbits and Velocity Amplitude Profiles

(c) SHALLOW WATER

201

Ld

<


Day1AM 44

Sample Calculation of Linear Wave Properties

Water depth, d = 100 ft Wave Height, H = 20 ft Wave Period = 10 sec Elevation, y = 0 ft Deep water wave length

2 20 / 2 5.1248 512.48L gT T ftπ= = =

k0 = 0.01226

d/L0 = 0.1951

d/L = 0.2210

k = 0.01389

sinh kd = 1.88; cosh kd = 2.13

um = π.20/10 x 2.13/1.88 = 7.12 ft/sec

vm = π.20/10 = 6.28 ft/sec

mu& = 2π/10 x 7.12 = 4.47 ft2/sec

mv& = 2π/10 x 6.28 = 7.12 ft2/sec


Day1AM 45

Time History of Linear Wave Properties

Note the phase relationships among variables


Day1AM 46

Exercise 1 Consider a regular wave of 20 ft and 10 sec. in 400-ft water depth. Using linear theory compute the amplitude of velocity and acceleration profiles with depth. Calculations are carried out on an excel spreadsheet

constants unit finite depth unit 2π = 6.2832 d = 200 ft g = 32.2 ft^2-s H = 20.0 ft ρ = 1.98 slug/ft^3 T = 10 s

dy = -10 ft ω = 0.6283 1/s Lo = 512.48 ft

ko = 0.0122604 1/ft Lprev = 504.93 ft L = 504.93 ft k = 0.0124436 1/ft d/L = 0.3961 gkH/2ω = 6.377 ft/s

Const Term = 1.052

amplitudes amplitudes y u v u dot v dot ft ft/s ft/s ft/s^2 ft/s^2

0.0 6.38 6.29 4.01 3.95 -10.0 5.64 5.54 3.54 3.48 -20.0 4.99 4.88 3.14 3.07 -30.0 4.42 4.30 2.78 2.70 -40.0 3.92 3.78 2.46 2.37 -50.0 3.48 3.32 2.19 2.08 -60.0 3.09 2.91 1.94 1.83 -70.0 2.75 2.55 1.73 1.60 -80.0 2.46 2.22 1.54 1.40 -90.0 2.20 1.93 1.38 1.21 -100.0 1.98 1.67 1.24 1.05 -110.0 1.78 1.44 1.12 0.90 -120.0 1.62 1.23 1.02 0.77 -130.0 1.48 1.04 0.93 0.65 -140.0 1.36 0.86 0.85 0.54 -150.0 1.26 0.70 0.79 0.44 -160.0 1.18 0.55 0.74 0.34 -170.0 1.13 0.40 0.71 0.25 -180.0 1.08 0.26 0.68 0.17 -190.0 1.06 0.13 0.67 0.08 -200.0 1.05 0.00 0.66 0.00


Day1AM 47

Kinematic amplitudes with depth

d = 200H = 20T = 10

-200

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

0.0 2.0 4.0 6.0 8.0u, v

y

uv

d = 200H = 20T = 10

-200

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

0.0 1.0 2.0 3.0 4.0u, v

y

u dotv dot


Day1AM 48

Non-dimensional horizontal particle velocity

kddskd

HTuu

sinh)/(cosh

)2/(20

0 ==π

Non-dimensional vertical particle velocity

kddskd

HTvv

sinh)/(sinh

)2/(20

0 ==π

Non-dimensional dynamic pressure

kddskd

Hgpp

cosh)/(cosh

)2/(0

0 ==ρ

0.0

0.5

1.0

1.5

1.0 2.0 3.0 4.0 5.0kd

ubar

s/d = 1.0s/d = 0.9s/d = 0.8s/d = 0.7s/d = 0.5s/d = 0.2

horizontal particle velocity


Day1AM 49

0.0

0.4

0.8

1.2

1.0 2.0 3.0 4.0 5.0kd

vbar

s/d = 1.0s/d = 0.9s/d = 0.8s/d = 0.7s/d = 0.5s/d = 0.2

vertical particle velocity

0.0

0.4

0.8

1.2

1.0 2.0 3.0 4.0 5.0kd

pbar

s/d = 1.0s/d = 0.9s/d = 0.8s/d = 0.7s/d = 0.5s/d = 0.2

linear dynamic pressure


Day1AM 50

Comparison of Linear Wave Pressures with Tank Measurements

Water depth = 10 ft Wave Periods = 1.25 – 3.0 sec


Day1AM 51


22

1 ΦεΦεΦ +=

Wave profile

)(2cos]2cosh2[sinhcosh

8

)cos(2

3

2

tkxkdkdkd

LH

tkxH

ωπ

ωη

−++

−=

Horizontal particle velocity

)(2cossinh

2cosh43

)cos(sinhcosh

4

2

tkxkdks

TH

c

tkxkdks

THu

ωπ

ωπ

−

+

−=

Vertical particle velocity

)(2sinsinh

2sinh43

)sin(sinhsinh

4

2

tkxkdks

TH

c

tkxkdks

THv

ωπ

ωπ

−

+

−=

Dynamic pressure

[ ]

2

2

2

cosh cos[ )]2 cosh

3 1 cosh 2 1 cos 2( )4 sinh 2 sinh 3

1 1 cosh 2 14 sinh

H ksp g kx tkd

H ksg kx tL kd kd

Hg ksL kd

ρ ω

πρ ω

πρ

= −

+ − −

− −


Day1AM 52

First and second order components

Wave profile over one cycle

-1.25

0

1.25

0 90 180 270 360

Angle, deg.

Ampl

itude

1st2nd1st + 2nd

NOTE:

The higher order components yield the vertical asymmetry in the wave

profile.


Day1AM 53

Example of First and Second Order

Horizontal Particle Velocity Components

Wave Height = 4 ft (1.22m)

Wave Period = 2 sec

Water Depth = 8 ft (2.44m)

The second order componet is less than 4 % of the first order velocity in this example.


Day1AM 54

Stokes Fifth Order Theory

55

44

33

22

1 ΦεΦεΦεΦεΦεΦ ++++=

)(coscosh5

1tkxnnksuu

nn ω−= ∑

=

first term: )cos(cosh tkxks ω−

second term: )(2cos2cosh tkxks ω−

third term: )(3cos3cosh tkxks ω−

fourth term: )(4cos4cosh tkxks ω−

fifth term: )(5cos5cosh tkxks ω−

Note the presence of five frequencies in the profile.


Day1AM 55

Example Given d=100 ft. H= 50 ft. T = 10 sec. Calculated L=502.219 ft.

θ = 0 20 40 60 80 100 120 140 160 180 η= 32.21 27.21 16.36 5.44 -2.36 -8.36 -12.65 -15.26 -17.05 -17.79

0 0 45 90 180 45 90 0 90 180 s u u u u v v p p p

132.21 28.73 125.00 25.77 343 113.93 11.87 15.4 110.00 20.74 1034 100.00 18.09 10.29 12.14 1511 94.06 -2.8 11.24 82.21 -10.37 80.00 14.07 8.44 -1.98 8.44 9.05 2518 1028 60.00 11.35 7.1 -1.23 -8.92 5.67 6.36 3596 2426 1621 40.00 9.6 6.19 -0.8 -8.01 3.5 4.04 4738 3774 3008 20.00 8.63 5.67 -0.59 -7.46 1.67 1.96 5938 5088 4351 0.00 8.31 5.5 -0.52 -7.28 0 0 7192 6379 5652

Note: The wave crest is at 132.21 ft while the wave trough is at 82.21 ft. The asymmetry in the horizontal velocity between the crest and the trough is remarkably pronounced.


Day1AM 56

STREAM FUNCTION THEORY

Two Types:

(1) Symmetric Regular (theoretical)

(2) Irregular (when the measured profile is known)

Coordinate System

moves with wave speed, c in the direction of the wave propagation

Horizontal velocity = u - c

If current is present, add the current velocity, U to the above

Horizontal velocity = u – c + U

For irregular waves,

The measured wave profile is used as an additional constraint

Solution taken in a series form:

∑=

+=N

nnkxnksnXcyyx

1cossinh)(),(Ψ

where N is the order of Stream Function wave theory

The unknowns X(n) are solved by using the dynamic free surface boundary condition


Day1AM 57

Order of Stream Function NORDER

Based on Wave Parameters d, H, T

This chart is useful in determining a suitable order of the stream

function theory in a particular wave case.


Day1AM 58

Nonlinear Regular Stream Function Wave Profile

compared with

Linear Wave Profile

Wave Height = 195.4 ft (59.6m)

Wave Period = 15 sec

Water Depth = 1152 ft (351.2m)

NOTE: in this figure h represents d = water depth


Day1AM 59

Irregular Stream Function Wave vs. Experimental Pressure Amplitude

d = 10 ft; T > 3.00 sec


Day1AM 60

Comparison of Computed Velocities with Measurements

Irregular Stream Function Theory

Water depth = 2.5 ft H/d = 0.7


Day1AM 61

Region of application of wave theories


Day1AM 62

PRACTICAL APPLICATION OF WAVE THEORIES

Wave Theories

Linear Theory

Low seastates (1 yr storm)

Swell

Large inertia dominated fixed and floating structures

Linear radiation damping

Fatigue analysis

Long term statistics


Slow drift oscillation of soft moored structures

TLP tendon analysis

Stokes Fifth Order or Stream Function Theory

Storm waves

Drag dominated structures

High wave tank data (irregular stream function)

Air gap or run-up

Deck waves

wave theory offshore structures

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