Probabilistic stability design of a submerine pipeline system

ABS TRACT

This

paper

presents a method for probabilistic

stabil.ty design of submarine pipelines related to SIS

design criteria.

The procedure as such may also cover

[JIS

criteria

by

including

strain

response.

The

probability

distribution

of

accumulated

long-term

response is evaluated by

integration

of

conditional

short-term

distributions

over

the environment

para-meter range.

The

exceedance

probability

for given

displacement

levels

is determined from the resulting

cumulative distribution function, which typically

has

a-

slowly

decreasing upper tail.

The method is

illu-strated by application to a specific pipeline

system.

Varying the submerged pipe weight will affect both the

shape and location of the density function of

accumu-lated

displacements significantly. Sensitivity of the

response distribution to the main

sources

of

uncer-tainty is investigated.

The relative contributions to

the variability

of

long

term

displacement

arising

respectively

from

short-

and

long-term statistical

uncertainties

are

of

particular

concern.

Model

uncertainties are represented by increasing the single

sea State response variance based on

experience

from

simulation studies.

1. rNTROIIJCTIQN

Present

pipeline

stability

design is normally

based on a simple stability

check

requiring

balance

between

hydrodynamic forces and pipe-soil resistance.

Such

a

method

has

several

shortcomings:

Static

stability

calculations

are

based

on

a traditional

Munson

hydrodynamic

force

formulation

with

force

coefficients

based

on

steady flow conditions (DnV,

197G). Pipe soil resitance is modelled

by a

simple

friction

term.

Hence,

the

physical mechanisms of

these forces

are

not

properly

represented

by

the

mathematical

models.

Consequently,

significant

uncertainties are attached to such a design procedure.

Recently

methods

and

tools

for

more

sophisticated response analyses

have

been

developed

(Wolfram et al., Hoithe et al, Verley et al, Wagner et

al, Slaattelid et al, Tryggestad et al, Labrakos et al,

Presented at the Eghrh International Conference on

Offshore Mechanics and Arctic Engineering

The Hague - Mach 19-23, 1989

March 19-23

OMAE Europe '89

The Hague

TECHNISCHE UNIVERSITEIT

Laboratorium voor

ScheepshydrOmeChaflI

Archief

Mekelweg 2, 2628 CD Deift

015- 786873 - Fax: 015 - 781838

PROBABILISTIC STABILITY DESIGN OF A SUBMARINE

PIPELINE SYSTEM

T. Sotberg and B. J. L&ra

SINTEF Division of Structural Engineering

R. L. P. Verley

STATO I L

Trondheim, Norway

Myrhaug

et al,

all

1967).

Realistic simulation of

pipeline

response

can

be

performed

_accordingly,

accounting

for

such

effects

as

directional

wave

spreading,

boundary

layer

reductions,

accurate

description

of

hydrodyriamic

forces,

and

soil

resistance dependence on

pipe

penetration

into

the

soil.

Since

each

simulation is rather costly, the

number of governing parameters has been reduced

by

a

factor of 3 by utilization of dimensional analysis.

A

response data base was established fro. numerous

indi-vidual

simulations

(Lambrakos et al, 1987), see also

the Appendix of this paper.

By

interpolation,

the

movement

and

strain

response can be predicted for a

given

set

of

para.eters

defining

a

short-term

stationary sea state.

A

number of the parameters essential for

deter-mination of long-term

pipeline

response

are

random

quantities.

Information

about

their probabilistic

properties is also available, and should

be

properly

represented

by

any

rational design method. In this

paper, focus is placed on a probabilistic

method

for

lateral

stability

analysis

of

pipeline

systems,

corresponding

to

SLS

(Serviceability

Limit

State)

design

criteria.

The

_{objective of such a method is}

twofold:

By

rpecification

of

target

exceedance

probabilities

for

given lateral displacement limits,

required pipe weights can be evaluated.

Additionaly,

the

method

may

be

used as a tool in the prct-ess of

developing

new

codes.

Both

calibration

against

existing

codes and derivation of partial coefficients

for a specified

overall target

reliability

can

be

facilitated.

Probabilistic methods for different limit states

than

those

considered

here

have

been

outlined by

Larsen et al (1986).

Lambrakos (1982)

used

a

wave-by-wave

approach

to

calculate

a

probabilistic

cumulative movement for pipelines. The

statistics

of

the

displacements

for

individual waves are combined

with the wave statistics.

Chao

(1988)

has

developed

a

procedure

for

calculating

the

long

term

_{cumulative displacement}

response of

the

pipeline

by

assuming

a

Poisson

distribution of storms over the lifetime and applying

a Monte Carlo simulation technique to derive the

long

derived from _{the ]oint density of the same two} vari-ables In our application st is approximated by a

lognormal distribution. The criterion adopted for identification of physica) unrealistic sea states with breaking waves is T I 3.2/H, Sea states outside ths5 region arc left out, and the joint density of R, and T,, has been renormalized.

The marginal density of significant wave height, H,, is given by a lognormal distribution for H, truncated at an upper limit, n. For larger H, a

Weibull distribution LS employed (Haver et al, 3986) Wave _{directionality d&ta are}

given by a

two-dimensional table specifying the relative occurrence of waves as a function of Ha and mean wave propagation direction, 8, an _{in Table 1. Spreading of wave} ener-gy _{around the mean direction is described by} applica-tion of a cosine wave spreading tuncapplica-tion. The single sea state directional wave spectrum is expressed as follows

S (w,9) S (w)-g(9,8

n q w (2)

where S (w) is the one-dimensional wave energy spectrum (Jonswap type with y 3.3), and )8,8,) is

a cosine spreading function:

i(8,B (

= Ccos(e -8)

w w (3)

where C is a normalized factor.

The pipeline response in a single sea state is

described by a set of nc.n-dimensional parameters as outlined in the Appendix. The following 3 parameters describe the bottom flow velocity statistics and determine the wave and current loading on the

pipe-line:

Significant velocity normal to the pipeline: U,

Zero up-crossing period of flow velocity : T,,

Current velocity normal to the pipeline :

The last of these is assumed to be determin-istic, while the first are random variables. Their

joint density is determined by finding the (U,, T,,)

combinations corresponding to the (Rs, Tz) pairs, weighting over 8, in each case. To achieve this, a transformation of the directional spectrum into a normal velocity spectrum is required for fixed values of (H, , T,, ). Some of the steps in this process _are (Sotberg et al, 1988):

The one-dimensional water particle velocity spectrum at the pipe level is calculated by multiplying S (w)

with a depth-attenuation factor which scales ith frequency.

The normal velocity spectrum for a given 8, is found

by applying a reduction factor, also accounting for wave spreading. A weighting over different 8,

is

subsequently performed.

The significant normal velocity is derived by integration of the corresponding spectral density. The zero up-crossing period is expressed through

moments of the same spectral density function

4. LONG TERM PIPELTNE RESPONSE

The long term distribution of accumulated pipe-line displacement for a reference period corresponding to the duration of a single sea state (3 hours), is

calculated as a weighted sus of the conditional single

sea state distributions. The weight function is the

long tCrm density function in the form given in the previous chapter.

.-jf(y(z)f(z)dz (4)

where

f )YJz) - _{conditional distribution of accumulated} lateral displacement Y given bottom velocity

Statistics Z

f(z) - probabilistic density of bottom velocity data n.

The conditional statfstical parameters required

fr specification

o f(YIz) are evaluated by inter-polating between points in a data base. _{Each point is} determined from simulations as described in Section 2.1. Given specific values for the set of dimension-less parameters defined in the Appendix, sufficient second-moment statistics for specification of the conditional density function are hence provided (Sotberg et al 1988).

The integral in Eq. (4) is discretized and

cal-culated numerically as a weighted sum of conditional probability density functions for a number of

discre-tized values of the environmental data vector z. The long term distribution of lateral

displace-ment is hence evaluated from a Gaussian distribution

of short-term response in a single sea state and a long term distribution of sea states as outlined. lt is believed that the long-term distribution of wave

height is the critical factor with respect to the

upper tail in the long term response distribution, and thus also for the calculated probability of failure.

The long term displacement probability density will have a peak around zero, with an expectation also

located close to that peak. The distribution will be near symmetric around the mean value. The properties of this long term distribution are studied ir. more

detail in connection with the numerical examples in Chapter 6, where some typical plots are given.

A compact, second-moment statistical characteri-zation of the long term displacement is provided by

the expectation, E)y ), and the variance, o2 - These

may simply be calculated from the density function in

Eq. (4). This calculation requires a very high

reso-lution in the numerical integration, however, due to

irregularities of the integraiìd. An alternative

approach, which also gives some more insight, is to compute these quantities via their shcrt-term

counter-parts:

EEl',]

= f E[y)zjf(r)dr (5) z and 02 = J E[y(z]2f(z)dz + /02 tz 1m)' y1 - E[YL]S (6)

where E[ylz], are the conditional expectation and

variance, respectively. The first arid last terms in Eq. (6) are the contributions from the long term wave environmental variation and the second term is contribution from the short term response variances.

For a relatively light pipeline with a large number of displacements throughout its lifetime, the first and last terms in Eq. (6) are expected to dominate. For a heavy pipeline only moving for the

are expected to be increasingly important, and the second term will be significant.

The second-moment Statistics of accumulated

displacements for long reference periods (e.g. 1, 5, 20 or 100 years), can readily be derived from_{those ir}

Eqs (5) and (6) by assuming indpendence _{between the} response in each sea-state. _{Denoting the total number} ct sea states

itt

the pend L N the expressions become

E[Y.j _NT.E[YLJ

(7)

ou =ì

'T

TYL

If desired, the _{distrubutiøn function of long}

term accumulated response can be evaloatd _by convo-lutir,g that in Eq. (4) a number of (N1 -1) times,

)Starisberg, 1986).

5. _{RELIABILITY ANALYSIS}

For sand soil, the SIS design method should allow limited pipeline movements. Criteria for accumulated displacements will in general be site

specific, depending e.g. on distance to neighbouring structures and width of the pipeline corridor. The

corresponding serviceability limit state function is

expressed as g(z)

T

-where Y _{is a deterministic displacement limit, and} Y is the actual random lateral displacement. The _vector

z contains basic random variables describing

uncertainties in long term wave environment, _{sea state} description (realization of wave time series),

hydro-dynamic load and soil resistance modelling. For a ULS (Ultimate Limit State) design criterion, the limit State function would involve strains as well as displacements.

The probability of failure P1 is the

probabi-lity that the vector z has a value for which g(z) 0:

Pf - P(g(z( 0)

= - Y

0) (9)

The reliability is very often expressed through the reliability index 8R defined as

= _(Pe)

(10)

where

(.)

designates the cumulative standard normal

distribution function.

Implicitly, both reliability measures are re-ferred to a specific time period, since the statis-tical properties of the variable Y will be completely different for a single sea state, Eq. (4), as compared

to a reference duration of one or more years. The relation between probability of exceedence P,, and the so-called return period, R5 is ex-pressed as

p

ex

p

where N = 2920 is the annual number of sea states of duration 3 hours; N5 is the number f sea states for

the given reference period. Putting N5 = 1, we get:

p = ex R U p (8) (12)

w

and with a return period of e.g. R9 100 years, the corresponding probability becomes P,,

_{0.34 10.}

Based on Eqs. (li) nd 12), the transition from single sea state probabilities to those for a one-year reference becomes:

N.

P (1 year) 1 - (I - p (3 hrsj) - ₍₁₃₎

ex ex

row with N5 U 2920. Implicit in Eq. (13) is a specific choice of R9. All of Eqs. (11), (12) and (13) are also based on the assumption of independence between excursions from sea-state to sea-state.

These probabilities can be employed in two ways. For a given pipeline, the exceedence probability for a

specific T can be found. Conversely, by specifying a

target P1 _{in advance, calculation of the pipe weight}

required to satisfy the displacement restriction can be performed. Alternatively, given P1 and pipe

weight, a corresponding Y1 will result.

6. EXAMPLES OF APPLICATION

The methods for probabilistic stability analysis are illustrated through some examples for an actual pipeline system. Determination of the submerged pipe weight W, to meet the specified criteria is the main design task. Sensitivity of the probability of failure to variation of submerged pìpe weight is examined. Also, effects of model uncertainties and sea state statistical uncertainties are touched upon.

6.1 Long terni environment and pipeline dafa

The statistical parameters defining the condi-tional density of T9 on H, in Eq. (1) and the marginal density of H, , are identical to those specified by Haver et al (1986). wave directionality is accounted for by the distribution specified in Table 1. Spreading of wave energy is represented by a fixed exponent of n = 4 in Eq. (3). A constant effective

current velocity V = 0.57 m/s is included. The

boundary layer reduction is accounted for by the method of Slaatelid et al (1987). The current is assumed to be perpendicular to the pipeline with no distribution over direction. The water depth is 80 m, and some of the remaining key parameters are:

Saturated soil density p, = 2000

kg/&

Outer steel pipe diameter D, = 0.97 m

Steel thickness t = 23.5 mm

Natural gas density = 94.7 kg/r& The probabilistic design calculation was first performed by inclusion of statistical response varia-tion only due to differences between wave realira-tiens, in addition to long term wave environment statistics.

The effect of model uncertainties was

subse-quently taken into accout by increasing the standard

deviation in the single sea state response based on results from earlier sensitivity studies (Sotberg and

Remseth, 1986, a and b). This approach was adopted for both model uncertainties related to the hydrodynamic force and soil resistance modelling.

The basis _{for determination of the effect from}

model uncertainties on the response variance is rather limited. Reliability analyses can, however, give valuable qualitative information on the sensitivity of the stability design process to these uncertainties.

6.2 Lesigr ezairpcs

The design _{crLterion ddopted is a}

ateraJ

dis-p1&;.mrt Lmit Y r 2) m for free pipeline scuon

awiy from obstac1e _{or other structure}

Cortespondin to the SF.S Condition. _{The correpondjn}

target prob.3-bility of exceedence, for t _{reference period oÍ coe}

year, is evaluated for R _{100 years, ie. P 102} This _{desLgn check ensures safe operation}

cf the

pipe-i pipe-inc.

The calculated long term probabiltty

density and the cuiulattve distribution functLon are given in

Figure 2 for a _{subnerged pipe weight equa'}

to 4300 N/rn, whLch gives _{n Outer pipe diameter O}

1.15 n.

The tai's of both functions have

been magnified due to

sa11ness o function va'ues.

It is _{observed that the distribution}

is a1ost

sysmetric around a eean value very cloie to zero for this case, and the characteristic

upper tail for large displacement levels is illustrated. _{From the}

distri-bution function the annual probability of exceeding the target displacement

level Y 20 s, is found to be 0.0082.

The corresponding reliability index is 2.40. Even a lateral displacement equal _{to ¿bout 18.5} in _{would satisfy the target probability}

of exceedence

equal to 10-2 within a single year.

Increasing the submerged pipe weight _{to 5000 N/s}

alters the tail in the distribution _{function as}

seen from Figure 3. _{The tail is moved to a lower}

displace-ment level and the shape has also been _{changed. Figure}

4 shows the results for a submerged pipe weight reduced to 3000 N/in, which makes the upper tail shift to _{displacements of about 70}

meters for the same probability levels.

The analyses reported above are all performed by including the statistical uncertainty due to differ-ences in wave realization. _{As a next step the} uncer-tainty on the response due to uncertainties in the soil model is also taken into account. This is done by increasing the standard deviation of the response by 50 overall and somewhat more for the heavier pipeline cases or the lower displacement levels, _where the sensitivity has been found to be largest, (Sotberg and Remseth, 1986, a and b).

Figure 5 gives a plot of the tail of the

distri-bution function similar to the one given in Figure 2, but now with soil model uncertainties included. The submerged pipe weight is still W! = 4300 N/rn. s

ex-pected, the tail in the distribution is moved to a

higher displacement level and the distribution is also

smoother, due to increased spreading of single sea state response. _{The probability of exceedence of the} target displacement level has increased to P1 = 0.013. The expectation value of long term response is unaltered, but the long term variance has increased

due to larger short term response variances.

The _{effect of also including hydrodynamic force}

model uncertainties is investigated by multiplying the response _{standard deviation by a factor of 2.5. This}

is a relatively rough and ad hoc _{representation, but} it should reflect _{a subjective, engineering type of}

feel for the influence from model uncertainties. Plots

based on these calculations are given in Figure 6 for a pipe weight W, 4300 N/in. The probability of

ex-ceedence of the target displacement is as high as 0.13

which corresponds to a reliability index = 1.13. A lateral displacement of about 37 in corresponds to the target reliability index = 2.33 (P1

= 10-2).

The reliability index 8 is shown as a

func-tion of submerged pipe weight by the solid lines in Figure 7. Also, the lateral pipe displacement,

id.,' as a function of the saine for 8R1 = 2.33 us depicted

by dashed lines. The cuives marked a) represent

results obtained by only accounting for otatistical

variation due _{to different wave realizations. Curves} b) and cl correspond to inclusion of soil model uncertainties and _{bot.h types of model uncertainties} respectively.

From _{this figure, the design weight}

correspond-ing to the target probability can _{be read}

from the

horizontal line marked by . It is seen that taking

all the uncertainties into account would increase the design weight from ¿bout 4200 N/rn, with only realization uncertainties included _{to above 6000 N/rn.} However, curve ci _{represents an}

upper estimate of total load effect uncertainties _{corresponding to the} factor 2.5.

The _{second-moment statistics of long-term}

accu-mulated displacements as computed from Eqs. (5) and

(6) are displayed in Figure 8 as a function of sub-merged pipe weight. _{Both expected value and}

standard deviation _{decrease for increasing weight,} coirespond-ing to improved stability. _{By including model} uncer-tainties for a fixed W, , the standard deviation is

succesively augmented as illustrated by _{curves b)} and cl.

The relative importance of different contribu-tions to the long term standard deviation, ci1 , can be

assessed by computing the ratio of the second term in term in Eq. (6) to the value of the complete expres-sion. This ratio is depicted in Figure 9, which demonstrates that the variance of short response becomes increasingly important for increasing pipe weights. This also implies that choice of

distri-bution function for accumulated short term response

will be crucial for the tail of the long term ::esponse distribution of heavier pipes.

Sensitivity of the estimated _{to variation in}

short term standard deviation, o , is studied _by recomputing the safety index for increasing _{values of}

the latter. Denoting the reference values by 8 and

o , respectively, the _{curves in}

Figure 10 are obtained. _{is the calculated reliability index with} the _{short term standard deviation equal}

to zero, i.e.

whereas o1 is the reference value of the short term _{standard deviation of displacement response when}

only wave realization uncertainty _{is included. The}

dashed line _{gives the ratio between the short term}

contribution (from sïngle sea state response) and the total long term _{variance as a function of the short}

term variance. The results are valid for a fixed _pipe

weight, W, = 4300 N/s.

The results indicate that by increasing the

standard deviation of response from zero, through the reference value and up to 2.5 times the reference the

reliabiLity index gradually decreases relative to BAs (=2.46) to half that value. The ratio between the

short term contribution and total long term variance similarly is a strongly increasing function o! the

short term variance.

The sensitivity factor with respect to model uncertainties, s, can generally be evaluated

from Figure 10 by the relation:

= 8R0

- e2)

The variation of a as a function of the short term variance is illustrated in Table 2. The

sensitivity to the short term response is increased from 16 per cent when only realization uncertainty is

included, to about 70 per

cent when

all model uncertainties (high estimate) are taken into account.

This result indicates that for reasonable values of model uncertainties the long term wave environmental

n = /

1

distrbntion is as important with respect t.')_stiahility desi.;n .' _iubmarin.' pipelines s th. us':ert.utnty in Ei 'Id'..O

iLort

_trrnt epririse 7. _DISCUSSION

A net hod or nrr..b nbili .t _{srah ¡1 t design}

if

auibr.rrne pipe1ne _{y:;t,'mn his been out1ind. F.mph.siia}

was pst ori thr' FLS design critericri _.iri'i pp$ ori sand soil.. _{The pro':edire was illustrated by}

.ippii:ation to

e specific pipeline

system.

'arying the sLibmorged

pipe

_{weight affects both the stirpe .ini location}

of the probability

density function of

ac-cumulated

displace-ments _{scgniii':antly. Displaement}

corresponding to

one-year

target

_excedsnce

_{probability of p 10-2}

ranges _{from 12 m for a wei.;ht of 5300 N/n}

to 7 ii for 3000 N/n.

Model _{uncertainties are}

nc1uUed by inereasin; the single icea state response vari.snce.

Fr

.t fined

pipe weight of 4300 N/rn and with p

1O, the

accumulated displacement _{var ies from li n for no model}

uncertainties,

through

22 ni with soil

_toiiStiflCC

models uncertainties,

ti)

37 n by als's accounting

rot

additronal

_{hydrodyne,miç force}

_model

_{uncertainties.}

Seond-rnoment statistics of long term

displace-ments depend heavily on pipe weight. _{HCdvy pipes will}

move Only during extreme sea states, implying

rela-tively small expectation values and standard

devia-tions. _{Concurrently,}

the single sea state response variance will contribute increasingly more to the

variance of the long term response.

In _{general, the probabilistic approach offers a} versatile tool for pipeline stability analysis when a

number of the governing design parameters are random.

The relative influence from different sources of uncertainty can be quantified and treated in a uniform way. _{Improving the statistical models by}

collection of data may reduce the uncertainties, resulting in more realistic and economic _{pipeline design. This}

also applies to _{progress in quantification of model} uncertainties, and possible refinement f the models as such.

In this _{process of advancing the general basis} for prOhsbilittc pipeline design, a number of specific topics need further clarification. A non-exhaustive list may be as follows:

Identification of the distribution function for short-term pipeline displacement to a higher confidence level by incredsing number of simulations.

Inclusion of uncertainties in statistical para

meters defining the long-term wave environment.

Inclusion of statistical models for current

magnitude and direction, and joint statistics of current and waves.

More detailed study of model uncertainty effects on response. tmproved data would give closer bounds on the reliability index.

Furthermore, similar ptobabilistic analyses for the iLS limit state should be undertaken by also

con-sidering strain response. Failure functions

corre-sponding to total collapse and local buckling could then be defined. The methods discussed above should still be relevant also in that context.

ACKNQWLEEXEt4ENT

The authcrz wioh te cYpress their _appreciation to Statoil and Esso Urge for permission to publish

this paper.

REFERENCES

Chao, J.C.: "Calculation of Long Term Cumulative Movements for Cubmarine Pipelines", Proc. Seventh

tnt. Syrrp. ori Offshore Mech. arid Artic Enqn.

Houston, Febr 7 - 12, 1988.

PnV Rules for Design, Construction and Inspection

of Submarine Pipelines and Pipeline Risers, 1976. Haver, 0. and Nyhus, K.A. "A Wave Climate

Description for Long Term Response Calculations",

Proc. Fifth Ent. Symp. on Offshore Mec-h, and Arctic Enqn, . Tokyo, April 13-18, 1986, Vol. IV pp. 27-35.

Holthe, K., Sotberg, T. and Chao, J.C.: "An

Efficient Computer Model for Predicting Submarine Pipeline Response to Waves and Current", Proc. of Nineteenth Offshore Technology Conference. Paper No. 5502, Houston, 1987.

Lambrakos,

KF.:

"Marine Pipeline Dynamic

Response to Waves from Directional Wave Spectra",

Ocean Eniqineerinq, Vol 9, No 4, pp 385-405, 1982. Lambrakos,

KF.,

Remseth, S., Sotberg, T. and

Verley, R.: "Generalized Response of Marine Pipelines",

Proc.

of Nineteenth Offshore

Con-ference, Paper No. 5507, Houston, 1987.

Larsen, E.U., Skjong, R. and Madsen, 11.0.:

"Assessment of Pipeline Reliability Under the Existence of Scour-Induced Free Spans", Proc. of Eighteenth Offshore Technology Conference, Paper No. 5343, Houston, 1986.

Myrhaucj, D.: "A Theoretical Model of Combined Wave and Current Boundary Layers Near a Rough Sea

Bottom", PrOC. 3rd. Int. Syap. on Offshore Mech. arid Arctic Eriqnq. , New Orleans, Feb. 12-16, 3984,

Vol. 3 pp. 559-568.

Slaattelid, 0.14., Myrhaug, D. and Lambrakos,

KF.: "North Sea Bottom Boundary Layer Study for Pipelines", Proc. of Nineteenth Offshore

Technology Conference, paper No. 5505, Houston, 1987.

Sotberg, T., Remseth, S.: "Pipe Response Sensitivity to the Wake Force Model", M.ARLNTEK Project Report SF 603455.60.01, The

SINTEF-Group, Trondheim, Norway, Dec 1986.

Sotberg, T., Remseth, S.: "Pipe Response

Sensitivity to the Pipe-Soil Interaction Models", MARINTEK Project Report SF 603455.60.02, The

SINTEF-Group, Trondheim, Norway, Dec 1986.

Sotberg, T., Lambrakos,

KF.,

Reaseth, S., Verley, R.L.P. , Wolfram, jr., W.R. : _"Stability

Design of Marine Pipelines", Proc. BOSS

Conference, Trondheim, June 1988.

Stansberg,

CT.:

"Estimation of Long Term

Pinelpe

Pro7ec Repor

APPENDIX

Calculation of lateral pipeline _displacement

- _{a non-dimensional description}

Genera I

The computer program PONDUS, (Holthe et al., 1987) has been employed _for

calculation of lateral displacements for _{pipelines during a sea state.}

The program includes _{itate-of-the-art models}

for the fluid-pipe-soil interaction.

A generalized decriptìon of _{the physical problem}

has several benefits. _{The pipeline}

response is ex-pressed in terms of a few non-dimensional

parameters representing combinations of a larger number of physical quantities describing the pipeline and the ocan environment. The generalized respone data constitute a data base for transformation _{from basic}

load parameters to _{response quantities in the} probabi-listic design method outlined in this piper.

Nsn-dimessional response description

Scaling _{of the response is performed by} employ-ing the equation of notion for the pipeline.

The equation of Sotion _{for pipeline Sections}

with no boundary disturbances

from end constraints, or

possibly a free-free section is

m-YrF 4F -F

D I S (15)

where s is the mass of the pipeline _{per unit length; y} and t denote pipeline _{displacement and time,}

respec-tively; F5 and F1 are the drag and inertia

hydro-dynamic forces per unit length, respectively _and F5

is the soil resistance force per unit length. The external forces (F0, F1 Fs I for the moving

pipe are defined as follows

F = 1/lp D

C0()Iu - 1I(u

-D w _lt lt [C

-'w T'

M lt (CMl) J F t F . f(W -F ) ₊

R' and the lift force _L is

5 s L

FL = 1/2 0D C0))(u -Ft

Here, Fe is a remaining nonlinear soil resistance force depending on soil density and pipe penetration, (Wagner et aL, 1987) . The flow velocity u, which

consists of wave and current components, represents the effective flow velocity calculated by the Wake Force Model, (Verley et aL. 1987). The coefficients C5 (drag) and C, (lift) are functions of the distance,

s, the water part:cles travel from the instant of flow

reversal (making these coefficients tine dependent). Tryggestad, S folistad, T. Nilien, _{J. and}

Gelanger, F.: Measuring Technique

lsd Field Data for Pipe.)ine _{Ctahility Studies', Proc.}

of Nine teenth Orfshore Technology _Conferencp

Paper No. 550G, Houston, 1987.

Verley, R. Lambrakos,

KF,

and Reed, K.

'Prediction of _{Hydrodynamic Forces}

on Seabed Pipelines' Proc. of _Nineteenth

Offshore Technology Conference, Paper No. 5503, Houston, 1987.

Wagner, D., Murff, D., Brennodden, H. and Sveggen, O. _{'Pipe-Soil Interactive Model', Proc.}

if _{Nineteenth Offshore}

Technoloìy ('inference, Paper No. 5504, Houston, 1987.

Wolfram Jr., W.R., Getz, JR. and Verley, R.L.P. PIPESTAB Project: Improved Design Baits for Sub-marine Pipeline Stability', Proc. of Ninetnth Offshore Tochnoloqy Conference. Paper No. 5501, HoustOn, 1987.

The coefficients C' _{(inertia) and f (soil friction)} are assumed constant

The various quantities in the equation _{of motion} are scaled .,,a follows:

j' y/D, t' = tíT, U' = u/U, s' = s/D ₍₂₀₎

where U, _{is the significant particle velocity normal} to the pipe. _{substituting the scaled}

quantities and collecting terms lead to the following dimensionless

equation of motion:

[2LKN

+ C j

----2C0(s')

(Ku' -

-rI{Ku'

_{- --i) +}

lt n M Ti 2f CL (s [Ku' - Ï_j2 _{+ g.0} - f r lt M lt T

This dimensionless equation illustrates that the

relative pipeline displacement (y') depends on the

quantitites K, L, N (which are defined below), and _u', s'. Analysis of the wake model equations and _response

simulations has shown that u' and s' scale with the

parameters 1< and M (a parameter representing the steady current in the flow) - _{Although other forms of}

the dimensionless equation are possible, equation (20) is quite convenient since the influence from the para-meter N is greatly reduced by the _{term 2LKN/m being} small compared to C for most cases of interest.

Thus, for a given sea state and sar,d density,

the four dimensionless groups governing the pipeline displacement are: [IT K = (22)

L-

₍₂₃₎ D W s 1/2pDIJ2 V M c U N = (25)

where T,, and V _{are zero up-crossing wave period and}

steady current for the sea state, respectively; D and

W, are the pipeline outer diameter and submerged

weight per unit length; g and p, are the acceleration

of gravity and mass density of water, respectively.

The wave velocity U,, and the current V refer to the (flow) components normal to the pipeline.

The scaling parameters K, L, M, N can be interpreted as follows: K is a Keuleqan-Carpenter

number, L is a ratio between pipe weight and

hydrodynamic forces, M is a current to wave velocity

ratio, _{and N is a representative acceleration for the} ea state.

(21)

.-. i-:.:i±:ñ--±±-'-i-'-'---»----

. .

h--.--

-Er.-f,

Figure 4

Probability density and distribution function,

W= 3000 N/rn, realization uncertainty included

-2

I D 1 2

510'

10 .08 .06 .04- .02-.00

Figure

1

Distribution of lateral displacement

xlO 4 .35 30-b. .20 .10 00 P( Y) 3 P(Y)J .08 4 .00

sia

Figure

alO .04 .03 .02 .01 .00

Figu'-e

-T, 0 5 20 30 40 DISPLACEMENT Y (a)

Probability density and

IV

4300 N/ni, soil model

10 510 .35 30. .20-10 00

distribution

-4 20 30 40 DISPLACEMENT Y (a)

function,

uncertainty included

Fure

1.0.. F(s)

0.8.

06

0.4 0.2 0.0 -4

Figure

-2 0 2 4 DISPLACEMENT Y (m) 2(a)

Probability density

4300 N/n', realization

of lateral

5 10 15 20 25 30 35 DISPLACEMENT Y (mI

pipe displacement

uncertainty included

-4 -1 .30 l-FIT .2') 10 t - .00 . 20 6 20 40 60 80

DISPLACEMENT Y (m) _{DISPLACEMENT T (a)}

Probability density and distribution function,

5V

4300 N/rn, ail model uncertainties included

-2 0 2 4 10 15 20 25 30 35

DISPLACEMENT T Im) DISPLACEMENT Y Im) 2(b)

Cumulative distribution function.

4300 N/rn

.20 - lo .00

Figure

35 30-.10 .00

N

3 10 15 20 25 30 5 10 15 20 25

DISPLACEMENT

(a)

DISPLACEMENT Y (m)

Probability density and distribution function,

5000 N/rn. realization uncertainty included

30

s'a_4

35 30- 10-.00 .04 .03 02-.01 .00 60 80 100 120 DISPLACEMENT Y (a) 40 60 80 100 120 40 DISPLACEMENT Y (a)

(al Realization uncertainty

b) Realization -

oiI

mode! uncertainty

cl Realization * ali model uncertainty

Figure 8

Standard deviation of the long term displacement

and long term expected displacement versus

Figure 9

Ratio between short term contribution and total

long term variance versus

0.5

0.0

0.0

0.5 1.0 1.5 2.0 2.5 °y '0YO ll R0

ratio

Table

_{Wave directionality data}

/

i

TabLe

2

_{Sensïtivity factor as a function of the}

_short

term standard deviation of response

li 8

(deg)

(a)

0 45

90

135 180

225

270

315 1 482

2433

1520 ₆₈₆ 783 407 591 3013 2 3723

3780

2640

3731 5838

5410

3152

2242

3 3356

822

750

2691

7332

5622

31322

2652

4

2377

159 90 1226

4730

4266

2918

2040

5 1214 27 9 423

2365

2133

1592 1428 6

583

7 3 171 1301

970

756

816

7 282 2

0

50 710

388

292

377 8 122 0 0 16 355 136 100 1134 . 61 0 0 5 142 39 32 87 10 31 0 0 1 _{64 13 11 44} 11 13 0

0

0 19 5 3 14 12 4 _{0 0 0 9} 1 1 5 13 2 0 0 0 2 0 0 2 14 0 0 0 0 0 0 0 1

ay/ayo

_13fBo

1

0.975

_0.16

1.5

0.91

0.31

2.5

0.46

0.74

Figure 7

_{Reliability index 13R versus pipe weight, and design}

_{ligure 10}

Sensitivity of 6R to short term variance

displacement for 13Rt

_{2.33 versus pipe weight}

3R

ratio

LII