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ABSTRACT

In this paper uncertainties related to stability design calculations of submarine pipelines are

in-vestigated. The pipeline considered rests on the sea bottom consisting of sand soil and is exposed to wave and current loading. Important sources of uncertain-ttes are identified, and the sensitivity of the

pipe-line reliability to various design parameters is

studied.

Extensive computer simulations have been performed as the basis for the sensitivity study and reliability calculations. Three different approaches for

reliabi-lity evaluation are illustrated. The method used in

this study is based on a hybrid formulation applying

nuertcally generated response data, in addition to

linearization of response relations at the design point and utilization of a multipurpose computer code

to determine the structural reliability. The latter allows a general formulation of the failure surface to be given as a function of the basic random variables.

Application of the method is illustrated through a case study. Lateral displacement of the pipeline is

considered. The probability of violating the actual limit state is calculated and sensitivity factors of

the pipeline reliability to the basic random parameters are calculated.

INTROEIJCTION

Recent years cf research on the stability of sub-marine pipelines has lead to a redefinition of the

design practice applied for on-bottom pipelines. Earlier pipeline stability design was based on a

simple balance between the hydrodynamic forces and the soil reaction acting ori the pipeline. The shortcomings of such a procedure have been mentioned in an earlier paper, (Wolfram Jr. et al., 1987). It is desirable that a refined design procedure should be consistent with the design practice applied for other offshore structural components, i.e. by calibration of partial safety factors based on reliability methods at some

Offshore Mechanics and Arctic En9ineering 127 OMAE Houston, Texas, Feb 18-23, 1990

TECHNISCHE UNIVERSITEIT

Laboratorium voor

ScheepshydromechafliCa

Archief

Mekelweg 2, 2628 CD Dem

Tel.: 015- 786873- Fax: 015-

781836

ON THE UNCERTAINTIES RELATED TO STABILITY

DESIGN OF SUBMARINE PIPELINES

T. Sotberg, B. J. Leira, and C. M. Larsen

SINTEF Structural Engneeríng

Troridheim, Norway

R. L. P. Vertey

STATO IL

Trondheim, Norway

higher level. This is in accordance with application of rules for offshore str4.ictures as by DnV (1982), see also the paper by Fjeld (1977).

Most of the governing parameters for prediction of the pipeline response are random quantities. Model and parameter uncertainties in the design procedure ¡ay be included through a probabilistic evaluation of the

design. This procedure provides a sound method to eva-luate the safety of the pipeline structural design. The papers by Ellinas and Williams (1989) arid Ellirias

et al. (1987) are important contributions to the app-lication of reliability methods in the pipeline design process and states the need for such procedures to secure a uniform safety level for all relevant limit

states.

An earlier paper, (Sotbeig et al., 1989), investi-gated the uricertaint.y in pipeline response due to the

variability of the ocean wave process and, in a simple manner, the uncertainty in the hydrodynamic force and soil resistance modelling. In the present paper, a thorough investigation of uncertainty in design due to statistical variability in the wave process, parameter uncertainties and model uncertainties is performed. The objective of this work is thus to obtain a better assessment of the safety level in the design of sub-marine pipelines. A representative base case has been chosen to illustrate the uncertainty aralysis and sen-sitivity study. It is noted that some uncertaintiy sources are disregarded in this work. This is, howe-ver, discussed more detailed in the next chapter.

The reliability calculations have been performed by the computer program ISPUD (Borgund and Bucher,

1986), based on linearization of the mean value re-sponse relations. This program employs an optimization procedure for calculation o: the design point and an importance sampling simulation for calculation of the failure probability.

For the example study, design according to the Recimme.nded Practice RP E305 (1988), have been com-pared with that obtained through the present method. In particuLar, the inherent safety in RP E305 is assessed

(2)

-UNCERTAINTIES IN DYNAMIC RESPONSE ANALYSIS OF A SL1.L4RINE PIPELINE

enera i

The behaviour of a submarine pipeline ori the sea bottom exposed to wave and current loading is a rather complex function of a large number of parameters. These parameters describe tTe ocean envjronent, water depth, sail conditions and pipeline properties.

Prediction of these parameters for a certain design Case is obstructed by inherent uncertainties. AdditLonal uncertainties will be introduced by the

mathematical models applied in the pipeline response analysis. These uncertainties are due to simplifica-tion of the physical nature of the soil-pipe-water interaction process.

In the following parameter uncertainties, statis-tical uncertainties and model uncertainties will be

described.

Parameter uncertainties are represented by the significant wave height, H,, the peak wave period,

as well as the current veloclty, V among the sea tate parameters . Site specific parameters are water .iepth, d, and relative soil density, Dr, for sand soil (only sand soil is considered in the present paper).

Among the pipeline parameters considered are the outer pipe diameter, D, and the subrerged pipe weight, W,

Statistical uncertainties are present in the de-scription of the pipeline response due to the inherent variability in the description of the ocean wave

pro-cess. Different wave realizations with identical (first order) statistical properties, will give diffe-rent prediction of pipeline response, i.e. a statisti-cal variability in the pipeline response.

Model uncertainties are caused by simplifications of and limitations to the numerical models applied in

the response analysis. The models applied for hydro-dynamic force and soil resistance are assumed to be without bias, i.e. give a correct prediction of the expected force levels (Verley and Reed, 1989,

Brennod-den et al., 1989). The statistical variability between predictions from the model and the real behaviour is

taken into consideration in the design procedure, i.e. by applying a probabilistic model for the disper-s ion.

In this paper only the lateral movement of the

pi-peline is considered and uncertainties related to

pa-'ameters not relevant for the displacement response .re excluded. The steel pipe diameter and wall thick-ness as well as the yield strength are parameters which need to be taken into account if strain or

stress in the pipeline wail is considered. Other random parameters may be required for other limit states.

The linear Airy wave theory has been applied in the transformation of surface waves to flow kinematics close to the pipeline. The uncertainty in the near-bottom wave kinematics is not included in this study.

This uncertainty could be of significance with respect to pipeline stability according to the work by Grace

(1976) and Jacobsen and Bryndum (1984).

The pipeline is initially assumed to be laying on the sea bed without any significant penetration into

the soil. This is a conservative assumption as sea bed erosion and small oscillatory movement of the pipe will cause larger embedment than that due to the

static pipe weight and thus increase soil capacity.

Probabilistic rodellinq of uncertainties

The statistical variability of lateral pipeline

jnj

displacement is determined by including inherent sta-tistical uncertainties, the effect of basic parameter variability as well as uncertainties related to the mathematical models. The specific statistical models pertaining to each source of uncertainty are described

below.

Modelling of ocen environient. The wave

conditi-on is modelled as a Gaussian process assumed to be stationary within a 3 hour period. Each short term condition is defined by the Jonsuap wave spectrul, with significant wave height, H,, and peak wave period T as the main parameters. The spectral parameters for a given short term sea state are applied deter-ministically. Uncertainties in the spectral for. and peakedness have been found to be unimportant wlth respect to the design of an on-bottom pipeline. The

specific detailed shape of the spectrum at higher fre-quencies is rather insignificant especially for medium to deep water conditions. The significant flow

veloci-ty at the sea bottom, which is an important loading

parameter, is not sensitive to the choice of wave spectrum. In the transformation of wave energy from

sea surface to the sea bottom, the high frequency com-puuent.3 will be filtered out relative to the low fre-quency part of the wave spectrum. The filtering effect is enhanced for increasing water depth.

The long t'r, wave environment is normally descri-bed by a joint probability density function for signi-ficant wave height and peak period. In the present application the joint probability density function is defined in terms of the marginal distributions and the related correlation coefficient. The marginal

pro-bability density function for significant wave height, H,, is given by a Weibull distribution which is fitted to the upper range of H, values, and the peak period is approximated by a lognor.al.

distri-bution (Haver and Nyhus, 1986). These parameters are usually correlated to some extent, and different levels of correlation have been applied in the relia-bility calculations.

The current velocity and direction is expected to be constant within a 3 hour period. The long ter. di-stribution of current magnitude is here given by the Weibull distribution which has been fitted to measured

data. The effect of correlations between wave and current is studied.

Modelling of pipeline

and $oil pro.ertie.

The pipeline diameter, the submerged pipe weight and the water depth are expected to be normally distributed. It is presumed that the sensitivity to the statistical variability of these parameters is low, due to a rela-tively narrow interval of admissible parameter values. They are included as random parameters, however, to verify this low importance with respect to uncertainty in response. The relative soil density is limited to

positive values only and a lognormal distribution is applied. The variability of the soil density is

related to the physical variation of the soil

porosi-ty. The choice of distribution function is not based

on any comprehensive data analysis, however, further comments related to the choice of distribution are

given later in the paper.

Model uncertainties. Uncertainties in the hydro-dynamic force and soil resistance formulation have been included by applying scaling factors to the cal-culated forces. These scaling factors are assumed to

be adequately represented by the normal distribution with expectation values of 1.0, which means that cal-culations of sensitivity factors and failure probabi-lities have been performed assuming no bias in the

(3)

hydrodynamic force and soil resistance models. The central limit theorem and measured scatter in the data supports the specific choice of distribution function. The sensitivity to the choice of distribution function is, however, evaluated by also applying a lognormal distribution.

MEflJÛS

FOR SENSITIVITY ANALYSIS ANO RELIABILITY CALCULATIONS

Desiqn criteria and the reliability measure

For sand soil, the design method for the Service-ability Limit State (SLS) may allow limited pipeline movements, (R? E305, 1988). The failure function is expressed as:

g(z) = 'T

- Y(z) (1)

where Y is a deterministic displacement limit, and Y is the actual random lateral displacement. The vector z obtains the basic random variables describing un-certainties in the long term ocean environment, the short term (3 hours) state description, the

hydro-dynamic load and soil resistance modelling as well as Statistical uncertainty.

The probability of failure P, is the probability

that the vector z has a value for which g(z) O:

Pf = P(g(z. O)

= - Y 0)

(2)

Implicitly, the reliability measure is referred to a specific time period, since the statistical proper-ties of the variable Y will be completely different for a single sea state, as compared to a reference duration of one or more years.

The transition from single sea state probabilities to those for a one-year reference is given by:

N P(1 year) = i - (i - P(3 hrs.))

S

(3)

where N = 2920 is the annual number of sea states of duration 3 hours. The equation is based on independ-ence between excursions from sea state to sea state.

Method of analysis

A unified solution procedure for reliability ana-lysis of general types of structures under random loading is lacking. This is particularly the case when there are significant nonlinearities in the design calculations.

For the design of submarine pipelines, general re-liability procedures can not be applied. This is

pri-marily due to the failure function not being expli-citly defined in terms of the basic random variables.

In turn, this is caused by a complex transformation from the basic load and resistance parameters to the

random response. It is hence pertinent to develop al-ternative procedures for sensitivity and reliability calculations of the submarine pipeline design. Three different approaches are outlined below. The third approach is the method utilized in the work reported here.

Response data base method. The method applied by

Sotberg et al. (1989) is based on the development of a limited number of nondjmensional parameters describing the pipeline response to a satisfactory level of accu-racy, and utilization of a generalized response data base given in terms of these parameters.

It is thus possible to avoid time consuming

simu-lations when performing the reliability calculations, requiring only interpolation in the generalized data base. This is particularly effective when the response

is a function of a large number of basic parameters and the transformation fro. basic load and resistance parameters to the response is complex.

However, in such a procedure it is difficult to

include all kinds of parameter and model uncertainties

as well as statistical uncertainties due to variabi-lity of the wave process. Problems will also occur when the mathematical models used are updated, as a. new response data base has then to be generated.

In-terpolation in the response data base will also intro-duce additional uncertainties which may be difficult

to quantify.

esponse surface iethod. An alternative approach to the generalized description is to develop numerical response distributions, for a specific case through extensive simulations. The response simulations should

be concentrated to the area where the design point is

assumed to be. Location of the design point is relati-vely well predicted based on earlier experience (Sot-berg et al., 1989) and design methods such as those outlined in R? E305, (1988).

À base case parameter set ïs chosen, and the re-sponse sensitivity is identified by varying one random parameter at a time while keeping all others fixed at the base case values. This will give the first order effect on response magnitude due to a variability of

each parameter, neglecting all possible coupling

ef-fects.

For each fixed set of basic parameters a relative-ly large number of simulations, (say 100), is required

to obtain a sufficiently accurate description of the distribution function for pipeline response. Each

si-mulation corresponds to a different realization of the

wave process.

Based on the generated response distribution matrix, which should be located close to the design point, a numerical algorithm can be used to integrate over the failure domain to calculate the failure pro-bability. If the parameters are independent this will involve a series of one-dimensional integrations. The design value of each random parameter is found as the value of the parameter where the contribution to

failure probability is largest.

When the design point is found, the chosen linea-rization point (i.e. base case) can be double-checked.

The linearization point should ideally be very close to the design point, in particular for the most sensi-tive parameters.

Through such a numerical procedure, the actual di-stribution function of the response and the sensiti-vity to a variation of a certain basic parameter are

directLy taken into account. However, any correlations between the random parameters will reduce the simpli-city of the calculation algorithm.

ybrid method. This approach is based on a formu-lation of the failure function similar to that above, and utilization of a general reliability analysis com-puter program. Correlated variables are then readily included. This method is the one used in the study re-ported here.

The raudom displacement in the failure function Eq. (1) is expressed in the following form:

(4)

In general Y(z) is expressed in terms of the basic variables z and the regression coefficients , for

each parameter and a scaling function f, to account for the statistical uncertainty due to variability of the wave process. Y5 is the mean displacement given the base case parameter set z0

Two different formulations of the displacement function Y have been applied in this study. Both

represent a linearization of the mean

response

rela-tion at

the point given by the random variable vector z=z0 (base case). Generation of the regression coeffi-cients and the function f, is based on the same kind

of response simulations as described for the method above. The accuracy of the formulation

illustrated here is dependent on the choice of linearization point. This should be close to the design point, in

particular for the most sensitive parameters. However, the choice of linearization point can be checked against the final design point as found from the

re-Liability calculations.

Computation of the design point and the probabi-lity of failure is performed using the computer pro-gram ISPUD. An optimization procedure is applied for

determination of the design point. The failure proba-bility is then calculated by using an important samp-ling procedure concentrated around the design point. The integration is accordingly performed in the region where the contribution to the failure probability has

its maximum. The design point calculation is based on

formulation of the failure function in the original space, i.e. no transformation to the normalized space is required. The combination of optimization and im-portance sampling seems to be an effective procedure for the present reliability analysis.

APPLICATION OF THE METHOD

The hybrid formulation presented above for sensi-tivity analysis and reliability calculation will be

applied to a relevant example. Determination of the submerged pipe weight, W , to meet the specified

cri-teria is the main design task. Sensitivity of the failure probability to variations in the basic random purameters is examined. Also, effects of model uricer-tainties and sea state realization uncertainties are studied.

Design data

The statistical parameters defining the long term distribution of H and T are calculated from a scatter diagram recorded in the southern part of the North Sea. Wave directionality and spreading is not taken into consideration in this study. The current is assumed to be perpendicular to the pipeline with no

distribution over direction. The water depth is 30 s and the sea bed consists of sand soil.

A summary of the design data is given in Table 1. The parameters in the Weibull distribution of H, and

V are given in the table. The values corresponding to the 1 year and 100 years return periods for H, are 5.8

n and 1.1 n, respectively. Similarly, current veloci-ties for the 1 year and 10 years return period are

0.54 s/s and 0.6 m/s, respectively. The outer pipe diameter is 1.25 m and the relative sand soil density

is 0.35.

A lognormal distributLon is applied for the rela-tive soil density, with a mean value of 0.35 (medium condition). The uncertainty in the estimate of the pa-rameter is represented by a standard deviation of

Table 1. Design data and probability distributions

£ = 0.4

0.15. This relatively Large variability of the soil density accounts for both the spatial variation and

the uncertainty in the specific porosity of the sand

sod. The uncertainty in the hydrodynamic force is

included by a coefficient of variation, CDV, of 0.1 applied to the magnitude of the force according to

findings by Verley and Reed, (1989). Uncertainties related to soil resistance force are taken to be

so-mewhat higher, and the CDV is set equal to 0.15 n the reliability calculations, (Brennodden et al.,

1989).

With respect to the uncertainty related to the soil resistance, it is noted that the model is assumed

to give a correct representation of the expected soil capacity for a given penetration level, i.e. no bias is included. It is, however, not fully verified that the model is without bias for general applications, in particular for cases with realistic lift forces which are more complex than the situation modelled in the Laboratory study (Brennodden et al.,

1989). A

model bias would of course change the reliability levels calculated in the following case study.

Distribution of the factor f, to account for statistical uncertainty due to variability of the wave process is discussed later in the paper.

Base case araieters

To generate the distribution of the short term response (displacement), the following base case has been chosen.

Base case: H = H = 7.1 n, T = T H = 10.75 mec

s P F

V

= V

= 0.6

m/s, D, = 0.35

c

c5

W, = 6225 N/m, D = 1.25 i, d =

30 i

f14 = f5 = 1.0

where f14 and f5 are scaling factors for the hydro-dynamic force and soil resistance force respectively.

The above combination of design values for H, and are n accordance with reco.mendations in RP £305,

(1988),

and comply with experience from earlier stu-dies. Use of 100 years return period for H, and 10

years for and exoectation values for the other pa-rameters reflects a wave dominated situation and some degree of correlation between wave heights and cur-rent. The above design values represent an appropriate choice if the target annual probability of exceedance

of design criteria is 10.

Para- Mean St. dey. Distribution Distribution

meter value parameters

H(m)

2.04

0.96

Weibull = 2.307

= 2.253

e T(sec) 5.73 1.76 Lognorm.al V(m/s) 0.211

0.089

Weibull

o = 0.238

= 2.525 e D 0.35 0.15 Lognormal

W,(N/)

6225 50 Normal D (s) 1.25 0.02 4orsal 1 (z) 30 0.5 Normal f14 1.0 0.1 Normal f, 1.0 0.15 Normal f,, 1.0 0.3 Weibull Location

(5)

Response

siiulatjon

The

pipeline

response sensitivity to a variation

in each of the basic

randoc

variables

is

found

by

varying

one

parameter

at

a

time while holding all

others fixed at the base case value. Table 2

summari.

zes

the

simulation

matrix and gives the mean value,

and the COy

of displacement response.

Case no. 01

base case parameter values

For each of the 10 cases in the table,

100

simula-tions

were

performed

using

the

computer

program

PONDUS,

(Holthe

et

aL.

1987), to establish the

di-stribution function for the lateral pipeline

displace-ment

in

a short term sea state condition. Due to the

statistical variation of the sea elevation, a

variati-on

in the lateral displacement of the pipeline is

ex-perienced. This variability is classified as

statis-tical uncertainty.

Both the cumulative distribution and density

func-tion of the pipeline displacement are

illustrated

in

Figures

1-5,

together with fitted distributions. The

figures illustrate the spreading of

maximum

pipeline

displacement

during

a sea state of

3 hours duration.

Each figure is related to one of the 10 cases in Table

2

and

based

on

100 simulations with different wave

realìzatjons. Estimates of the distribution

functions

are obtained by the computer

program STARTIMES

(Skjâ-atad and Fames, 1989, Hoen and Brathaug, 1987).

Fin-dings

from

this study are: A three parameter Weibull

distribution seems to fit

the

numerically

generated

displacement

distribution quite well when

is about

20 m (Figure 1). For increasing

displacement

Levels

the

data approaches a normal distribution (Figure 5),

whrch could be expected as the total displacement

con-sists

of a sum of a large number of small

contributi-ons from single wave eventS (central limit theorem)

The coefficient of variation increases for

decrea-sing displacement level, as say be expected. For

app-lication

in

the reliability calculation it is

impor-tant to have a good fit

r the tail of the

distributi-on.

It

is seen that when

j

is low

(p5 (

5-10 m)

arr

exponential distribution fits the tail well (Figure

2

and

Figure 3), whereas for higher displacement levels

25-30 rn),

the normal distribution

provides

the

best

model.

A Weibull distribution with a COy of 30'.

has been chosen to represent

the

statistical

varia-bility

of

the

displacement, i.e. the function f, in

Eq.

(4) .

The Weibull distribution

fits

well

to

the

data

and

is

the

best among the candidates selected

here when the mean displacement is equal to the design

criteria

applied

(Y5=20 rs). A COV of 30

will

under-estimate the variability tor mean

displacements

less

than

about

10 m, and overestimate for mean

displace-ments larger than, say

30 m.

131

Para..eterization of response

For

application

of the reliability computer

pro-gram,

ti-- failure function g(z) has

to

be

expressed

directly

in

term.s of the basic random variables. Two

different formulations of the failure function in

Eq.

(1)

have

been applied here irr order to provide

inde-pendent calculations of P

The first expression for 1(z) is based on a sum of

first order contributions from the basic variables.

n

Y (z)

= [Y

+ E (z1

-z° )]f,

(5)

j=1

where

Y

= random displacement of the pipeline

Y

= base case mean displacement value

= regression coefficient for parameter no.

i

z,

= basic parameter no.

i

base case value of variable no.

i

f,

= Weibull

distributed

variable

to

take

into

account the

uncertainty

in response due to the

variability of the wave process

The

factors are

calculated from sensitivity

study results.

Response effects due to a coupling

of

the

basic variables

are not included in the failure

function above.

In

the

second formulation, the pipeline response

may be

described

as

a

function

of

nondimensional

scaling

groups expressed in terms of the basic random

variables

(Lambrakos et al., 1987,

Sotberg

et

al.

1988). The following scaling groups are used:

UT

W V T

,M=,T=

(6)

0.5DU,2

s u

where U, and T,, are the significant water velocity and

zero uperossing period, respectively.

Ti is

the

sea

state

duration (3 hours) and

is the water density.

T is the number of waves in the sea state.

Linearization of the mean

esporìse relation is

ex-pressed in terms of these scaling groups

irr

addition

to

the

soil density parameter and model and

statis-tical uncertainty factors.

The

random

displacement

function has the following fori:

3 z1 6 z

1(z) = [Y

+ L

(-1)T-f,

'

[

(-1)+1)

i=1

z,0

j=4

zj°

The

summation with index i covers the sensitivity

to the scaling parameters K,

L

and

M,

whereas

the

product

with index j represents the effect due to

re-lative soil density, Dr ,

and

the

model

uncertainty

factors,

H

and

f,

.

Et is expected that the effects

due to a variation of the scaling parameters are

addi-tive, while the hydrodynamic force and soil resistance

uncertainty as well as the variability of soi],

proper-ties represent a multiplicative effect on responme.

This

formulation is expected to constitute a mure

stable form in the vicinity of the

design

point

and

will

also

take

possible

coupling effects into

con-sideration through the scaling groups K, L and M.

How-ever,

the

sensitivity of the probability of failure

estimate will be checked by application of both

formu-lations.

Table 2.

Simulation matrix

Case

no.

H, m T

sec

V

rn/s

Dr -D rs W, N/m d f m -f5 -j5 rs Coy

-01 03 04 05 06 07 02 08 09 10

7.1

8.0

10.75

12.0

0.6

0.75

0.35

0.45

1.25

1.20

6225 7500 30 1 35

0.85

1

1.15

25.3

60.1

54.0

135.0

44.7

14.0

4.9

9,3

16.5

19.5

.38

.25

.38

.15

.32

.49

.42

.74

.53

.51

(6)

Reliability calculation results

The design criterion adopted is a lateral displa-cement of Y1 = 20 m for a free pipeline section away from obstacles or Other structures, corresponding to the SLS condition. The Corresponding

target probabi-lity of exceedance for a reference period of one year is P = l02 which is equivalent to a target probabi-lity P1 0.34 iO for a reference period of 3 hours.

A summary of the result for the nominal base case

data is given in fable 3 below.

Probability of failure, P1 = 0.5 iO

The probability of failure P1 is calculated to be

0.5 iQ-. The coefficient of correlation between H, and T1, Mf is equal to zero for this case. Note that application of Equation (5) or (7) in the failure function gives no significant difference in the esti-mate of

P,

i.e. the formulations coincide with respect to calculation of failure probabilities for this application.

Only a few of the random parameters give signifi-cant contributions to the variability of the response.

In this case the submerged pipe weight, outer pipe diameter and the water depth could be taken as deter-ministic quantitites. This is in accordance with

earLier findings, and due to the low uncertainty in

these parameter values.

The total uncertainty of response can be classifi-ed into two main groups. The long term uncertainty represented by the ocean environment is seen to

produce a large contribution. Among the short term un-certainties the soil density variation and the statis-tical variability of the response due to the uncer-tainty in the wave record are the most significant contributions.

According to the a vector, the sensitivity to the

peak period, T1 , is seen to be Lower than for H, . This is expected to be due to the low water depth in this case. Increasing water depth will increase the impor-tance at the peak wave period relative to the signifi-cant wave height, K,. The flow velocity amplitude at the sea bed is a linear function of wave height, whereas the wave period is a parameter in the

ex-ponential depth attenuation function. Hence, for inc-reasing water depth, an incinc-reasing period is more sig-nificant to the flow velocity than an increase in wave

height, Sotberg et al. (1988).

The uncertainties in the hydrodynamic force and soil resistance models are not significant with respect to the uncertainty in pipe displacement provi-ded that the models are without any bias. This is

il-lustrated through the importance factors, and in addi-tion by the fact that an increase of the COV of the scaling factors by a factor of 3 gave no significant

effect on P1 . Application of a lognormal distribution for the scaling factors similarly gave a negligible

effect.

The long term uncertainties represent about 8O' of the total variability of response for the base case,

while the remaining 2O is caused by soil parameter and statistical uncertainty.

By increasing the correlation between H, and T1,

the P1 is found to increase significantly. A reduction

of displacement criteria will of course increase the probability of failure or exceedarice. Reduction of the pipe weight will increase the area of contribution to failure and hence also increase the probability of

failure. Figs. 6 and 7 illustrate these effects for a certain variation of

T and W, for different vaLues of the coefficient of correlation between H,-T and H, -V . By analysing the scatter diagram which was the

basis for determination of the long term distributions for H, and T , a coefficient of correlation of about

0.6 was found.

The relative importance of the peak wave period compared to the significant wave height or current ve-locity increases for increasing correlation

ç'1,

between wave and current. Table 3 and 4 illustrate this effect on the actual design point for increasing correlation between H, and T1

W,6225

'Hf°6

y1=ZO

Probability of failure, P1 = 0.3 10-2

In the recommendations given in RP ElOS, (1988) a return period of the combined wave and current loading equal to 100 years is proposed. If no information on

the joint probability of wave and current is available the following combination is recommended:

If wave forces dominate

H,100

and

V10

If current forces dominate: H,10 and Vtoo

The design point identified from the reliability calculations reflects a wave dominated situation as

the return period for current is low (< 1 year) compa-red to that for H, (lO years), see Table 4. Both para-meters are related to return periods less than the re-commendations given above. This is, however, due to

the additional ran.ìom parameters included in the present reliability calculation. Selection of these additional parameters to their expectation values would bring the wave parameters up to larger return periods related to the target probability level. Table 3. Results for base case data with p=O

Random variable a vector Importance factor Design vslue H, (s) 0.61 37.6 5.62 T1 (sec) 0.50 25.0 11.08 V (mis) 0.45 19.8 0.45 D, 0.33 11.2 0.55 W, (N/m) -0.02 0.0 6218 D (s) 0.08 0.6 1.26 d (s) -0.05 0.3 29.85 0.12 1.4 1.07 f1 -0.12 1.4 0.89 0.16 2.7 1.30

Table 4. Results for base case data with HT°6

Random variable a vector Importance factor Design value 0.53 28.0 6.31 T1(sec) 0.80 64.0 17.59

V(m/s)

0.20 4.1 0.36 0.11 1.3 0.44 W,(N/m) -0.01 0.0 6222 D )m) 0.01 0.0 1.25 d (s) -0.02 0.0 29.93 0.06 .4 1.05 f1 -0.06 .4 0.92 0.09 .8 1.22 W, = 6225 N/s Y1 = 20 s QHT = O

(7)

A design calculation based on the generalized method in the code RP E305 (988) gave a design weight

= 6300 N/rn

using the base case design data. This result is seen to be close to but above that found from the reliability calculations. A consistent

com-parison is based on a correlation

PHT = 0.6 which was

estimated from data in the scatter diagram of K, and T . From Figure 6 it LS

seen that a pipe weight of

atout 5500 N/rn would satisfy the target probability level uf 10 when p = 0.6. Application of RP E305 (1988) is hence conservative for this case. However,

for a lower correlation between

H, and T an excessive conservatism seems to be present.

Increasing the correlation between the ocean envi-ronmental variables, tends to decrease the relative importance of the statistical uncertainty and the model uncertainty, see Table 4, i.e. the short term

uncertainty is less important for increased correlati-on between the lcorrelati-ong term envircorrelati-onmental parameters. For fully correlated variables the design values should be based on equal probability levels related to each mar-ginal distribution.

The sensitivity to the long term ocean environment is investigated for a case with pipe weight equal to 5000 N/sr

Y = 20 sr, g = O and v15

= 0.6

rn/s.

Inc-reasing H,100 from 7.1 in to 8.0 s changes P from

0.2. 10 to 0.8 10 P

was further increased to 4.3 10-2, a factor of about 50, by increasing V10

to

0.83

sr/s a factor of about 50, which confirms the high Sensitivity to the long term ocean environmental parameters

CONCLUSIONS

Analysis and ethods.

Uncertainty analysis related to the stability design of submarine pipelines has been performed

inc-luding parameter uncertainties, model uncertainties as well as statistical uncertainties related to the wave process.

Three different methods of sensitivity analysis and reliability calculation procedures relevant for

the present problem have been outlined. The method based on a response surface description ar,d utili-zation of an important sampling procedure have been found efficient and applied in the present case.

Application of the method has been illustrated through a case study on 30 meter water depth.

Comprehensive simulations related to the case study provided a basis for generation of the response surface in the area of the design point. The

numeri-cal simulation illustrated the statistinumeri-cal variability

of lateral displacement during a 3 hour sea state, which was fitted to a Weibull distribution with a

co-efficient of variation as high as 30 '.

Two different formulations of the failure surface have been applied which gave identical results with

respect to the probability of failure.

Reliability calculations.

The sensitivity of P to the various random para-meters is briefly summarized in the following.

The ocean environment gives a large contribution to the total uncertainty (80-90'.).

Uncertainty in the relative soil density and sta-tistical uncertainty are the dominating short term contributions amounting to about

10-20'.,

which s in accordance with findings by Sotberg et al. (1989).

lodel uncertainties have a relatively srnall. effect ori the calculated P - This is based on the assuptiOfl

that the bias in the models are negligible. In the earlier paper by Sotberg et al. (1989) a relatively simple representation of model effects was included by magnifying the short term variance, which gave a

larger effect on the total response. Physically, this implies a correlation between the model uncertainty factors and f, and the statistical uncertainty factor f equal to 1.0, which represents a much too conservative assumption.

Increased correlation between the ocean environ-mental parameters will in general increase the proba-bility of failure.

From Figures 6 and 7, it can be seen that with no correlation between wave height and period, a

sub-merged weight as low as 3200 N/rn could be allowed for the given design data and displacement limit.

By introducing a typical level of correlation,

oHrO.6,

the pipe weight has to be increased to 5500 N/rn to satisfy the given criteria. The pipe weight has to be increased to W,

= 6200

{/m when applying a

cor-relation coefficient equal to 0.5 between current and wave heights, which is generally conservative.

Application of RP E305 would give a conservative but comparable result, w, = 6300 P4/rn. The

recoefl-dations given in the mentioned code are in general conservative with respect to design values for the en-vironmental parameters. This is particularly true when the correlations are low.

DISWSSUW ANORECOINDATION.S

A method for reliability calculations of submarine pipeline systems has been outlined. The SLS design criterion for pipes on sand soil has been considered. The procedure has been illustrated by application to a

realistic pipeline case in 30

t

water depth.

The inherent safety in the Recommended Practice RP F305, (1988) is found to be above the design criteria

of 10-2. When correlation between the ocean environ-mental parameters is low, the conservatism seems to be unnecessarily large.

A full probabilistic design gives arr overall quan-tification of the different uncertainty sources and the effect on the probability of violating the rele-vant design criteria. The total uncertainty can be

classified into two main groups; long ter. uncertainty represented by the ocean environment and short term uncertainty due to soil parameter uncertainty, model uncertainty and statistical variability of the sea

elevation process.

It is found that a refined probabilistic design would imply a reduction of design weight compared to

more simplified methods and lower level reliability calculations. Application of such a method would hence be cost effective.

Further work on this topic may be related to cali-bration of rules. The method can be used to perform a more refined calibration of the procedures given in RP

F305. It is noted that calibration should be perfor-med based ori a compromise between accuracy in descrip-tion and simplicity irr use. Further refinements irr

codes related to pipeline stability design are ex-pected to be required. Extension of the present work say include additional sources of uncertainties such as variability of wave kinematics at the sea bottom as well as a further investigation of any bias in the soil model when applied for realistic hydrodynamic

(8)

Reliability calculations considering the strain response and corresponding formulation of the failure

function for that case is another area where this app-roach should be applied.

The probability of fatigue failure for an on-bottom to free spanning pipeline as well as for a constrained on-bottom section, could be given a tho-rough evaluation based on the reliability methods used in the present paper.

ACKfrOWLEfJ3E(ENT

The authors wish to thank Statoil for permission to publish this paper.

REFERENCES

Brerinodden, H. Lieng, J.T. and Sotberg, T. Àn

Energy-Based Pipe-Soil Interaction Model', Proc. of 21st Offshore Technology Conference, Paper No. 6057, Houston, 1989.

Borgund, 1(. and Bucher, C.G. : Imporance Sampling Procedure Using Design Points (ISPUD) - A Users

Manual', Institute of Engineering Mechanics,

Uni-versity of Innsbruck, Report S-86.

Det

norske Ventas

(DnV) : 'Rules for the Design,

Construction and rnspection of Offshore Structures', HØvik, Norway, 1982.

Ellinas, C.P. and Williams, K.A.J. : Reliability Engineering Techniques in Subsea Pipeline Design'

proc. of the Eight

mt,

Conference on Offshore Mechanics and Artic Enqn.. The Hague, March 19-23,

1989, Vol. V.

Ellinas, C.P. et al.: 'Limit State Philosophy in Pipeline Design', Journal of Offshore Mechanics and

Artic Engineering, Vol. 109, pp. 9-22, 1987.

Fjeld, S.: 'Reliability of Offshore Structures', Proc. of the 9th Offshore Technology Conference, Paper No. 3027, Houston, 1977.

Grace, RA.: 'Near-bottom motion under ocean waves.

Proc. of the 15th

mt.

Conference on Coastal Engineering, Honolulu 1976, pp. 2371-2386

dC.'i. V.

rv. E 9

---..,LLOf.4.

I I I I

-

L. -

J

-i I I I I -1

/

T

Hi

-

.L__l__J_._J

_L__

I, I I -2 -, O Z S sr. V. 9

.:

0-sc ro. O

t

P/miT OIS

.:

so. o( V. o(

o,t.,.a* ,--ç.. ..:

O

p0,'_.*

- 2 0Oi

s..a.v...:

S

,,..:

Ftt.d

2.i,Ct

s

Tr--4

o,.'-cr, .I.

2.1Oi

54.

v..:

9 .SS4+O3

F:'pre 1 CDF of pipeline displacement, base case

No. 01, weibull fit

Haver, S. and Nyhus, KA. : 'A Wave Climate Description for Long Term Response Calculations',

Proc. Fitth

mt.

SYmP. on Offshore Mach. and Arctic Engn. , Tokyo, April 13-18, 1986, Vol. IV pp. 27-35.

Hoen, C. and Brathaug, H-P.: STARTIMES: ?roaot User Manual', SINTEF Report STF71 A87047, Trondheim, Norway.

Holtbe, K., Sotberg, T. and Chao, J.C.: 'An Effi-cient Computer Model for Predicting Submarine Pipeline Response to Waves and Current', proc. of Nineteenth Offshore Technology Conference, Paper No. 5502,

Houston, 1987.

Jacobsen, V. and Bryndum, MB.: 'Determination of

flow kinematics close to the marine pipelines and their use in stability calculations. groc. of 16th Offshore Technology Conference, Paper Mo. 4833,

Houston, 1984.

Lambrakos, K.F., Remseth, S., Sotberg, T. and

Verley, R.: 'Generalized Response of Marine Pipelines', Proc. of Nineteenth Offshore Technology Conferenc.,.. Paper No. 5507, Houston, 1987.

Recommended Practice, RP E305: 'On'Bottom Stability Design of Submarine PipeLines', Veritec, OsLo, Oct. 1988.

Skjâstad, O. and Fames, KA. : 'STARTII4ES: Promot Theory Manual , SINTEF Report, STF7I A89012,

Trondheim, Norway.

Sotberg, T., Lambrakos, KF., Remseth, S., Verley, R.L.P., Wolfram, jr., W.R.: Stability Design of

Marine Pipelines', l'roc. BOSS Conference, Trondheil, June 1988.

Sotberg, T., Lema, ß.J. and Verley, R.L.P. r Proba-bilistic Stability Design of Submarine Pipelires, Froc. of the

Eight mt.

Conference on Offshore Mechanics and Attic Erigo., The Hague, March 19-23,

1989, Vol. V.

Verley, R.L.P. and Reed, K.: 'lise of Laboratory Force Data in Pipeline Response Simulations' Proc. of

the Eiqht mt. Conference on Offshore Mechanics and

Arctic Engn. , The Hague, March 19- 23, 1989, Vol. V. Wolfram Jr., W.R., Getz, J.R. and Verley,. R.L.P.: P[PESTAB Project: Improved Design Basis for Submarine Pipeline Stability', proc. of Nineteenth Offsriore

Tech.nolov Conference, Paper No. 5501, Houston, 1987.

P,.p.1,.ro d,.roLos,4. 40-' W.

- - -

E.n4..oL2 ft'.

I I I

Z.9-.---I.---4---I-4

-L.,.ò.---.4

I I I I I I I

r

I I

IL -

J i L

-P4T OIS PI.JTI4

.:

NO.

o.' .&...,:

Mo.O.'

o.,L.,,d. ,-. .:

O

S4..4.004. -.

4

S4..,'...:

I

mm

Kso-4.t..:

4.I8

e,.:

I I *O4 F.tI_ U.oro4.'.00l. - 2

*

SooI.u'q. 2 O144E'$

4.?E'I

2.O144E.

Figure 2 CDF of pipeline displacement, case Mo.

02,

Exponential fit

-2 -, 2 5 4

ST. s.

p.r SDE

(9)

0.99- £AOAPO4.s..

-

- - T

fU.

igure 3 CDF of pipeline displacement case No. 02,

Wejbull fit 0.5 0.0

d.oL4. 0.1Sø3 MI.

PLp.y. dt.pLfl$. rS ps,.

L -2.6-2.2-4.S-'.0-6.S 0.0 S.S '.0 4.5 2.3 2.5 ST. V. PÑ(T Sc42S o,._. 4J, 50 . c.

PT OISTRIJTI4

. : Fill. No. o(

L....:

joe

No. o" .L.

o r..

S4.ots.tooL

-.

i

Sko...

2.37446,02 T .3rO K,.r4.o.,..T

4.48

l.ZO4

f.4...d tJ.o..tcoL SLUTQ..: 4 .I.

4 .*02

2.34446,02

PfT DISRlJTrf

E.f.,.ot'_o"

..:

FU.I. No. of .QL... . T 432 No. of

L.

o.A.,.d. T 3 S4.o& L -.

I.4+OI

K.r*.o.,..

2.T4

41u...T

2. 41,O4

4,,

9.I1+OI

F,.4.f..d 4JTo'n.L n

6 .,C 4

S.4..d.v...T

4.i4Ol

20 't'5[m)

Figure 7 P versus the displacement criterion Y1

with V = 6225 Nf and Q14T = 0.6 4.0 3.0 2.0 1.0 0.0 d,..O.4.o.-T Po'.*n( . L: jo.,LL-3

- -

StoL,..4.,.L

4 .lEt02

- -

.1. : L_ S&.d...'...T 2.37+46+02 1 . f .3rT2 I I K,.rto.,.,.:

i.8

- -4

- - -

I

- -

pj,.n...:

i

Ii

U i . i

Noo,..:

I S

Figure 4 PDF of pipeline displacement, case No. 02,

Weibull fit

- ST. TV. pn(T 5ES

yo'-. TNoU' OC - 'o'. OC

PPfT 0IS1RIJTI04 E.11o

.o'..:

LL c.

of *. .

No. of .i..

o'A.,4. ro'Q. . . T O

F,.tf..d 4j-r.L

L000&o'

I..02

5ocL.rnQ..:

i.

igure 5 CDF of pipeline displacement, case No. 03, 3000 4000 6225

Gaussian fit W, IN/mi

Figure 6 Pf versus w0 for different levels of

correlation between H,-T0 and H,-V

P'

5 10 15

= 20m

5.0

Cytaty

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