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Three-Dimensional Finite Element Modeling for Wave-Seabed Pipeline Interaction

T.N. Chen

Department of Civil Engineering National Chung-Hsing,Taichung,402,ROC Taiwan

K.F. Cai

Department of Civil Engineering National Chung-Hsing,Taichung,402,ROC Taiwan

Y.S. Lin

Department of Civil Engineering National Chung-Hsing,Taichung,402,ROC Taiwan

Abstract - The wave-seabed-pipeline interaction problem is particularly important for coastal geotechnical engineers involved in the design of submarine pipelines. However, most previous investigations have been only limited to two-dimensional cases, any directional wave forced on buried pipeline cannot be taken account. We will examine the problem of wave-seabed-pipeline interaction from the aspect of marine geotechnical engineering in three-dimensions.

In this study, we will establish a three-dimensional finite element model to investigate the basic mechanism of wave-seabed-pipeline interaction. In the model, the boundary between soil and pipeline will be considered. Unlike previous models, the oblique waves are also considered in the new model. Based on the new three-dimensional finite element model, effects of wave characteristics (including wave direction), soil behavior and properties of the pipeline (such as dimension of pipeline and size of pipeline etc) on the wave-induced soil response will be examined. Moreover, the internal stresses of buried pipeline change significantly with different properties of the pipeline, especially for the case of oblique waves. This research validates the accuracy and effectiveness of our mathematical model and simulation program. Hopefully, it is expected to be provided for the underwater engineering practice.

I. INTRODUCTION

Submarine pipelines have been a type of common used offshore installations. When gravitational waves propagate over the ocean, they cause fluctuating pressure upon the seabed, which will further induce excess pore pressure and effective stresses within seabed soil. Marine structures caused by liquefaction and erosion of the seabed soils in the vicinity of the structure, resulting in collapse of the structure as a whole. (Maeno and Nago, 1988)Under such conditions, the pipelines will lose stability. Therefore, the evaluation of the wave-induced soil response around a buried pipeline is important for coastal geotechnical engineers.

To date, many researchers have developed theories for the wave-induced soil responses in an elastic medium.

Yamamoto et al.[1] and Madsen[7] proposed analytical solutions for ocean wave-seabed interaction problem with hydraulically isotropic an anisotropic uniform seabed of in finite thickness, respectively.

Although the importance of wave-seabed pipeline interaction penomenon has been addressed in the literature,this problem has not been fully understood

because of the complication of behaviors of soil,wave loading and geometry of pipeline. However, all aforementioned investigations have only examined the soil response in seabed under the action of two-dimensional progressive waves with one directional incident wave.

Based on Biot’s theory [5], the wave-induced pore pressure around a buried pipeline has been studied through a boundary integral equation method and a finite element method. Among these, Cheng and Lin considered a buried pipe in a region that is surrounded by a two impermeable walls. Magda[12,13] considered a similar case with a wider range of the degree of saturation. All these have only discussed the wave-induced pore pressure and uplift forces around the buried pipe. Other soil responses in the vicinity of a buried pipeline, such as effective stresses, were not discussed. Lin [15-17] investigated the wave-induced pore pressure, effective stresses along the buried pipeline in the seabed with variable permeability and shear modulus. But, the author has only concerned with one direction incident angle in two dimensions, ignoring the influence of wave direction. Therefore, it is necessary to examine the influence of different wave directions upon the wave-pipe-line-seabed interaction.

This paper is aimed at investigating the distributions of three dimensional wave-induced pore pressures along the surface of pipeline and the internal stresses within the pipeline. In this study, the three dimensional finite element model proposed from two dimensional finite element models by the author will be adopted.

II. GOVERNING EQUATIONS AND FINITE ELEMENT FORMULATION

In this study, we consider a fully buried pipeline (with radius of R) in a porous seabed of finite thickness h laid upon an impermeable rigid bottom, as depicted in Fig. 1.

The three-dimensional finite element is modeled to investigate the basic mechanism of wave-seabed-pipeline interaction. The wave is assumed to propagate in the positive y-direction, while the z-direction is upward from the seabed bottom; the positive x-direction is defined right hand theory (see Fig. 1).

A. Governing Equations

Based on conservation of mass, Biot’s consolidation equation (Biot, 1941) is used as the governing equation for the wave-induced seabed response (Madsen, 1978 and so

(2)

Fig. 1. Finite element mesh on), that is

(2.1)

where p is the wave-induced pore pressure, γωis the unit weight of pore fluid,n′ is soil porosity, and t is time.

In (2.1), the compressibility of pore fluid β and soil volume strain ε are defined by

(2.2)

(2.3)

where Kω is true modulus of elasticity of water (2×109N m2 ),Pωo is absolute water pressure, S is degree of saturation, and u ,

v

and ω are the soil displacement in the x, y and z directions.

The equations of equilibrium in a poro-elastic material can be expressed in terms of effective stresses and pore pressure as

(2.4)

(2.5) (2.6) Under conditions of three dimensional plane strains, the

equations of equilibrium are

(2.7)

(2.8)

(2.9)

(2.10)

(2.11)

(2.12) The relationship between the shear modulus (G ),the Young modulus (E) and the possion ratio(µ) are defined by

Substituting (2.7) ~ (2.9) into (2.4) ~ (2.6) yields:

(2.13)

(2.14)

(2.15) The governing equations of pore pressure and soil

displacements for the problems of the interaction between wave, seabed and pipeline are (2.1), (2.13), (2.14) and (2.15).

B. Finite Element Analysis

Since the wave-induced oscillatory soil response is periodically fluctuating in the temporal domain, the wave-induced pore pressure, effective stresses and soil displacements can be expressed as

(2.16)

where subscripts ”r ” and ” c ” represent the real and imaginary parts of the soil. Substituting (2.16) into (2.13),(2.14) and (2.15), directly applying the Galerkin’s method, to these equations, the finite element analytical formulations can be expressed in a matrix form as

(2.17) ,

, , , ,

, , , ,

, ,

, , , , , , , ,

, ,

, , , , , , , (

, , ,

t i

yzc xzc xyc z yc xc c c c c

yzr xzr xyr zr yr xr r r r r

yz xz xy z y

x e

z) (x,y T

z) (x,y T

z) (x,y T

z) (x,y S

z) (x,y S

z) (x,y S

z) (x,y W

y,z) (x V

y,z) (x U

z) (x,y P

i

z) (x,y T

z) (x,y T

z) (x,y T

z) (x,y S

z) (x,y S

z) (x,y S

z) (x,y W

y,z) (x V

y,z) (x U

z) (x,y P

z;t) (x,y

z;t) (x,y

z;t) (x,y

z;t) (x,y

z;t) (x,y

z;t) (x,y

z;t) x,y w

z;t) v(x,y

z;t) u(x,y

z;t) p(x,y

ω

τ τ τ σ σ

σ





















































+

























=

























z .

w y v x u

∂ +∂

∂ +∂

=∂ ε

, x p z y

x xy xz

x ∂ +∂ ∂ +∂ ∂ =∂ ∂

∂σ′ τ τ

1 , 1

P o

S Kω ω

β = + −

, y p z y

x y yz

xy ∂ +∂ ′ ∂ +∂ ∂ =∂ ∂

∂τ σ τ

. z p z y

x yz z

xz ∂ +∂ ∂ +∂ ′ ∂ =∂ ∂

∂τ τ σ

2 , 1

2









∂∂

=









∂ +∂

∂ +∂

∂ ∂

+∂

∂ +∂

∂ + −





∇ 

x Px P

z W y V x

U z

W y V x U x G U

G U

c r

c c c

r r r

c r

µ

2 , 1

2









∂∂

=









∂ +∂

∂ +∂

∂ ∂

+∂

∂ +∂

∂ + −





∇ 

y Py P

z W y V x

U z

W y V x U y G V

G V

c r

c c c

r r r

c r

µ

2 . 1

2









∂∂

=









∂ +∂

∂ +∂

∂ ∂

+∂

∂ +∂

∂ + −





∇ 

z Pz P

z W y V x

U z

W y V x U z G W

G W

c r

c c c

r r r

c r

µ ,

) ( 22 22 22

t t n p z

p y

p x

k p

= ∂

′ ∂

∂ − +∂

∂ +∂

∂ β ε

γω

[ ] [ ]

[ ] [ ]

,

3 2 3

2 2 2

1 1 1

∫ +

∫ +

=

A T

A T

A T

s

dA dA dA tdS

Ni

U B D B

P B D B

P B D B Qe

. )) 1 ( 2 ( +µ

= E G

[

( 1 2 )

]

,

2 µ µε

σ′x = Gux+ −

[

( 1 2 )

]

,

2 µ µε

σ′y = Gvy+ −

[

( 1 2 )

]

,

2 µ µ ε

σ′z = Gwz+ −

[ ]

yx,

xy G u y v x τ

τ = ∂ ∂ +∂ ∂ =

[ ]

zx,

xz G u z w x τ

τ = ∂ ∂ +∂ ∂ =

[ ]

zy.

yz G v z w y τ

τ = ∂ ∂ +∂ ∂ =

(3)

(2.18)

where,

where ne is the number of nodes per element, Ni is the shape function of ith node,and coefficient matrices Bi and Di can be derived from the governing equations, and Fb is the force matrix acting on the pipeline.

C. Boundary Conditions

Wave-induced boundary conditions, as shown in Fig. 1, the wave crests are assumed to propagate in the x-direction, while the z-direction is upward from the seabed bottom, will be discussed in detail:

1)Boundary conditions of seabed

For the soil resting on an impermeable rigid base, zero displacements and no vertical flow occur at the impermeable horizontal bottom, i.e.

at

2)Boundary conditions of surface of seabed

one assumes that the bottom frictional stress is small

Fig. 2. Finite element mesh in the vicinity of pipeline.

and negligible. The vertical effective normal stress and shear stress vanish and pore pressure is equal to the wave pressure at the surface of the seabed,i.e.

at

The pore pressure at the surface of seabed is equal to the wave pressure induced by the progressive wave.Therefore, pore pressure is equal to the wave pressure at the surface of the seabed,i.e.

wherepοϖH 2coshkd , denotes the amplitude of the wave pressure at the surface of the seabed, d =water depth, H =wave height, k =wave number, ω =wave frequency,Re represents the real part of the function in the brackets.

3)The surface boundary conditions of buried pipeline The buried pipeline is regarded as impermeable. There is no pore pressure around the shell of pipeline.i.e.

at

where x0,y and0 z0are global coordinate of the pipeline center , x , y and z are the global coordinate of pipeline shell, n is normal direction of pipeline surface.

4)The lateral boundary conditions of seabed

Since the existence of the pipeline only affects the wave-induced soil response near the pipeline,the ‘disturbed pressure’ from the pipeline should vanish at poits far away from the pipeline. Thus, the lateral boundary conditions at this points are given by the solution without pipeline ,which will be described later.

Once the lateral boundary conditions are obtained, the local refinement of the FE mesh always has to be taken into account in the region near a pipeline as shown in Fig.

2.

III. Numerical Results and Discussions To have a basic understanding of the mechanism of the wave-seabed interaction in the vicinity of a buried pipeline, we examine the effects of wave and soil characteristics in the distribution of the wave-induced pore pressure at this section. The general input data for numerical examples are tabulated in Table I.

=0

=

=

=v w p z u

[ ]

[ ]

,

∫ +

= +∫

A A

S S

dV dV

dS dS

P B B U B D B

F B F

B

T 1 4 5

T 4 4

T b e 5

T5

) . ( )

(

) ( )

(

cosh 3 4

2 1









× +

×

× +

× ×

=

σ σ

σ γ σ

ω ω

ω ω ω

f e f

e

f e f

e kd p H

t i t

i

t i t

i

=0

=

′ = xz yz

z τ τ

σ z=h.

. ) ( ) ( )

(xx0 2+ yy0 2+ zz0 2 =R ,

=0

p n ,

i z

zc zr yzc yzr xzc xzr

y

yzc yzr yc yr xyc xyr

x

xzc xzr xyc xyr xr xr

zc zr yc yr xc xr

n

S S T T T T

n

T T S S T T

n

T T T T S S

f f f f f f





























+

















+













=













,

i c z r c y

r c x

r

c i

nr n

z Pz P n K

y Py P n K

x Px P K q

q

















∂∂

∂ +









∂∂

 +







∂∂

 =





ω ω

ω γ γ

γ

( ) ( )

( )

0

( )

,

0

0 0

1

1

 

=

nc ne nc

nr ne T nr

e q q

q Q q

"

"

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )

0

( )

0

( )

,

0

0 0

0

0 0

0 0

0 0

0

1 1 1

1 1 1





=

zc ne yc ne

xc ne

zr ne yr ne

xr ne

zc yc

xc

zr yr

xr eT

f f

f

f f

f

f f

f

f f

f F

"

"

( ) ( )

( )

0

( )

,

0

0 0

1

1

 

=

c ne c

r ne T r

p p

p p

"

"

P

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( )

0

( )

0

( )

,

0

0 0

0

0 0

0 0

0 0

0

1 1

1

1 1

1





=

c ne c ne

c ne

r ne r ne

r ne

c c

c

r r

T r

w v

u

w v

u

w v

u

w v

u

"

"

U

.

=0 z

(4)

The effect of wave-induced seabed response in the vicinity of a buried pipeline is considered. Fig. 3 illustrates contour of the cross section wave-induced pore pressure in the seabed. It is almost the same with the two-dimensional model solution[17].

Fig.4 illustrates vertical distributions of wave -induced pore pressure versus the soil depth. We found that with increasing seabed depth, the wave-induced pore pressure decreases. Fig.5 illustrates distributions of wave-induced pore pressure with pipeline and without pipeline.It is observed that the pore pressure near the pipeline changes more obviously with pipeline than those without pipeline.

Fig.6 illustrates distributions of wave-induced maximum effective principal stress with pipeline and without pipeline.The figure shows that there are concentration stresses around the pipeline and the normal stress at the surface of pipeline enlarges obviously.

A. Effects of Wave Direction

To havae a better understanding of the vertical distributions of wave-induced pore pressure, we choose two different wave directions. Fig. 7 shows distributions of wave-induced pore pressures with different wave incident

Fig. 3. Contour of the cross section wave-induced pore pressures in the seabed (incident at 45 degree).

Fig.4 Vertical distributions of wave -induced pore pressure versus the soil depth.

Fig. 5. Distributions of wave-induced pore pressure with pipeline and without pipeline.

angles θ at 15, 30, 45, 60, and 75 degrees around pipeline.

The wave-induced pore pressures change slightly with wave incident angles.

B. Effects of Buried Pipeline

Fig. 8 shows distributions of wave-induced(incident angle π/4) pore pressure around pipeline with different buried pipeline depths are 1.5m, 2.0m, 2.5m, 3.0m in three dimensions. The wave-induced pore pressure upon the pipeline decreases with buried pipeline depth increasing.

Fig. 6. Distributions of wave-induced maximum effective principal stress with pipeline and without pipeline.

TABLE I

INPUT A FOR NUMERICAL EXAMPLES Wave characteristics

Water period T 12 sec

Water depth d 40 m

Wave length L 193.6 m

Wave incident angle π/4 Soil characteristics

Porosity n′ 0.4

Poisson ratio µ 0.4

Seabed thickness hs 40 m

Young's module E 7×107N/m2 Coefficient of permeability 1×102m/sec Degree of saturation S 0.98

Pipline characteristics Pipeline radius R 12 sec Pipeline thickness t 40 m Young's module pE 3×1010N/m2

Poisson ratioν 0.2

Densityρ 2320 kg/m3

Burial pipeline depth d 2m

-0.5 0 0.5 1 1.5

P / P0 0

5 10 15 20 25 30 35 40

Z(m)

0.2642

0.2223

0.2642

0.2223 0.2642

0.1990 0.2223 0.2642

0.1990 0.1990

0.1685

0.1339

0.0680

0.0680

0.0272

0.2223 0.2223

Y(m)

Z(m)

95 100

30 35 40

(5)

Fig.7 Distributions of wave-induced pore pressures along pipeline surface for various wave incident angles.

Fig.9. Vertical distributions of wave-induced pore pressure versus the soil depth at different seabed permeability.

The closer seabed the pipeline is the larger the wave-induced pore pressure upon the pipeline is.From the analysis, the wave effects on soil and buried pipeline responses become more obviously with the decreasing burial pipeline depth.

C. Effect of Seabed Permeability

Fig. 9 shows vertical distributions of wave-induced pore pressure with variable seabed permeability

sec / 10 2m

K= andK=104m/sec.It is observed that the pore pressure around the pipeline is significantly affected by variable permeability. the pore pressure sharply changes around the surface of the seabed.

Fig. 10 shows distributions of normal stresses along the pipeline surface by variable permeability, respectively. It is observed that the stresses around pipeline in sand are larger than those in rubble.As seen in Fig. 10, the stress concentration is obvious at the top of the pipeline and at the bottom of the pipeline than that at other parts.

D. Effects of Pipeline Radius

Figs. 11-14 present distributions of wave-induced pore pressure,normal stress,cyclic stress and shear stress along the pipeline surface for various pipeline radiuses, respectively. It is observed that the pore pressure at top of pipeline increases with pipeline radius increasing. The pore

Fig.8. Distributions of wave-induced pore pressure along pipeline surface for various of burial pipeline depths.

Fig. 10. Distributions of wave-induced normal stress along pipeline surface for different seabed permeability.

pressure at bottom of pipeline slightly decreases with pipeline radius increasing. The normal stress does not affect by pipeline radiuses obviously, but also increases gradually with pipeline radius increasing. As shown in Figs.

13 and Fig. 14, the shear stress and cyclic stress both increase with pipeline radius increasing. However, the wave-induced cyclic stress are almost the same in all directions for larger pipeline radius.

IV. CONCLUSIONS

This paper presents a three-dimensional finite element model based on Galerkin’s method for investigating wave-induced seabed response, including effective maximum principal stresses, pore pressures and seepage.forces in the vicinity of a buried pipeline. Based on the numerical results presented above, the following conclusions can be drawn :

The pore pressure slightly changes for various incident angles. The pore pressure is not affected by various incident angles,as shown in Fig. 7.

The wave-induced pore pressure penetrates downward from seabed surface while wave crest is on the top of pipeline. Therefore, the wave-induced pore pressure decreases with seabed depth increasing.

The pore pressure is obviously related with seabed

0 0.7854 1.571

2.356

3.142

3.927

4.712

5.498 0.3927 1.178

1.963

2.749

3.534

4.32 5.105

5.89

θ(rad)

0 0.2 0.4 0.6 0.8 1

cita=15 cita=30 cita=45 cita=60 cita=75

P/Po 0

0.7854 1.571

2.356

3.142

3.927

4.712

5.498 0.3927 1.178

1.963

2.749

3.534

4.32 5.105

5.89

θ(rad)

0 0.2 0.4 0.6 0.8 1

b=1.5 b=2.0 b=2.5 b=3.0

P/Po

0 0.7854 1.571

2.356

3.142

3.927

4.712

5.498 0.3927 1.178

1.963

2.749

3.534

4.32 5.105

5.89

θ(rad)

-8 -4 0 4 8

K=10^-2 K=10^-4

σrr/Po

-0.5 0 0.5 1 1.5

P / Po 0

5 10 15 20 25 30 35 40

Z(m) K=10-2

K=10-4

(6)

Fig. 11. Distribution of wave-induced pore pressure along pipeline surface for various pipeline radiuses.

Fig. 13. Distribution of wave-induced cyclic stress along pipeline surface for various pipeline radiuses

permeability. The pore pressure penetrates hardly in sand.

The wave-induced pore pressure in sand is less than in rubble, especially at the top of the seabed.

The wave-induced pore pressure at the bottom of pipeline decreases with pipeline radius increasing .However, the pore pressure changes slightly at the top of pipeline. The normal stress,cyclic stress and shear stress along the pipeline surface obviously increase with pipeline radius increasing.

Acknowledgments

The authors are grateful for invaluable comments from reviewers. This study was supported by the National Science Council (NSC) of Taiwan under project NSC 90-2211-E-005-046.

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[3] Yamamoto, T., Trevorrow, M. V., Badiey, M. and

Fig. 12. Distribution of wave-induced normal stress along pipeline surface for various pipeline radiuses.

Fig. 14. Distribution of wave-induced shear stress along pipeline surface for various pipeline radiuses.

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Geophysical Journal International, vol. 98,No. 1,pp.

177-182, 1989.

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155-164, 1941.

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28, pp.377-393, 1978.

[8] Lin, Y. S. and Jeng, D. S., “The effects of variable permeability on the wave-induced seabed response,”

Ocean Engineering, vol. 24, No. 7,pp. 623-643.

[9] Jeng, D. S. and Lin, Y. S., “Non-Linear wave-induced response of porous seabed: a finite elment analysis,”

International Journal for numerical and analytical Method in Geomechanics, vol. 21, pp. 15-42.

[10] Mcdougal, W. G. Davidson, S. H., Monkmeyer, P. L.

and Sollitt, C. K., “Wave-induced forces on buried pipelines,” J, Waterway Port Coast. Ocean Engng

0 0.7854 1.571

2.356

3.142

3.927

4.712

5.498 0.3927 1.178

1.963

2.749

3.534

4.32 5.105

5.89

θ(rad)

0 0.2 0.4 0.6 0.8 1

r=0.50 r=1.00 r=1.50 r=2.00

P/Po 0

0.7854 1.571

2.356

3.142

3.927

4.712

5.498 0.3927 1.178

1.963

2.749

3.534

4.32 5.105

5.89

θ(rad)

-4 0 4 8

r=0.50 r=1.00 r=1.50 r=2.00

σrr/Po

0 0.7854 1.571

2.356

3.142

3.927

4.712

5.498 0.3927 1.178

1.963

2.749

3.534

4.32 5.105

5.89

θ(rad)

-80 -60 -40 -20 0 20

r=0.50 r=1.00 r=1.50 r=2.00

σss/Po 0

0.7854 1.571

2.356

3.142

3.927

4.712

5.498 0.3927 1.178

1.963

2.749

3.534

4.32 5.105

5.89

θ(rad)

-80 -40 0 40 80

r=0.50 r=1.00 r=1.50 r=2.00

σrs/Po

(7)

Div., ASCE, vol. 114, pp. 220-236, 1988.

[11] Lennon, G.. P., “Wave-induces forces on buried pipelines, “J. Waterway Port Coast. Ocean Engng Div., ASCE, vol. 111, pp.511-524, 1985.

[12] Magda, W., 1996, “Wave-induced uplift force acting on a submarine buried pipeline: finite element formulation and verification of computation,”

Computers and Geotechics, vol. 19, No. 1, pp. 47-73.

[13] Magda, W., “Wave-induced uplift force acting on a submarine buried pipeline in a compressible seabed,”

Computers and Geotechics, vol. 24, No. 6, pp.

551-576, 1997.

[14] Lin,Y. S. and Jeng, D. S., “Wave-induced pore pressure around a buried pipeline in Gibson Soil:

Finite element analysis,” International Journal for Numerical and Analytical Methods in Geomechanics, vol.23, 1559~1154,1999.

[15] Lin,Y. S. and Jeng, D. S., “Pore pressures on a submarine pipeline in a cross-anisotropic non-homogenous seabed under wave loading,”

Canadian Geotechnical Journal,, 1999.

[16] Lin,Y. S. and Wang, X. and Jeng, D. S. 2000, “ Effects of a cover layer on wave-induced pore pressure around a buried pipe in an anisotropic seabed,” Ocean Engineering, vol. 26, 43-64. (EI, SCI)

[17] Lin,Y. S. and Zhang, Qi Ming, “Finite Element Model for Wave-Induced Force Buried Pipeline,” a Master's Paper. Department of Civil Engineering, National Chung-Hsing University, 1999.

References

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