Wave-induced Uplift Force Acting on a Submarine Buried ...wmag/publikacje/publikacja_1996_1_(CG-1996-19-1... · considered in the pipeline stability analysis. ... induced excess pore

Computers and Geotechnics, Vol. 19, No. 1, pp. 41-73, 1996 Couvrieht 0 1996 Elsevier Science Limited

ELSEVlER

P&&d-in &eat Britain. All rights reserved 0266-352X(95)00036-4 0266-352X/96 %15.00+0.00

Wave-induced Uplift Force Acting on a Submarine Buried Pipeline: Finite Element Formulation and Verification of

Computations

W. Magda

Marine Civil Engineering Department, Hydro-Engineering Faculty, Technical University of GdaLsk, ul. G. Narutowicza 1 l/12, 80-952 Gdarisk, Poland

(Received 28 July 1994; revised version received 28 August 1995; accepted 12 September 1995)

ABSTRACT

This paper describes a two-dimensional finite element model developed to simulate the wave-induced hydrodynamic upltjii force acting on a submarine pipeline buried in sandy seabed sediments subjected to continuous loading of sinusoidal surface waves. Neglecting inertia forces, a linear-elastic stress- strain relationship for the soil, and Darcy’s law for the JEow of pore$uid are assumed. The model takes into account the compressibility of both components (i.e. pore fluid and soil skeleton) of the two-phase medium. The governing equations are discretized using the GalerkinJinite element method. Due to the geometry of the problem, four-node isoparametric elements are chosen and the Gaussian quadrature formulae is used in a numerical integration procedure to compute the element stifSness matrices. Several verification problems are presented to demonstrate the model utility and check numerical accuracy influenced by the time- and space-discretization as well as the quadrature rule of numerical integration. Copyright 0 1996 Elsevier Science Ltd

INTRODUCTION

A wave-induced excess pore pressure cyclically generated in the vicinity of a buried submarine pipeline constitutes one of the main factors that has to be considered in the pipeline stability analysis. The hydrodynamic uplift force acting continuously on the pipeline is comparable to the displaced water weight [l-4] if the pipeline is located in the pore pressure boundary layer. An inadequate design can cause flotation of the pipeline, very often leading subsequently to costly failures and environmental catastrophies. Therefore, it

47

48 W. Magda

is essential to improve the knowledge on interactions among waves, seabed and pipeline-like marine structures.

It is a very complex and challenging task to properly define the wave- induced excess pore pressure around a submarine pipeline buried in a porous seabed. For many purposes in soil mechanics, it is permissible to uncouple the soil and fluid parts of any analysis in order to treat two simpler analyses separately. Such a treatment leads to a solution of only one partial differential equation (‘potential’ problem constituted by the Laplace equation) describing the pore fluid pressure changes in the soil skeleton when both the pore fluid and soil skeleton are assumed to be incompressible [.5].

Most of the analytical [ 1,6,7] and numerical [8] solutions to the uplift force acting on submarine pipelines buried in seabed sediments, including the per- turbation pore pressure effects, are devoted only to the case of incompressible media. The only publications concerning a compressible seabed sediment two-phase system are those of Bobby et al. [9] and Cheng and Liu [4] who used the finite element method (FEM) and the boundary integral equation method (BIEM), respectively, in their analyses.

In the numerical study presented by Bobby et al. [9], however, the geometry of the problem does not seem to be so relevant for the coastal engineering practice because, for the analysis, a concrete-coated submarine pipe was assumed to be buried 7.5 m below the sea bottom. Practically, pipelines located in a water depth up to 60 m are buried but the cover layer has a thickness ranging from 0.5 to 1.0 m [lo]. Moreover, Bobby et al. [9] and Cheng and Liu [4] performed their numerical analyses for only a few selected values of wave conditions and relative compressibility of the two-phase medium. When searching for the possible maximum wave-induced hydrodynamic uplift force acting on a buried submarine pipeline, the scope of their analyses must be extended by using wider ranges of input data.

Some cases of laboratory studies reported in the literature [ 1,7] evidently show that there is a difference between theoretically computed values of oscillating wave-induced pore pressures and those observed in experiments. The reported differences between theoretical and experimental results could have been caused by the following three main reasons, namely: (1) the potential theory used in the computations does not contain all the important and decisive soil and pore fluid parameters (incompressibility of pore fluid and soil skeleton are assumed); (2) boundary conditions applied to the computations are not realistic compared with laboratory model tests where a soil layer always has a finite thickness; (3) input data required for computations are not adequate with in situ values from laboratory modelling.

Implementation of certain soil and pore fluid parameters, e.g. saturation (indirectly: compressibility of pore fluid), soil permeability and compressibility, does not lead to the Laplace equation, which makes the problem

Wave-induced Uplift Force 49

independent of these parameters, but to a much more complex system of three coupled equations [l l-151. The solution to this equations system can be formed by a combination of solutions to both the Laplace equation and the diffusion equation [15]. An analytical solution of the diffusion equation, for the geometry of the problem shown in Fig. 1, is possible using the tech- nique of separation of variables. However, the complexity of the mathema- tical formulation as well as the difficulties to solve it (it is not an easy task to handle with special functions, i.e. Hankel’s functions, the use of which is incorporated into the solution to the Helmholtz equation in the polar coordinates system [3]) have motivated the look for another solution to the wave- induced hydrodynamic uplift force acting on a submarine pipeline buried in porous and elastic seabed sediments.

This time, a numerical method has been chosen to perform the analysis of the true coupled performance of a composite continuum, in which the two phases interact, and the wave-induced oscillations of pore pressure are completely dependent upon the relative stiffness of the components of the system. In order to get a better explanation and more insight into the important details involved in the governing problem, a wide range of wavelengths as

Pore-pressure distribution h around the pipeline

Sea bottom

Rigid, impermeable layer

Fig. 1. Definition sketch.

50 W. Magda

well as pore fluid and soil skeleton compressibilities are assumed for the analysis. However, before performing any set of numerical computations, a test analysis is required in order to check the influence of time- and space- discretizations on the accuracy of the solution; this paper pertains to this particular problem.

Figure 1 illustrates schematically the geometry of the governing problem; a submarine pipeline of diameter D is embedded in a sandy soil up to a depth of burial b (measured from the seabed surface to the top of the pipe) and loaded hydrodynamically by simple two-dimensional harmonic waves. The pipeline wall is assumed to be made of steel and thereby it is non-deformable compared with deformations in the soil skeleton.

GOVERNING PLANE STRAIN EQUATIONS

Seabed two-phase medium

Under plane strain conditions, the following two equations, describing elastic deformations of the soil skeleton, together with the ‘storage’ equation constitute the coupled problem and can be written as:

where p is the wave-induced excess pore pressure, u, and u, are the horizontal and vertical displacements of the soil skeleton, respectively, G is the shear modulus of soil, p is the Poisson ratio of the soil, k is the coefficient of the isotropic permeability of soil, yW is the unit weight of the pore fluid, ,B’ is the pore fluid compressibility, n is the soil porosity, x and z are the horizontal and vertical components of the Cartesian coordinates system, respectively, and t is time.

A linear stress-strain relationship for the soil is represented by eqns (la) and (1 b), which are formed from the equilibrium condition in the x- and z- directions, respectively. Equation (lc) reflects the continuity principle incor- porating Darcy’s law of fluid flow through a porous medium. The inertia forces are small [4] and therefore the body forces are set to zero and neglec- ted in the present analysis.


Under realistic conditions, the pore fluid is represented by a two-phase medium where the water and air components can be distinguished. However, in order to simplify the calculation procedure, the compressibility of this two-phase medium is defined by a very convenient, from the engineering point of view, formula proposed by Verruijt [I l] and applied by many other researchers (e.g. Madsen [12], Yamamoto et al. [13], Okusa [15]). The formula, describing the pore fluid compressibility with respect to saturation conditions represented by the degree of saturation, has the following form:

(2)

where ,B’ is the compressibility of pore fluid, p is the compressibility of pure water, S is the degree of saturation and pa is the absolute hydrostatic pressure (Pa = Pat + ph, where Pat is the atmospheric pressure and ph is the hydrostatic pressure). Table 1 shows how strong the dependence of the pore fluid compressibility on the saturation conditions is.

Introducing the following auxiliary constants, i.e:

G G, =p

l-21.1’ B=,h, A=$

grouping the terms containing the same variable functions all the function terms separately and bringing all the terms the equations, the system of eqns (la) to (lc) becomes:

Pa, b, 4

together, writing to the left side of

Ga d2U

&+(G+G++G~---= d2Ux ap ()

axaz az ( w TABLE 1

Dependence of the pore fluid compressibility on the degree of saturation (assumed: pa E pat + ph = 100 + 100 = 200 kPa, p = 4.2 x low7 m2 / kN)

Degree of saturation

SC--_)

Compressibility of pore fluid

P’ h21kN)

1.00 4.2 x 1O-7 0.999 5.4 x 10-6 0.99 5.0 x 10-5 0.98 1.0 x 10-4 0.95 2.5 x 10“ 0.90 5.0 x lOA

52 W. Magda

Pipeline

The system of coupled equations describing the governing problem within the pipe-wall material is even simpler than in case of the soil skeleton. Still in the range of elastic deformations and assuming that the pipeline solid wall (e.g. made of steel or concrete) creates an impermeable boundary condition, only two equilibrium equations are required. They can easily be obtained from the first two equations [i.e. eqns (4a) and (4b)], describing the governing process in the soil, after neglecting the pore fluid terms and replacing the values of G and p, assumed for the soil skeleton, with proper values char- acteristic for the pipe-wall material. The third equation [eqn (4c)], describing the pore fluid flow in a porous medium, has no physical meaning as far as the pipe-wall domain is concerned and therefore disappears from the equations system.

The coupled system of the second-order multivariable partial differential equations is now formulated. In the next step, the Galerkin method of weighted residuals will be applied to obtain a numerical formulation of the coupled system of eqns (4a) to (4~).

FINITE ELEMENT FORMULATION

In the finite element method the continuous functions uX, u, and p are replaced by the values of the function specified at a finite number of nodes. Therefore, the analytical region is divided into a finite number of elements. Using the functions of time and space corresponding to a finite number of nodes, the pore pressure and displacements in the element are represented as follows:

z&(x, z, t) = 5 aj(t)*jUx)(x, z) j=l

U,(X, Z, t) = 5 bj(t)!Pj"')(X, Z) j=1

p(X, Z, t) = 5 Cj(t)Qy)(X, Z) j=l


where N is the number of nodes per element and aj, bj and Cj are the nodal values of ux, U, and p, respectively. These are functions dependent only on

time t while @‘“) !I$“‘) and Q,@’ are the shape functions dependent only on the x,‘z-coordinates.

Discretizing eqns (4a) to (4c) using the Galerkin method, the final finite element formulation of the equations describing the governing process in the seabed two-phase medium is obtained in the following form:

=G f

n au, qjE!“‘)& + z a

x ’

+G f ,., du, Q!“‘)d) Xax ’

(G + G1) f n, 2 Q~“Z)dl

54 W. Magda

where $‘x), @I’ and @” are the weight functions of u,, u, and p, respectively, IZ, = cos(x$) is the direction cosine of the angle between the positive x-direction and the unit vector B and n, = cos(z, fi) is the direction cosine of the angle between the positive z-direction and the unit vector ii. The Galerkin method of weighted residuals assumes the weight functions to be identical with the shape functions, i.e. !l?i = !l!j.

The two adequate equations, formulated for the pipeline domain, can be easily obtained from eqns (6a) and (6b) by assuming cj = 0.

A general matrix representation of the above presented system of eqns (6a) to (6~) can be written as:

where K is the global stiffness matrix, 9 is the vector of unknowns and F is the force vector.

APPROXIMATION OF THE INTEGRAL EQUATIONS

Isoparametric elements

The geometry of the problem (see Fig. 1) induces that not just rectangular elements are required for constructing a finite element mesh. For element shapes more complicated than rectangular, the isoparametric elements serve as a very good solution. The four-node quadrilateral elements were used to construct the finite element mesh. Figure 2 shows the parent element in <, n- space and the parent element mapped onto a real element in x, y-space. Using this type of element no integrals need to be integrated by hand; in this case it is usual to perform a numerical integration.

From the governing equations [eqns (la) to (Ic)] it is clear that p is of the same (second) order differentiable as u, and u,. In the variational formulation of eqns (la) to (lc) [see eqns (6a) to (6c)] over an element all the unknowns are one order less differentiable, i.e. of the first order. Therefore, the interpolation used for all three of these unknowns should be of the same degree (for consistency of approximation) and, at least, linear interpolation functions have to be used. Therefore, for this simple element, the parent shape functions can be written as follows:

Wave-induced

(4

j3

(=l

Uplift Force 55

lb) ? rl=l

4

t= -1

Y t

l@

i

.-;/i,

X

@I nodes with u,, u,, and p

Fig. 2. (a) The parent element for the four-node C’-linear isoparametric quadrilateral; (b) a real element with the parent element mapped onto it [16].

~3(bl) =;p +uu +77) w

QSK>77) =& -r>(l+77) VW

In most cases the number of shape functions used per element is equal to the number of nodes in this element. It is also convenient to use the same type of shape functions in the approximation of all the unknowns, that is:

(9)

Numerical integration

In order to construct the element stiffness matrix for the problem described by integral eqns (6a) to (6~):

where Si for i = 1 ,..., 10 are the coefficients resulting from the time approximation, the following terms have to be computed by numerical integration:

56 W. Magda

It is difficult to provide a general guideline, applicable to all types of problems, on whether to use the so-called reduced integration [(mr - m, + l)- point quadrature rule that has an accuracy of O(/Z~(“P-“~+‘))] or the higher- order integration [(mP - m, + 2)-point quadrature rule that has an accuracy of at least 0(h2(mp-m~+1)+1 )] where 2m,, denotes the order of the governing differential equation and w+, denotes the degree of the complete polynomial in the element trial solution. It depends on the type of physical problem being solved (i.e. the nature of the governing differential equation), the type of element being used and the degree of mapping distortion in each element. As a rough rule-of-thumb guide, reduced integration is generally better sui- ted to undistorted elements, whereas higher-order integration tends to do a better job as the element becomes more distorted [16].

Thus, for the problem considered in this paper and for the linear quadrilateral, one has m, = 1 and m,, = 1, so reduced integration calls for a quadrature accuracy of 0(h2) which is the one-point rule. Higher-order integration calls for an accuracy of at least 0(h3), which requires the two-by-

Wave-induced Up@ Force 57

two-point rule. The question of choosing between reduced and higher-order integration will be analysed with respect to the quality of the pipeline uplift force solution, and is discussed in detail in the following.

APPROXIMATION OF THE FIRST-ORDER (TIME) DERIVATIVES

In time-dependent (unsteady) problems, the undeterminated parameters ai, bi and cj in eqns (6a) to (6~) are assumed to be functions of time, while Qj are assumed to depend on spatial coordinates. The e-family of approximation is introduced to approximate a weighted average of the time derivative of a dependent variable at two consecutive time steps by linear interpolation of the values of the variable at the two time steps. For the terms with respect to time, a finite difference method is applied in the following form:

aQj _ *F*t - G$

at- At (12a)

!B, = Be+*’ + (1 - e,(a,! J J ww

where @j represents discrete values for the horizontal displacement, ux, or vertical displacement, u,, or pore pressure, p, t = t, = EYE, At refers to the enclosed quantity of time, At = tn+l - tn is the nth time step.

Introducing eqns (12a) and (12b) into the governing matrix equation [eqn

aj r+At

J

! (e - I)$? (e - 1)x$) (e- I)x$) ’

(e - i)xf) (e - 1)x$) (e - 1)x$)

$q &Yp &q + (e - l)$!“’ ‘J 1

(13)

where the coefficients J$’ (k = I,... ,lO) are given in eqns (1 la) to (1 lj), and the force vector:

58 W. Magda

F!‘) = (G + Gi) I f

.$Qj”)dl+ G 4

,,zOI”)dl+ G1 f

n du, QKgP4a) “dz ’

F!*) = G, I f

n %!l!!‘)dl+ (G+ Gi) Zdx ’ f

n,%Q!z)d/+ G az ’ f

n,~Q%@4b) ax ’

where repeated indices i, j = l,...,N indicate the sumation. From eqn (13) one can obtain a number of well-known difference schemes

of time approximation by choosing the value of 8. Looking for a suitable method of time approximation in the governing problem, the forward difference scheme can at once be eliminated. It is clear that for 0 = 0 some coefficients of the main diagonal of the element stiffness matrix are set to zero and, consequently, the global stiffness matrix is not non-singular any more.

After Burnett’s [16] recomendation, the backward difference scheme (0 = 1) has been chosen for application in computations of the governing problem. The backward difference method is ‘well-behaved’ and it can be used in a variety of problems, both linear and non-linear. Although not always optimal from a computational efficiency standpoint, it is nevertheless a reli- able and robust algorithm and relatively easy to program. The backward difference method shows no result oscillations at all.

The above system of coupled equations [eqns (13) with eqns (14a) to (14c)] can be used for the seabed domain and also the pipeline domain. It is obvious that in the case of pipeline domain only the first two equations are valid. The above equations are solved every small incremental time step after the formulation (assembly) of the global linear system of equations and imposition of boundary conditions.

A noticeable feature of the global stiffness matrix assembled for the governing problem is that it is banded; that is all the non-zero terms are clus- tered within a narrow band about the main diagonal. However, the global stiffness matrix in the present analysis is not symmetric due to the presence of eqn (1~). The asymmetric character of the stiffness matrix induces the neces- sity to put all the terms from the full bandwidth into a solution procedure.

IMPOSITION OF THE BOUNDARY CONDITIONS

Using the previously described partial differential equations, the problem of the uplift force acting on a submarine pipeline buried in seabed sediments can be solved when a proper set of boundary conditions is applied to the solution procedure. All the boundary conditions assumed for the governing problem are schematically illustrated in Fig. 3.

U -0 z-

ap - &I ax

Wave-induced Uplift Force

p = pb = &, COS(aZ - Wt)

TP Pipeline h ap Q

-_=o an

21, = u, = 0 (alternatively)

I

2 1

U -0 z-

ap - =o ax

u z=o,g=o

Fig. 3. Boundary conditions (primary and natural) assumed in the analysis of uplift force acting on a buried submarine pipeline.

Primary BCs

The main primary boundary condition relates to the seabed surface where certain pressure oscillations are induced as the consequence of water parti- cles motion due to the cyclic loading of surface waves. It is assumed that this load is a travelling two-dimensional harmonic wave. The wave-induced hydrodynamic bottom pressure, pb, that constitutes the boundary condition at the seabed surface (z = 0 and 0 < x < B), can be calculated using the linear wave theory:

’ = Pb = 2 cosh(Xh) ei(Xx-wt)

where the fraction term denotes the amplitude PO of the bottom pressure pb, yW is the unit weight of water, H, is the wave height, X is the wave number (A = 27r/L, L is the wavelength), w is the wave frequency (w = 27r/T, T is the wave period), h is the water depth, x is the horizontal coordinate, t is time and i is the imaginary unit.

If the soil layer of a finite thickness H, overlaying a stiff and impermeable base is considered, the primary boundary conditions to be specified are as follows:

60 W. Magda

- at the bottom of the permeable soil layer (z = H and 0 < x < B)

uz = 0 Wb)

u, = 0 (for completely rough base) (154

- to the left (x = 0 and 0 < z < H) and to 0 < z < H) of the vertical ‘walls’ of the seabed

2.4, = 0

the right (x = B and domain

(154

where u, and u, are the horizontal and vertical displacements of the soil, respectively.

As far as the pipeline structure is concerned, optional boundary conditions can be studied where the nodes in all pipeline elements are constrained due to zero displacement u, = u, = 0. This can be a realistic case when a system of pipeline anchors is installed in the seabed. These boundary conditions can also be assumed when performing calculations for the pipeline embedded in the totally incompressible soil skeleton.

Natural BCs

If the permeable soil layer of width B and finite thickness H, overlaying a stiff and impermeable base is considered, the natural boundary conditions to be specified are as follows:

- pressure gradient in the z-direction, i.e. normal to the base, is zero which means that there is no flow normal to the horizontal boundary; therefore, at the bottom of the permeable soil layer (z = H and ‘0 < x < B)

- due to artificial limitation of the horizontal dimension of the seabed domain by introducing two impermeable vertical ‘walls’ on the left and right side of the seabed domain (x = 0 or x = B, and 0 < z < H)

dp 0 ax= P-9

The last natural boundary condition is formed by an impermeable pipeline wall.

- pressure gradient in the n-direction, i.e. normal to the pipeline surface, is zero which obviously means that there is no flow through the pipeline wall; therefore, on the pipe surface (r = r,, and 0 < cx < 27r, where


r and Q are the polar coordinates with the origin in the centre of the pipe cross-section, and rr, is the pipe outer radius)

dp 0 an= (133)

After imposition of primary and natural BCs, the global stiffness matrix and the force vector are ready to be implemented into the solution procedure.

MESH GENERATION

Many different possible solutions of dividing the domain into a set of finite elements around a pipe-like structure, depending on the problem concerned, are described in the literature (e.g. Burnett [16], Desai and Abel [17], Reddy [18], Zienkiewicz [19]).

When dividing the domain into smaller elements, the local refinement of the FE-mesh always has to be taken into account in all zones where a con- centration of stresses (strains) is to be expected; mostly, this happens in the vicinity of a considered structure. This improves the accuracy of the solution in this region. A sufficiently acceptable discretization of the seabed domain in the vicinity of the buried submarine pipeline is quite important considering the fact that the pore pressure values computed in all nodes from the pipeline outer surface constitute the main result of the present analysis.

Due to the non-rectangular boundaries formed by the pipe-like structure (see Fig. 1) involved in the problem, rectangular elements are not recommended in selecting a mesh. On the basis of the commonly accepted rules used in the discretization process (a maximum aspect ratio of 5 to 1 or 10 to 1, and interior corner angles no smaller than 20 to 30”) a proposal of the FE- mesh containing the four-node isoparametric elements, suitable for the governing problem, has been made. Figure 4 illustrates the example of space- discretization within the rectangle zone of the pipeline nearest proximity, for a given number of divisions in the tangential (ndJ and radial (nd,) directions. The two parameters (i.e. nd, and nd,) control the mesh refinement and the influence of mesh discretization around the pipeline on the solution quality will be discussed in the following.

TEST CALCULATIONS AND DISCUSSION

The pre-processing PIPELINE-MESH computer program [20] has been prepared to perform a quasi-automatic FE-mesh generation that can be

62 W. Magda

Fig. 4. Example of FE-mesh discretization in the pipeline proximity, used in the uplift force analysis.

applicable to the problem of buried submarine pipelines. The above-mentioned finite element alogorithm has been put into the main PIPELINE- FEM2D computer program [20] which calculates the wave-induced hydrodynamic uplift force acting on a submarine pipeline buried in porous and elastic seabed sediments. For simplicity of the presentation, the uplift force is given in its relative form:

Fz = FzIPo (kN/m/kPa)

where F, is the maximum relative wave-induced hydrodynamic uplift force per metre of pipeline length, Fz is the maximum absolute wave-induced hydrodynamic uplift force per metre of pipeline length, PO is the amplitude of the hydrodynamic bottom pressure pb [see eqn (15a)]. The term ‘maximum’ denotes the highest value of wave-induced hydrodynamic uplift force found during one wave cycle period after a sufficient number of loading cycles required for obtaining the numerical stability of computed results; the term ‘maximum’ will be omitted in the following text.

Input data

The input data required for the automatic FE-mesh generation program and the main FE program consists of the following parameters: B, width of the

Wave-induced UpI@ Force 63

soil domain; H, height of the soil domain; D, outer diameter of the pipeline; b, depth of burial of the pipeline (measured from the bottom surface to the top of the pipe); h, water depth; L, wavelength; T, wave period; n,,, number of time-steps within one period of wave loading; pL,, Poisson’s ratio of soil; G,, shear modulus of soil; p’, compressibility of pore fluid; ppLp, Poisson’s ratio of pipe-wall material (e.g. concrete-coated steel); Gp, shear modulus of pipe-wall material; n, porosity of soil; k, coefficient of isotropic permeability of soil; E, accuracy of solution (in the end-test procedure used for evaluating the stability of the results after each wave cycle); 19, coefficient in the time-dependent scheme; and N,, indicator for the N,-point rule of numerical integration.

In all test computations, the following values were assigned to some of the above-mentioned parameters: D = 1 m; b = 1 m; h = 10 m; L = 30 or 120 m; T = 4.45 or 12.65 s (respectively for L and h); pL, = 0.3 (loose sand) or pu, = 0.2 (dense sand); pp = 0.28 (steel, pp = 0.3; concrete, ,LL~ = 0.2); II = 0.4; k = 1O-4 m/s (coarse sand, k = lop3 m/s; fine sand, k = 1 Oe5 m/s); E = 1O-3 kN/m/kPa; 19 = 1; and Nq = 1 or 2.

Additionally, four different models of the relative compressibility of the two-phase system (pore fluid and soil skeleton) were assumed for the numerical analysis:

(4

(b)

Cc)

(4

INCOMP (pore fluid, soil skeleton and pipe-wall incompressible) G, = lOlo (21 co) kN/m2, GI, = lOlo (E 00) kN/m2 and ,L?’ = 4.2 x lo-lo (21 0) m2/kN. The value of coefficient of the isotropic permeability of soil, k, has no influence when the incompressible two-phase system is introduced.

COMP-P (only pore fluid compressible) G, = lOlo (E 00) kN/m2, Gp = lOi (E 00) kN/m2, /3’ = 4.2 x lop7 _ 5.0 x lop4 m2/kN (S = 1 .OO-0.90, respectively, see Table 1) and k = 0.0001 m/s. COMP-S (only soil skeleton compressible) G, = 2.6 x lo4 kN/m2, Gp = 1O’O (- 00) kN/m2, p’ = 4.2 x 10plo (21 co) m2/kN and k = 0.0001 m/s. COMP-PS (pore fluid, soil skeleton and pipe-wall compressible) G, = 2.6 x lo4 kN/m2, Gp = 2.6 x lo8 kN/m2, ,D’ = 4.2 x 10e7-5.0 x lo-” m2/kN (S = 1.00-0.90, respectively, see Table 1) and

k = 0.0001 m/s.

Influence of time-discretization and stability of the results

Assuming five different numbers of time-steps within one cycle of wave loading (n,, = 10, 20, 40, 60 and loo), the test computations were performed to check the accuracy of the solution depending on the time-discretization.

64 W. Magda

For this type of test computation, the mesh discretization of type n, = ~1, = 2 (see p. 67) in the pipeline proximity was used. The computations were performed for L = 30 m and the results are presented in Fig. 5.

For all compressibility models there is almost no difference in the results of wave-induced uplift force obtained for II 1s > 60 (the lines in Fig. 5 are almost

horizontal for this range of the number of time-steps). As far as the compressible two-phase system is concerned (models: COMP-S, COMP-P, COMP-SP), it was found that by introducing 20 time-steps within one period of wave loading, the difference in the results, compared to the case of

nt, = 100, is equal to 12.8, 6.9 and 14.6%, respectively. The respective values for the comparison between n,, = 40 and nt, = 100 are: 4.9, 2.8 and 6.2%. This accuracy can be estimated as satisfactory and relevant for engineering practice.

For the totally incompressible two-phase system (INCOMP model) the number of time-steps does not play any role because the system reacts immediately (no damping effects) on the imposition of the inducing bottom pressure. For all the studied compressibility models except the INCOMP model, smaller numbers of time-steps lead to an underestimation of the uplift force.

The results obtained from the calculations with the above-assumed end- test (E = 0.001 kN/m/kPa) become stable after only few cycles, i.e. from 2 (totally incompressible system - INCOMP) to 5 (strong compressible

I- ~~ -. Compressibility model:

++ INCOMP

* COMP-P

-++- COMP-S

-+ COMP-PS

20 40 60 80 100

Number of time-steps, nfr [-]

Fig. 5. Influence of the number of time-steps on the convergence of uplift force results obtained for different relative compressibilities of the two-phase medium.

Wave-induced Uplif Force 65

system - COMP-SP), of the regular wave loading. Generally, the more compressible the system the more cycles of wave loading are needed to obtain stable results; however, even then, when the pore fluid is strongly compressible (e.g. /3’ = 2.504 x lop4 m2/kN for S = 0.95) the number of required cycles is not large and can be executed on a PC-unit without any special time-demands.

Influence of space-discretization

The second part of the test computations was performed in order to check the influence of the size of the main domain with respect to the wavelength. For this type of test computations, the same pattern of mesh-discretization in the pipeline proximity (i.e. n, = n, = 2, see p. 67) was used.

The FE-mesh proposal was verified by comparison of the numerical results with the analytical solution (incompressible two-phase medium) proposed by MacPherson [5], McDougal et al. [6] and Monkmeyer et al. [l]. For the assumed geometry of the problem (i.e. L = 30 m), the results of the wave- induced hydrodynamic uplift force F,, obtained analytically (note that only the knowledge of wavelength is required) and numerically (in addition to the wavelength, the dimensions of the seabed domain, B and H, are required), are compared in Table 2.

Testing different sizes of the main domain, it was found that the domain width B = L and the domain height H = L/3 are enough to obtain uplift force results with an accuracy satisfying engineering practice. Further

TABLE 2 Comparison of the uplift force, F, (kN/m/kPa), computed for different sizes of the seabed

domain, and different compressibility models of the two-phase medium

HIL

Compressibility

of two-phase medium

Solution

Numerical Analytical

(see p. 63) B/L = I B/L = 3 B/L = 00

INCOMP 0.236 0.237 0.242 l/3 COMP-S 0.335 0.335

COMP-P 0.229 0.233 COMP-SP 0.193 0.193

INCOMP 0.245 0.245 0.251 1 COMP-S 0.335 0.335

COMP-P 0.227 0.228 COMP-SP 0.190 0.190

00 INCOMP 0.250

66 W. Magda

increase of both dimensions of the main domain has no significant practical meaning for the quality of the results. For example, if the domain dimensions are increased by a factor of 3 (i.e. B = 3L and H = L) the differences in the uplift force, computed with respect to the solution with the domain dimensions B = L and H = L/3, are only equal to 3.8, 0.0, - 0.4 and - 1.6%, respectively to the compressibility model (see p. 63). The largest difference was obtained for the totally incompressible two-phase system (TNCOMP model) that reacts immediately (no damping effects) even in relatively distant points of the main domain. That is why the artificial impermeable ‘walls’, defining the borders of the seabed domain, have a meaningful disturbing effect even when they are placed far away from the pipeline. In more compressible systems (e.g. models: COMP-S, COMP-P or COMP-SP) the damping effects start to play an important role and, therefore, the extension of the domain size is not so meaningful for the quality of the results as it was the case with the totally incompressible system (INCOMP model).

The numerical solution underestimates the analytical results obtained for the incompressible two-phase medium (INCOMP model); the differences between these two solutions were found to be c. 2.5 %.

In addition to the question of space-discretization of the seabed domain, the effect of different mesh-discretizations in the pipeline proximity on the quality of the results was investigated in particular. The FE-mesh in the pipeline proximity was built using the four-node isoparametric elements. Even though, the discretization error is not zero because of the approximation of the circular arc by straight lines. The only way to improve the accuracy was to use different refinements of the mesh in the pipeline region. A set of different combinations of divisions in both the tangential, n, (with respect to 45”-sector, see Fig. 4), and radial, n,, directions was studied for n,, ~1, = 1, 2, 3 (Fig. 6). The results of the computations are presented in Fig. 7.

It is shown evidently that the refinement of the FE-mesh around the pipeline improves the convergence of the results. The best convergence was obtained when the refinement was equal in both the tangential and radial directions to the maximum values (i.e. nt = n, = 3) assumed for the study. Further refinement would certainly bring additional improvements of results but, for practical purposes, it was not necessary to use a finer mesh (the lines in Fig. 7, the inclination of which indicates the rate of convergence, are almost horizontal when II, = 2-3).

Comparing mesh-discretization pattern (e) (nt = n, = 2) with pattern (a) (nt = 12, = l), the differences in the uplift force are equal to 7.2, -- 47.8, - 37.6 and - 143.5%, respectively to compressibility models: INCOMP, COMP-S, COMP-P and COMP-SP. If the comparison is made between mesh-discretization pattern (e) and pattern (i) (nt = H, = 3) the differences decrease significantly and are equal to - 1.7, 5.7, 2.6 and 19.7%, respectively.

Fig.

Wave-induced Uplift Force

n, = 1

(4 nt = 1 (b) nt = 2 (c) nt = 3

67

nF = 2

(4 nt = 1 (e) nt = 2 (f) nt = 3

nF = 3

k> nt = 1 (h) nt = 2 (9 nt = 3

6. Mesh-discretization patterns in the pipeline proximity studied in the test computations.

68 W. Magda

Number of radial divisions, n, [-]

Tangential division:

s nt = 1 / INCOMP

-s-n*=2

*nt=3

-x- nt = 1 / COMP-P

--•- nt = 2

--X-- nt = 3

0.4 -

+ nt = 1 / COMP-S

0.3 - -s. nt = 1 / COMP-PS

1 2 3

,

Number of radial divisions, n, [-]

Fig. 7. Influence of different patterns of mesh-discretization is the pipeline proximity on the results convergence, for different compressibility models.


Numerical integration

All the above-presented results were obtained from the test computations performed with the two-point quadrature rule applied in the numerical integration procedure. The difference between the one-point and two-point quadrature rule will be studied carefully in the following.

First of all, the totally incompressible two-phase system was considered (INCOMP model). The numerically computed results of the wave-induced hydrodynamic uplift force are presented in Table 3. For comparison purposes, the uplift force values computed analytically for both the infinite (H= cc) and finite (H = L/3) thickness of the seabed layer, using the ‘image- pipe’ theory [l], are also given in Table 3.

The differences in uplift force, computed using the two-point and one- point quadrature rule of numerical integration, range from -2.5% for L = 30 m to -2.8% for L = 120 m. Using the one-point quadrature rule, the differences are even smaller: 0.4 and 1.4%, respectively.

In the next step, the compressibility of pore fluid (due to unsaturated soil conditions S < 1 .OO) was assumed, whereas the soil skeleton was still treated as incompressible (COMP-P model). Additionally, a wide range of soil saturation conditions was assumed: p’ = 4.2 x 10m7 - 5.0 x lop4 m2/kN. This variation in ,8’ reflects the variation in the degree of saturation from S = 1 .OO to S = 0.90, respectively (see Table 1). The results of computations are shown in Table 4, for fully saturated soil conditions (S = 1 .OO) and the real value of pure water compressibility (i.e. ,B’ = 4.2 x lop7 m2/kN), and illustrated in Fig. 8, for a wider range of soil saturation conditions (S = 0.90-l .OO).

For the whole range of variation of the degree of saturation, assumed for the present analysis, the one-point quadrature rule makes the results higher than in the case of numerical integration using the two-point quadrature rule. The differences in the results are quite small for fully saturated soil conditions (S = 1 .OO) and increase gradually when the degree of saturation falls down. For L = 30 m, the differences are equal to 2.7% (S = 1.00) and

TABLE 3 Comparison of the uplift force F, computed analytically and numerically for an incompres-

sible two-phase medium (INCOMP model)

Wavelength Uplift force, F, (kN/m/kPa)

L(m) Analytical Numerical (H = L/3)

H=cc H = L/3 l-point q.r. 2-point q.r.

30 0.251 0.242 0.243 0.236 120 0.073 o.p71 0.072 0.069

70 W. Magda

TABLE 4 Comparison of the uplift force F, computed numerically for an incompressible (INCOMP

model) and compressible (COMP-P model, S = 1 .OO) two-phase medium

Wavelength Uplift force. F, (kN/m/kPa)

L(m) INCOMP COMP-P (S = 1.00)

l-point q.r. 2-point q.r. l-point q.r, 2-point q.r.

30 0.243 0.236 0.264 0.257 120 0.072 0.069 0.142 0.138

0.7, ~ I I

-+- L = 120~1 / l-point

0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.90 0.99 1

Degree of saturation, S [-]

Fig. 8. Influence of the Gaussian quadrature rule for numerical integration, compressibility of pore fluid, defined in terms of soil saturation conditions (model COMP-P) and wavelength on

the wave-induced hydrodynamic uplift force.

27.9% (S = 0.90). Longer waves reduce these differences and, e.g. for L = 120 m, they equal 2.9 and 7.1%, respectively to the soil saturation conditions.

The same comparison was made for the compressible two-phase medium where the compressibility of the soil skeleton was additionally introduced to the system (COMP-SP model). Also here, the higher values of uplift force obtained for the one-point quadrature rule than for the two-point quadrature rule were indicated. In the case of relatively loose sandy sediments (G, = 2.6 x lo4 kN/m2) and fully saturated soil conditions (S = 1 .OO), the differences between both quadrature rules oscillate very closely around the above-mentioned for the COMP-P model and are equal to: I .7% (L = 30 m)


and 3.2% (L = 120 m). Assuming the lower degree of saturation (e.g. S = 0.90), a slight increase of the differences, compared to the COMP-P model, was found: 34.8 and 10.8%, respectively to the wavelength.

CONCLUSIONS

Assuming four different models of the relative compressibility of the two- phase system (see p. 63), the influence of time- and space-discretization, as well as the type of quadrature rule applied to the numerical integration, on the results of the wave-induced hydrodynamic uplift force acting on a submarine pipeline buried in sandy sediments, was analysed and discussed.

The analysis of time-discretization has shown that the number of time- steps within one cycle of wave loading should be at least equal to c. n,, = 40 if the discretization error is to be kept in the order of only few per cent. Taking into account that the computations on a PC-unit are time-demand- ing, it seems wise and permissible to use an even smaller number of time- steps (e.g. nt, = 20) in computations of illustrative character; in this case the computed result will underestimate the correct result but the difference is expected to be less than c. 15%.

The analysis of mesh-discretization of the seabed domain has proved that the size of the seabed domain B x H = L x L/3 (B and H are the width and height of the soil domain, respectively, and L is the wavelength) is sufficient when performing wave-induced hydrodynamic uplift force analysis, espe- cially with realistic values of the relative stiffness of the two-phase medium.

The quality of the solution due to the different mesh refinements in the pipeline proximity was analysed for both the incompressible and compressible two-phase system. The results of computations performed for nine different mesh-discretization patterns have shown that the pattern n, = n, = 2 [(see Fig. 6 (e)] is satisfactory and can be recommended for the true numerical analysis of the pipeline uplift force.

Generally, the relatively large differences between the uplift force results computed with the one-point and two-point quadrature rules of numerical integration were indicated. It was shown, however, that the maximum uplift force, which constitutes the most important result to be used in pipeline stability analysis, seems to be related to higher values of the degree of saturation (c. S = 0.98-1.00). On the other hand, the difference between the one- point and two-point quadrature rules becomes smaller for higher values of the degree of saturation and almost vanishes when fully saturated soil conditions are introduced. This finding confirms a rather small meaning of the choice between both rules of numerical integration as far as the maximum value of wave-induced uplift force is concerned.

72 W. Magda

The above-presented analysis of computational errors, generated by time- and space-discretization, have supplied the acceptable values of discretization parameters to be used in the true analysis of the wave-induced hydrodynamic uplift force acting on a submarine pipeline buried in a compressible two-phase medium, the results of which will be presented in due course.

ACKNOWLEDGEMENT

The above summarized work was carried out at the Marine Civil Engineer- ing Department, Technical University of Gdatisk, thanks to the financial support obtained from KBN - Grant No. 772629102 “Pore-pressure gen-

eration in seabed sediments due to surface wave action - a method of computation of the hydrodynamic loading acting on a buried submarine pipeline “. The author very much appreciates the help of KBN.

1.

2.

3.

8.

9.

REFERENCES

Monkmeyer, P. L., Mantovani, P. & Vincent, H., Wave-induced seepage effects on a buried pipeline. Proc. of the Coastal Structures ‘83 Conference, pp. 5 19-53 1,1983. Magda, W., Wave-induced pore pressure distribution in sandy seabed sediments, Ph.D. Thesis, Technical University of Gdansk, Poland, 1992. Magda, W., Wave-induced pore water pressure acting on a buried submarine pipeline. Proc. of the 23rd International Conference on Coastal Engineering (ICCE’92), Venice, Italy, 4-9 October 1992, pp. 3135-3148. Cheng A. H. D. & Liu, P. L.-F., Seepage force on a pipeline buried in a por- oelastic seabed under wave loading. Appl. Ocean Res., 8 (1986) 22-32. Putnam, J. A., Loss of wave energy due to percolation in a permeable sea bottom. Trans. Am. Geophys. Union, 30 (1949) 349-356. MacPherson, H., Wave forces on pipeline buried in permeable seabed. J. Waterway Port Coast. Ocean Engng Div. AXE, 104 (1978) 407-419. McDougal, W. G., Davidson, S. H., Monkmeyer, P. L. & Sollitt, C. K., Wave- induced forces on buried pipelines. J. Waterway Port Coast. Ocean Engng Div., ASCE 114 (1988) 220-236. Lennon G. P., Wave-induced forces on buried pipelines. J. Waterway Port Coast. Ocean Engng Div., ASCE, 111 (1985) 511-524. Bobby, W., Arockiasamy, M., Haldar, A. K. & Reddy, D. V., Finite element analysis of pipe-soil-wave interaction. Proc. of the 2nd International Conference on Behaviour of OH-shore Structures (BOSS’79), London, England, 28831 August 1979, Paper No. 39, pp. 503-506.

10. Dursthoff, W. & Mazurkiewicz, B., Problems related to subsea pipeline-seabed interaction. Mitt. Franzius-Znst. Wass. Ktisteningen. Univ. Hannover, 61 (1985) 1726-238.

11. Verruijt, A.,Elastic storage of aquifers. In Flow Through Porous Media (Edited by de Wiest), pp. 331-376, Accademic Press, New York, 1969.


12. Madsen, 0. S., Wave-induced pore pressures and effective stresses in a porous bed. Gtotechnique, 28 (1978) 377-393.

13. Yamamoto, T., Koning, H. L., Sellmeijer, H. & Hijum, E., On the response of a poro-elastic bed to water waves. J. Fluid Mech., 87 (1978) 193-206.

14. Nago, H. & Maeno, S., Pore water pressure in sand bed under oscillating water pressure. Mem. Sch. Engng, Okayama Univ., 19 (1984) 13-32.

15. Okusa, S., Wave-induced stresses in unsaturated submarine sediments. Gdotechnique, 35 (1985) 517-532.

16. Burnett, D. S., Finite Element Analysis; From Concepts to Applications. Addi- son-Wesley, Reading, MA. 1987.

17. Desai, C. S. & Abel, J. F., Introduction to the Finite Element Method. A Numerical Method for Engineering Analysis. Van Nostrand Reinhold, New York, 1972.

18. Reddy, J. N., An Introduction to the Finite Element Method. McGraw-Hill, Sin- gapore, 1986.

19. Zienkiewicz, 0. C., The Finite Element Method in Engineering Science. McGraw-Hill, London, 197 1.

20. Magda, W., Mazurkiewicz, B. K., Marcinkowski, T., Szurowski & Dobrzy- kowski, A., Pore-pressure generation in seabed sediments due to surface wave action - a method of computation of the hydrodynamic loading acting on a buried submarine pipeline (in Polish). Manuscript in preparation.

Documents

Wave-induced Uplift Force Acting on a Submarine Buried ...wmag/publikacje/publikacja_1996_1_(CG-1996-19-1... · considered in the pipeline stability analysis. ... induced excess pore