Wave-induced cyclic pore-pressure perturbation effects in hydrodynamic uplift force acting on submarine pipeline buried in seabed sediments

Coastal Engineering 39 Ž2000. 243–272 www.elsevier.comrlocatercoastaleng

Wave-induced cyclic pore-pressure perturbation effects in hydrodynamic uplift force acting on submarine pipeline buried in seabed sediments W. Magda

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Marine CiÕil Engineering Faculty, Technical UniÕersity of Gdansk, ´ G. Narutowicza 11r12, 80-952 Gdansk, ´ Poland Received 21 April 1999; received in revised form 3 September 1999; accepted 5 November 1999

Abstract Among all environmental loads usually considered in the design procedure, the most critical problem in determining a vertical stability of a submarine pipeline buried in permeable soils under progressive surface-water-wave loading is the prediction of the wave-induced cyclic pore-pressure response of a seabed in the vicinity of a submarine pipeline. A study of the hydrodynamic Ži.e., wave-induced. uplift force acting on a submarine pipeline buried in sandy seabed sediments is presented, under the assumption of different compressibility models of the two-phase seabedrpore-fluid medium. Introducing an uplift-force perturbation ratio, the question of perturbation effects affecting the wave-induced pore-pressure field by the presence of a stiff and impermeable body of the submarine pipeline is analysed thoroughly. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Submarine pipelines; Progressive waves; Pore pressure; Uplift force; Soil saturation

1. Introduction When a submarine pipeline is buried in a porous seabed, the problem of wave-induced pore-pressure oscillations in the pipeline vicinity becomes rather complicated. However, for permeable and relatively stiff soils Že.g., sands and very dense silts., soil deformations are small and do not seem to influence the pore pressure significantly. Therefore, it is suggested that the full equations from the consolidation theory of Biot )

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0378-3839r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 3 9 Ž 9 9 . 0 0 0 6 3 - 0

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Ž1941. are not required to determine the wave-induced pore-pressure cyclic response in seabed sediments. For this reason, many studies related to submarine buried pipelines were based on the potential pressure model, derived under the condition of incompressibility of both the pore fluid and the soil skeleton. Under this assumption MacPherson Ž1978., Monkmeyer et al. Ž1983., and McDougal et al. Ž1988. analysed the problem analytically, whereas Lai et al. Ž1974., Liu and O’Donnel Ž1979., Lennon Ž1985., and Spierenburg Ž1986. investigated this problem numerically. The boundary value problem, defined by the Laplace equation and appropriate boundary conditions related to a stiff and impermeable pipeline body embedded in seabed sediments, is rather difficult to be analysed in rectangular coordinates. However, this difficulty can be overcome by transforming to a space in which the variables become separable. MacPherson Ž1978. and McDougal et al. Ž1988., using conformal mapping methods, presented analytical solutions for the case of infinite depth of the seabed, whereas Monkmeyer et al. Ž1983., using the method of image pipes, found an equivalent solution, the advantage of which is that it can also be applicable for the case of finite depth of the permeable seabed layer. Although Monkmeyer et al. Ž1983. pointed out some computational difficulties encountered when the pipeline is located near either the upper or lower geometric boundary, the character of the ‘image pipe’ method of Monkmeyer et al. Ž1983. enables Žafter introducing proper numerical techniques. computations including very extreme cases in which the pipeline can approach the sea floor orrand the impermeable base underlying the permeable seabed layer. McDougal et al. Ž1988. treated the pore pressure in total, without distinguishing and superpositioning the two pore-pressure components obtained for the case of pipeline absence and presence in the seabed sediments. MacPherson Ž1978. and Monkmeyer et al. Ž1983. were the first whose algorithms allowed to extract a pore-pressure solution component responsible for local perturbations in the pore-pressure field around the pipeline. Therefore, the algorithms proposed by MacPherson Ž1978. and Monkmeyer et al. Ž1983. enable the study of the non-perturbed component and the correction component of the hydrodynamic uplift force separately. Liu and O’Donnel Ž1979. considered two different types of waves acting on the seabed, namely, monochromatic and solitary, and introduced the integral equation method to solve the resulting integral equation. In a numerical solution procedure developed by Lennon Ž1985., the pressure distribution on the pipeline was calculated using also the boundary integral equation method. The common feature of all the studies mentioned above is that the effect of compressibility of both the pore fluid and the porous medium was neglected. Comprehensive numerical studies of the hydrodynamic force acting on a submarine pipeline buried in compressible seabed sediments were performed by Cheng and Liu Ž1986., using the boundary integral equation method, and by Magda Ž1996; 1997., and Magda et al. Ž1998., using the finite element method. In all these works, careful analyses of the sensitivity of the pore-pressure response to different soil and pore-fluid parameters were performed. In the following, two basic types of computational methods for the calculation of the wave-induced pore-pressure cyclic oscillations in the pipeline vicinity and the hydrody-

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namic uplift force acting on a pipeline buried in seabed sediments are presented and discussed with respect to some decisive soilrpore-fluid mixture parameters. The succeeding analysis is mainly focused on the problem of pore-pressure perturbation effects, affecting the wave-induced pore-pressure field by the presence of a stiff and impermeable body of the submarine pipeline. Introducing a hydrodynamic uplift-force perturbation ratio, the contribution of pore-pressure perturbation effects in a global solution for the hydrodynamic uplift force acting on a submarine pipeline buried in seabed sediments will be thoroughly studied.

2. Hydrodynamic pore pressure and uplift force on pipeline Considering a porous permeable seabed Že.g., consisted of sandy sediments., the bottom pressure fluctuations have a direct and continuous influence on changes in pore pressure within the seabed medium ŽFig. 1.. An overpressure Žwith respect to the initial hydrostatic pressure distribution defined by the still water level. generated in the soil due to a passage of a wave crest creates seepage forces acting downwards. Contrary to that, an underpressure induced by a passage of a wave trough is responsible for seepage forces directed upwards. Should one assume an existence of a submarine pipeline buried in a seabed, this pipeline will be exposed to seepage forces induced in seabed sediments due to a passage of surface water waves. The resultant seepage force acting on a submarine pipeline has a

Fig. 1. Distributions of wave-induced pore pressure, p, with depth under two characteristic phases Ži.e., wave crest and wave trough. of progressive surface-water-wave oscillations, and wave-induced uplift force, Fz , acting on a submarine pipeline buried in sandy seabed sediments.

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magnitude and direction alternating cyclically with the same frequency as the frequency of surface-water-wave oscillations. For each moment of wave oscillations, it is possible to compute a vertical component of the resultant hydrodynamic force, and to distinguish two special cases thereof, namely: a vertical force acting upwards Žcase B in Fig. 1., trying to suck the pipeline out of the seabed, and a vertical force acting downwards Žcase A in Fig. 1., pressing the pipeline down. As far as the first case of loading is concerned, the resultant force acting on a pipeline is called the wave-induced uplift force or hydrodynamic uplift force. This force plays a very important role and has to be taken into consideration in any vertical stability analysis. When the hydrodynamic uplift force exceeds the pipeline effective Ži.e., with respect to the buoyancy of water. gravity force together with restraint forces acting in a soil layer covering the pipeline, the stability condition is not satisfied. This can lead to large displacements of the pipeline onto the seabed floor, even to pipeline floatation up to the sea level, and sometimes pipeline breakout which normally results in serious failures of the pipeline and severe environmental catastrophe. The hydrodynamic uplift force Žper unit length of pipeline., Fz , can be obtained by integrating the wave-induced pore pressure along the pipeline circumference Ža negative sign implies a positive value of the uplift force when acting upwards, as denoted in Fig. 1.: Fz s y

2p

H0

p cos c rp d c

Ž 1.

where, Fz is the hydrodynamic uplift force, p is the wave-induced pore pressure, rp is the outside radius of the pipeline, and c is the angular coordinate of the polar coordinates system Žwith the origin placed in the centre of the pipeline cross-section.. For further analysis and comparison purposes, in order to introduce independence of the wave height, it is convenient to present a relative Žand dimensionless. form of the wave-induced pore pressure: ps

p

Ž 2.

P0

where, p is the relative Žand dimensionless. wave-induced pore pressure, p is the wave-induced pore pressure, P0 is the amplitude of the hydrodynamic bottom pressure Že.g., derived from Airy’s small-amplitude water wave theory.. Similarly, by dividing both sides of Eq. Ž1. by the amplitude of hydrodynamic bottom pressure, P0 , the hydrodynamic uplift force can also be presented in its relative Žstill dimensional but independent of the wave height. form: Fz s

Fz P0

2p

H0

sy

p cos c rp d c

Ž 3.

where, additionally, Fz is the relative hydrodynamic uplift force. Of course, the solution to the hydrodynamic uplift force given by Eq. Ž3. can be found either analytically or using one of the numerical integration methods. In many cases, however, a performance of simple algebraic summation of the component forces

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acting along the pipeline circumference could be of particular interest, especially when the wave-induced pore-pressure function is described by an expression which is mathematically too complicated to be treated analytically.

3. Compressibility models for pore-pressure computations The hydrodynamic uplift force acting on a pipeline buried in seabed sediments is directly related to the hydrodynamic pore-pressure field around the pipeline. Therefore, a value of the hydrodynamic uplift force will be different depending on a model describing the wave-induced pore-fluidrseabed interaction. Treating seabed sediments as a two-phase medium Žconsisting of pore fluid and soil skeleton., Magda Ž1996; 1997. distinguished the following four models that can be assumed for the hydrodynamic uplift force analysis. The main feature of these models relates to the compressibility of the single media. And thus: Ža. INCOMP model Žpotential problem. — pore fluid and soil skeleton incompressible, Žb. COMP-P model Ždiffusion problem. — pore fluid compressible whereas soil skeleton incompressible, Žc. COMP-S model — pore fluid incompressible whereas soil skeleton compressible, Žd. COMP-SP model Žstorage problem. — pore fluid and soil skeleton compressible. Respecting the above depicted models of the relative compressibility of the two-phase seabed medium, a basic mathematical formulation of the wave-induced pore-pressure response in seabed sediments is given in the succeeding sections. 3.1. INCOMP model If the two-phase system is treated as totally incompressible, the problem is pure geometric Žindependent of any pore-fluid and soil parameters. and the equation to be solved is the Laplace equation: E2 p Ex2

E2 p q

Ez2

s0

Ž 4.

where, p is the wave-induced poreŽ-water. pressure, and x and z are the horizontal and vertical coordinates of the Cartesian coordinates system, respectively. Assuming the hydrodynamic bottom pressure oscillations in the form: p b s P0 cos Ž ax y v t . s

H 2

gw

1 cosh Ž ah .

cos Ž ax y v t .

Ž 5.

where, p b is the bottom pressure, P0 is the amplitude of the bottom pressure, a is the wave number Ž a s 2prL., L is the wavelength, v is the wave angular frequency Ž v s 2prT ., T is the wave period, H is the wave height, gw is the unit weight of sea

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water, and h is the water depth, an analytical ‘finite-thickness’ solution to the potential problem was initially given by Putnam Ž1949.: cosh a Ž d y z .

p s P0

cosh Ž ad .

cos Ž ax y v t .

Ž 6.

where, additionally, d is the thickness of the permeable seabed layer. 3.2. COMP-P model If the pore fluid is treated as compressible whereas the soil skeleton stays as an incompressible phase, the equation to be solved is the so-called consolidation, or diffusion, or heat conduction equation: E2 p Ex

2

E2 p q Ez

2

1 Ep s cd Et

Ž 7.

with cd s

kn b X

Ž 8.

g

and

b X s bw q

1yS Ph

for 1 y S < 1

Ž 9.

where, p is the wave-induced poreŽ-fluid. pressure, cd is the coefficient of diffusion, k is the coefficient of permeability for isotropic soil, n is the porosity of soil, b X is the compressibility of pore fluid, b w is the compressibility of pure water, S is the degree of saturation, Ph is the absolute hydrostatic pressure Ž Ph s pat q p h ., pat is the atmospheric pressure, p h is the hydrostatic pressure, g is the unit weight of pore fluid, x and z are the horizontal and vertical coordinates of the Cartesian coordinates system, respectively, and t is the time. An application of formula Ž9. is practically restricted to the degree of saturation from the range S s 0.85–1.0. The compressibility of pure water is as low as b w , O Ž10y7 . m2rkN. The implementation of the pore-fluid compressibility into the theoretical description creates a more realistic picture of the wave-induced pore-pressure response in porous seabed sediments. It was proved that partly saturated seabed sediments can exist under natural conditions of coastal environment ŽMagda, 1998.. It is also well-known that the relative compressibility of the two-phase seabed medium is strongly dependent on soil saturation conditions, represented by the degree of saturation ŽVerruijt, 1969; see Eq. Ž9... For example, if S s 0.99 Ži.e., 1% of air is entrapped and dissolved in pore water. and water depth h s 10 m are assumed, the pore-fluid compressibility can be steeply increased by a factor of O Ž10 2 .. Assuming the bottom pressure function as given in Eq. Ž5., an analytical ‘finite-thickness’ solution to the diffusion problem was presented by Moshagen and Tørum Ž1975..

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3.3. COMP-PS model If both phases Ži.e., pore fluid and soil skeleton. of the seabed medium are assumed to be compressible, the mathematical two-dimensional formulation of the governing problem consists of three coupled partial differential equations Ži.e., two linear elastic equilibrium equations and the so-called storage equation governing the flow of compressible pore fluid through a porous and compressible medium according to the Darcy law.: G G

ž ž

E2 u x Ex

2

E2 u z Ex

2

k x E2 p kz Ex

2

q

E2 u x Ez

q

2

E2 u z Ez

2

/ /

Ez

2

1 y 2n E x G

q

E

E

1 y 2n E z

g nb X Ep

E2 p q

G q

s

ž ž

g q

kz

Et

kz

Eu x

q

Ex Eu x

Ez q

Ex

ž

Eu z

Eu x Ex

Eu z Ez

q

/ /

Eu z Ez

Ep s Ex Ep s Ez

/

Ž 10a. Ž 10b. Ž 10c.

where, p is the wave-induced poreŽ-fluid. pressure, u x and u z are the soil displacements in the x- and z-directions, respectively, G s Erw2Ž1 q n .x is the shear modulus for isotropic soil, E is Young’s modulus of elasticity for isotropic soil, n is Poisson’s ratio for isotropic soil, k x and k z are the coefficients of soil permeability in the x- and z-directions, respectively, g is the unit weight of pore fluid, n is the porosity of soil, b X is the compressibility of pore fluid, x and z are the horizontal and vertical coordinates of the Cartesian coordinates system, respectively, and t is the time. Compressibility properties of the soil skeleton are represented by the shear modulus, G. Generally, the shear modulus depends on the type of soil and its density. For loose sands, it is reasonable to assume G , O Ž10 3 –10 4 . kPa, whereas for dense and very dense sands G , O Ž10 6 –10 7 . kPa. A typical value of Poisson’s ratio for sands that undergo elastic deformations is n s 0.3–0.33. The system of Eqs. Ž10a., Ž10b. and Ž10c. was solved analytically by, e.g., Madsen Ž1978. and Yamamoto et al. Ž1978., for infinitely deep permeable seabed layer. The ‘finite-thickness layer’ solution to the storage problem was derived analytically by Magda Ž1994; 1998. and obtained numerically by, e.g., Magda Ž1992., using one-dimensional approximation. In the present analysis, the most general COMP-PS model is used to perform a comprehensive parameter study of the contribution of pore-pressure perturbation effects in the hydrodynamic uplift force with respect to soil compressibility and permeability characteristics. The numerical analysis is based on the same 2D finite-element formulation of the governing equations ŽEq. Ž10a. through Eq. Ž10c.., assuming hydraulic isotropy, and Eq. Ž9. which was already presented and used by Magda Ž1996; 1997.. 4. Methods of hydrodynamic uplift force computation Assuming that a submarine pipeline is an impermeable and stiff submerged body, buried parallel to wave fronts, the governing problem of the wave-induced pore pressure

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in seabed sediments in the vicinity of the submarine pipeline is expressed by the same partial differential equationŽs. and water-wave loading conditions as in the case of pure wave-seabed interaction and, additionally, by the pipeline impermeable surface condition in the form: Ep s 0 for r s rp Ž 11 . Er where, p is the wave-induced pore pressure, r is the linear coordinate Žnormal to the pipeline surface. of the polar coordinates system, and rp is the outside radius of the pipeline. The solution of the governing problem can be found by means of superposition of two separate solutions: one is the wave-induced pore-pressure solution with the absence of the pipeline structure in the seabed, and the second one is a kind of pore-pressure correction of the wave-induced pore-pressure solution, such that the gradient of the total wave-induced pore pressure at the pipeline surface is zero. An appearance of a rigid and impermeable structure Žhere: pipeline. inserted into a porous medium, where the wave-induced pore pressure undergoes cyclic oscillations, disturbs a virgin picture Ži.e., without any structure in seabed sediments. and changes the pore-pressure field in the structure vicinity due to the so-called scattering, or perturbation, or disturbance effects. Therefore, the pore pressure affected by the described phenomenon is very often called the scattered pore pressure or perturbed pore pressure. And thus, the perturbed pore pressure can be defined as: p ' p Ž p. s p Ža. q p Žc . or, according to Eq. Ž2., in relative Žand dimensionless. form: p p' ' p Ž p. s p Ža. q p Žc . P0

Ž 12 . Ž 13 .

where, p Žp. is the perturbed Ždue to the pipeline presence in the seabed. wave-induced pore pressure, p Žp. is the relative perturbed wave-induced pore pressure, p Ža. is the non-perturbed Ždue to the pipeline absence in the seabed. wave-induced pore pressure, p Ža. is the relative non-perturbed wave-induced pore pressure, p Žc. is the wave-induced pore-pressure correction, and p Žc. is the relative wave-induced pore-pressure correction. By simply adding p Ža. and p Žc. Žor p Ža. and p Žc. ., the perturbed wave-induced pore-pressure variations around the submarine pipeline buried in seabed sediments can be easily obtained. Adequately to Eqs. Ž12. and Ž13., the hydrodynamic uplift force acting on a submarine pipeline buried in seabed sediments, taking into account the presence of the pipeline structure in the seabed, can be also presented as a sum of two separate solutions: Fz ' FzŽ p. s FzŽa. q FzŽc . or, in relative Žstill dimensional but independent of the wave height. form: Fz Fz ' ' FzŽ p. s FzŽa. q FzŽc . P0

Ž 14 .

Ž 15 .

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where FzŽp. is the perturbed hydrodynamic uplift force, FzŽp. is the relative perturbed hydrodynamic uplift force, FzŽa. is the non-perturbed hydrodynamic uplift force, FzŽa. is the relative non-perturbed hydrodynamic uplift force, FzŽc. is the hydrodynamic upliftforce correction, FzŽc. is the relative hydrodynamic uplift-force correction. In general, the inclusion of the pore-pressure perturbation effects into the global solution magnifies the non-perturbed pore pressure acting on the pipeline circumference Ži.e., the Žover.pressure increases and the Žunder.pressure decreases., leading thereby to the increase of the hydrodynamic uplift force. In order to get a better insight into the influence of pore-pressure perturbation effects on the hydrodynamic uplift force, the following uplift-force perturbation ratio will be introduced and analysed: rF s

Ž F Žzc . . max ,Q Ž F Žzc . . max ,Q s Ž F Žza . . max ,Q Ž F Žza . . max ,Q

Ž 16 .

where, additionally, r F is the hydrodynamic uplift-force perturbation ratio, and subscript Žmax,Q . denotes the maximum value of the hydrodynamic uplift force obtained for the optimum phase Qopt of the water-wave loading oscillations. 4.1. Non-perturbed hydrodynamic uplift force — a simplified method For the first estimation of the hydrodynamic uplift force, the wave-induced pore-pressure correction, p Žc., and therefore the hydrodynamic uplift-force correction, FzŽc., are assumed to be zero. It implies that the perturbed parameters are equal to the non-perturbed ones. An exemplary definition sketch of such analysis is shown in Fig. 2. In order

Fig. 2. Wave-induced pore-pressure distribution with depth Žpotential solution., and along the circumference of the pipeline located just under the wave trough.

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to carry out this type of calculation, the knowledge of two elements is required, namely: a geometry of the system and an analytical description of the wave-induced pore-pressure distribution with depth in the permeable seabed layer. 4.1.1. INCOMP model (potential solution) Assuming, for example, the permeable seabed layer to be of infinite thickness, the potential problem Žgoverned by the Laplace equation. has the following wave-induced pore-pressure solution ŽPutnam, 1949.: p s exp Ž yaz . cos Ž ax y v t .

Ž 17 .

where, p is the relative Žand dimensionless. wave-induced pore pressure, a is the wave number, v is the wave angular frequency, x and z are the horizontal and vertical coordinates of the Cartesian coordinates system, respectively, and t is the time. Assuming now the most inconvenient situation for the potential solution, in which the pipeline is located directly under the wave trough Ži.e., x s Lr2 or ax s p , and t s lT for l s 0,1,2, . . . or v t s mp for m s 0,2,4, . . . , for which the phase of wave loading oscillations Q t ' Ž ax y v t . t s p and therefore cos Q t s y1; see Figs. 1 and 2 and Eqs. Ž5., Ž6. and Ž17.., the relative hydrodynamic uplift force Žsee Eq. Ž3.. reaches its maximum value and can be defined in the following integral expression: 2p

Ž Fz . t s yH0

exp  ya b q rp Ž 1 y cos c .

4 cos

a

ž

L 2

q rp sin c

/

cos c rp d c

Ž 18 . where, Ž Fz . t is the relative hydrodynamic uplift force, obtained for the case of wave trough above the pipeline, for which the hydrodynamic uplift force reaches its maximum Žin the potential problem only., a s 2prL is the wave number, L is the wavelength, b is the depth of burial of the pipeline Žmeasured from the sea floor to the pipeline top., rp s Dr2 is the outside radius of the pipeline, D is the outside diameter of the pipeline, and c is the angular coordinate of the polar coordinates system. The solution to the integral given in Eq. Ž18. is surprisingly simple:

Ž Fz . t s p arp2 exp

ya Ž b q rp .

Ž 19 .

In the case of the finite thickness of the seabed layer, the method given by Monkmeyer et al. Ž1983. has to be used. 4.1.2. COMP-PS model (storage solution) 4.1.2.1. Input data. Input data used in the analysis of the hydrodynamic uplift force, performed according to the storage model, is as follows: water depth h s 10 m, wave period T s 12.65 s, wavelength L s 120 m, outside diameter of pipeline D s 1.0 m, depth of burial b s 0.5 m, thickness of seabed layer d s 1.5 m Žpipeline adjoins the impermeable and rigid base., d s Lr10 s 12 m, and d s L s 120 m, Young’s modulus of soil E s 10 4 , 10 7 kPa Žloose, dense sediments., Poisson’s ratio n s 0.3, porosity n s 0.4, coefficient of soil permeability k s 10y4 mrs, degree of saturation S s 0.9–1.0,

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compressibility of pure water b s 4.2 = 10y7 m2rkN, and atmospheric pressure pat s 101.325 kPa. 4.1.2.2. Results of computations. In the potential problem, it was enough to get results for only one case of the relative pipeline position Ži.e., just under the wave trough., which is the most critical with respect to the hydrodynamic uplift-force magnitude. However, considering the diffusion model or the storage model of the seabed response to progressive surface water waves, one has to remember that this very special location of the pipeline with respect to the surface water wave may not always be adequate to the most unfavourable case of the pore-pressure gradient in the vicinity of submarine pipeline buried in seabed sediments and, therefore, it may not be synonymous with obtaining a maximum value of the hydrodynamic uplift force. The phase-lag phenomenon, characterizing pore-pressure oscillations in the seabed, is responsible for a certain phase lag in the hydrodynamic uplift-force oscillations with respect to the phase of surface-water-wave oscillations. It means that a precise detection of a maximum value of the hydrodynamic uplift force requires a wide analysis from the whole period of oscillations. Taking the above into account, the following notation will be used in the analysis of the relative hydrodynamic uplift force considered in the storage problem ŽCOMP-PS model.:

Q s Qopt ' Ž ax y v t . opt ´ Fz s Ž Fz . max ,Q

Ž 20 .

where, Q is the phase of the water-wave loading oscillations, uopt is the optimum phase of the water-wave loading oscillations, Fz is the relative hydrodynamic uplift force, Ž Fz . max,Q is the relative hydrodynamic uplift force, maximum with respect to the phase of the water-wave loading oscillations. Fig. 3 shows the wave-induced pore-pressure distributions with depth Žup to z s D q b s 1.5 m, obtained — as an example — for S s 0.99 and the wave trough positioned just above the pipeline, i.e., Qopt ' Q t ., whereas Fig. 4 illustrates the maximum relative hydrodynamic uplift force Ž FzŽa. . max,Q influenced by the different values of the degree of saturation from the range of S s 0.9–1.0. Simultaneously, two different values of Young’s modulus of soil skeleton are introduced, in order to represent either a dense state Ž E s 10 7 kPa. or a loose state Ž E s 10 4 kPa. of seabed sediments. It has to be emphasised that the maximum hydrodynamic uplift force was computed using the undisturbed pore-pressure distributions Žsee Fig. 3., which were obtained for the pipeline absence in the seabed; it means that Ž FzŽp. . max,Q s Ž FzŽa. . max,Q . Assuming a constant value of the degree of saturation Že.g., S s 0.99., Fig. 3 indicates a stronger influence on the pore-pressure response Ži.e., higher values of the pore-pressure gradient are obtained., in the whole vertical profile Ž0 - z - 1.5 m. considered in the example, for the dense sandy seabed sediments than for the loose ones. This particular case can be generalized for a wider range of the degree of saturation. A simple and direct consequence of this fact is reflected in the results of the hydrodynamic uplift-force analysis shown in Fig. 4. Using the above presented computational example, it becomes very clear that the hydrodynamic uplift force is very sensitive on the variation of soil saturation conditions,

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Fig. 3. Definition sketch for the hydrodynamic uplift-force analysis Žpore-pressure gradient influenced by soil skeleton compressibility and seabed layer thickness; degree of saturation Ss 0.99, pipeline located just under wave trough, i.e., Qopt ' Q t ..

especially for the degree of saturation from the upper range S , 0.98–1.0. For dense sandy sediments, the hydrodynamic uplift force increases much stronger than for loose sediments when the degree of saturation drops from S s 1.0 to S , 0.98. It can be also

Fig. 4. Hydrodynamic uplift force versus different soil saturation conditions and compressibility of soil skeleton Žpipeline located just under wave trough, i.e., Qopt ' Q t ..

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recognized Žfor dense sediments more evidently than for loose ones. that for a certain value of the degree of saturation, Sopt , the hydrodynamic uplift force, Ž FzŽa. . max,Q , reaches its absolute maximum. It is clear now that the wave-induced pore-pressure gradient can be strongly influenced by both the phase of water-wave loading oscillations and soil saturation conditions at the same time. By overlapping of these two effects, one can write the following relation:

Q s Qopt and S s Sopt ´ Fz s Ž Fz . max ,Q ,S

Ž 21 .

where, S is the degree of saturation, Sopt is the optimum degree of saturation, Fz is the relative hydrodynamic uplift force, Ž Fz . max,Q ,S is the relative hydrodynamic uplift force, maximum with respect to the phase of the water-wave loading oscillations and soil saturation conditions. Especially for dense seabed sediments, the optimum degree of saturation is rather close to S s 1.0 which denotes fully saturated soil conditions. From the practical point of view it is very probable that sandy seabed sediments in the coastal zones are characterized by the degree of saturation from the range of S , 0.97–1.0. However, it is very difficult to measure the degree of saturation of the upper layer of the sandy seabed sediments with a high accuracy required by sophisticated mathematical models Že.g., COMP-PS model.. Man-made errors in seabed sampling and spatial inhomogeneity of the soil saturation conditions make it almost impossible to give one correct value of the degree of saturation that could be used as an input data for the design procedure of a certain engineering structure. Therefore, taking the above into account, it seems to be very reasonable to perform a kind of parameter study in order to detect the most critical and relevant case for the analysis of the vertical stability of a submarine pipeline buried in seabed sediments, in which the hydrodynamic uplift force becomes absolutely maximal. Additionally, it becomes possible also to define a phase lag, d , in appearance of the maximum hydrodynamic uplift force with respect to the time-point when the wave trough reaches the position just above the centre of the pipeline:

d ' Qopt y Q t s Qopt y p

Ž 22 .

The most important results indicated in Fig. 4 can be summarized as follows: - dense sandy seabed sediments Ž FzŽa. . max,Q ,S s 0.376 kN my1 kPay1 for d s 1.5 m and Sopt s 0.97, d s 28 Ž FzŽa. . max,Q ,S s 0.408 kN my1 kPay1 for d s 120 m and Sopt s 0.98, d s 98 - loose sandy seabed sediments Ž FzŽa. . max,Q ,S s 0.175 kN my1 kPay1 for d s 1.5 m and Sopt s 0.93, d s 648 Ž FzŽa. . max,Q ,S s 0.156 kN my1 kPay1 for d s 120 m and Sopt s 0.94, d s 578 The effect of Sopt can be visualized even better if different wavelengths are considered. For comparison purposes, assuming a four-times-shorter wavelength Ži.e., L s 30 m. and keeping d s L, the hydrodynamic uplift force does not alter significantly Ži.e., Ž FzŽa. . max,Q ,S s 0.402 for dense sediments, Ž FzŽa. . max,Q ,S s 0.133 for loose sediments. but the optimum degree of saturation Ž Sopt s 0.992 for dense sediments, Sopt s 1.0 for loose sediments. moves evidently towards the full-saturation point Ž S s 1.0.. Inde-

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pendence of the hydrodynamic uplift force in dense sediments from the wavelength was discussed by Magda Ž1995.. The obtained maximum values of the relative hydrodynamic uplift force are much higher than those obtained for fully saturated Ži.e., S s 1.0. near-incompressible Ži.e., b X s b , 0. soils, where the pore-pressure response can be described by the potential solution. For the present computational example, it appears that the maximum hydrodynamic uplift force obtained for the storage problem Ži.e., COMP-PS model. is: Ža. for dense sediments, from 350 Žfor d s 1.5 m. to 10 Žfor d s L s 120 m., and Žb. for loose sediments, from 160 to 4, respectively, times higher than the respective values computed according to the potential problem Ži.e., INCOMP model.. On the other hand, a rather minor influence of the ‘finite-thickness layer’ boundary condition on the pore-pressure gradient Žsee Fig. 3. and the hydrodynamic uplift force Žsee Fig. 4. is observed. The proximity of the lower boundary condition seems to have a similar influence irrespective of whether they are dense or loose sediments. 4.2. Perturbed hydrodynamic uplift force — an adÕanced method 4.2.1. INCOMP model (potential solution) Using the potential solution of Monkmeyer et al. Ž1983. to the wave-induced pore-pressure response in the vicinity of a submarine pipeline buried in seabed sediments, the influence of the geometric parameters Ži.e., wavelength, depth of burial, and seabed layer thickness. on the hydrodynamic uplift force will be studied. First of all, assuming D s 1.0 m, b s 0.5 m, and the infinite thickness of permeable seabed layer as an example of input data, the influence of the wavelength on the maximum hydrodynamic uplift-force components and the hydrodynamic uplift-force perturbation ratio was examined and illustrated in Fig. 5. It can be easily indicated that for a certain optimum

Fig. 5. Influence of wavelength on the hydrodynamic uplift force and the perturbation ratio ŽINCOMP model; infinite thickness, Ds1.0 m, bs 0.5 m..

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wavelength, the uplift-force components reach their maxima. For the case presented, it happens already for the wavelength L , 6 m, or — in terms of a relative parameter — LrD , 6. The perturbation ratio increases sharply and becomes almost constant for LrD ) Ž LrD . opt , 6. Introducing an additional geometric parameter, which is the depth of burial of the pipeline in seabed sediments, Fig. 6 indicates the following trend: decreasing depth of burial induces higher values of the maximum hydrodynamic uplift force, and smaller values of the optimum wavelength. If the pipeline is placed just under the sea floor Ži.e., brD s 0., the maximum uplift force Ž FzŽp. s 0.806. is approximately four times larger than in the case of brD s 2 Ž FzŽp. s 0.223.. Still considering the seabed as a permeable half-space, the influence of the relative and dimensionless geometric parameters Ži.e., brD and LrD . on the hydrodynamic uplift-force perturbation ratio is shown in Fig. 7. The behaviour observed in Fig. 5 is confirmed again: increasing values of the wavelength enlarge the uplift-force perturbation ratio, which tends to its limiting values Žpractically, already for LrD ) 10.. Additionally, the perturbation ratio increases with depth of burial. If the pipeline is located in a distant place with respect to the sea-floor level, the influence of the upper Ži.e., sea floor. boundary condition becomes smaller and the perturbation ratio approaches unity; practically, it happens already for brD ) 10. If the pipeline is shifted far away from the sea floor, the wave-induced pore pressure undergoes a full reflection from the stiff and impermeable pipeline-wall, inducing thereby the perturbed pore pressure as large as twice the non-perturbed pore pressure acting along the pipeline circumference. Using the method of Monkmeyer et al. Ž1983., it was possible to study the influence of finite thickness of the permeable seabed layer on the uplift-force perturbation ratio.

Fig. 6. Influence of wavelength and depth of burial Žrelative. on the hydrodynamic uplift force ŽINCOMP model; infinite thickness, Ds1.0 m..

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Fig. 7. Influence of depth of burial Žrelative; in logarithmic scale. and wavelength Žrelative. on the uplift-force perturbation ratio ŽINCOMP model; infinite thickness..

Assuming D s 1.0 m, b s 0.5 m, and L s 10 m, Fig. 8 illustrates the hydrodynamic uplift-force perturbation ratio as a function of the seabed-layer thickness and the depth of burial Žrelative. of three different values: brD s 0.25, 0.5, and 1. If the seabed

Fig. 8. Influence of seabed layer thickness and depth of burial Žrelative. on the uplift-force perturbation ratio ŽINCOMP model; finite thickness, Ds1.0 m, Lr Ds10..

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thickness is small and comparable to the outside diameter of the pipeline Ž drD - 5., the hydrodynamic uplift-force perturbation ratio increases sharply; deep-soil conditions make the perturbation ratio constant. Fig. 9 illustrates the most general results obtained from the parameter study of the uplift-force perturbation ratio performed for the INCOMP model. Assuming large values of the wavelength with respect to the outside diameter of the pipeline Ži.e., LrD ) 10., which is relevant for coastal and offshore engineering, the influence of all other geometric parameters Ži.e., brD and drD . describing the governing problem is presented. Considering the case of brD ) 1, the half-space solution Žpractically, already denoted by the case drD s 10. creates a lower-limit case Ž r F , 1.0.. An upper-limit solution Ž r F , 1.42; see dashed line in Fig. 9. is obtained when the pipeline rests on the impermeable base. In both the limiting solutions, the uplift-force perturbation ratio decreases sharply for smaller depths of burial Ži.e., brD - 1.. 4.2.2. COMP-PS model (storage solution) Using the finite-element formulation of the governing problem ŽMagda, 1996. it was possible to perform computations of the hydrodynamic uplift force acting on a submarine pipeline buried in seabed sediments, according to the COMP-PS model that enables to take into account the following principal soilrpore-fluid parameters: soil permeability, compressibility of soil skeleton, and compressibility of pore fluid Žor soil saturation., strongly influencing the wave-induced pore-pressure oscillations in the seabed sediments ŽMagda, 1998.. Using the COMP-PS model, the cases respective to INCOMP, COMP-P, and COMP-S models can be easily studied. If the pipeline structure is modelled to be buried in seabed sediments, the finite-element modelling of the governing problem results directly in the perturbed wave-induced

Fig. 9. Influence of seabed layer thickness Žrelative. and depth of burial Žrelative. on the uplift-force perturbation ratio ŽINCOMP model; finite thickness, relatively long wave Lr D)10..

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pore pressure and, consequently, the perturbed hydrodynamic uplift force. In order to evaluate the hydrodynamic uplift-force perturbation ratio, it was necessary to conduct additional computations of the non-perturbed Ži.e., with pipeline absence in the seabed domain. wave-induced pore pressure and hydrodynamic uplift force. This can be achieved using one of the previously mentioned analytical solutions Že.g., Madsen, 1978; Magda, 1994, 1998. for the wave-induced pore pressure in seabed sediments. However, in order to eventually omit even the small discrepancies between the analytical and numerical solutions Že.g., due to side-wall effects in the finite-element seabed domain., it was decided to compute the non-perturbed pore pressure, using the same finite-element numerical modelling and, consequently, the non-perturbed hydrodynamic uplift force, using a simple summation technique mentioned in Section 2. 4.2.2.1. Input data. In the finite-element numerical analysis, the following input data was used: water depth h s 10 m, wave period T s 4.45 s, wavelength L s 30 m, outside diameter of pipeline D s 1.0 m, depth of burial b s 0.25, 0.5, 1.0, 2.0 m, seabed thickness d s Lr2 s 15 m, Young’s modulus of soil E s 10 4 , 10 7 kPa Žloose, dense sediments., Poisson’s ratio n s 0.3, porosity n s 0.4, coefficient of soil permeability k s 10y3 , 10y4 mrs, degree of saturation S s 0.95–1.0, compressibility of pure water b s 4.2 = 10y7 m2rkN, and atmospheric pressure pat s 101.325 kPa. The wavelength Ž L s 30 m. assumed for the FE-modelling of the problem is not long compared with the wave conditions existing in coastal zones. However, the main reason for using such minor value was to reduce costs Ži.e., time and memory. of demanding numerical computations. According to Magda’s Ž1996. study, it is rather required to have the soil domain as long as, at least, one wavelength of surface-wave oscillations in order to keep correct the influences of the loading function on the seabed response. This statement is true for both dense and loose sediments. On the other hand, if L s 30 m and D s 1 m are assumed, the dimensionless ratio LrD is still greater than 10, for which the loading conditions become similar to water-table vertical oscillations. A further increase of the wavelength seems to have no significant influence on the relative hydrodynamic uplift force and the uplift-force perturbation ratio, which has been already proved in the case of the potential problem ŽINCOMP model; see Section 4.2.1.. Magda Ž1995; 1997. also showed that the wavelength does not influence the maximum hydrodynamic uplift force when the pore fluid is treated as compressible Ži.e., COMP-P model.. This case is controlled by the diffusion coefficient Žsee Eq. Ž8.., and the wavelength influences only the optimum degree of saturation for which the hydrodynamic uplift force obtains its maximum. 4.2.2.2. Finite-element mesh. Global dimensions of the finite-element seabed domain are related to the wavelength. A detailed study of the influence of the soil domain dimensions on the quality of the hydrodynamic uplift force results was given by Magda Ž1996.. And thus: width of seabed domain Bd s L s 30 m, and height of seabed domain Hd s d s Lr2 s 15 m. Taking into account the shape of a pipe-like structure placed within the seabed domain, it was desired to create the finite-element mesh using isoparametric linear four-node elements. Fig. 10 shows an example of the FE-mesh used in the analysis for

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Fig. 10. Example of FE-mesh extracted from pipeline vicinity Ž bs 0.5 m; total mesh: 1894 elements, 1999 nodes..

b s 0.5 m. The FE-mesh designed for the computations of non-perturbed pore-pressure values consists only of four-node rectangular elements, and is principally based on the FE-mesh used for computations of the perturbed pore pressure, of course, after excluding some elements from the pipeline vicinity and replacing them by simple rectangular elements. 4.2.2.3. Results of 2D-FEM analysis. The results of numerical analysis are presented in two groups distinguished with respect to compressibility of the soil skeleton. And thus, loose sandy sediments are modelled using Young’s modulus E s 10 4 kPa, whereas for dense and very dense sediments E s 10 7 kPa was assumed. Fig. 11 Žfor E s 10 4 kPa. and Fig. 12 Žfor E s 10 7 kPa. illustrate the results of the perturbed hydrodynamic uplift force, Ž FzŽp. . max,Q , acting on a submarine pipeline buried in seabed sediments, influenced by soil permeability ŽŽa. k s 10y3 mrs and Žb. k s 10y4 mrs., soil saturation conditions modelled by the degree of saturation from the practically possible range S s 0.95–1.0, and the depth of burial b s 0.25, 0.5, 1.0, and 2.0 m. Analysing the results obtained for loose seabed sediments Ž E s 10 4 kPa; see Fig. 11., it is observed that slightly higher values of the perturbed hydrodynamic uplift force appear for higher permeable soils Žcompare Fig. 11a and b.. The influence of soil saturation conditions is rather mild. The decrease in the degree of saturation induces the increase of the hydrodynamic uplift force when the depth of burial is relatively small Ži.e., b s 0.25 m.; for greater depth of burial Ži.e., b s 2.0 m. the hydrodynamic uplift force decreases slightly. Taking into account dense and very dense seabed sediments Ž E s 10 7 kPa; see Fig. 12., a behaviour very similar to the case of non-perturbed uplift force can be seen, where the maximum perturbed hydrodynamic uplift force Ž FzŽp. . max,Q reaches the absolutely maximal value of Ž FzŽp. . max,Q ,S for the optimum degree of saturation, Sopt . The hydrodynamic uplift force is also influenced by the depth of burial of the pipeline. And thus, the absolutely maximal perturbed hydrodynamic uplift force Ž FzŽp. . max,Q ,S increases from 0.27 Žfor b s 2.0 m and Sopt s 0.999. through 0.443 Žfor b s 1.0 m and Sopt s 0.998., and 0.626 Žfor b s 0.5 m and Sopt s 0.994. to 0.757 Žfor b s 0.25 m and

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Fig. 11. Perturbed hydrodynamic uplift force acting on submarine pipeline buried in seabed sediments influenced by depth of burial, soil saturation, and permeability: Ža. k s10y3 mrs, Žb. k s10y4 mrs ŽFE-modelling, COMP-PS model, loose sand Es10 4 kPa..

Sopt s 0.99.. In all these cases, the values of the optimum degree of saturation denote near-fully saturated soil conditions. These results confirm the need for a very careful analysis Ži.e., with a very small increment of the degree of saturation, at least in the upper range of soil saturation conditions. in order to detect properly a maximum value of the hydrodynamic uplift force. Additionally, it can be stated that the maximum value of the perturbed hydrodynamic uplift force does not depend significantly on soil permeability. However, soil permeabil-

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Fig. 12. Perturbed hydrodynamic uplift force acting on submarine pipeline buried in seabed sediments influenced by depth of burial, soil saturation, and permeability: Ža. k s10y3 mrs, Žb. k s10y4 mrs ŽFE-modelling, COMP-PS model, dense sand Es10 7 kPa..

ity influences the optimum value of the degree of saturation, Sopt , for which the perturbed hydrodynamic uplift force obtains its absolute maximum, Ž FzŽp. . max,Q ,S . Considering the case b s 0.25 m as an example, it can be found that in the case of k s 10y3 mrs Žsee Fig. 12a. Ž FzŽp. . max,Q ,S , 0.72 for Sopt , 0.95, while Ž FzŽp. . max,Q ,S , 0.76 for Sopt s 0.99 in the case of k s 10y4 mrs Žsee Fig. 12b.. When E s 10 7 kPa Ži.e.,

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COMP-P model, practically. is assumed, the coefficient of diffusion Žsee Eq. Ž8.. is responsible for a one-term modelling of the physical characteristic of the two-phase soil-skeletonrpore-fluid seabed medium. Therefore, a maximum value of the hydrodynamic uplift force is related to the optimum value of this controlling parameter. Of course, a certain optimum value of this coefficient can be created by several combinations of the values of the single parameters appearing in the formula defining the coefficient of diffusion Žsee Eq. Ž8... The next results pertain to the perturbation ratio, r F Žsee Eq. Ž16.., of the maximum Žwith respect to one oscillating cycle. hydrodynamic uplift force. Fig. 13 shows the COMP-S case, in which the pore fluid is treated to be incompressible Ž b X s 0 m2rkN. whereas the soil skeleton is assumed to be compressible Ž E s 10 4 kPa.. An increasing depth of burial of the pipeline in seabed sediments increases the perturbation ratio up to approx. r F s 1 for b s 2.0 m. It is only for the smaller values of the burial depth, where the influence of soil permeability can be recognized. Higher permeability of the soil relates to lower values of the perturbation ratio where r F , 0.42 and 0.54 for k s 10y3 and 10y4 mrs, respectively. The results obtained for the INCOMP model Ž b X s 0 m2rkN and E s 10 7 kPa. are presented in Fig. 14. Comparing with the COMP-S case, the influence of the depth of burial is not so pronounced here. For the depth of burial b s 0.25–2.0 m, the perturbation ratio varies from r F , 0.8 to 1, respectively. Additionally, as anticipated for the potential problem, soil permeability has no other meaning. The results obtained for the ‘full’ COMP-PS model are presented in Fig. 15 Ž E s 10 4 kPa — loose sediments. and in Fig. 16 Ž E s 10 7 kPa — dense sediments., where the hydrodynamic uplift-force perturbation ratio is shown as a function of the depth of burial, soil saturation and permeability. If a high permeable soil Ži.e., k s 10y3 mrs. is

Fig. 13. Hydrodynamic uplift-force perturbation ratio influenced by soil permeability and depth of burial ŽFE-modelling, COMP-PS model; loose sand: Es10 4 kPa, and incompressible pore fluid: b X s 0 m2 rkN — COMP-S case..

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Fig. 14. Hydrodynamic uplift-force perturbation ratio influenced by soil permeability and depth of burial ŽFE-modelling, COMP-PS model; dense sand: Es10 7 kPa, and incompressible pore fluid: b X s 0 m2 rkN — INCOMP case..

assumed, the pore-fluid compressibility represented by the degree of soil saturation has rather mild influence on the perturbation ratio. The perturbation ratio obtains maximum values for fully saturated soil Ži.e., S s 1.0., and decreases when the pore fluid becomes more compressible; the decrease is a little bit stronger for dense sediments Žsee Fig. 16a. than for loose sediments Žsee Fig. 15a.. Concerning the pipeline vertical position, the following phenomenon can be observed: the deeper the pipeline, the higher the hydrodynamic uplift-force perturbation ratio. The introduction of the lower soil permeability Ži.e., k s 10y4 mrs. complicates the relationship of the perturbation ratio, where the perturbation ratio decreases strongly with the degree of saturation for loose sediments and deeper pipeline positions Ži.e., b s 1.0 and 2.0 m; see Fig. 15b.. Assuming dense sediments, the perturbation ratio decreases sharply when the degree of saturation drops from S s 1.0 to approximately S s 0.99; a smaller saturation causes the perturbation ratio in all the cases considered to approach the same value of r F , 0.5–0.55 for S s 0.95 Žsee Fig. 16b.. It is also very interesting to illustrate the time-history Žwithin one cycle of water-wave loading oscillations. of the hydrodynamic uplift-force components Ži.e., FzŽa. — non-perturbed, FzŽc. — corrected. with respect to the phase of bottom-pressure loading oscillations, p b . Fig. 17 shows the results of computations performed with the COMP-PS model, assuming dense sediments Ž E s 10 7 kPa., k s 10y4 mrs, and S s 0.95. Comparing the phases of oscillations, the following two observations can be made, namely: Ža. a certain phase lag, with respect to the bottom pressure oscillations, exists in the perturbed hydrodynamic uplift force, FzŽp., when partly saturated soils are considered; and Žb. there is no difference in the phases of oscillations of the hydrodynamic uplift force components Ži.e., non-perturbed, FzŽa., and correction FzŽc. .. The phase lag of

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Fig. 15. Hydrodynamic uplift-force perturbation ratio influenced by depth of burial, soil saturation, and permeability: Ža. k s10y3 mrs, Žb. k s10y4 mrs ŽFE-modelling, COMP-PS model, loose sand Es10 4 kPa..

the perturbed hydrodynamic uplift force is a natural consequence of the phase lag appearing in the wave-induced pore-pressure oscillations in seabed sediments. No phase lag between the non-perturbed and correction hydrodynamic uplift force means that the perturbed hydrodynamic uplift force, being the sum of the non-perturbed component and the correction component Žsee Eq. Ž14. or Eq. Ž15.., obtains its maximum exactly at the time when the non-perturbed hydrodynamic uplift force is maximum.

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Fig. 16. Hydrodynamic uplift-force perturbation ratio influenced by depth of burial, soil saturation, and permeability: Ža. k s10y3 mrs, Žb. k s10y4 mrs ŽFE-modelling, COMP-PS model, dense sand Es10 7 kPa..

Simultaneously, Fig. 17 indicates the time-history of the hydrodynamic uplift-force perturbation ratio computed using Eq. Ž16.. Generally, the perturbation ratio is constant Ži.e., r F , 0.55. within the whole period of water loading oscillations, although a certain time lag in the oscillations of the hydrodynamic uplift-force components with respect to water loading oscillations exists. Incompressibility ŽINCOMP model. or near-incompressibility ŽCOMP-PS model with S s 1.0 and E s 10 7 kPa. would not induce any

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Fig. 17. Time-history of the hydrodynamic uplift-force components and the perturbation ratio within one cycle of water loading oscillation ŽFE-modelling, COMP-PS model, dense sand Es10 7 kPa, k s10y4 mrs, Ss 0.95..

significant time lag; only partly saturated soil conditions can cause the time lag of a meaningful value Ž d , 0.2, T s 728 for S s 0.95; see Fig. 17..

5. Conclusions The problem of the hydrodynamic uplift force acting on a submarine pipeline buried in sandy seabed sediments was treated analytically and numerically. The perturbation effects in the wave-induced pore pressure and hydrodynamic uplift force were studied in order to extract the disturbing influence of a stiff and impermeable pipeline-wall structure on the wave-induced pore-pressure field in the pipeline vicinity. Introducing the so-called perturbation ratio of the hydrodynamic uplift force, which is a kind of magnification ratio, the contribution of the perturbation effects into the global value of the hydrodynamic uplift force was investigated with respect to some decisive physical properties Ži.e., relative compressibility and permeability. of the two-phase seabed medium as well as geometric parameters of the system Ži.e., wavelength, depth of burial of the pipeline, and thickness of the seabed layer.. The importance of the contribution of the perturbation pore pressure in the calculations of the hydrodynamic uplift force was shown for the different models of the wave-induced pore-pressure response in seabed sediments. Assuming real and practically important wave conditions Ži.e., LrD ) 10., and considering the potential problem ŽINCOMP model. of the wave-induced seabed response, the analytical analysis has indicated that the perturbation ratio can vary from r F , 1 for the pipeline close to the

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seabed floor, to r F , 1.4 for the pipeline close to the impermeable base under the permeable seabed layer, when the relative depth of burial brD G 1; smaller depth of burial induced smaller perturbation ratio. The results of numerical 2D FE-modelling performed for the more realistic storage problem ŽCOMP-PS model. of the wave-induced seabed response have shown that partly saturated seabed sediments Ži.e., S - 1.0. contribute to the decrease in the hydrodynamic uplift-force perturbation ratio, compared to fully saturated sediments Ži.e., S s 1.0.. The upper limit Ži.e., safe estimation. of the perturbation ratio computed for the compressible two-phase seabed sediments is always set by the potential solution. The hydrodynamic uplift force is strongly and directly influenced by the wave-induced pore-pressure vertical gradient in seabed sediments. Therefore, more compressible soil skeleton Ži.e., E s 10 4 kPa. makes the perturbation ratio changes more mild than in the case of dense and near-incompressible sediments Ži.e., E s 10 7 kPa.. Simultaneously, the influence of soil permeability on the perturbation ratio is more pronounced in dense sediments than in loose ones. Additionally, it was shown that the perturbation effects do not make any phase lag between the oscillations of the non-perturbed and correction components of the hydrodynamic uplift force, even if certain compressibility is introduced to the two-phase seabed medium. It implies that the perturbed hydrodynamic uplift force obtains its maximum exactly at the same time when the non-perturbed uplift force is maximum. Of course, partly saturated sediments cause a certain phase lag between the oscillations of the hydrodynamic uplift force and the surface-water-wave loading. Taking the above into account, the knowledge of the hydrodynamic uplift-force perturbation ratio can be very helpful for an easy determination of the hydrodynamic uplift force computed for real conditions Ži.e., assuming the existence of wave-induced pore-pressure perturbation effects.. In such case, instead of time-demanding numerical FE-computations, it is enough to compute the non-perturbed absolutely maximal hydrodynamic uplift force, Ž FzŽa. . max,Q ,S Žbased on the pore-pressure results obtained from a certain wave-induced pore-pressure theory., and then — using the perturbation ratio, r F , obtained from the present study — to compute the correction component of the absolutely maximal hydrodynamic uplift force in order to get the absolutely maximal perturbed hydrodynamic uplift force, Ž FzŽp. . max,Q ,S :

Ž FzŽ p. . max ,Q ,S ( Ž 1 q r F . Ž FzŽa. . max ,Q ,S

Ž 23 .

In order to assume the perturbed hydrodynamic uplift force safely, the perturbation ratio, r F , should follow the potential solution ŽINCOMP model. and varies from 1 to 1.4 depending on the relative position of the pipeline with respect to the lowerrupper boundary Ž r F , 1 when the pipeline approaches the sea floor, and r F , 1.4 when the pipeline approaches the impermeable base; see Fig. 9.. The above-presented computational analysis has proved evidently that neglecting the disturbing function of the impermeable pipeline structure on the pore-pressure field around can cause a serious design fault Ži.e., the hydrodynamic uplift force calculated excluding the perturbation pore pressure is approximately one half of what is expected when the perturbation effects are incorporated into the computational procedure..

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6. Notations a b cd d D E Fz FzŽa. FzŽc. FzŽp. Fz FzŽa. FzŽc. FzŽp. Ž Fz . t Ž Fz . max,Q Ž Fz . max,Q ,S G h H k k x ,k z L n p pat pb ph p Ža. p Žc. p Žp. p p Ža. p Žc. p Žp. Ph P0 r rF rp

wave number depth of burial of the pipeline coefficient of diffusion thickness of the permeable seabed layer outside diameter of the pipeline Young’s modulus of elasticity for isotropic soil hydrodynamic uplift force non-perturbed hydrodynamic uplift force hydrodynamic uplift-force correction perturbed hydrodynamic uplift force relative Ždimensional. hydrodynamic uplift force relative non-perturbed hydrodynamic uplift force relative hydrodynamic uplift-force correction relative perturbed hydrodynamic uplift force relative hydrodynamic uplift force respective to the wave trough Žpotential problem only. relative hydrodynamic uplift force, maximum with respect to the optimum phase of water-wave loading relative hydrodynamic uplift force, maximum with respect to the optimum phase of water-wave loading and degree of saturation shear modulus for isotropic soil water depth wave height coefficient of permeability for isotropic soil coefficients of soil permeability in x- and z-directions, respectively wavelength porosity of soil wave-induced pore pressure atmospheric pressure bottom pressure hydrostatic pressure non-perturbed wave-induced pore pressure wave-induced pore-pressure correction perturbed wave-induced pore pressure relative Ždimensionless. wave-induced pore pressure relative Ždimensionless. non-perturbed wave-induced pore pressure relative Ždimensionless. wave-induced pore-pressure correction relative Ždimensionless. perturbed wave-induced pore pressure absolute hydrostatic pressure amplitude of the hydrodynamic bottom pressure linear coordinate of the polar coordinates system hydrodynamic uplift-force perturbation ratio outside radius of the pipeline

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S Sopt t T u x ,u z x, z

bw bX g gw d Q Qopt Qt n c v

271

degree of saturation optimum degree of saturation time wave period soil displacements in x- and z-directions, respectively horizontal and vertical coordinates of the Cartesian coordinates system, respectively compressibility of pure water compressibility of pore fluid unit weight of pore fluid unit weight of sea water phase lag of the hydrodynamic uplift force with respect to the wave trough phase of water-wave loading optimum phase of water-wave loading phase of water-wave loading respective to the wave trough Poisson’s ratio for isotropic soil angular coordinate of the polar coordinates system wave angular frequency

References Biot, M.A., 1941. General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164. Cheng, A.H.-D., Liu, P.L.-F., 1986. Seepage force on a pipeline buried in a poroelastic seabed under wave loadings. Appl. Ocean Res. 8 Ž1., 22–32. Lai, N.W., Dominguez, R.F., Dunlap, W.A., 1974. Numerical solutions for determining wave-induced pressure distributions around buried pipelines. Texas A&M University, Sea Grant Pub. No. TAMU-SG-75-205. Lennon, G.P., 1985. Wave-induced forces on buried pipelines. J. Waterway, Port, Coastal, Ocean Eng. Div. 111 Ž3., 511–524. Liu, P.L.-F., O’Donnel, T.P., 1979. Wave-induced forces on buried pipelines in permeable seabeds. Proc. of the 4th Conference on Civil Engineering in the Oceans, San Francisco, Vol. II, 10–12 September 1979, pp. 111–121. MacPherson, H., 1978. Wave forces on pipeline buried in permeable seabed. J. Waterway, Port, Coastal, Ocean Eng. Div. 104 ŽWW4., 407–419. Madsen, O.S., 1978. Wave-induced pore pressures and effective stresses in a porous bed. Geotechnique 28 Ž4., ´ 377–393. Magda, W., 1992. Finite-element approximation of one-dimensional model for wave-induced pore pressure and displacements within seabed sediments. Arch. Hydroeng. XXXIX Ž3., 58–77. Magda, W., 1994. Analytical solution for the wave-induced excess pore-pressure in a finite-thickness seabed layer. Proc. of the 24th International Conference on Coastal Engineering ŽICCE’94., Kobe, Japan, Vol. III, 23–28 October, 1994, pp. 3111–3125. Magda, W., 1995. Wave-induced uplift force acting on a buried submarine pipeline — a practical engineering approach. Proc. of the 4th International Conference on Coastal and Port Engineering in Developing Countries ŽCOPEDEC IV., Rio de Janeiro, Brazil, Vol. III, 25–29 September 1995, pp. 2295–2310. Magda, W., 1996. Wave-induced uplift force acting on a submarine buried pipeline. Finite-element formulation and verification computations. Comput. Geotech. 19 Ž1., 47–73. Magda, W., 1997. Wave-induced uplift force on a submarine pipeline buried in a compressible seabed. Ocean Eng. 24 Ž6., 551–576.

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