Dynamic response of porous seabed to ocean waves

Computers and Geotechnics 28 (2001) 99±128 www.elsevier.com/locate/compgeo

Dynamic response of porous seabed to ocean waves D.S. Jeng a,*, T.L. Lee b a School of Engineering, Grith University Gold Coast Campus, QLD 9726, Australia Department of Architecture, Nanya Institute of Technology, Chung Li, Taiwan 320, Republic of China

b

Received 19 April 2000; received in revised form 28 August 2000; accepted 27 September 2000

Abstract In this paper, dynamic response of a seabed to ocean waves is treated analytically on the basis of a poro-elastic theory. The seabed is modelled as an isotropic homogeneous material of ®nite thickness. Most previous investigations for the wave-seabed interaction problem have treated the problem in the well-known quasi-static state. However, the dynamic response of the porous seabed cannot be predicted in the quasi-static solutions, because the acceleration generated by pore ¯uid and soil particles are excluded in the previous solutions. This paper proposes a semi-analytical solution for the dynamic response of seabed to waves. Based on the newly solution, the relative di€erences between dynamic and quasi-static solutions and the wave driven seepage ¯ux at the water-sediment interface will be examined. The wave-induced pore ¯uid displacement is about ten times of soil displacements in gravelled seabed. The wave driven seepage ¯ux is important in a coarser unsaturated seabed of thicker thickness under the action of a longer wave in shallow water. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Dynamic response; Seepage ¯ux; Pore pressure; Elastic waves

1. Introduction Recently, considerable e€orts have been devoted to the phenomenon of wave±seabed interaction. The reason for the growing interest is that some marine structures have possibly failed due to seabed instability, rather than constructional causes [1,2]. Also, the poro-elastic model for the wave±seabed interaction has been further

* Corresponding author. Tel.: +61-7-5594-8683; fax: +61-7-5594-8065. E-mail address: [email protected] (D.S. Jeng). 0266-352X/01/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0266-352X(00)00026-4

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applied to estimate the wave driven seepage ¯ux exchange at the water±sediment interface, which is related to mass transfer rate [3]. When gravity ocean waves propagate over the ocean, they exert signi®cant dynamic pressures on the sea¯oor. These pressure ¯uctuations further induce the variations in e€ective stresses and pore water pressure within non-cohesive marine sediments. When excess pore-pressure and diminishing vertical e€ective stresses, part of the seabed may become unstable or even lique®ed. Once liquefaction occurs, the soil particles are likely to be carried away by the prevailing bottom current or mass transports owing to the action of ocean waves. Besides the wave-induced soil response, the wave-driven seepage ¯ux is another important issue. Sediments in bays, estuaries, and in the seabed near river inlets are often contaminated. Many inorganic contaminants (notably heavy metals) do not decompose. Under certain conditions, these accumulated substances can be released back into the receiving body of water through mass transfer processes at the seabed. The mass transfer rate is largely controlled by the seepage ¯ux exchange between the sediment and the seawater [4]. Increased wave action and higher sediment hydraulic conductivity generally cause larger transfer rates. Clearly, quanti®cation of the mass transfer rate is a key factor in water quality modelling. Numerous investigations on the wave-induced soil response have been carried out in the last few decades [5±9, 11±13]. The assumptions of either non-deformable seabed [5,7,11] or incompressible pore ¯uid [6] have been used as the ®rst approximation in the area of wave-seabed interaction. Based on Biot's consolidation equation [14], several investigations have been presented [8,9,12,13]. The seabed has been modelled as being in®nite [8,9], ®nite [13] and layered [12,13]. Later, more complicated soil behaviour such as an isotropic non-homogeneous seabed have been considered with similar framework [15±17]. Additionally, the potential for waveinduced seabed instability due to a generalised three-dimensional wave system has also been explored [18]. Limitations and applications of previous investigations was reviewed recently [19]. However, all aforementioned theories have been limited to quasi-static solution, i.e. the acceleration generated by pore ¯uid and soil particles were not included in the models. Besides the quasi-static solutions mentioned previously, Yamamoto [20] has been the ®rst to investigate the e€ects of Coulomb-damped friction on the wave-induced seabed response. Yamamoto [20] demonstrated that the Coulomb-damping friction signi®cant a€ects the wave-induced seabed response in clay, rather than sandy seabed. However, his solution is only limited to the seabed of in®nite thickness. Also, some parameters involved in his solution have diculties to be determined in neither laboratory nor ®eld measurements. Since we focus on the soil response in sandy seabeds in this study, a simpler solution is desired. Later, based on Biot's poro-elastic theory [10], Mei [21] re-derived a set of governing equations for the wave-seabed interaction problem with dynamic soil behaviour. However, to simplify the complicated mathematical procedure, Mei [21] proposed a boundary-layer approximation to solve the problem, rather than the exact close-form solution. Based on Mei's formulations [21], Yuhi and Ishida [23] further proposed an analytical solution for the wave-induced seabed response. Their solution was limited

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to an in®nite seabed, although the thickness of seabed has been reported to signi®cantly a€ect the wave-induced seabed response [22]. Furthermore, only the characteristics of elastic waves were examined in their study. Neither e€ects of dynamics soil behaviour nor wave-driven seepage has been investigated. Recently, an analytical solution for the wave-induced seabed response with inertia forces was derived by the author and co-workers [24,25]. It was found that the inertial e€ects signi®cantly a€ects the wave-induced soil response in shallow water. However, the accelerated term considered in the model [24,25] was only the acceleration generated from soil particles, excluding the acceleration from pore ¯uid. Thus, the solution [24,25] is not a complete dynamic solution. The main purpose of this paper is to derive an analytical solution for the dynamic response of porous seabed to ocean waves. The seabed is considered as an isotropic homogeneous material of ®nite thickness. Based on the analytical solution derived here, the relative di€erence between dynamic and quasi-static solution will be investigated. Also, the wave-driven seepage ¯ux at water±sediment interface will be discussed in detail. 2. Boundary value problem In this study, we consider a gravity wave propagating over a porous seabed. The wave crests are assumed to propagate in the positive x-direction, while the z-direction is upward from the seabed surface, as shown in Fig. 1.

Fig. 1. De®nition of wave±seabed interaction problem.

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The proposed model for wave-seabed interaction is based on combining incompressible irrotational ¯ow for the water waves and the Biot's poro-elastic theory [10] for the porous ¯ow within the soil skeleton. 2.1. Governing equations Marine sediment is a mixture of up to three phases: a solid phase that forms an interlocking skeletal frame, a liquid phase that occupies a major portion of pore space, and a gas phase that sometimes occupies a small portion of pore space. Thus, it is reasonable to assume both pore ¯uid (a continuum comprising both liquid and gas) and soil skeleton to be compressible. Based on the assumption of linear poro-elasticity [10], Mei [21] presented a set of linearised governing equations for dynamic variations from the static equilibrium state, i.e. a storage equation, an equation of motion and a conservation of momentum equation of the soil-skeleton and the pore-water. They are summarised as the follows: 1. Storage equation: This equation is based on conservation of mass, originally derived by Verruijt [26] without dynamic terms generated from acceleration of soil particles and pore ¯uid. Mei [21] re-derived the storage equation with dynamic terms as k 2 k : : r p ÿ ne p ˆ r
u ÿ w r
w ;

w

w

…1†

2. Equation of motion: Again, the accelerations generated by soil particles (uÈ) and pore ¯uid (wÈ) are also included in the equation of motion, as shown on the right-hand-side of Eq. (2). These terms have not been included in quasi-static models [8,13].   1 : 2 r…r
u† ÿ rp ˆ ne w w ‡ …1 ÿ ne †s u …2† G r u‡ 1 ÿ 2 3. Momentum equation: In previous quasi-static solutions [8,22], the momentum equation was not included in the governing equations. However, it is necessary to have momentum equation in dynamic model, because the relative motion between pore ¯uid and soil particles needs to be considered. The linearised momentum equation of soil particles and pore ¯uid can be expressed as [21] w w ˆ -rp ÿ

w ne : : …w ÿ u†; k

…3†

In Eqs. (1)±(3), u(ux,uz) denotes the displacements of soil particles, while w(wx,wz) represents the displacements of pore ¯uid. The dots (
) denote di€er-

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entiation respect to time (i.e. @/@t). w is the unit weight of pore water, p is pore water pressure, k is permeability of the seabed, ne is the soil porosity, G is the shear modulus of the soil,  denotes the material density and is the compressibility of pore ¯uid, which is de®ned as ˆ

1 1ÿS ‡ ; Kw Pwo

…4†

in which Kw is the true bulk modulus of pore water, S is the degree of saturation and Pwo is the average absolute pore water pressure in the sediment. To simplify the rather complicated governing Eqs. (1)±(3), Mei [21] used a boundary-layer approximation to obtain a closed-form solution for wave-induced soil response. However, the boundary-layer approximation is limited to the case of low permeability [22]. Here, we directly solve (1)±(3) without any further assumptions [23]. Since all equations are linear, we can further decompose the displacement vectors of the soil and the pore water, u and w into two parts: the irrotational part and the non-divergent part. This decomposition has been used widely for elastic analysis in consolidation problems. By using scalar potentials (s and w) and vector potentials ( s and w) the decomposition can be expressed in the form: u ˆ rs ‡ r  s and w ˆ rw ‡ r  w :

…5†

By substituting Eq. (5) into Eqs. (1)±(3), we obtain : k 2 k :  w ˆ 0; r p ÿ r2 s ÿ ne p ‡ w r2 

w

w

…6†

2…1 ÿ † Gr2 s ÿ p ÿ ne w w ÿ …1 ÿ ne †s s ˆ 0 1 ÿ 2

…7†

ne w : ne w : w ÿ s ˆ 0; w w ‡ p ‡ k k

…8†

Gr2 s ÿ ne w w ÿ …1 ÿ ne †s  s ˆ 0

…9†

and

w ne :

w ne : w ÿ s ˆ 0 w w ‡ k k

…10†

Under the condition of plane strain, the stress±strain relationship is given as   @ux  r
u …11† ‡ x0 ˆ 2G @x 1 ÿ 2

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z0 ˆ 2G

  @uz  r
u ‡ @z 1 ÿ 2

…12†

and ˆG

  @ux @uz ‡ ; @z @x

…13†

where ux and uz are the components of soil displacements (u) in the x- and z-directions, respectively, x0 and z0 are the e€ective normal stresses in the x- and z-directions, respectively, and  is the shear stress. It is noted that the compressive stress is taken as negative sign in this study. 2.2. Boundary conditions Appropriate boundary conditions are required to solve the above governing Eqs. (1), (2) and (3). For a porous ¯ow in a seabed, the boundary conditions at the impermeable rigid bottom requires that the dynamic ¯uctuations of all the physical quantities vanish, i.e. @p ˆ 0 at z ˆ ÿh @z u ˆ w ˆ p ˆ 0 as z ! ÿ1

uˆwˆ

…14†

At the seabed surface, the vertical e€ective stresses and shear stress vanish. The ¯uid pressure is transmitted continuously from the sea to the pores in the seabed, z0 ˆ  ˆ 0 and p ˆ

w H cos…k0 x ÿ wt† ˆ po Refei…ko xÿwt† g; 2coshld

…15†

where k0 (=2/L, L is the wavelength) is the wave number, ! (=2/T, T is the wave period) is the wave frequency, and H is the wave height. 3. General solutions In general, the mechanism of the wave-induced seabed response can be classi®ed into two categories, depending upon how the pore water pressure is generated [27]. One is caused by the residual or progressive nature of the excess pore pressure, which appears in the initial stage of cyclic loading. This type of soil response is similar to that induced by earthquakes, caused by the build-up of the excess pore pressure. The other, generated by the transient or oscillatory excess pore pressure, accompanied by the damping of amplitude in the pore pressure, appears as a periodic response to each wave. The residual soil response is normally important for a soft

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seabed (such as ®ne sand and clay, etc.), while the transient soil response is important for a non-cohesive sandy bed [28]. We focus on the transient soil response in this paper. Since the transient soil oscillatory response under harmonic wave loading will be spatially and temporally periodic, the wave-induced soil response can be expressed as [8] 9 8 9 8 P…z† > p > > > > > > > > > > > > > = < Sc …z† > = < s > w ˆ po Wc …z†  ei…k0 xÿ!t† ; …16† > > > > > > > > …z† S > > > > > > > ; : s ; : s> w Ws …z† where the subscripts c and s denote the compressive elastic wave components and shear wave components, respectively. The various functions in braces on the right side of Eq. (16) are de®ned in the Appendix. Following the procedure proposed by Yuhi and Ishida [23], the governing Eqs. (6)±(10), together with the boundary conditions (14) and (15) are solved for the wave-induced soil displacements, pore pressure and pore ¯uid displacements (details are given in the Appendix)  ÿ
ÿ
ux ˆ po ik0 a1 eik1 z ‡ a2 eÿik1 z ‡ a3 eik2 z ‡ a4 eÿik2 z ÿ ik3 a5 eik3 z ÿ a6 eÿik3 z ei…k0 xÿ!t† ; …17†  ÿ
uz ˆ po ik1 a1 eik1 z ÿ a2 eÿik1 z ‡ ik2 …a3 eik2 z ÿ a4 eÿik2 z † ‡ ik0 …a5 eik3 z ‡ a6 eÿik3 z † ei…k0 xÿ!t† ;

 ÿ
ÿ
p ˆ po 1 a1 eik1 z ‡ a2 eÿik1 z ‡ 2 a3 eik2 z ‡ a4 eÿik2 z ei…k0 xÿ!t† ; n   wx ˆ po ik0 1 …a1 eik1 z ‡ a2 eÿik1 z † ‡ 2 …a3 eik2 z ‡ a4 eÿik2 z o ÿ ik3 3 …a5 eik3 z ÿ a6 eÿik3 z † ei…k0 xÿ!t† ; n wz ˆ po ik1 1 …a1 eik1 z ÿ a2 eÿik1 z † ‡ ik2 2 …a3 eik2 z ÿ a4 eÿik2 z † ÿ ik0 3 …a5 eik3 z ‡ a6 eÿik3 z †gei…k0 xÿ!t† ; where k21 ˆ ÿk20 ÿ

k22 ˆ ÿk20 ÿ

d2 ‡

q d22 ÿ 4d1 d3

2d1 q d2 ÿ d22 ÿ 4d1 d3 2d1

…18†

…19† …20†

…21†

…22†

…23†

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k23 ˆ ÿk20 ‡

 !2 ine e ‡ …1 ÿ ne †vs ; G…v ‡ ine †

e ˆ ne w ‡ …1 ÿ ne †s ; v ˆ

w !k ;

w

…24† …25†

where the constants di (i=1±3) are given in the Appendix. In Eqs. (17)±(21), the six ai coecients can be obtained by numerical methods. Once the ai coecients are obtained. the wave-induced pore pressure and displacements can be obtained by inserting the coecients into (17)±(21), then the e€ective normal stresses and shear stresses can be calculated from (11)±(13). For a special case with in®nite seabed, ai (i=2,4, 6) are zeros. Thus, S1 …z† ˆ a1 eik1 z ‡ a3 eik2 z ;

…26†

W1 …z† ˆ 1 a1 eik1 z ‡ 2 a2 eik2 z ;

…27†

P…z† ˆ 1 a1 eik1 z ‡ 2 a2 eik2 z ;

…28†

S2 …z† ˆ a5 eik3 z

…29†

W2 …z† ˆ 3 a5 eik3 z

…30†

where ai coecients are given as a1 ˆ ÿ a3 ˆ

2 ; 1 2 ÿ 2 1

…31†

1 ; 1 2 ÿ 2 1

a5 ˆ ÿ

…32†

2ik0 …k1 2 ÿ k2 1 † ; …k23 ‡ k20 †…1 2 ÿ 2 1 †

…33†

in which 1 ˆ k21 ÿ …k21 ‡ k20 † ÿ

2k2 k1 k3 …1 ÿ 2† ; k23 ‡ k20

…34†

2 ˆ k22 ÿ …k23 ‡ k2 † ÿ

2k2 k2 k3 …1 ÿ 2† ; k23 ‡ k2

…35†

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Substituting (26)±(30) into (5), we have the expressions for displacements of pore ¯uid and soil skeleton as  ux ˆ po ik0 …a1 eik1 z ‡ a3 eik2 z † ÿ ik3 a5 eik3 z  ei…k0 xÿ!t† ;

…36†

 uz ˆ po k1 a1 eik1 z ‡ k2 a3 eik2 z ‡ ik0 a5 eik3 z  ei…k0 xÿ!t† ;

…37†

 wx ˆ po ik0 …1 a1 eik1 z ‡ 2 a3 eik2 z † ÿ k3 3 a3 eik3 z  ei…k0 xÿ!t†

…38†

 wz ˆ po 1 k1 a1 eik1 z ‡ 3 k2 a3 eik2 z ‡ ik0 3 a5 eik3 z  ei…k0 xÿ!t† ;

…39†

4. Numerical results and discussions As mentioned previously, Yamamoto [20] proposed another dynamic solution with some experimental work. It is necessary to perform a comparison between two solutions as well as veri®cation with experimental data (Section 4.1). Based on the analytical solution presented previously, the in¯uences of dynamic terms on the wave-induced seabed response will be investigated in Section 4.2. Then, the wavedriven seepage ¯ux at the water-sediment interface will be discussed in Section 4.3. The input data for the parametric study are tabulated in Table 1.

Table 1 Parameter values used for the case study Wave characteristics Wave period, T Wave depth, d Wave height, H Soil characteristics Seabed thickness h Poisson's ratio,  Porosity, e Degree of saturation, S Density of soil skeleton, s Density of pore water, w Soil conductivity, k Shear modulus, G

15 s or various 20 m or various 2m 20 m or various 1/3 0.35 98% or various 2650 kg/m3 1000 kg/m3 10ÿ1 m/s (gravel) 10ÿ2 m/s (coarse sand) 10ÿ4 m/s (®ne sand) 5107 N/m2 (gravel) 107 N/m2 (coarse sand) 5106 N/m2 (®ne sand)

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4.1. Comparison with Yamamoto (1983) The major di€erence between Yamamoto [20] and the present solution is that Yamamoto [20] considered the coulomb-damping friction in his model, which is important for the evaluation of pore pressure in clay. Thus, Yamamoto's model [20] can be applied to the softer seabed such as clay, while the present model is applicable to the sandy seabed. However, Yamamoto's work [20] has been limited to the fact that some of their parameters cannot be directly determined from experiments, which comes with some assumed values. In the present model, all parameters are arti®cial no much diculties to be determined in laboratory experiments. this is the strength of the present solution. Based on the input data provided in Yamamoto [20], results from both Yamamoto [20] and the present solution are presented in Fig. 2. The experimental data is also included in the ®gure. Since Yamamoto's solution [20] is only for in®nite seabed, we further extend his model to ®nite thickness following the same framework outlined in his paper. As seen in the ®gure, the present solution overall agree with Yamamoto's solution and experimental data [20] in ®ne sand. Since the present model is not applicable to the seabed of clay, it is meaningless to have a comparison for such a case.

Fig. 2. Comparison of Yamamoto's solution and the present solution. The Present solution in solid lines, Yamamoto's solution in dashed lines, and experimental data in symbols (input data: d=0.9 m, h=0.5 m, =0.333, ne =0.4, S=0.98, k=10ÿ4 m/s, G=106N/m2).

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4.2. E€ects of dynamic terms Referring to the governing equations for the porous ¯ow, (1)±(3), the major differences between the present dynamic model and previous quasi-static model are the pore ¯uid and soil skeleton acceleration terms. It is of interest to examine the e€ects

Fig. 3. Vertical distribution of (a) pore pressure (|p|/po), (b) vertical e€ective normal stress (| z0 |/po) and (c) shear stress (/po) versus burial depth (z/L) in a gravelled bed for various seabed thickness (h/L). The present (dynamic) solution in solid lines, and the quasi-static solution in dashed lines.

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of these additional dynamic terms on the wave-induced soil response, including pore pressure, e€ective normal stresses, shear stress and soil displacements. In this sections, we consider the e€ects of seabed thickness (in terms of h/L), degree of saturation (S) and soil types (gravel, coarse sand and ®ne sand). 4.2.1. Seabed thickness The thickness of a seabed is one of dominant factors in the analysis of waveinduced seabed response [13]. It is of interest to examine the the in¯uence of seabed thickness on di€erence of the wave-induced soil response obtained from quasi-static and dynamic solution. Based on input data given in Table 1, Figs. 3±7 illustrate the vertical distribution of the wave-induced seabed response (including pore pressure, vertical e€ective normal stress and shear stress), and soil and pore ¯uid displacements versus burial depth (z/L) for various seabed thickness in three di€erent seabed (coarse sand, ®ne sand and gravelled seabed).

Fig. 4. Vertical distribution of soil and pore ¯uid displacements (a) 2Gko|ux|/po and 2Gko|wx|/10 po, and (e) 2Gko|uz|/po and 2Gko|wz|/10 po versus burial depth (z/L) in gravelled bed for various seabed thickness h/L). Solid lines are for soil displacements, and dashed lines are pore ¯uid displacements.

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As seen in Figs. 3(a), 5(a) and 7(a), the wave-induced pore pressure (|p|/po) is almost una€ected by dynamic terms [i.e. acceleration terms in Eqs. (1)±(3)]. However, the relative di€erence of the wave-induced vertical e€ective normal stress (| z0 |/po) and shear stress (||/po) between the present dynamic solution (in

Fig. 5. Vertical distribution of (a) pore pressure (|p|/po), (b) vertical e€ective normal stress (| z0 |/po) and (c) shear stress (/po) versus burial depth (z/L) in coarse sand for various seabed thickness (h/L). The present (dynamic) solution in solid lines, and the quasi-static solution in dashed lines.

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solid lines) and quasi-static solution (in dashed lines) is observed from Figs. 3(b) and (c), 5(b) and (c) and 7(b) and (c). It is found that the relative di€erence of | z0 |/po and ||/po between two solutions increases as the seabed thickness decreases. This implies that the acceleration term is more important in a shallow soil depth under the same wave conditions. Figs. 4, 6 and 8 present the distribution of the waveinduced soil and pore ¯uid displacements for various seabed thickness. In the ®gures, the solid lines denote the results of the wave-induced soil displacements, and dashed lines represent the pore ¯uid displacements. it is noted that the wave-induced pore ¯uid displacements cannot be obtained from quasi-static solution [8,13], because the acceleration terms were not included. Although the author's recent work [24] considered the inertial e€ects on the wave-induced soil response, it is only the acceleration term generated from solid components, not from pore ¯uid components. Fig. 4 indicates that the pore ¯uid displacements is about 10 times soil

Fig. 6. Vertical distribution of soil and pore ¯uid displacements (a) 2Gko|ux|/po and 2Gko|wx|/po, and (b) 2Gko|uz|/po and 2Gko|wz|/po versus burial depth (z/L) in coarse sand for various seabed thickness (h/L). S lid lines are for soil displacements, and dashed lines are pore ¯uid displacements.

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displacements in a gravelled seabed, while they are at the same order in both coarse sand ®ne sand (Figs. 6 and 8: both solid lines and dashed lines are identical). Thus, the dynamic soil behaviour cannot always be ignored in the evaluation of displacements.

Fig. 7. Vertical distribution of (a) pore pressure (|p|/po), (b) vertical e€ective normal stress (| z0 |/po) and (c) shear stress (/po) versus burial depth (z/L) in ®ne sand for various seabed thickness (h/L). The present (dynamic) solution in solid lines, and the quasi-static solution in dashed lines.

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4.2.2. Degree of saturation Gas is common to be observed within marine sediments. The degree of saturation is de®ned by the ratio of volume of gas and total volume of pore. It has been reported that the degree of saturation plays an important role in the evaluation of the wave-induced transient soil response [13,30]. It is of interest to examine the in¯uence of degree of saturation on the relative di€erences between dynamic and quasi-static solutions. Figs. 9, 11 and 13 illustrate the distribution of wave-induced soil response for various degree of saturation in di€erent seabeds. General speaking, the e€ects of dynamic soil behaviour on the wave-induced pore pressure will increases as the degree of saturation S decreases in gravelled seabed. However, its in¯uence on vertical e€ective normal stress and shear stress will increases as S increases. For coarse sand and ®ne sand, the in¯uence of dynamic soil behaviour is insigni®cant.

Fig. 8. Vertical distribution of soil and pore ¯uid displacements (a) 2Gko|ux|/po and 2Gko|wx|/po, and (b) 2Gko|uz|/po and 2Gko|wz|/10 po versus burial depth (z/L) in ®ne sand for various seabed thickness (h/L). Solid lines are for soil displacements, and dashed lines are pore ¯uid displacements.

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The wave-induced pore ¯uid and soil displacements decreases as the degree of saturation decreases, as seen in Figs. 10, 12 and 14. In general, the in¯uence of degree of saturation on the pore ¯uid displacement is more signi®cant than on soil displacements, especially in the vertical directions.

Fig. 9. Vertical distribution of (a) pore pressure (|p|/po), (b) vertical e€ective normal stress (| z0 |/po) and (c) shear stress (/po) versus burial depth (z/L) in a gravelled bed for various degrees of saturation (S). The present (dynamic) solution in solid lines, and the quasi-static solution in dashed lines.

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4.2.3. Soil types Soil types (in terms of di€erent permeability and shear modulus) is another key factor which directly a€ects the wave-induced seabed response. In this study, we consider three di€erent materials: coarse sand, ®ne sand and gravelled seabed. The soil characteristics of di€erent materials are given in Table 1. Comparing Figs. 3±14, it can be found that the dynamic soil behaviour in the evaluation of the wave-induced seabed response in a gravelled seabed is more important than that in sandy bed. Furthermore, the wave-induced pore ¯uid displacements in gravelled is about ten times of that in a sandy seabed. 4.3. Wave driven seepage ¯ow Water wave over a porous seabed drives a seepage ¯ux into and out of the sediment. As noted above, the volume of ¯uid exchanged per wave cycle directly relates to the mass transport rate of contaminants in the sediment, an important quantity in water quality modelling.

Fig. 10. Vertical distribution of soil and pore ¯uid displacements (a) 2Gko|ux|/po and 2Gko|wx|/10 po, and (b) 2Gko|uz|/po and 2Gko|wz|/10 po versus burial depth (z/L) in gravelled bed for various degrees of saturation (S). Solid lines are for soil displacements, and dashed lines are pore ¯uid displacements.

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117

The net seepage ¯ux over one wave cycle is zero. However, for the mass transport caused by the cyclic wave motion, the relevant quantity is the volume of water pumped into over one-half wave period (T) and one-half wavelength (L). This volume (per unit width of sea bottom) can be expressed as a function of depth (z) [3]

Fig. 11. Vertical distribution of (a) pore pressure (|p|/po), (b) vertical e€ective normal stress (| z0 |/po) and (c) shear stress (/po) versus burial depth (z/L) in coarse sand for various degrees of saturation (S). The present (dynamic) solution in solid lines, and the quasi-static solution in dashed lines.

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… T=4 … L=4 V…z† ˆ ÿ ÿT=4

k @p dxdt ÿL=4 w @z

…40†

In this study, we will focus on the volume of seepage exchange between seawater and seabed driven by the water waves, that is, the volume exchange at the water and sediment interface, Vo=V(0). 4.3.1. E€ects of soil parameters In this section, we will examine the e€ects of three common soil parameters (degree of saturation, soil types and seabed thickness) on the wave-driven seepage ¯ux at the water±sediment interface. Fig. 15 illustrates the distribution of the seepage ¯ux at the water-sediment interface (Vo) versus wave period for various seabed thickness (h/d) in di€erent seabeds. In general, the seepage ¯ux Vo increases as the seabed thickness increases in gravelled and coarse sandy beds [Fig. 15(a) and (b)]. However, two trends have been observed

Fig. 12. Vertical distribution of soil and pore ¯uid displacements (a) 2Gko|ux|/po and 2Gko|wx|/po, and (e) 2Gko|uz|/po and 2Gko|wz|/po versus burial depth (z/L) in coarse sand for various degrees of saturation (S) Solid lines are for soil displacements, and dashed lines are pore ¯uid displacements.

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119

in ®ne sand, those are: h/d=0.5, 1.0 and h/d=2.0, 3.0. For a shorter wave period (T49 s), the maximum Vo occurs at h/d=1.0. For a longer wave period (T514 s), the maximum Vo occurs at h/d=3.0. However, the seepage ¯ux in a ®ne sand is much smaller than in other materials.

Fig. 13. Vertical distribution of (a) pore pressure (|p|/po), (b) vertical e€ective normal stress (| z0 |/po) and (c) shear stress (/po) versus burial depth (z/L) in ®ne sand for various degrees of saturation (S). The present (dynamic) solution in solid lines, and the quasi-static solution in dashed lines.

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In reality, the marine sediment is not completely saturated. Practically, the decomposition of organic materials can produce methane bubbles in the pore water. Since our interest lies in the pollution of estuaries and near-shore water, the existence of gas in the sediment is common. When the pore water is not completely saturated, the compressibility of gas±water continuum increases drastically [29]. The ¯ux into the sediment will then be a€ected. The in¯uences of the degree of saturation on the seepage ¯ux at water±sediment interface (Vo) is signi®cant as seen in Fig. 16. The seepage ¯ux (Vo) increases as the degree of saturation decreases. Soil type (in terms of soil permeability and shear modulus) is another important parameter in the evaluation of the wave-induced seepage ¯ux. Increasing soil permeability generally enhances the transfer rate of water between seawater and sediments. Comparing Figs. 15 and 16, it can be concluded that the seepage ¯ux in gravelled seabed is much larger than that in coarse sand. For example, Vo in gravelled is about 10 times of that in coarse sand, and Vo in coarse sand is about 5 times that in ®ne sand.

Fig. 14. Vertical distribution of soil and pore ¯uid displacements (a) 2Gko|ux|/po and 2Gko|wx|/po, and (e) 2Gko juz j=po and 2Gko jwz j=10po versus burial depth (z/L) in ®ne sand for various degrees of saturation (S). Solid lines are for soil displacements, and dashed lines are pore ¯uid displacements.

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121

4.3.2. E€ects of wave parameters It is well known that the wavelength is determined by wave period (T) and water depth (d) through wave dispersion relation [31]. The wave period in ocean environments ranges from 5 to 15 s for gravity ocean waves [32]. For shallow water, the

Fig. 15. Distribution of the wave driven seepage ¯ux at the water±sediment interface Vo versus wave period (T) in (a) gravelled seabed, (b) coarse sand and (c) ®ne sand for various seabed thickness (h/d).

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relative water depth (d/L, L=1.56 T2 tanh kd is the wavelength of deep water) is less than 0.1, while d/L50.5 de®nes deep water. In this section, we examine the e€ects of relative water depth and wave period on the volume of water exchange at the seabed surface (Vo) in di€erent materials.

Fig. 16. Distribution of the wave driven seepage ¯ux at the water±sediment interface Vo versus wave period (T) in (a) gravelled seabed, (b) coarse sand and (c) ®ne sand for various degrees of saturation (S).

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123

Fig. 17 shows the in¯uences of water depth on the wave driven seepage ¯ux (Vo). Basically, seepage ¯ux increases as water depth d/L increases. This implies that the seepage ¯ux will be larger in shallow water than in deep water. Also, comparing Figs. 15±17, it can be found that Vo increases as wave period T increases, implies the action of a longer wave will enhance the seepage ¯ux.

Fig. 17. Distribution of the wave driven seepage ¯ux at the water±sediment interface Vo versus wave period (T) in (a) gravelled seabed, (b) coarse sand and (c) ®ne sand for various relative water depth (d/L).

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5. Conclusions The aims of this paper are to derive an analytical solution for dynamics response in a porous seabed coupling the wave motion and porous medium ¯ow. Based on the solution, the wave-driven seepage ¯ux into and out of the marine sediment is investigated. From the numerical examples presented, the following summarising points can be made: 1. A comparison between Yamamoto's solution [20], the present solution and experimental data has been performed. The present solution overall agree with Yamamoto's solution and experimental data in ®ne sand. 2. The dynamic soil behaviour is important in the analysis of the wave-induced e€ective stress, shear stress and displacements in a gravelled seabed than sandy beds, and shallow soil depth. The wave-induced pore ¯uid displacements is about ten times of soil displacements in gravelled seabed. 3. The magnitude of the seepage ¯ux, Vo, is a€ected by the soil type, degree of saturation. In general, Vo is larger in gravelled seabed than in coarse and ®ne sands. Vo increases as degree of saturation decreases. Also, Vo increases as seabed thickness increases. 4. The wave driven seepage ¯ux is also a€ected by wave periods and water depth. Vo is larger in either shallow water or under the action of longer wave period waves. Appendix. Mathematical derivation of closed-form solution for porous ¯ow The mathematical derivation of Eqs. (17)±(21) is given in this appendix. Firstly, substituting (16) into the governing Eqs. (6)±(8), we have a characteristics equation for the compressive waves as d1

@4 Sc …z† @2 Sc …z† ‡ d2 ‡ d3 Sc …z† ˆ 0 4 @z @z2

…A1†

where d1 ˆ ÿ

2iGk!ne …1 ÿ † ;

w …1 ÿ 2†

…A2†

  s 2 d2 ˆ ne ! ‰1 ‡ 2m…1 ÿ †Š ÿ iv! 2m…1 ÿ † ‡ …1 ÿ ne † ‡ ne …1 ÿ ne † ; w 2

2

d3 ˆ ne !4 ‰ne e ÿ iv…1 ÿ ne †s Š; in which e and  are given in Eq. (25), and m is de®ned by

…A3†

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mˆ

Gne 1 ÿ 2

125

…A4†

In this case, the characteristic equation is in quadratic form, from which two pairs of complex solutions, k1 and k2 [see Eqs. (22) and (23)], can be obtained. Then, the general forms of p, Fs and Fw for a seabed of in®nite thickness can be speci®ed as ik1 z ÿik1 z ik2 z ÿik2 z P…z† ˆ P…‡† ‡ P…ÿ† ‡ P…ÿ† ‡ P…ÿ† ; 1 e 1 e 2 e 1 e

…A5†

ik1 z ÿik1 z ik2 z ÿik2 z ‡ S…ÿ† ‡ S…‡† ; ‡S…ÿ† Sc …z† ˆ S…‡† 1 e 1 e 2 e 2 e

…A6†

ik1 z ÿik1 z ik2 z ÿik2 z ‡ W…ÿ† ‡ W…‡† ‡ W…ÿ† ; Wc …z† ˆ W…‡† 1 e 1 e 2 e 2 e

…A7†

where k1 and k2 correspond to Biot [10] ®rst and send kind of compressive waves. Similarly, we have the characteristic equation for shear waves arising from (9) and (10) as @2 Ss …z† ÿ k23 Ss …z† ˆ 0 @z2

…A8†

where the wave number of shear waves k3 is given by Eq. (24). Thus, we have ik3 z ÿik3 z ‡ S…ÿ† ; Ss …z† ˆ S…‡† 3 e 3 e ik3 z ÿik3 z ‡ W…ÿ† ; Ws …z† ˆ W…‡† 3 e 3 e

…A9† …A10†

The coecients Pi, Si and Wi in Eqs. (A5)±(A7) and (A9)±(A10) are not independent, and the relations can be determined by substituting these equations into (6)± (10). As a result, Wi and Pi can be expressed in terms of Si as …† …† …† P…† 1 ˆ 1 S1 ; P2 ˆ 2 S2 ; …† …† …† …† …† W…† 1 ˆ 1 S1 ; W2 ˆ 2 S2 ; W3 ˆ 3 S3

…A11†

where 1 ˆ ne w !2 1 ‡

2G…1 ÿ † 2 …k1 ÿ k20 † ‡ …1 ÿ ne †s !2 ; 1 ÿ 2

…A12†

2 ˆ ne w !2 2 ‡

2G…1 ÿ † 2 …k2 ÿ k20 † ‡ …1 ÿ ne †s !2 ; 1 ÿ 2

…A13†

1 ˆ

ine ! ‡

2Gk…1 ÿ † 2 s …k ÿ k20 † ‡ …1 ÿ ne †! w

w …1 ÿ 2† 1  ; ! ine ‡ …1 ÿ ne †

…A14†

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2 ˆ 3 ˆ

ine ! ‡

2Gk…1 ÿ † 2 s …k2 ÿ k20 † ‡ …1 ÿ ne †! w

w …1 ÿ 2†  ! ine ‡ …1 ÿ ne †

ine ; ine ‡ 

…A15† …A16†

Then we have the displacements of soil and pore ¯uid as @s @ ÿ @x @z  ˆ po ik0 …a1 ek1 z ‡ a2 eÿk1 z ‡ a3 ek2 z ‡ a4 eÿk2 z † ÿ k3 …a5 ek3 z ÿ a6 eÿk3 z † ei…k0 xÿ!t† ;

ux ˆ

…A17† @s @ s ‡ @z @x  ˆ po k1 …a1 ek1 z ÿ a2 eÿk1 z † ‡ k2 …a3 ek2 z ÿ a4 eÿk2 z † ‡ ik0 …a5 ek3 z ‡ a6 eÿk3 z † ei…k0 xÿ!t† ;

uz ˆ

…A18† @w @ w ÿ @xn @z  ÿ k1 z
ÿ
 ˆ po ik0 1 a1 e ‡ a2 eÿk1 z ‡ 2 a3 ek2 z ‡ a4 eÿk2 z ÿ
o ÿ k3 3 a5 ek3 z ÿ a6 eÿk3 z ei…k0 xÿ!t† ;

…A19†

@w @ w ‡ n@z ÿ @x
ÿ
ˆ po k1 1 a1 ek1 z ÿ a2 eÿk1 z ‡ k2 2 a3 ek2 z ÿ a4 eÿk2 z ÿ
o ÿ ik0 3 a5 ek3 z ‡ a6 eÿk3 z ei…k0 xÿ!t† ;

…A20†

wx ˆ

wz ˆˆ

Now, we have six unknowns (coecients Si) with six boundary conditions (14) and (15). This leads to the coecient matrix as 32 3 2 3 2 A1 A1 A2 A2 A3 A3 a1 0 ÿ
ÿ
6 2ik k 2 2 2 2 76 6 7 ÿ2ik k 2ik k ÿ2ik k ÿ k ‡ k ‡ k ÿ k 0 1 0 1 0 2 0 2 6 07 0 3 0 3 76 a2 7 7 76 7 6 6 7 6 7 6 1    0 0 6 7 1 2 2 6 p a 76 4 7 6 o 7 6 7 ˆ …‡† …ÿ† …‡† …ÿ† …‡† 7 6 ik e…ÿ† 7 6 7 ÿik0 e1 ik0 e2 ÿik0 e2 ik0 e3 ik0 e3 76 6 0 1 6 a3 7 6 0 7 7 6 7 6 7 76 6 k e…ÿ† ÿk1 e…ÿ† k2 e…ÿ† ÿk2 e…‡† k3 e…ÿ† ÿk3 e…‡† 54 a5 5 4 0 5 4 1 1 1 2 2 3 3 a6 0 1 k1 e…ÿ† ÿ1 k1 e…‡† 2 k2 e…ÿ† ÿ2 k2 e…‡† 0 0 1 1 2 2 …A21†

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127

where Ai ˆ …1 ÿ †k2i ÿ k0 ; i ˆ 1; 2

…A22†

A3 ˆ ik0 k3 …1 ÿ 2†

…A23†

ˆ eiki h e…† i

…A24†

We solve the above linear equations by numerical method. Then, the waveinduced soil response can be obtained. References [1] Lundgren H, Lindhardt JHC, Romhild CJ. Stability of breakwaters on poor foundation. In: Proceedings 12th International Conference on Soil Mechanics and Foundation Engineering, 1989. p. 451±4. [2] Silverter R, Hsu JRC. Sines revisited. Journal of Waterways, Port, Coastal and Ocean Engineering. ASCE 1989;115(3):327±44. [3] Mu Y, Cheng AHD, Badiey M, Bennett R. Water wave driven seepage in sediment and parameter inversion based on pore pressure data. International Journal for Numerical and Analytical Methods in Geomechanics 1999;23:1655±74. [4] Oldham CE, Lavery PS. Porewater nutrient ¯uxes in a shallow fetch-limited estuary. Marine Ecology Progress Series 1999;183:39±47. [5] Liu PLF. Damping of water waves over porous bed. Journal of Hydraulic Division, ASCE 1973;99:2263±71. [6] Moshagen H, Torum A. Wave-induced pressure in permeable seabeds. Journal of Waterway, Harbour, and Coastal Engineering, ASCE 1975;101:49±57. [7] Massel SR. Gravity waves propagated over permeable bottom. Journal of Waterway, Harbour, and Coastal Engineering, ASCE 1976;102:111±21. [8] Madsen OS. Wave-induced pore pressures and e€ective stresses in a porous bed. GeÂotechnique 1978;28:377±93. [9] Yamamoto T, Koning HL, Sellmejier H, Hijum EV. On the response of a poro-elastic bed to water waves. Journal of Fluid Mechanics 1978;87:193±206. [10] Biot MA. Theory of propagation of elastic waves in a ¯uid-saturated porous solid, I. Low frequency range. Journal of the Acoustical Society of America 1956;28:168±78. [11] Badiey M, Jaya I, Magda W, Ricinine W. Analytical and experimental approach in modeling of water±seabed interaction. In: Proceedings of the 2nd International O€shore and Polar Engineering Conference, 1992. p. 398±402. [12] Rahman MS, El-Zahaby K, Booker J. A semi-analytical method for the wave-induced seabed response. International Journal for Numerical and Analytical methods in Geomechanics 1994;18:213±36. [13] Jeng DS. Wave-induced seabed response in front of a breakwater. PhD thesis, The University of Western Australia, 1997. [14] Biot MA. General theory of three-dimensional consolidation. Journal of Applied Physics 1941;12:155±64. [15] Jeng DS. Soil response in cross-anisotropic seabed due to standing waves. Journal of Geotechnical Engineering, ASCE 1997;123(1):9±19. [16] Jeng DS, Seymour BR. Response in seabed of ®nite depth with variable permeability. Journal of Geotechnical and Geoenvironmental Engineering, ASCE 1997;123(1):902±11.

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