Wave-induced response of seabed: Various formulations and their applicability

Applied Ocean Research 31 (2009) 12–24

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Wave-induced response of seabed: Various formulations and their applicability M.B.C. Ulker a , M.S. Rahman a,∗ , D.-S. Jeng b a

Department of Civil Engineering, North Carolina State University Raleigh, NC, USA

b

Division of Civil Engineering University of Dundee, Dundee, UK

article

info

Article history: Received 17 June 2008 Received in revised form 6 January 2009 Accepted 27 March 2009 Available online 9 May 2009 Keywords: Seabed Water wave Formulation Analytical Dynamics Response

abstract In this study, a set of generalized analytical solutions are developed for the wave-induced response of a saturated porous seabed under plane strain condition. When considering the water waves originating in deep water and travelling towards the shore, their velocities, lengths and heights vary. Depending on the characteristics of the wave and the properties of the seabed, different formulations (fully dynamic, partly dynamic, quasi-static) for the wave-induced response of the seabed are possible. The solutions for the response with these formulations are established in terms of non-dimensional parameters. The results are presented in terms of pore pressure, shear stress and vertical effective stress distributions within the seabed. For typical values of wave period and seabed permeability, the regions of applicability of the three formulations are identified and plotted in parametric spaces. With given wave and seabed characteristics, these regions provide quick identification of the appropriate formulation for an adequate evaluation of the wave-induced seabed response. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction The evaluation of the wave-induced response of a porous seabed is a key factor in the design of foundations around marine structures. Depending on the nature of wave loading vis-à-vis the characteristics of the media, different formulations for fully dynamic (FD), partly dynamic (PD) and quasi-static (QS) conditions are possible. In quasi-static analysis, common assumptions of drained or undrained behavior are made depending on the rate of loading and drainage and for intermediate cases a full consolidating behavior is considered. In partly dynamic analysis, the mixed or u-p formulation in which both displacement, u, and pore water pressure, p, are the field variables, is adopted considering the acceleration of soil skeleton. In fully dynamic analysis, the formulation is further complicated by the presence of inertial terms associated with both the motion of soil skeleton and that of the pore water. Three formulations are summarized as:

• Fully dynamic: In this case, the coupled equations of flow and deformation are formulated including both the acceleration of soil skeleton and the acceleration of pore water relative to that of soil skeleton.

∗ Corresponding address: North Carolina State University, Department of Civil Engineering, Stinson Dr. Mann Hall, 27695, Raleigh, NC, United States. Tel.: +1 919 515 7633; fax: +1 919 515 7908. E-mail address: [email protected] (M.S. Rahman). 0141-1187/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.apor.2009.03.003

• Partly dynamic: In this case, the coupled equations of flow and deformation are formulated considering only the acceleration of soil skeleton and not that of pore water. This is also called u-p dynamic form in which soil displacement, u, and pore water pressure, p are used as field variables. • Quasi-static: Here, both inertial terms (associated with soil skeleton and pore water) are ignored resulting in quasi-static coupled flow and deformation formulation. The equations governing the response of saturated porous media incorporating the fluid-solid skeleton interaction were first established for quasi-static case by Biot [1] who then extended them to include dynamics [2,3]. Truesdell [4,5] introduced ‘‘mixture theory’’ to formulate this case which provided a new basis for such coupled equations. The wave-induced stresses and pore pressure in a porous seabed have been studied by various authors. Early studies were based on the assumption of incompressible pore-fluid and soil skeleton as well as the flow in porous seabed governed by Darcy’s law [6–8]. Madsen [9] and Yamamoto et al. [10] were the first to develop analytical solutions for the problem using Biot’s theory of coupled flow and deformation. Okusa [11] has developed a simple analytical solution for wave-induced stresses in unsaturated sediments. Zienkiewicz et al. [12] analytically investigated one-dimensional (1D) situation and studied the response of saturated porous media with all the above three formulations and examined the ranges of their validity in a non-dimensional parametric space. Later, more complete formulations of the problem, including the waveseabed interactions, the inertial effects and vertical variation of

M.B.C. Ulker et al. / Applied Ocean Research 31 (2009) 12–24

13

L = Shallow water wavelength

L0 = Deep water wavelength

η(x,t) = H ei(kx–ωt) 2

H0

H Shallow water wave height

Deep water wave height

d = Water depth

q(x,t) = +z

ρwgH 2cosh (kd)

ei(kx–ωt)

-z

h = Seabed thickness

Fig. 1. Water waves in deep and shallow water.

soil properties, were developed by Yamamoto [13] and Yamamoto and Schuckman [14]. Mei and Foda [15] studied the waveinduced response by developing a general approximation and the associated existence of a boundary layer system. Rahman [16] studied the mechanism and described the conditions leading to wave-induced instability of seabed. Later, Rahman et al. [17] developed a semi-analytical approach for the wave-induced seabed response for a layered seabed. Jeng et al. [18] determined the effects of inertia forces on wave-induced seabed response and later Cha et al. [19] and Jeng and Cha [20] have obtained a two dimensional analytical solution for porous seabed and also studied the effect of inertial terms on the wave-induced pore pressures and stresses in the seabed. In this study, the generalized two dimensional (2D) analytical solutions for the wave-induced seabed response (for the three formulations) are developed for plane strain case in terms of non-dimensional parameters. Then the regions of applicability of the formulations are plotted and identified in parametric spaces.

In this study, we consider waves that originate in deep water and travel toward the shore during which length, velocity and height vary, (Fig. 1). The sea surface elevation of a wave field at a fixed point may be represented by, H i(kx−ωt ) e . (1) 2 In deep water, wavelength, L0 , depends on the wave period, T , through,

η ( x, t ) =

gT 2

.

(2)

2π Here, we assume that the wave period does not change as the wave propagates over the surface of water. Hence, for typical periods of T = 5 s, 10 s, 15 s and 20 s, we calculate the deep water wave height, H0 using the relation for deep water steepness from the range, 0.008 ≤ HL o ≤ 0.055 proposed by Horikawa [21]. o An average steepness value is considered here. The wavelength in shallow water, L, is then calculated from,

 L = L0 tanh

2π d L



,

"( H = HO

1+

(3)

which is obtained from the dispersion relationship, ω2 = gk tanh (kd) of the linear wave theory. Here k is the wave number

4π d L sinh 4πL d

)




tanh

2π d L

#−1/2

,

(4)

which is obtained from the consideration of energy conservation. As the depth ratio decreases, the shallow water wave steepness, H /L increase until it reaches a critical value at which the wave breaking occurs. Following criterion developed by Wiegel [22] is used for wave breaking, H

1



2π d



. (5) L 7 L When water waves propagate over the seabed, they exert a hydrodynamic pressure which for linear waves can be written as, ≤

tanh

q (x, t ) =

2. Water waves propagating on the surface of seabed

L0 =

and d is the water depth for which the corresponding depth ratios, d/L, satisfying the wavelengths from (3) are obtained. Then wave heights in the shallow water regime are calculated from the relation,

ρw gH ei(kx−ωt ) . 2 cosh (kd)

(6)

In summary, it should be noted that for all analyses presented in later sections, we choose a wave period (which fixes a deep water wavelength according to Eq. (2)) and deep-water wave steepness (from a typical range of observed values), and then propagate the wave into shallow water in which the wave steepness ‘H /L’ is uniquely defined by normalized water depth ‘d/L’ (see Eqs. (3) and (4)) and is restricted to non-breaking wave Eq. (5). 3. Dynamic response of seabed: General formulation The equations governing the general dynamic response of saturated porous seabed are presented in this section which follows the formulation by Zienkiewicz et al. [12]. The overall equilibrium for a unit total volume of soil can be written as,

¨¯ i . σij,j + ρ gi = ρ u¨ i + ρf w

(7)

Considering the flow to be governed by Darcy’s law, the equation for the equilibrium of fluid phase can be written as,

− p,i + ρf gi = ρf u¨ i +

¨¯ i ρf w

+

ρf gi ˙ w ¯ i.

(8) n ki Finally the continuity condition due to mass balance can be stated as, ˙¯ i,i = − n p˙ . u˙ i,i + w (9) Kf

M.B.C. Ulker et al. / Applied Ocean Research 31 (2009) 12–24

Kf =

ρw gdKw , ρw gd + Kw (1 − S )

σij = σij0 − δij p,

(11)

where σij is effective stress and δij is Kronecker delta. Here tension is taken as positive. For the constitutive behavior of the soil, a linear elastic relationship is assumed and for a soil under plane strain condition, stress–strain relationship in terms of Lame’s parameters λ and G, is written as, 0

σij0 = λεkk δij + 2Gεij .

(12)

Here, εkk and εij are the volumetric and deviatoric components of strain, respectively. The Eqs. (7)–(9) and (11), (12) provide the complete formulation for the coupled flow and deformation response in a saturated medium subjected to a general dynamic loading. 4. Analytical solution In this section, an analytical solution to the wave-induced response of seabed at a site in a localized region (Fig. 2) is presented. From (9) we can write: ui,i + w ¯ i ,i

n


i

= −p,i ,

(13)

Kf

ui,i + w ¯ i ,i

n

σij0,j +

Kf n


i

= ρf u¨ i +

ui,i + w ¯ i ,i


i

¨¯ i ρf w n

ρf gi ˙ + w ¯ i,

-z

Saturated porous seabed

h

Fig. 2. Two dimensional seabed under harmonic wave load.

by Zienkiewicz et al. [12]. Considering all the response variables to be of the form, u (x, z , t ) = U (z ) ei(kx−ωt ) , we can obtain a harmonic complex form of the governing equations (see Appendix) where U (z ) represent the amplitudes of the response variables which in this case are the solid and relative fluid displacements ¯ x, W ¯ z . The equations, in terms of in x and z directions, Ux , Uz , W the parameters defined above form a linear system that can be written in a matrix form as in Box I, where DD and DD2 denote, ∂/∂ z¯ and ∂ 2 /∂ z¯ 2 respectively. This matrix yields a characteristic equation of order 6 written as α1 DD6 + α2 DD4 + α3 DD2 + α4 = 0 leading to a linear system of equations with a 6 × 6 matrix of coefficients aj with elements ψij (i, j = 1, 2, 3, 4, 5, 6) in terms of eigenvalues ηi (i = 1, 2, 3, 4, 5, 6) that are obtained from the characteristic equation. In order to solve this linear system, a set of boundary conditions for the physical problem illustrated in Fig. 2 is introduced as; at z = −h, the three displacements go to zero, ¯ z = 0) and at z = 0, vertical effective stress and (Ux = Uz = W shear stress vanish, (σzz0 = τxz = 0) and p = q0 ei(kx−ωt ) where, ρw gH q0 = 2 cosh is the amplitude of the load. This linear system is (kd) readily solved and aj coefficients and ψij are evaluated. Then, the normal and shear stresses as well as pore pressure response are evaluated as,

σxx = 0

" 6  X " 6  X

ikK + bj λ

ikλ + bj K

" 6  X ηj

(14b)

K+ n

as the ratio of the volumetric part of Lame’s constants to the bulk G modulus of the system; κ2 = Kf , as the ratio of the shear K+ n

modulus to the bulk modulus of the system; m = kh, as the spatial variation of the loading where k = 2π /L is the wave number and ρ

k V2

x c h is the layer thickness; β = ρf , as the density ratio; Π1x = g βω , h2 as the ratio of time for pore fluid flow in x direction to time for compression wave to travel, ω being the frequency of load and

as the speed of compression wave; Π1z =

kz Vc2

g βωh2

,

same as Π1x but accounting for the flow in z-direction; Π2 = ω 2 h2

σzz = 0

τxz = G

¨¯ i . = ρ u¨ i + ρf w K /n

ρ

L

x

(14a)

ki

simultaneous solution of (14). These are: κ = K +fK /n , as the ratio of f the bulk modulus of fluid to the bulk modulus of skeleton, K , plus λ the pore fluid and is close to 1 for most of the soils; κ1 = Kf ,

K +Kf /n

k=

2cosh (kd)

-

j =1

A number of non-dimensional parameters are defined for the

Vc2 =

q=

+

j=1

and if we substitute (13) into (8) and use the effective stress relation of (11), we obtain two final coupled equations as:

Vc2

L = wavelength

(10)

where Kw is the bulk modulus of water and d is the water depth. For partly dynamic case, the inertial terms associated with the motion ¨¯ i , are ignored and for the quasi-static case, both of pore water, w ¨¯ i , are neglected. In addition, the effective inertial terms, u¨ i and w stress relation is written as,

Kf

+z

8

In these equations, σij is the total stress, p is the pore water ˙¯ i and w ¨¯ i are pressure, u¨ i is the soil skeleton acceleration, w the average relative water velocity and acceleration, gi is the appropriate component of the gravitational acceleration, ki is the component of permeability, n is the porosity, Kf is the bulk modulus of pore water, ρ is the total density and ρf is the density of the pore water. Kf is related to the saturation, S, through,

8

14

, as the ratio of the rate of dynamic loading to the speed of

compression wave, Vc . Here, κ1 , κ2 , m and Π1x are the new nondimensional parameters in addition to the others defined earlier

j =1

p=−

Kf n

h

" 6  X j =1

ηj  h

ηj  h

# aj e

ηj hz

ei(kx−ωt ) ,

(15)

ei(kx−ωt ) ,

(16)

# ηj hz

aj e

#



+ ikbj aj e




ηj hz

ik 1 + cj +

ηj h

ei(kx−ωt ) ,

bj + dj




(17)

# aj e

ηj hz

ei(kx−ωt ) , (18)

where bj , cj , dj are the elements of the eigenvectors associated with the eigenvalues, ηi of Box I. The closed form solutions for the above response variables (also for the displacements which are not shown here) for simplified formulations (partly dynamic and quasi-static forms) follow the same steps as described above. However, for the simplified cases, the linear system is reduced due to the removal of inertial terms. Besides, the coefficients of the characteristic equation of Box I, and hence the eigenvalues will be different leading to the changed values of ψij and aj . The details of the solution are presented in Appendix. 5. Wave-induced seabed response: Results The results for the wave-induced seabed response are obtained from the analytical solutions presented in the previous section. A parametric study is carried out and the resulting response from the

M.B.C. Ulker et al. / Applied Ocean Research 31 (2009) 12–24

      



β Π2

i

−m κ

β Π2 − m κ

imκ DD

imκ DD

β Π2 + κ DD

imκ DD

im (κ + κ1 + κ2 ) DD
Π2 − m2 κ2 + DD2

β Π2 − m2 κ imκ DD

2

n


2

Π2 − m2 + κ2 DD im (κ + κ1 + κ2 ) DD


2

+

Π1x

2



imκ DD




15



  Ux   i β Π2 2   Uz  + + κ DD   ¯  = 0,  Wx n Π1z  W ¯z imκ DD
2 β Π2 + κ DD

Box I.

three formulations, FD, PD and QS are presented in a set of figures (Figs. 3–7) for various representative seabed and wave parameters. The seabed parameters that affect the response significantly are the permeability (kz ), degree of saturation (S), and the thickness of the seabed (h/L); whereas for the wave loading, wave steepness (H /L) and the wave period (T ) are the dominant factors. The range of numerical values of the parameters used here is presented in Table 1. In this study, the seabed is assumed primarily to be fully saturated (S = 1), however the effect of slight unsaturation on the dynamic response is also investigated. Additionally, the seabed is assumed to be isotropic (in terms of permeability) and the permeability is taken as kz = 0.002 m/s except for Fig. 3 where its effect on the response is presented. From the response presented in Fig. 3, we see that as the permeability increases the FD formulation yields significantly different response especially for pore pressure and effective vertical stress. However, the evaluated response from PD and QS formulations are almost identical. Fig. 4 illustrates the effect of degree of saturation on the seabed response. For only slight unsaturation (i.e. S close to 1), the pore fluid can still be treated as of single phase, and its effect can be captured by simply modifying the fluid compressibility, β¯ , [11,17]. The compressibility of pore fluid, is modified as,

β¯ =

1 Kf

=

1 Kp

+

1−S p0

,

(19)

where p0 represents absolute pore pressure and Kp is the bulk modulus of the fluid itself (in case of water, Kp = Kw ). Jeng and Cha [20] also makes this assumption. Here, as the saturation increases, the need to account for the inertial terms associated with the pore water flow slightly increases without significantly changing the difference between PD and QS for all three response variables. Furthermore, we should note that as the saturation decreases, pore pressures dissipate more and quickly within the depth while increasing the effective stresses as expected. Overall, the waveinduced seabed response significantly depends on the degree of saturation which was also investigated by Okusa [11]. In Fig. 5, the effect of the normalized seabed layer thickness (h/L) to the wave-induced response is presented. It can be readily observed that, as the thickness increases pore pressures attenuate faster at larger depths within the seabed. As the seabed thickness decreases, the FD and PD formulations become more significant around the mid-depth of the seabed. For seabed with shallow depths, it is necessary to account for the relative acceleration of water in the evaluation of the pore pressure response and so FD formulation should be used. It should also be noted that for very large seabed thicknesses (h/L → ∞), the results converge to one type of distribution for all three formulations, which are presented for h/L = 1.25. Fig. 6 examines the effect of wave steepness (H /L) to the response variables for T = 15 s wave. As discussed earlier, we consider the waves traveling toward the shore during which the wavelength and the water depth decreases while the wave steepness increases. It can be observed from the pore pressure response that as the steepness increases, we need to use the FD formulation. The increase in the steepness, however, does not

Fig. 3. Effect of permeability (kz ) on the pore pressure (p), effective vertical stress

(Szz ) and shear stress (Sxz ) responses, T = 10 s; d = 25 m; L = 130.38 m; H /L = 0.0344; S = 1; h = 30 m. Table 1 Numerical values of the parameters used in analyses. Depth of seabed, h (m) Permeability, kz (m/s) Elasticity modulus, E (kPa) Poisson’s ratio, ν Bulk modulus of water, Kw (MPa) Saturation, S Porosity, n Density of soil skeleton, ρs (t/m3 ) Density of water, ρf (t/m3 ) Wavelength, L (m) Wave period, T (s) Water depth, d (m)

10–200 10−5 –0.01 14000 0.35 2000 0.9–1.0 0.333 2.0 1.0 90–142 5–20 6–60

significantly affect the difference between the PD and QS solutions suggesting solely the use of QS formulation. This behavior could also be seen in shear and effective vertical stress responses where at the same depth, the stresses are higher as the wave steepness increases. Fig. 7 illustrates the effect of wave period on the wave-induced seabed response. In the pore pressure response, as the wave period increases leading to an increase in the wavelength, higher pore pressures are obtained at the same depth. This behavior is observed up to z /h = −0.5. Then for z /h > −0.5, pore pressures start to

16

M.B.C. Ulker et al. / Applied Ocean Research 31 (2009) 12–24

Fig. 4. Effect of degree of saturation (S) on the pore pressure (p), effective vertical stress (Szz ) and shear stress (Sxz ) responses, T = 10 s; d = 25 m; L = 130.38 m; H /L = 0.0344; kz = 0.002 m/s; h = 30 m.

increase as the wave period decreases (especially for T = 5 s). For the selected wave periods (T = 5, 10, 15 s), and for the indicated seabed parameters inertial terms should be considered in evaluating the responses. 6. Domains of applicability of formulations: Non-dimensional parametric space In this section, the analytical solutions for the response variables (shear stress, effective vertical stress and pore pressure) presented in the previous section are used to determine the domains of applicability of the three formulations (QS, PD, FD) in a non-dimensional parametric space. Of course it should be recognized that the validity of these results should ultimately be verified through experimental results. For various values of seabed permeability and wave period, the discrepancies in response variables evaluated from the three formulations may become significant. The locations of the boundaries separating the domains of formulations depend on the maximum discrepancy allowed in the solution. In this study, we considered a maximum discrepancy of 3% (between the solutions of response variables from three formulations) as the criterion to determine the domains. The choice of 3%, although arbitrary, was first suggested by Zienkiewicz et al. [12] and considering the nature of our problem is also used in this study. In Figs. 8 and 9, the regions of applicability of the three formulations are identified in the space of parameters, Π1 , Π2 and m, assuming isotropic seabed soil (Π1x = Π1z = Π1 ). Using the excitation frequency, ω = 2π/T and compression wave velocity, Vc = 2h/T˜ , we can rewrite the non-dimensional parameters as,

Π1x = Π1z =

kVc2 g βωh2

 =

2 g βπ



T T˜ 2



k ,

(20a)

Fig. 5. Effect of depth of seabed (h/L) on the pore pressure (p), effective vertical stress (Szz ) and shear stress (Sxz ) responses, T = 10 s; d = 25 m; L = 130.38 m; H = 4.485 m; kz = 0.002 m/s.

Π2 =

ω2 h2 Vc2

=π

2

T˜ T

!2 ,

(20b) ∗

where, T is the excitation period and T˜ = 2h is the representative Vc natural period of the system where h∗ is the characteristic length of the physical problem which is taken as half wavelength for the problem of wave-induced seabed response. The lines separating the formulations divide the solution space into three regions of FD, PD and QS. In the region of FD, only fully dynamic formulation yields the correct response. This is the region of high frequency loading and/or high permeability porous medium describing ‘fast phenomena’. In the region of PD, a partly dynamic formulation (neglecting the inertial effects associated with the relative acceleration of water) will be adequate. In the region of QS, all the inertial effects can be neglected and quasistatic consolidation formulation of the response can be used. This is the region of low frequency and/or low permeability where the process of flow and deformation is ‘slow’. The regions of applicability of formulations can be used for a specific site with known characteristics of the wave loading and those of the porous seabed. Here, the areas corresponding to various soil types for the wave-induced seabed response problem are identified for two different degrees of saturations. In Fig. 8, seabed is assumed to be fully saturated and a typical wave period of T = 5 s is taken leading an m value of about 10 which is calculated for a representative ratio of seabed thickness to the wavelength, h/L of 0.15. Using (20b), we calculate a Π2 range of 1.58 × 10−4 ≤ Π2 ≤ 6.31 × 10−4 for S = 1 in the dynamic response of seabed under water wave loading. Then the regions corresponding to

M.B.C. Ulker et al. / Applied Ocean Research 31 (2009) 12–24

Fig. 6. Effect of wave steepness (H /L) on the pore pressure (p), effective vertical stress (Szz ) and shear stress (Sxz ) responses, T = 15 s; d/L = 0.101, 0.128, 0.182; L = 197.52, 234.21, 286.26 m; kz = 0.002 m/s, h = 30 m.

different seabed soil types are illustrated depending on Π1 values calculated from different ranges of permeability. It can be observed from Fig. 8 that for m = 10, the FD formulation should be used for most sands and some gravels in the response of seabed. Besides, fully drained formulation is adequate for most gravels. For clays, however, the undrained formulation is adequate and QS solution is sufficient for the soils with intermediate permeability neglecting all the inertial terms leading to a consolidation formulation. For other wave periods (i.e. T = 10–20 s) for which the results are not presented here, PD solution will suffice. Fig. 9 shows the effect of slight unsaturation (S = 0.98) on the regions of applicability of formulations. Here, we calculate 1.74 × 10−2 ≤ Π2 ≤ 7.37 × 10−2 . As the permeability of seabed is increased, inertial terms become crucial in the response. For all silts and most sands, QS will be adequate whereas for some sands and all gravels FD solution will be needed for the evaluation of the response. Earlier Jeng and Cha [20] proposed the following simple yet useful equation, for the boundary between QS and FD solutions assuming that for most cases in the wave-induced seabed response, PD and FD solutions yield almost identical results;


Π2 = 0.0298kz0.5356 Π1−0.5356 .

(21)

Using this relation the boundary between PD and FD can be shown on the regions. For a T = 5 s wave (m = 10) with an average wavelength of 30 m calculated from (3), if we take ρf = 1.0 t/m3 for the density of water, ρs = 2.0 t/m3 as the density of soil solids and n = 0.33 as porosity, the density ratio is calculated as, β = 0.6. We then obtain Π1 = 600kz using (20a) where T˜ = 0.03 with h∗ being 15 m and Vc = 1000 m/s for S = 1 for a water depth of d = 30 m. If we substitute this Π1 into (21), we get constant Π2

17

Fig. 7. Effect of wave period (T ) on the pore pressure (p), effective vertical stress

(Szz ) and shear stress (Sxz ), d/L = 0.101; H /L = 0.055; kz = 0.002 m/s; h = 30 m.

of about 0.001. The boundary defined by this equation is shown as a dashed line in Fig. 10. This boundary matches well with that obtained in this study for 2.7 × 10−4 ≤ Π1 ≤ 0.1. However, for other values of Π1 , this simple equation does not provide the complete classification of the domains of validity for the waterwave induced seabed response problem. 7. Domains of applicability of formulations: Actual variable space In this section, we investigate the domains of validity of the three formulations in an actual parametric space in terms of wave period, seabed permeability and water depth. In the shallow water range, as the wave travels towards the coast, for various water depths and wave periods, we determine the permeability at which the maximum discrepancies are obtained. The regions covering the applicability of the three formulations are plotted in permeability vs. water depth graphs for typical wave periods, T = 5–20 s. These regions are presented in semi-logarithmic figures as shown in Figs. 11–14 and the seabed soil types according to the corresponding range of permeability are also identified in the figures. The boundary identified as ‘deep water limit’ shows the value of the water depth beyond which is considered as deep water in linear wave theory and below which corresponds to the intermediate water regime. Also the boundary of ‘wave breaking limit’ shows the d/L value at which wave reaches its maximum steepness of H /L = 0.068 value and breaks. It should also be noted that FD can be used in all domains with all dynamic terms. In the domain identified as QS, there is no need to consider any of the inertial effects as stated before due to the differences between the responses evaluated from QS and those from PD (and also

18

M.B.C. Ulker et al. / Applied Ocean Research 31 (2009) 12–24

Fig. 8. Regions of applicability of formulations for the wave-induced seabed response for S = 1.

Fig. 9. Regions of applicability of formulations for the wave-induced seabed response for S = 0.98.

FD) being less than 3%. Similarly in the domain identified as PD, one needs to include only the inertial effects associated with soil skeleton. In Figs. 11–14, we see that as the wave period decreases, boundaries move towards lower permeability soils and the areas covering FD and PD formulations increase while decreasing the areas for QS formulation. This is due to more pore flow taking place in seabed soil under lower period wave action where the inertia associated with the relative water displacement to that of soil skeleton becomes significant. The PD region occurs as a band between the other two formulations for T = 20 s wave. This yields the necessity of including the acceleration of soil skeleton only. For shorter wave periods, overall need to include the inertial terms increases. In all four figures, it can also be seen that as the permeability of seabed increases, it becomes necessary to include the inertial terms regardless of the water depth. 8. Distribution of discrepancies between the formulations in wave-induced seabed response In this section, the distribution of the maximum discrepancies (error ratios) in the evaluated response from the three formulations of FD, PD and QS are examined within the seabed depth for various seabed and wave parameters. We should note that the discrepancies in response variables evaluated from the formulations depend on the parameters, T , k, d/L and h/L or z /h. The discrepancies quantify how much error is introduced in the solution by

Fig. 10. Comparison of the domains of applicability as proposed by Jeng and Cha [20] in relation to this study.

neglecting the inertial terms and are defined as;

|FD (p, σzz , τxz ) − PD (p, σzz , τxz )| × 100, (22a) |FD (p, σzz , τxz )| |PD (p, σzz , τxz ) − QS (p, σzz , τxz )| = × 100,(22b) |PD (p, σzz , τxz )| |FD (p, σzz , τxz ) − QS (p, σzz , τxz )| = × 100. (22c) |FD (p, σzz , τxz )|

E1 = ErrorFD−PD = E2 = ErrorPD−QS E3 = ErrorFD−QS

M.B.C. Ulker et al. / Applied Ocean Research 31 (2009) 12–24

19

Fig. 11. Domains of applicability of formulations for T = 20 s.

Fig. 12. Domains of applicability of formulations for T = 15 s.

A detailed error analysis including all the response variables has been carried out and the distributions of errors E1 , E2 and E3 were evaluated for T = 5, 10, 20 s and d/L = 0.5, 0.25, 0.08 for the waves generated in deep water and travel towards shore until breaking occurs. The seabed permeability is also considered as a determining factor. Here, only a few important results are presented in terms of error distributions with respect to the shear stress and pore pressure. It is determined from the analyses that inclusion of the inertial terms associated with the relative water displacement is more significant in pore pressure response whereas for the shear stress response, mostly inertial terms associated with the motion of soil skeleton are important. However, for some values of wave and seabed parameters, relative water accelerations should also be considered in the shear stress response. Figs. 15 shows the distributions of discrepancy with the depth in pore pressure response between FD–PD and PD–QS formulations. The distributions of discrepancy are obtained for various seabed permeabilities for T = 5, 10 s waves at a water depth of

d/L = 0.25. In Fig. 15(a), the necessity of including all the inertial terms is presented in terms of the differences between FD and PD in the pore pressure response. Here, as the permeability increases, maximum error occurs in deeper locations and exceeds 3%. Fig. 15(b) illustrates the significance of PD formulation in pore pressure for T = 10 s. Even though the maximum discrepancy is less than 3%, as the behavior gets closer to the undrained extreme, motion of soil solids gain significance closer to the surface. Fig. 16 presents the discrepancy distribution in shear stress between the formulations for different permeability values with T = 10 s and d/L = 0.25. In Fig. 16(a) all the inertial terms become crucial in the response with differences reaching a maximum at locations closer to the seabed surface where the pore water flow is permitted. In Fig. 16(b) where the error made in neglecting the motion of soil skeleton in the shear stress response is presented, permeability does not affect the distribution of error significantly making the QS solution adequate for all permeability. If we consider the depths at which the maximum discrepancies in the formulations occur in the responses and plot them versus the

20

M.B.C. Ulker et al. / Applied Ocean Research 31 (2009) 12–24

Fig. 13. Domains of applicability of formulations for T = 10 s.

Fig. 14. Domains of applicability of formulations for T = 5 s.

corresponding values of permeability, we obtain Figs. 17 and 18. We can see that as the permeability increases pore flow takes place deeper in the soil and so the inertial effects become more crucial. For lower period waves, lower depths are affected more and the discrepancy between the FD–PD solutions reaches a maximum in the pore pressure response, (Fig. 17(a)). This behavior is also observed in the discrepancy between PD–QS formulations until soil permeability reaches a value of about 8 × 10−4 m/s, (Fig. 17(b)). For coarser grained soils, motion of soil skeleton becomes significant at the seabed bottom for low period waves. Here a maximum discrepancy of less than 3% is also considered in the solution. Fig. 18(a) shows the locations of maximum discrepancies in shear stress response for the difference between FD and PD. The inertial terms in the shear stress response are significant only in z /h ≤ −0.2 for the three wave periods and as the period decreases, inertial terms become significant at locations closer to the surface mostly for the soils with permeability in the range of 10−6 < kz < 10−4 m/s. For permeability values higher than 10−4 m/s,

however, T = 5 s wave has more effect on the location of the maximum discrepancy. In Fig. 18(b), the locations of maximum discrepancy between the PD and QS do not change significantly as the permeability increases. However, for lower period waves, the depths where the error obtained in the QS solution are significantly higher resulting from the higher frequencies associated with the motion of soil solids. It should be noted here that even though Figs. 17 and 18 show the locations of maximum error in the solutions, they cannot serve as exact boundaries between the formulations, since the actual error distribution is not uniform within the seabed depth, (Figs. 15 and 16). As the wave travels in shallow water, the locations of the maximum discrepancy in response between FD and PD (for pore pressure in Fig. 19(a)) and PD and QS (for shear stress in Fig. 19(b)) are plotted for each wave period. In the pore pressure response, almost at same depths, FD becomes crucial as the wave travels toward the shore (Fig. 19(a)). For T = 5 s wave, difference between FD and PD becomes maximum at the surface when the

M.B.C. Ulker et al. / Applied Ocean Research 31 (2009) 12–24

a

b

21

a

b

Fig. 15. Distribution of discrepancy in pore pressure response between (a) FD–PD and (b) PD–QS solutions for different values of permeability, T = 10 s; d/L = 0.25.

wave breaks. As the period increases, the maximum discrepancy in pore pressure is obtained in deeper soil. In shear stress response, as the wave travels toward the shore, the motion of soil skeleton should be considered at deeper locations (Fig. 19(b)). For each wave period, the deepest location of maximum discrepancy is obtained when the wave breaks. These depths increase as the period decreases allowing more pore water flow in deeper soil. Here permeability of seabed is taken as constant and does not significantly affect the location of maximum error in the seabed.

9. Closure In this study, analytical solutions are presented for the waveinduced seabed response under plane strain condition for three different formulations. These formulations are: fully dynamic (FD), partly dynamic (PD) and quasi-static (QS). The last two are the idealized cases of the first one. FD considers both of the inertial terms associated with the motion of soil skeleton and the pore fluid. PD includes only the inertial term for the soil skeleton while QS ignores both of the inertial terms. The wave-induced seabed response is presented in terms of the absolute values of the effective vertical stress, shear stress and pore pressure within the seabed. A maximum discrepancy of 3% is considered between the results obtained from the three formulations and then the regions where each formulation is valid are identified

Fig. 16. Distribution of discrepancy in shear stress response between (a) FD–PD and (b) PD–QS solutions for different values of permeability, T = 10 s; d/L = 0.25.

in both non-dimensional and actual variable spaces. The areas for different soil types corresponding to the wave-induced seabed response problem are identified in the graphs for regions in the non-dimensional space. The effect of slight unsaturation on these regions is also examined which was found to be significant. Then the regions showing the domains of applicability of the formulations are presented in the shallow water range in terms of permeability of seabed, depth of water and period of water wave. The domains of applicability may help the practicing engineers to identify the necessary formulation to be used in a given situation. The locations where the maximum error made by neglecting the inertial terms in the idealized solutions are obtained for typical wave periods and seabed permeabilities in the shallow water regime. From the results of the numerical experiments we conclude the following: (i) The inclusion of inertial terms associated with both the relative water displacement and the soil skeleton displacement may have significant influence on seabed response depending on the wave and seabed parameters. (ii) For mostly clay soils, QS formulation would suffice except for very low period waves. (iii) For silty soils PD will be needed for periods less than 10 s (see Fig. 14 for T = 5 s wave).

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M.B.C. Ulker et al. / Applied Ocean Research 31 (2009) 12–24

a

b

Fig. 17. Locations of maximum discrepancies in pore pressure response between (a) FD–PD, (b) PD–QS solutions for various permeability and wave periods, d/L = 0.25.

a

b

a

b

Fig. 19. Locations of maximum discrepancies in (a) pore pressure response between FD–PD and (b) shear stress response between PD–QS within the seabed in shallow water kz = 0.001 m/s.

(iv) For a sandy seabed, the formulation that is adequate will depend on the permeability of seabed and the period of the wave. (v) For highly permeable soils such as gravels, FD formulation should be used to evaluate the wave-induced seabed response. (vi) Water depth also affects the response such that for permeabilities between 10−3 –10−5 m/s, inertial terms associated with the pore water become significant as the wave approaches to the breaking limit. (vii) A slight unsaturation clearly affects the domains of applicability of formulations. Here, as the overall domains of the formulations shift towards higher Π2 and lower Π1 values in the non-dimensional space, the QS region becomes larger affecting the regions for PD and FD. (viii) As the permeability of soil increases, more flow takes place in deeper soil leading to the necessity of the inclusion of inertial terms in the wave-induced seabed response. Appendix The harmonic complex form of the governing equations yields the following system in terms of the displacement amplitudes,

ρf h2 ω2 − k2 h2 K+

Kf

!

n

Kf

Ux

n K

+h

ρf hω2 /n + iωρf hg /kx − k2 h nf K+

Fig. 18. Locations of maximum discrepancies in shear stress response between (a) FD–PD, (b) PD–QS solutions for various permeability values and wave periods, d/L = 0.25.

Kf

+ ikh



n

K+

Kf n

dUz dz¯

+

Kf

! ¯x W

n

¯z dW dz¯

 =0

(23a)

M.B.C. Ulker et al. / Applied Ocean Research 31 (2009) 12–24 K K ¯z ¯x ikh nf dUx ikh nf dW d2 Uz d2 W n + + + Kf K K dz¯ 2 dz¯ 2 K+ n K + nf dz¯ K + nf dz¯ ! ρf ω2 h2 ρf ω2 /n + iωρf g /kz ¯ 2 + Uz + h Wz = 0 K K K + nf K + nf ! ! Kf + λ + G dUz ρω2 n 2 2 h − k Ux + ikh K K dz¯ K + nf K + nf ! K K ¯z ρf ω2 − k2 nf ikh nf dW ¯x + + h2 W Kf Kf K+ n K + n dz¯ Kf



Kf

K+ d2 Uz dz 2

¯

ikh

+



n

dz¯

n

ρω − k G 2

+ h2

Kf

x

Kf

K+

2

K+

(23b)

Kf

!

n

+

K+

Then the desired displacements are written as: Ux = a1 eη1 z¯ + a2 eη2 z¯ + a3 eη3 z¯ + a4 eη4 z¯ + a5 eη5 z¯ + a6 eη6 z¯ ,

n

Kf

+

dz 2

¯

n

ikh K+

Kf n Kf

¯x dW

n

dz¯

2

A B A 31 D

B C D A42

A13 B A B

Ux B A24   Uz  ¯ x  = [M] {X } = 0, B  W ¯z C W





(24)

where A = β Π2 − m κ, A13 = A24 =

+

n

β Π2

+

n

i

Π1x i

Π1z

− m κ, 2

C = β Π2 + DD κ, (25)

+ DD2 κ,

− m κ, 2

ξ1j = 1, ξ2j = bj ,  η j + ikbj , ξ5j = G h

Π1z

+ DD κ, 2

(26)

C˜ = DD2 κ,

A˜ 31 = −m2 + κ2 DD2 ,

(34)

ξ3j = dj ,

K

ξ4j = bj ηj + ikλ, h

ηj bj + d j . ξ6j = ik 1 + cj +

(35)

h

  1     bi

  ci  

.

(36)

Using (36), one can obtain the coefficients, bi , ci , di from [M] {Vi } = 0 as: bi =

A42 = Π2 − m2 κ2 +DD2 .

A˜ 24 =

(33)

Also bi , ci , di form the elements of Vi which are the eigenvectors corresponding to the eigenvalues of the characteristic equation. Vi is written as,

A31 = Π2 − m2 + κ2 DD2 ,

i

(32)

di

2

di =

and for the quasi static form additionally A31 and A42 are modified as: A˜ 42 = −m2 κ2 + DD2 .

(27)

Then the solution can be obtained by writing the characteristic equation from det [M] = 0 as,

α1 DD6 + α2 DD4 + α3 DD2 + α4 = 0.

(28)

The roots of the characteristic equation are evaluated as:

s

η3 z¯

where aj are obtained from the elements of 6 × 6 coefficient matrix, ξij (i, j = 1, 2, 3, 4, 5, 6). These are;

ci =

Π1x ˜A = −m2 κ,

η2 z¯

+ a4 d4 eη4 z¯ + a5 d5 eη5 z¯ + a6 d6 eη6 z¯ ,

For the partly dynamic form, A13 , A24 , A and C are modified as; A˜ 13 =

η1 z¯

(31)

¯ z = a1 d1 eη1 z¯ + a2 d2 eη2 z¯ + a3 d3 eη3 z¯ W

B = imκ DD,

D = im (κ + κ1 + κ2 ) DD,

i

η3 z¯

+ a2 c2 e + a3 c3 e + a4 c4 eη4 z¯ + a5 c5 eη5 z¯ + a6 c6 eη6 z¯ ,

Vi = 2

β Π2

η2 z¯

(23d)

The matrix system given in Box I for the fully dynamic formulation can be written as,



η1 z¯

+ a2 b2 e + a3 b3 e + a4 b4 eη4 z¯ + a5 b5 eη5 z¯ + a6 b6 eη6 z¯ ,

Uz = a1 b1 e

ρf h ω ¯ Wz = 0. K K + nf 2

Uz +

¯z d2 W

(30)

ξ = 36α1 α3 α2 − 108α12 α4 − 8α23 ζ = 4α1 α33 − α22 α32 − 18α1 α2 α3 α4 + 27α12 α42 + 4α4 α23

(23c)

 + λ + G dU

Kf

 p 1/3 ϕ = ξ + α1 12 3ζ

=0

dz¯ 2

n

where,

¯ x = a1 c1 e W

d2 Ux

G

+

23

ϕ 2 + 2α2 (2α2 − ϕ) − 12α1 α3 η1,2 = ± 6α1 ϕ v  u √  √ 
u −1 + i 3 ϕ 2 − 4α2 ϕ + 4 1 + i 3 3α1 α3 − α 2 2 t , (29) η3,4 = ± 12α1 ϕ v    u √ √
u −1 − i 3 ϕ 2 − 4α2 ϕ + 4 1 − i 3 3α1 α3 − α 2 2 t η5,6 = ± 12α1 ϕ

!

B3 + B [C (A31 − A) − DB − A31 A24 ] + DA24 A B2 (A42 − C ) − BD (C − A24 ) + A C 2 − A24 A42

−B2 A42 + 2BCD − D2 A 24 − A31 C 2 − A24 A42

,(37)




DD=ηi


!

B2 (A42 − C ) − BD (C − A24 ) + A C 2 − A24 A42

, (38)




DD=ηi

BD (D − B) + BA42 (A − A31 ) + C (A31 B − AD) B2 (A42 − C ) − BD (C − A24 ) + A C 2 − A24 A42




! . (39) DD=ηi

where the eigenvalues, ηi , are substituted in place of DD, implicitly. References [1] Biot MA. General theory of three dimensional consolidation. J Appl Phys 1941; 12:155–64. [2] Biot MA. Theory of elasticity and consolidation for a porous anisotropic solid. J Appl Phys 1955;26:182–5. [3] Biot MA. Mechanics of deformation and acoustic propagation in porous media. J Appl Phys 1962;33:1482–98. [4] Truesdell C. Sulle basi della thermomeccanica. Rend. Lincei, Series 1957; 22(33–38):158–66. [5] Truesdell C. Mechanical basis of diffusion. J Chem Phys 1962;37:2336. [6] Putnam JA. Loss of wave energy due to percolation in a permeable sea bottom. Trans Am Geophys Union 1949;30:349–56. [7] Reid RO, Kajiura K. On the damping of gravity waves over a permeable seabed. Trans Am Geophys Union 1957;38(5):662–6. [8] Liu PLF. Damping of water waves over porous bed. J Hydr Din ASCE 1973; 99(HY12):2263–71. [9] Madsen OS. Wave-induced pore pressure and effective stresses in porous bed. Geotechnique 1978;28(4):377–93.

24

M.B.C. Ulker et al. / Applied Ocean Research 31 (2009) 12–24

[10] Yamamoto T, Koning HL, Sellmeijer H, von Hijum E. On the response of a poroelastic bed to water waves. J Fluid Mech 1978;87(1):193–206. [11] Okusa S. Wave-induced stresses in unsaturated submarine sediments. Geotechnique 1985;35(4):517–32. [12] Zienkiewicz OC, Chang CT, Bettess P. Drained, undrained, consolidating and dynamic behavior assumptions in soils. Geotechnique 1980;30(4):385–95. [13] Yamamoto T. Wave-induced pore pressure and effective stresses in homogeneous seabed foundations. J Ocean Engg 1981;8:1–16. [14] Yamamoto T, Schuckman B. Experiments and theory of wave–soil interactions. J Engg Mech ASCE 1984;110(1):95–112. [15] Mei CC, Foda MA. Wave-induced responses in a fluid-filled poro-elastic solid with a free surface-a boundary layer theory. Geophys J Roy Astro Soc 1981;66: 591–637. [16] Rahman MS. Wave-induced instability of seabed: Mechanism and conditions. Mar Geotech 1991;10:277–99.

[17] Rahman MS, El-Zahaby K, Booker J. A semi-analytical method for the wave induced seabed response. Int J Numer Anal Methods Geomech 1994;18: 213–36. [18] Jeng D-S, Rahman MS, Lee TL. Effects of inertia forces on wave-induced seabed response. Int J Off Polar Engg 1999;9:307–13. [19] Cha DH, Jeng D-S, Rahman MS, Sekiguchi H, Zen K, Yamazaki H. Effects of dynamic soil behavior on the wave-induced seabed response. Int J Ocean Engg Tech 2002;16(5):21–33. [20] Jeng D-S, Cha DH. Effects of dynamic soil behavior and wave nonlinearity on the wave induced pore pressure and effective stresses in porous seabed. Ocean Engg 2003;30:2065–89. [21] Horikawa K. Coastal Engg. Tokyo, Japan: University of Tokyo Press; 1978. [22] Wiegel RL. Oceanographical Engg. International Series, Englewood Cliffs, NJ: Prentice Hall; 1964. p. 1205–7.