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SHORT-CRESTED WAVE-INDUCED SEABED RESPONSEVARIABLE PERMEABILITYDong-Sheng Jeng a & Brian Richard Seymour ba Center for Offshore Foundation Systems, The University of Western Australia,Nedlands, WA, 6907, Australiab Department of Mathematics, The University of British Coulombia, Vancouver, V6T1Z2, CanadaVersion of record first published: 26 Mar 2012.

To cite this article: Dong-Sheng Jeng & Brian Richard Seymour (1997): SHORT-CRESTED WAVE-INDUCED SEABEDRESPONSE VARIABLE PERMEABILITY, Journal of the Chinese Institute of Engineers, 20:4, 377-388

To link to this article: http://dx.doi.org/10.1080/02533839.1997.9741843

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Journal of the Chinese Institute of Engineers, Vol. 20, No. 4, pp. 377-388 (1997) 377

SHORT-CRESTED WAVE-INDUCED SEABED RESPONSE WITH

VARIABLE PERMEABILITY

Dong-Sheng Jeng* Center for Offshore Foundation Systems

The University of Western Australia Nedlands, WA 6907, Australia

Brian Richard Seymour Department of Mathematics,

The University of British Coulombia Vancouver, V6T JZ2, Canada

ABSTRACT

The soil permeability of marine sediments varies with burial soil depth because of consolidation under overburden pressure. However, most previous investigations available for water wave-seabed interac­tion problem have simply assumed seabed with uniform permeability. This paper proposes an analytical solution to investigate the short­crested wave-induced soil response in a porous seabed with variable permeability. The pore pressure and effective stresses generated by a three-dimensional short-crested wave system are obtained from a set of equations incorporating a variable permeability. The numerical results indicate that the effect of variable permeability on the wave­induced soil response is significant.

Key Words: short-crested waves, soil response, variable permeability.

I. INTRODUCTION

Most shore protection installations and large harbor breakwaters have been constructed at an angle to incident waves. A short-crested wave system is then resulted in front of these structures. This wave system has a free surface which does not only fluctu­ate periodically in the direction of propagation, but also in the direction normal to the former. It has been reported that the wave forces acting on marine struc­tures subjected to a three-dimensional short-crested wave system are larger than two-dimensional progres­sive and standing waves [4, 18]. Furthermore, the seabed response caused by short-crested wave fronting breakwaters and seawalls has attracted great attentions among the marine geotechnical and coastal

*Correspondence addressee

engineers, because it may lead to failure of some ma­rine structures [ 17].

The permeability of a soil is a measure of how fast fluid is transmitted through the voids between soil grains. In natural seabeds, marine sediments below the water-soil interface undergo consolidation due to both the overburden soil pressure and the wa­ter pressure. This causes the soil permeability to vary with the burial soil depth. An example of permeabil­ity varying with burial depth (z) has been reported for marine sediments in the Gulf of Mexico [3]. Simi­lar evidence for soil permeability versus depth can also be found in literature [ 1, 16].

To date, numerous theories of the wave-seabed interaction problem have been developed, based on various assumptions of the relative rigidity of solid

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378 Journal of the Chinese Institute of Engineers, Vol. 20, No.4 (1997)

and pore fluid [15, 21]. However, most previous in­vestigations have so far considered progressive waves propagating over a porous seabed with uniform per­meability. Only few works have treated this problem within a layered medium, but they are only valid for uniform soil characteristics in each sub-layer [6, 20]. The resultant soil response of layered solutions displays a discontinuity at the interface between two sub-layers. This discontinuity may cause substantial error in the stability analysis off foundations.

Recently, the first author has developed a series of studies for the wave-induced seabed response in a porous seabed of infinite [7], finite thickness [ 11] and layered medium [6] as well as a cross-anisotropic sea­bed [8, 9] with uniform permeability. Based on these general solutions, the wave-induced seabed instabil­ity has been further investigated [ 1 0]. The effects of variable permeability on the soil response due to two-dimensional waves have been investigated by the authors [ 13]. Here we further consider the three-di­mensional short-crested wave-induced soil response in a porous seabed with variable permeability in front of a breakwater.

The major mathematical difficulty in solving the wave-seabed interaction with variable permeability has been the governing partial differential equation with variable coefficients. It is not possible to find analytical solutions to even simple differential equa­tions with rapidly varying coefficients until Varley and Seymour [19]. Based on VS function (named by the authors [ 19]), this paper proposes an analytical solution for the wave-induced seabed response with variable permeability. The effect of variable perme­ability on the pore pressure and effective stresses are detailed. Based on the Mohr-Coulomb's criterion, the wave-induced shear failure in a seabed with variable permeability is also examined.

II. BOUNDARY VALUE PROBLEM

Herein, we consider a soil matrix with a semi­infinite thickness in a sandy seabed in front of a ver­tical reflecting wall, as seen in Fig. 1. A short-crested wave system is produced by incident progressive waves interacting with their reflected waves in front of a vertical wall. The obliquity angle e is measured between a wave orthogonal and the normal to the wall, or the wave crests to the wall. The resultant wave crest propagates parallel to the wall in the positive x­direction, while they-direction is normal to the wall. The z-direction is measured upwards from the sur­face of the seabed.

To simplify the problem, some necessary as­sumptions are made for deriving the analytical solu­tion as follows: 1. The seabed is horizontal, unsaturated, hydraulically

y

z

free surf?· ~ SWL

porous seabed

rigid Impermeable bottom

Fig. 1. Definition sketch of short-crested wave system produced by obliquereflection from a breakwater, showing cartesian co-ordinates used for theanalysis of wave-induced soil re­sponse.

anisotropic and of infinite thickness. 2. The soil skeleton and the pore fluid are compress­

ible. 3. The soil skeleton generally obeys Hooke's law. 4. The flow in the porous seabed is governed by

Darcy's law. 5. The permeability of seabed is assumed as

(1)

where Kz denotes the vertical permeability, Kzo is the permeability at the seabed surface and a(z) is a depth function.

Based on the above assumptions, the governing equation combines the three-dimensional consolida­tion with variable permeability, and related to pore pressure (p) and the volume strain (E) as

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D.S. Jeng and B.R. Seymour: Short-Crested Wave-INduced Seabed Response with Variable Permeability 379

in which Kx, Ky and K20 are the coefficients of soil permeability in the x-, y- and z-directions, respec­tively. Yw is the unit weight of the pore-water, n' is the soil porosity, f3 is the compressibility of the pore-fluid, and tis time. The volume strain E is de­fined as

(3)

where u, v and w denote the soil displacements in the x-, y- and z-directions, respectively.

The compressibility of the pore fluid can be re­lated to the apparent bulk modulus of the pore-water, K' and the degree of saturation Sr [21], such that

(4)

where Kw is the true bulk modulus of elasticity of wa­ter, and P wo is the absolute pore-water pressure. If a soil skeleton is absolutely air-free, i.e., fully saturated, then K'=Kw, since S,=l.

According to Hooke's law, force equilibrium within the soil provides the final equations of equi­librium as follows

G"iPu + _Q_ i1E =- dp 1-2,u dX dX' (5)

(6)

(7)

in the x-, y- and z-directions, respectively. The shear modulus of a soil (G) is related to Young's modulus (E) and Poisson's ratio (,U) in the form of E/2(1+,u).

The relationship between the effective stresses and soil displacement are expressed as

a:= 2G ( * + 1 !:2,u €), (8)

, dv .u crY = 2G ( dY + 1 - 2.U €) , (9)

, dw .u a,= 2G ( dZ + -2- E ) , (10) - 1- .u

rxy = G ( ~~ + ~) = ryx, (11)

Txz = G ( ~~ + CJ; ) = Tzx , (12)

Tvz = G ( f + ~w ) = rzv' (13) . z y .

where the Cauchy stress tensor on the adjacent faces of a stress element consists of three effective normal stresses and six shear stress components. The shear stresses are expressed in double subscripts r,s denot­ing the stress in the s-direction on a plane perpen­dicular to the r-axis. It is important to note that compressive stresses are taken as a positive sign here.

For a seabed of infinite thickness, the wave-in­duced pore pressure and soil displacements should vanish at the rigid bottom, i.e.,

u=v=w=p=O as z~-oo (14)

The vertical normal stress and shear stresses vanish at the seabed surface, hence

(15)

while the pore pressure on this upper soil boundary is given by the first-order short-crested wave pres­sure, i.e.

p= 2cr::::.kd cosnkycos(mkx-mt)

=p0 cosnkycos(mkx-mt) at z=O, (16)

in which "cosnky cos(mkx-mt) " represents the spatial and temporal variations within a three-dimen­sional wave field. In Eq. (16), Po is the wave pres­sure amplitude, Hs is the wave height of the short-crested waves, k is the wave number of the in­cident and reflected waves (k=2n!L, in which Lis the wavelength), w is the angular frequency of the wave (w=2n!T, T is the wave period), and d is the water depth above the seabed surface.

Two wave oblique parameters m and n are the components of the wave number k in the x- and y­directions, respectively, i.e.,

kx=2n1Lx=mk=ksin8 and ky=2n!Ly=nk=k cosO, (17)

where Lx and Ly are the crest length of the short­crested waves in the x- and y-directions, respectively (Fig. 1), and from Eq. (17), m2+n2=1.

The boundary value problem, including Eqs. (2), (5)-(7) and boundary conditions (14) and (15), de­scribes the wave-induced soil response in a porous seabed of infinite thickness subject to a short-crested wave system. The pore pressure and soil displace­ments can be obtained firstly, and the effective stresses can then be computed from Eq. (8)-(13).

III. GENERAL SOLUTIONS OF SEABED RESPONSE

In the analysis of the linear governing

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380 Journal of the Chinese Institute of Engineers, Vol. 20, No.4 (1997)

equation, it is expedient to perform the principal of superposition for functions expressed in com­plex variables. Thus, it is advantageous in handling problem with complicated wave conditions, such as the present short-crested wave system. Since the wave system is produced by the two identical linear progressive wave interactions at an angle to each other, the wave pressure at the seabed surface by Eq. ( 16) can be expressed as the sum of there fleeted (mkx+nky-mt) and incident (mkx-nky-mt) compo­nents, i.e.,

P = ~ Re {ei(mkx+nky-llJI) + ei(mkx-nky-llJI)}. (18)

where Re stands for the real part of function and i( =-I=T) denites the complex variable. Only the real parts of the complex solution are utilized. For the sake of simplicity, this is applied to the real part of all complex solutions in the following analysis, un­less specified otherwise.

By comparing with the analytical solutions for the short-crested wave-induced soil response of a semi-infinite porous seabed with uniform permeabil­ity [7], the components of solid displacement and the pore pressure can be expressed as:

where nj is given by

{n if j = 1

nj= -n if }=2

(19)

(20)

In the above equation, the subscript "1" denotes the contribution from reflected wave components and "2" is for incident wave components.

Substituting expression (19) into (5)-(7), the governing equation yields

(21)

(22)

where a; (i=l-4) are unknown coefficients. Introducing (21) and (22) into Eqs. (5) and (2),

produces two differential equations:

" 2 _ imk(l - 211) k kz ~ - k ~-- 2G(l- 11) Ej- 2(1- 11) ((nlal- a3)e

(23)

where the complex parameter (j and ~o are given by

s:2 k2( Kx 2 Ky 2) iOJYw ( '{3 1 - 211 ) (25) u = K m + K n - --rc- n + 2G(l - 11) '

zo zo zo ,...

(26)

Eq. (24) contains a variable coefficient depth func­tion a(z), which describes the variations of the soil permeability with depth. When a(z) is a rapidly vary­ing function, it is difficult to obtain exact solutions to differential equation of the form (24). Varley and Seymour [19] proposed a family of exact solution for any variable coefficient partial differential equation. Based on the approach of Varley and Seymour [ 19], it is possible to obtain an exact solution for such an equation (see Appendix).

The general solutions of Eq. (24) can be ex­pressed as

in which E 1(z) and E2(z) satisfy the homogeneous Eq. of (24), given by

E l(z) = e- 8R(zl[r2 + rl (j 1/(i\RJ] r 2 + r 1 (j lf(i(JJ) (28)

(29)

The function a(R) and R(z) in Eq. (28) and (29) are defined as

fz ds and R(z) = a(s) , (30)

in which a(R) is so called VS 1 function [ 19]. In Eq. (28) and (29), E 1(z) and E2(z) satisfy the

boundary conditions

In Eq. (27), F(z,s) is given by

F(z' s) = E 1 (z~E2(s)- E 1 (s)E2(z~ a 2(s) (E 1 (s)Ez(s)- E 1(s)E2(s))

=E1 (z) 1C2(s)-E2(z) 1C1 (s), (32)

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D.S. Jeng and B.R. Seymour: Short-Crested Wave-INduced Seabed Response with Variable Permeability 381

D = 28r?('2 - 82) 0 722 - 1j202/a(O) ·

(33)

For a porous seabed of infinite thickness, the bottom boundary condition, Eq. (14), requires a2

=a4=a6=0. Thus the pore pressure can be rewritten as

where S1(z) and S2(z) are defined as

S1(z) =I 1C2(s)e-ksds and S2(z) = f= IC1(s~ds. (35)

Substituting Eq. (34) into Eqs. (21)-(23), Uj, Vj and Wj can be derived as:

f1(z) = a7ekz + 4( 1 z_ J.l) (n ia1 - a3)ekz

+ M (ekzi e-ksfj(s)ds- e-kzf= eksfj(s)ds), (36)

\.j(z) = Jh a7ekz + ( 4m(~z_ J.l) (nia1 - a3)- ~ a 1) ekz

+ nM (ekzi e- ksfj(s)ds- e- kzf= ~fj(s)ds), (37)

w.(z)=-i_ {(a - kz+1 'n.ka -a )+la )ekz 1 m 7 4k(1 _ J.l) ~ 1 1 3 k 3

+ mM (ekzi e-ksp/s)ds + e-kzf= eksP/s)ds)}

(38)

M = i(1 - 2J.L) 4G(1- J.L). (39)

The four unknown coefficients, ab a3, a5 and a7

in Eqs. (34)-(38) for pore pressure and soil displace­ments can be solved by applying the boundary condi­tion (15) at the surface of the seabed.

Upon the concept of superposition, after some algebraic manipulations, the wave-induced soil dis­placements and pore pressure can be expressed as

{ ~ } = { 7n~~~ ~~: ~~} ei(mkx- mt) w Po W(z) cosnky ' P P(z) cos nky

(40)

in which

P(z)=B 1E 1 (z)+<l> ,A 1 (S 1 (z)-S2(z) ), (41)

U(z) = C1ekz + 4(1 z_ J.l) + M (ekzi e- ksP(s) ds

- e-kzf= eksP(s) ds), (42)

Three coefficients Ab BI and cl are given by

_ 8M(l-J.Li A1 - (1 - 2J.L) (3 - 4J.L) ' (44)

_ 8M(1- J.L)2

BI - 1 + (1 _ 2J.l) (3 _ 4J.l) S2(0) , (45)

_ 2M(1- J.l) C1 - MS3(0)B1 + (1 _ 2J.l) (3 _ 4J.l) ((1- 2J.L)

(46)

where

(47)

Once soil displacements and pore pressure are obtained, the effective stresses can be further com­puted through Hooke's law, Eq. (8)-(13).

IV. MOHR-COULOMB'S CRITERION

The wave-induced shear stress at a point within the sediment may become large enough to overcomes its shearing resistance, thus c·ausing seabed failure. The actual mode of such instability depends on the spatial distribution of the wave­induced shear failure and the shear strength of the sediment. Conventionally, predication of failure for soils has been based on Mohr-Coulomb's failure criterion, which remains the most widely used in geotechnical engineering practice. Although other criteria off ailure have been suggested in the literature [5], Mohr-Coulomb's criterion is used here because of its simplicity and conser­vatism.

In the previous section, only the wave-induced incremental changes in effective stresses and pore pressure within soils from the initial equilibrium have been considered. Thus, the effective normal stresses (a;, a; a;) and shear stresses (-:x'xz• ""'fyz and ""'fxy) are given by

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382 Journal of the Chinese Institute of Engineers, Vol. 20, No.4 (1997)

I

ax axO I

q, I

ay 0')0 I oy

I I I

az azO q = +

""fxz !' xzO ';z

Tyz !' yz() 'lYz

""'fxy ""'fx)O ';y

I

q, Ko I

Ko oy

I

=- (Ys- Yw) 1 q

(48) 0

z+ ';z

0 'lYz 0 ';y

where the subscript "o" denotes the stresses at the initial equilibrium, while Ys and Yw are the unit weights of soil and water, respectively.

In Eq. (48), K 0 is the coefficient of earth pres­sure at rest. The value of K0 for soils ranges from 0.4 to 1.0 and K 0 =0.5 is commonly used for marine sedi­ments.

For study of the general stress field that occurs in a complicated boundary value problem, it is con­venient to use a principal stress space. This leads to a convenient geometric representation of various fail­ure criteria. the effective principal stresses, at> a 2 and a3 can be expressed as [5]

where

- 1 (_I _I _I) S = J3 ax+ ay + az '

t=

(49)

(50)

as shown in Fig. 2, the limiting condition in a given soil may be expressed by

(56)

where lfltdenotes the angle of internal friction of soil, ""'f1 and a 1 represent the shear stress and effective normal stress on the failure plane, respectively. When the stress reaches the failure envelope, the stress angle

Instantaneous stress

0

Fig. 2. Diagram of Mohr's circle.

and

(52)

S ~ s: S -2 S -2 S -2 2-2-2 T 2 J = x'-' r z - x 1' yz- y 1' xz- z 1' xy + 1' xz 1' yz xy •(53)

2a;- a:- a; Sz= 3 (54)

Eq. ( 49) ensure that a 1:::;;a2:::;;a3. The stress state at a given location and instant

may be expressed by the angle 1ft between the tangent from the origin to the instantaneous Mohr' circle and the a-axis (Fig. 2). The stress angle 1ft is defined by

(55)

According to Mohr-Coulomb's failure criterion,

(51)

( lp') become sidentical to the internal friction angle of soil ( lflt)· Thus, based on elastic theory, the failure criterion at a specific location and time may be de­fined as

(57)

In general, the value of 1ft depends on the soil type, for example, 30-35 degrees for sand and 35-40

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D.S. Jeng and B.R. Seymour: Short-Crested Wave-INduced Seabed Response with Variable Permeability 383

~.1

.J ... N ~.3

Type1

~.4

a{z} Fig. 3. Three different cases of depth function a(z) numerically

examined forsoil matrix with variable permeability in the present study. Solid linesof Type 1 (a(z)=1), dashed lines for Type 2 (r1=-0.4124, rz=5.0, and r3=-0.3), and dotted lines for Type 3 (r1=-0.7413, r2=0.8, and r3=-0.3).

degrees for small gravel and shingle.

V. RESULTS AND DISCUSSION

Three different types of depth function a(z) are used as an example here (Fig. 3). Type I, a(z)=1, represents the conventional assumption with uniform permeability (solid line), together with the previous three-dimensional analytical solution [7]. Type 2 (dashed line) and Type 3 (dotted line) are for a gradual reduction in permeability, but with different slopes. The wave condition and soil characteristics are indi­cated in Fig. 4.

It is noted that integration has appeared in the general solutions presented above. The Gaussian in­tegration method has been used for numerical calcu­lation here. The influence of integrating points on the wave-induced pore pressure (with Type 3) has been examined through a preliminary study and if has been found that the solution seems to converge to five decimal points with 60-point gaussian integration. Thus, we use 60-point Gaussian integration in the fol­lowing numerical computation.

1. Effects of variable permeability

The vertical distribution of the wave-inpucj:!p pore pressure lpllp0 and effective normal stress I (jz I! Po are illustrated in Fig. 4. The results of the analyti­cal solution for a seabed with uniform permeability [7] are also included. Since it is difficult to directly compare two solutions term by term, we only present numerical comparisons. The results for a seabed with uniform permeability through the reduction of the present theory (Type 1, a(z)= 1) are in dentical with

..OA

0.0

-0.1

-0.2

..J .... N -0.3

-0.4

..0.5 ' 0.0

Type2

------ J T ... 3 ',

'•, ype ' ···· .. / ,'

..... , ______ ,

l~',' ,.;

.·• _.-;"

.. ·;''

' ' ' '

.. ·, .... -;~' ,.., .. , .. ,

' ' '

0.2 0.4 0.6 0.8

I cr.' 1/ p o

T= 12.5 sec

d=20m

L= 159.95 m G=10 7 N/m 1

n' = 0.3

11= 1/3

Kx=K,=Kao=10" 2 m/s

S,= 1.0 9=45.

h->-

Coarse Sand

T = 12.5 sec d=20m

L = 159.95 m G = 10 7 N/m'

n'=0.3

I'= 1/3 K 11 = K.,= K 20 = 10' 2 m/s

S,= 1.0 9=45'

h->-

Coarse Sand

1.0 1A

Fig. 4. Vertical distribution of the maximum (a) IPI/p0 and (b)/ a;/1 Po versus z/L in a saturated seabed.

the conventional solution (Fig. 4 ). In a fully saturated, hydraulically isotropic sea­

bed of infinite thickness with a uniform permeability (i.e., Type l), the wave-induced soil response has been reported to only depend on wave characteristics [7]. However, as shown in Fig. 4, the wave-induced pore pressure and effective stresses with variable perme­ability depend on the depth function (i.e., variable permeability) as well as the wave condition. These results are different from the conclusion obtained from the conventional solution with uniform perme­ability [7].

Figure 4 also illustrated the influences of variable permeability (in terms of depth function a(z) on the pore pressure and vertical effective normal stress. Generally speaking, the wave-induced soil response is affected significantly by variable permeability near the surface of the seabed. The difference between uniform and variable solutions becomes less in a deeper part of the seabed. based on the same input data, the numerical values are

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384 Journal of the Chinese Institute of Engineers, Vol. 20, No.4 (1997)

Table 1. Maximum pore pressure and effective stress amplitudes in a saturated coarse sand at various depths for Type 1 and Type 2 soils.*

ziL IPtypeliiPo IPtypeziiPo

(IPtype2-PtypeJI)Ipo

0.0 1.0 1.0 (0.0)

-0.05 0.7572 0.5213 (-0.2359)

-0.10 0.5545 0.2265 (-0.3280)

-0.15 0.3986 0.0674 (-0.3312)

-0.20 0.2850 0.0124 (-0.2726)

-0.25 0.2046 0.0238 ( -0.1808)

-0.30 0.1081 0.0240 (-0.1276)

-0.35 0.1481 0.0547 (-0.0934)

-0.40 0.0792 0.0110 (-0.0682)

-0.45 0.0581 0.0085 (-0.0486)

-0.50 0.0426 0.0064 (-0.0362)

+Input data is the same as the legend in Fig. 4.

e -~ _______________ ~-·-= 5.0 m

N 4

2.5m

2 2.5m

--- ---

Coarse Sand

---

Type2

---· Type1

--------------------------------------

g.9~s-~-o::-.~.9::-6-~-o..J..9--7 -~-o.l..9e-~-o . .L..ss-~--l,.oo

s, Fig. 5. Distribution of maximum shear failure depth Zsm versus

the degree of saturation for various values of wave height.

also tabulated in Table 1. As shown in the table, the maximum differences between two solutions may reach 44% of Po for the vertical effective normal stress. It is important to note that con­ventional solution with uniform permeability [7]

I a;,typell I Po I a;,type2l I Po

d a;, typezl-1 a;, type tl) I Po

0.0 0.0 (0.0)

0.2067 0.5580 (0.3513)

0.3199 0.7620 (0.4421)

0.3629 0.7237 (0.3608)

0.3597 0.5907 (0.2310)

0.3304 0.4673 (0.1369)

0.2893 0.3777 (0.0877)

0.2457 0.3066 (0.0609)

0.2043 0.2462 (0.0419)

0.1673 0.1959 (0.0286)

0.1355 0.1548 (0.0193)

overestimates pore pressure, but underestimates ver­tical effective normal stress in a seabed with variable permeability. At least from Fig. 4 and Table 1, the influence of variable permeability on the soil re­sponse cannot always be ignored without substantial errors if soil permeability of the seabed is not uni­form.

Based on the analytical solution presented in this study, together with three-dimensional Mohr­Coulomb's criterion, the influence of variable perme­ability on the wave-induced shear failure are presented in Fig. 5, based on the same input data of Fig. 4. In this example, only Type 2 (variable perme­ability) and Type 1 (uniform permeability) are used for different wave heights and degree of saturation. Basically, the maximum depth of shear failure (Zsm) in a seabed with variable permeability is larger than that with uniform permeability, as shown in Fig. 5. The trend becomes more significant in an unsatur­ated seabed.

2. Effects of wave height and degree of saturation

As shown in the previous investigations by the

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D.S. Jeng and B.R. Seymour: Short-Crested Wave-INduced Seabed Response with Variable Permeability 385

first author [ 1 0], the short-crested wave-induced seabed instability is affected significantly by the de­gree of saturation and wave heights in a porous sea­bed with uniform permeability. Herein, the influences of these two parameters are re-examined for a seabed with variable permeability.

As shown in Fig. 5, it is found that the influ­ences of variable permeability on the wave-induced shear failure (Zsm) increases as the degree of satura­tion decreases, but it increases as wave height in­creases. This implies that variable permeability has to be taken into consideration in the evaluation of the wave-induced seabed response under either unsatur­ated conditions or larger waves.

VI. CONCLUSIONS

This paper proposes an analytical solution for short-crested wave-induced soil response in a porous seabed with variable permeability. The present solution has been verified through it sreduction to the special case of uniform permeabil­ity [7].

At least from the examples presented in this study, the wave-induced pore pressure and effective stress depend on both wave and soil characteristics for a fully saturated, hydraulically isotropic seabed of infinite thickness with variable permeability. This result differs from the conventional solution [7] for such a condition but with a uniform permeabil­ity.

Three different types of burial depth function a(z) are used as an example in this study. The nu­merical results clearly show that the wave-induced seabed response is affected significantly by variable permeability. In general, the conventional solution [7] overestimates the pore pressure, but underesti­mates the vertical effective normal stress. Further­more, the influence of variable permeability on the seabed response cannot always be ignored, espe­cially under either unsaturated or large wave condi­tions.

In this study, only the wave-induced soil re­sponse in a seabed of in finite thickness has been considered. For more complicated cases such as a seabed of finite thickness, readers can refer to the au­thors' recent paper [ 12, 14].

ACKNOWLEDGEMENTS

The authors thank Dr. Y. S. Lin at Department of Civil Engineering, National Chung-Hsing Univer­sity, Taiwan and Dr. A. D. Barry at Department of Environmnetal Engineering, The University of Western Australia, Australia for their helpful com­ments. Part of this work has been done when the

first author worked at Department of Environmental Engineering, the University of Western Austrialia. The valuable comments from reviewers are also ap­preciated.

AI> A2, B~> C1

E Ei> E 2

F(z,s) G Hs Ko Kx,Ky Kz Kzo

K'

L M Pwo

R(z) S; (i=1-4)

s, T a; (i=1-4) a(z) d k m, n n'

U, V, W

VI VIJ Ys Yw /..l a,, a2. a3 a;, a;,a; 'l"xz• 'l"xy• 'l"yz ()

NOMENCLATURE

coefficients Young's modulus (N/m2

)

coefficients as a function of depth [see Eqs. (28) and (29)] parametric function [see Eq. (32)] shear modulus (N/m2

)

wave height coefficient of earth pressure at rest horizontal soil permeability vertical soil permeability reference soil permeability at the sur­face of the seabed apparent bulk modulus of pore-wa­ter true bulk modulus of elasticity of wa­ter wave length (m) coefficient [see Eq. (39)] absolute pore-water pressure depth function [see Eq. (30)] coefficients of a function of depth [see Eqs. (35) and (47)] degree of saturation wave period (sec) coefficients depth function for soil permeability water depth wave number (1/m) wave obliquity soil porosity pore pressure amplitude factor of wave pressures parameters variable in VSN parametric variable time soil displacements co-ordinates or subscripts coefficient volume strain of soil parametric function stress angle internal friction angle of soil unit weight of soil unit weight of pore-water Poisson's ratio principal stressed effective normal stresses shear stresses incident wave angle

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386 Journal of the Chinese Institute of Engineers, Vol. 20, No. 4 (1997)

REFERENCES

1. Bennett, R.H., Li, H., Lamber, D.H., Fischer, K.M., Walter, D.J., Hickox, C.E., Hulbert, M.H., Yamamoto, T. and M. Badiey, "In-situ Porosity and Permeability of Selected Carbonate Sedi­ments: Great Rahama Bank, Part I: Measure­ments," Marine Geotechnology, Vol. 9, pp. 1-28 (1990).

2. Biot, M.A. "General Theory of Three-Dimen­sional Consolidation," Journal of Applied Phys­ics, Vol. 12, pp. 155-164 (1941).

3. Bryant, W.R., W. Hottman and P. Trabant, "Per­meability of Unconsolidated and Consolidated Sediment, Gulf of Mexico," Marine Geote­chnology, Vol. 1, pp. 1-14 (1975).

4. Fenton, J.D. "Wave Forces on Vertical Walls." Journal of Waterway, Port, Coastal and Ocean Engineering, ASCE, Vol. 111, pp. 693-718 (1985).

5. Griffiths, D.V. "Failure Criteria Interpretation Based on Mohr-Coulomb Friction." Journal of Geotechnical Engineering, ASCE, Vol. 116, pp. 986-999 (1990).

6. Hsu, J.R.C., D.S. Jeng and C.P. Lee, "Oscillatory Soil Response and Liquefaction in an Unsatur­ated Layered Seabed," International Journal for Numerical and Analytical Methods in Geome­chanics, Vol. 19, pp. 825-849 (1995).

7. Hsu, J.R.C., D.S. Jeng and C.P. Tsai, "Short­Crested Wave-Inducedsoil Response in a Porous Seabed of Infinite Thickness." Interna­tional Journal for Numerical and Analytical Methods in Geomechanics, Vol. 17, pp. 553-576 (1993).

8. Jeng, D.S. "Wave-Induced Liquefaction Potential in a Cross-Anisotropic Seabed." Journal of the Chinese Institute of Engineers, Vol. 19, pp. 59-70 (1996).

9. Jeng, D.S. "Soil Response in Cross-Anisotropic Seabed Due to Standing Waves," Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 123, 9-19 (1997).

10. Jeng, D.S. "Wave-Induced Seabed Instability in Front of a Breakwater." Ocean Engineering Vol. 24, pp. 887-917 (1997).

11. Jeng, D.S. and J.R.C. Hsu, "Wave-Induced Soil Response in a Nearly Saturated Seabed of Finite Thickness." Geotechnique, Vol. 46, pp. 427-440 (1996).

12. Jeng, D.S. and Y.S. Lin, "Finite Element Model­ling for Water Waves-Soil Interaction." Soil Dy­namics and EarthquakeEngineering, Vol. 15, pp. 283-300 (1996).

13. Jeng, D.S. and B.R. Seymour, "Wave-Induced Pore Pressure and Effective Stresses in a Porous

Seabed with Variable Permeability." Journal of Offshore Mechanics and Arctic Engineering, ASME, Vol. 119 (1997) (in press).

14. Jeng, D.S. and B.R. Seymour, "Response in Seabed of Finite Depth with Variable Permeabil­ity." Journal of Geotechnical and Geoenviron­mental Engineering, ASCE, Vol. 123 (1997) (in press).

15. Liu, P.L.F. "Damping of Water Waves Over Po­rous bed". Jouranl of Hydraulic Division, ASCE, Vol. 99, pp. 2263-2271 (1973).

16. Samarasinghe, A.M., Y.H. Huang and V.P. Drnevich, "Permeability and Consolidation of Normal Consolidation Soils," Journal of Geo­technical Engineering Division, ASCE, Vol. 108, pp. 835-849 (1982).

17. Silvester, R. and J .R.C. Hsu, "Sines Revisited," Journal of Waterway, Port, Coastal and Ocean Engineering, ASCE, Vol. 115, pp. 327-344 (1989).

18. Tsai, C.P. and D.S. Jeng, "A Fourier Approxima­tion for Finite Amplitude Short-Crested Waves," Journal of the Chinese Institute of Engineers, Vol. 15, pp. 713-721 (1992).

19. Varley, E. and B.R. Seymour, "A Method for Ob­taining exact Solutions to Partial Differential Equations with Variable Coefficients," Studies in Applied Mathematics, Vol. 78, pp. 183-225 (1988).

20. Yamamoto, T. "Wave-Induced Pore Pressure and Effective Stresses in Inhomogeneous Seabed Foundations," Ocean Engineering, Vol. 8, pp. 1-16 (1981).

21. Yamamoto, T., H.L. Koning, H. Sellmejjer and E.V. Hijum, "On the Response of a Poro-elastic bed to Water Waves," Journal of FluidMechanics, Vol. 87, pp. 193-206 (1978).

APPENDIX: EXACT SOLUTIONS OF LINEAR VARIABLE COEFFICIENT EQUATIONS

The general solution of Eq. (24),

' 2 (a2(z)lj)'- 8 lj = Q(z), (Al)

depends on a result derived by Varley and Seymour [ 19] where it was shown that many linear, variable coefficient equations have exact solutions if their co­efficients satisfy a particular system of nonlinear or­dinary differential equation. They considered a general linear, second order partial differential equa­tion of the form:

(A2)

a 1 and a2 are arbitrary constants. Defining the new

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D.S. Jeng and B.R. Seymour: Short-Crested Wave-INduced Seabed Response with Variable Permeability 387

independent variable:

R-j·z dv - a(v)'

(A3)

and regarding a andfas function of Rand t, equation (A2) can be rewritten in the canonical form:

(A4)

It was shown that if the coefficient function a(R),

ei(R),fi(R), ... eN_1(R).fN-I(R) satisfy the system of 2N-1 independent nonlinear ordinary differential Eq. [19]

(AS)

(A6)

(A7)

where

(AS)

then any solution to equation (A4) can be written in terms of the corresponding solutions F to the con­stant coefficient equation,

(A9)

as

(AIO)

Using Eq. (A3), an ordinary differential equa­tion of the form (AI) transforms into

1 ' ' 5:2 71 (alj) - u lj = Q(z), (All)

and the solution to the corresponding homogeneous equation (Q=O) now has the simple form

(AI2)

where c 1 and c2 are arbitrary constants, and E 1 (R) and E2(R) are given by Eqs. (28) and (29) for N=l. (For convenience we normalize the Ei, so that Ei(O)=l.)

The general solution to the nonhomogeneous Eq. (AI) the takes the form

lj(R(z)) = c1E1(R) + c2E2(R) + JR K(R, S)Q(s(S)) dS,

(AI3) where

K(R' S) = E~(R)E2(S)- E 1(S)E;(R)'

E I (S)E2(S)- E I (S)EiS) (AI4)

and S= r dvla(v). It should be noted that for N=l the denomfnator in (AI4) is proportional to a(S).

Now, we further check solution (28) for Pj=E1(R(z)) (with similar calculation for E2(R(z))). According to Eq. (30), we have

Ja(R(z)) =- J"h. coth (..ffi.(R(z) + r 3)), (AI5)

(AI6)

Using Eqs. (A3) and (AI6), the ordinary differ­ential Eq. (24) can be transformed into

(AI7)

Eq. (28) can be written as

EI(R(z))= eb"R<zl[r2 + rl (j J.j(i{ji)] =EoeoR(2+-i). r2+r,o!Ja(O) rt JZi (AIS)

where E0 is a constant. Then

a dE - 1 _ E eb"R(r, + ().;a) (j dR - 0 I (AI9)

and

(A20)

satisfy Eq. (A17). It was shown by Varley and Seymour [ 19]

that many linear, variable coefficient equations will have exact solutions, if their coefficients (as functions of the independent variables R) satisfy the system of ordinary differential Eqs. (A5)-(A8). These coefficients are defined in terms of an Nth order VS function, VSn(R; ], P). Here J is a vector containing N integral entries, and P is a vector containing 2N+ 1 arbitrary constant entries. The main advantage of VS function is their ability to approximate a wide variety of functions that can vary rapidly. Here only VS1 is used; the reader is referred to Varley and Seymour [ 19] for the details for gen­eral N.

In principle, the general system (As)-(As) can be reduced to a (2N-1)st order nonlinear ordi­nary differential equation for fo(R), containing the

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388 Journal of the Chinese Institute of Engineers, Vol. 20, No.4 (1997)

two arbitrary parameters kN and lN. Quite remark­ably, the general solution to this system was con­structed by Varley and Seymour [19] for any value of N by solving linear algebraic equations. The solution, designed as fN(R) = VSN (R; ], P), is given as the ratio of two determinants whose elements are defined in terms of the z~\R; Am RN), j=jk, and the An.

Discussions of this paper may appear in the discus­sion section of a future issue. All discussions should be submitted to the Editor-in-Chief.

Manuscript Received: Mar. 14, 1996 Revision Received: Jan. 4, 1997

and Accepted: Mar. 10, 1997

~*~ 'ffi~~J+I:k~~~~~**iEliffJi;${,,

B. R. Seymour

m; ~

~~±~MW~-·~®*Zm*~·~±~·N®~~o~®·ttfiffi iijjJt~:k~1~~w®*Zm7]< 1*at.;¥.g~ )tf1ii o ;;f>:Jti'JHf:l~ IE!JifWtRfW*~M~ ~~513&~m*1*aw®*±fl.:t.J o ~~513&ZfL~~~:t.JW1H':l!:.:t.JJJ~~~ 1*•~)t:1J1i~*~ o ~:2f>:Jta1iUs*~~. ~•*1*•w~~si3&Z±JJJZ• Z~ll-tlit.;li=l!fo

iijj-~:~~~,~-*~·'±JJJZ.o

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