Dynamic response of a circular pipeline in a poroelastic medium

Dynamic response of a circular pipeline in a poroelastic medium

Mechanics Research Communications 36 (2009) 898–905 Contents lists available at ScienceDirect Mechanics Research Communications journal homepage: ww...

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Mechanics Research Communications 36 (2009) 898–905

Contents lists available at ScienceDirect

Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom

Dynamic response of a circular pipeline in a poroelastic medium Xiang-Lian Zhou a,*, Jian-Hua Wang a, Bin Xu a,b, Ling-Fa Jiang c a b c

Department of Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, China Department of Civil Engineering, Nanchang Institite of Technology, Nanchang 330099, Jiangxi, China Instutite of Rock and Soil Mechanics, the Chinese Academy of Science, Wuhan 430071, China

a r t i c l e

i n f o

Article history: Received 20 September 2007 Received in revised form 20 June 2009 Available online 8 August 2009 Keywords: Poroelastic medium Plane wave Pipeline Wave function expansion method

a b s t r a c t The focus of this contribution is to develop a semi-analytical method to solve the scattering of wave by a circular pipeline embedded in a poroelastic medium. The harmonic equations for the poroelastic medium are derived in the context of Biot’s theory. Then these equations are solved by reducing to Helmholtz equations that the potentials satisfy. The lining structure can use the elastic material and decouple into two Helmholtz equations. Utilizing the wave function expansion method, the general solutions of Helmholtz equations can be obtained. By using the boundary and continuity conditions between the poroelastic medium and the lining, the unknown coefficients can be determined. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Pipeline in soft ground is an increasingly common geotechnical activity for construction of urban transportation or water management facilities in many large cities around the world. For this reason, numerous researchers use various methods to investigate this problem. Among these, Gamer (1977) used the wave function expansion method to study the dynamic stress concentration at the surface of a semi-circular cavity in an elastic half-space excited by a plane harmonic SH wave. Varadan (1978) studied the scattering of P, SV and SH waves by a elliptic cylinder using the scattering matrix approach. Sancar and Pao (1981) gave the solution for the scattering of the plane harmonic pressure wave by two cylindrical cavities in an elastic solid by using the eigenfunction expansion method. Datta et al. (1984) studied the dynamic stresses and the displacements around the cylindrical cavities in an elastic medium by employing a combined approach of the finite element method and the eigenfunction expansion. Okumura et al. (1992) investigated the dynamic interaction of twin circular tunnels subjected to an incident SV wave by using the two-dimensional finite element method. Shi et al. (1996) used the conformal mapping method and the wave function expansion method to study the interaction of SH wave and the cavity with an elastic lining in an anisotropic medium. Moore and Guan (1996) studied the dynamic interaction of the lined tunnel subjected to a seismic loading in an infinite medium using the successive reflection method. The preceding review has primarily focused on research works involving cavities or tunnels in single-phase elastic medium. However, many geophysical applications require multi-phase model of the soil. In fact, plenty of literature suggests that, under certain conditions, there are significant differences between modeling soil as saturated poroelastic medium and as elastic single-phase medium. For the saturated porous media, several scholars have also addressed the scattering of the elastic wave by an embedded cavity. Bachmat and Bear (1983) used the principle of continuum mechanics and volumetric averaging approach to deduce the macroscoptic balance equation of extensive quantity of a fluid phase in a porous medium. Mei et al. (1984) investigated a circular cavity embedded in a poroelastic medium and subjected to P and SV waves.

* Corresponding author. Tel./fax: +86 21 62932915. E-mail addresses: [email protected], [email protected] (X.-L. Zhou). 0093-6413/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2009.08.002

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Berryman (1985) treated the scattering of an incident P1 wave by a spherical poroelastic inhomogeneity. Zimmerman (1993) used the boundary element method to study the problem of the wave diffraction by a spherical cavity in an infinite poroelastic medium. Degrande et al. (1998) studied the harmonic and transient wave propagation in a multilayered saturated and an unsaturated porous medium. Kattis et al. (2003) used the boundary element method to solve the problem of incident P and SV waves by tunnels in an infinite poroelastic saturated soil. Lee et al. (2004) developed the boundary integral equation technique to calculate the scattering of the elastic wave with an isotropic and an orthotropic cylindrical inclusion in the absence or the presence of a parallel cylindrical void in its vicinity. Wang et al. (2005) used the complex function method to solve the scattering of the plane wave by multiple elliptic cavities in a saturated medium. Lu et al. (2007) investigated the frequency domain response of a circular tunnel with prefabricated piecewise lining subject to a seismic wave by using the wave function expansion method. Zhou et al. (2008) solved the scattering of the plane wave by a circular-arc alluvial valley in a poroelastic half-space. In this study, the scattering of the elastic wave by a circular pipeline in a poroelastic medium is investigated. The saturated poroelastic medium is described by Biot’s theory (Biot, 1956, 1962). By introducing three potentials, the governing equations for Biot’s theory are decoupled and reduced into three Helmholtz equations. The lining is treated as an elastic single-phase medium. The two regions are coupled by the continuity conditions at the interface of the poroelastic medium and the lining. Utilizing the wave function expansion method, the general solutions of Helmholtz equations are obtained. By utilizing the boundary and continuity conditions between the poroelastic medium and the lining structure, the unknown coefficients in the potentials can be determined. Some numerical results are given in the paper. 2. Governing equations and general solutions of porous media In this study, the lining structure is considered to be infinitely long, while the incident plane wave has a direction perpendicular to the axis of the lining. Thus, the dynamic interaction between the lining and the surrounding medium can be reduced to a plane strain problem (Fig. 1). Based on Biot’s theory for two-phased material, the constitutive relations for homogeneous saturated porous medium are expressed as (Biot, 1956, 1962)

rij ¼ 2leij þ kdij e  adij pf

ði; j ¼ x; yÞ

ð1Þ

pf ¼ aMe þ M#

ð2Þ

e ¼ ui;i ;

ð3Þ

# ¼ wi;i

where rij denotes the total stress of the bulk material; eij and e are the strain component and the dilatation of the solid matrix, respectively; k and l represent Lamé constants; dij denotes the Kronecker delta; # is the variation of the fluid content per unit reference volume; a and M are Biot parameters; pf is the excess pore pressure; ui and wi denote the average solid displacement and the fluid displacement relative to the solid frame. The equations of motion for the bulk material and the pore fluid can be expressed in terms of displacements ui and wi as

€i lui;jj þ ðk þ a2 M þ lÞuj;ji þ aMwj;ji ¼ qu€ i þ qf w qf g €i þ w _i aMuj;ji þ Mwj;ji ¼ qf u€ i þ w n

ð4Þ ð5Þ

k

where q and qf denote the density of the porous medium and the density of the pore fluid, q = (1  n)qs + qf, qs is the density of the solid skeleton and n is the porosity of the porous medium; k is the permeability of the porous fluid; g is the fluid viscosity; the superimposed dot on a variable denotes the derivative with respect to time t. According to the Helmholtz theorem, any vector field can be expressed as the sum of the gradient of scalar field u and the curl of vector field w. Although two independent displacement vectors for the solid skeleton and the pore fluid are used in

y Incident wave r

O

θ x

a1

a2

β

Fig. 1. Incident wave by a circular pipeline in a poroelastic medium.

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Biot’s theory, there are only four independent variables in the two-phase porous medium. The displacement of solid skeleton and the pore pressure can be expressed as the sum of the scaler field u and the vector field w (Zimmerman, 1993).

^ k;j ¼ u ^ k;j ^i ¼ u ^ ;i þ eijk w ^ f ;i þ u ^ s;i þ eijk w u ^ ^ ^ pf ¼ Af uf ;ii þ As us;ii

ð6Þ ð7Þ

where uf and us denote the scalar displacement potentials corresponding to P1 and P2 wave, respectively; wk (k = 1, 2, 3) is the vector displacement potential; a caret denotes the Fourier transform with respect to time; Af and As are two constants; eijk is the Levi-Civita symbol. Substitution of the above to Eqs. (4) and (5) leads to (Bourbie et al., 1987)

  ^ k ;m ¼ 0 ^ k;jj þ b3 w ^ f ;jj þ b3 u ^ s;jj þ b3 u ^ f  þ ½ðk þ 2l  b2 As Þu ^ s  ;i þ eimk ½lw ½ðk þ 2l  b2 Af Þu     ^ f þ As u ^ s ;kk ¼ 0 ^ f ;jj þ ðb5 Af  b4 Þu ^ s;jj þ ðb5 As  b4 Þu Af u

ð8Þ ð9Þ

which is satisfied if

^ f þ k2f u ^ f ¼ 0; r2 u 2

^s þ r u 2^

r wk þ

2 ks ^ s 2^ kt wk

u ¼ 0; ¼ 0;

2

kf ¼ b3 =ðk þ 2l  b2 Af Þ ¼ ðb5 Af  b4 Þ=Af 2 ks 2 kt 2

ð10Þ

¼ b3 =ðk þ 2l  b2 As Þ ¼ ðb5 As  b4 Þ=As

ð11Þ

¼ b3 =l

where b1 ¼ qf x =n  igx=k;

ð12Þ b2 ¼ a þ qf x2 =b1 ;

b3 ¼ qx2 þ q2f x4 =b1 ;

b4 ¼ ab1 þ qf x2 ;

b5 ¼ b1 =M ð13Þ

It is seen that uf, us and wk satisfy the scalar and vector wave equations, respectively. Since the wave equations are much simpler than the original equations of motion, solutions for ui and pf will be constructed from Eqs. (6) and (7) in which the potentials satisfy the wave equations Eqs. (10)–(12) and the boundary and initial conditions. A question arises as to whether every solution of Eqs. (10)–(12) is included in the above procedure. This is answered by the completeness theorem stating that every solution of Eqs. (4) and (5) adimits a decomposition of Eqs. (6) and (7) with uf, us and wk satisfying the Eqs. (10)–(12) (Pao and Mow, 1973). Based on Eqs. (10) and (11), the following equation for Af and As is obtained

b3  ðk þ 2lÞb5  b2 b4 ðk þ 2lÞb4 Af ;s þ ¼0 b2 b5 b2 b5

A2f ;s þ

ð14Þ

The vector potential wk satisfies the following condition:

^ i;i ¼ 0 w

ð15Þ

The infiltration displacement of the pore fluid can be represented by

^i ¼ w

q f x2 1 ^i pf ;i  u b1 b1

ð16Þ

^ 3 of the vector potential w ^ k remains. For simplicFor the plane strain wave involved in this paper, only one component, i.e., w ^ is used to denote w ^ 3. ity, the symbol w The complex kf, ks and kt in Eqs. (10)–(12) are the complex wave numbers for P1, P2 and S wave of the poroelastic medium. In order to ensure P1 wave is faster than P2 wave, the inequality Re(kf) < Re(ks) should always hold. For the scattering of the elastic wave by a pipeline in an infinite domain, the total wave field is composed of the incident wave and the scattered wave

^¼w ^ ðIÞ þ w ^ ðSÞ ^ ðIÞ ^ ðSÞ u ^s ¼ u ^ ðIÞ ^ ðSÞ w ^f ¼ u u s þ us ; f þ uf ;

ð17Þ

where superscripts I and S denote the incident wave and the scattered wave, respectively. Because the potentials for the incident waves field satisfy the Helmholtz equations, accordingly, the potentials for the scattered field should satisfy corresponding the Helmholtz equations. Therefore, the general solutions for the scattered field can be expressed in terms of Hankel function as follows:

^ ðSÞ u f ¼ ^ ðSÞ u s ¼ ^ ðSÞ ¼ w

1 X n¼1 1 X n¼1 1 X

inh an Hð1Þ n ðkf rÞe

ð18Þ

inh bn Hð1Þ n ðks rÞe

ð19Þ

inh cn Hð1Þ n ðkt rÞe

ð20Þ

n¼1

where Hð1Þ n ðÞ denotes the first kind of Hankel function of order n; an, bn, cn are arbitrary coefficient to be determined by the boundary conditions.

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The incident fast or slow wave in the coordinate system of a cylindrical system are written in form of infinite generalized Fourier series

^ ðIÞ u f ;s ¼ u0

þ1 X

ðiÞn J n ðkf ;s rÞeinh

ð21Þ

n¼1

where Jn is the cylindrical Bessel function of the first kind; u0 represents the potential amplitude for the incident wave. For the plane strain problem of a poroelastic medium, the displacement of the solid skeleton, the stress and the pore pressure can be represented by three potentials uf, us and w. In a polar coordinate system (r, h), the displacements, the stresses and the pore pressure have the following expressions (Lu et al., 2007):

@ uf @ us 1 @w þ þ r @h @r @r   1 @ uf @ us @w uhI ¼ þ  r @h @r @h " ( " !#)  # 2 2 @ uf @ us 1 @w @ 2 us @ uf @2w @ 2 us @ uf þ r 2  apf rrrI ¼ 2 2l þ k þ þ r l l Þ þ þ rðk þ 2 þ r @h @[email protected] @r @r @r2 @r 2 @h2 @h2 (    @2u @ uf @ us 1 @w @ 2 u @2w rhhI ¼ 2 2l þ 2 s þ 2f þ r ðk þ 2lÞ þ  2r l r @h @[email protected] @r @r @h @h " !#) 2 2 2 2 @ us @ uf @ us @ uf þk  apf þ þ r2 þ @r 2 @r2 @h2 @h2 ! ( " #) 2 @ uf l @ us @ 2 w @w @ 2 us @ uf @2w r 2 rrhI ¼ 2 2 þ2 2 þ 2 þr þ @r @r r @h @h @[email protected] @[email protected] @h urI ¼

2

2

pfI ¼ Af kf uf  As ks us

ð22aÞ ð22bÞ ð22cÞ

ð22dÞ ð22eÞ ð22fÞ

where subscript I designates the functions in the poroelastic medium. Note that the infiltration displacement of the pore fluid can be derived from Eqs. (16) and (22). 3. Governing equations and general solutions of pipeline The equation of motion for a concrete pipeline can be expressed as

lui;jj þ ðk þ lÞuj;ji ¼ qu€ ði; j ¼ x; yÞ

ð23Þ

where q denote the density of the lining; k, l represent Lamé constants of the lining. There are four refracted waves in the lining structure: two inward propagating waves and two outward propagating waves excited by the incident plane wave.

uðf Þ ¼ wðf Þ ¼

1 X

ð2Þ inh ½dn Hð1Þ n ðkp rÞ þ en H n ðkp rÞe

ð24Þ

inh ½mn Hnð1Þ ðks rÞ þ nn Hð2Þ n ðks rÞe

ð25Þ

n¼1 1 X n¼1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffiffi 2 2 2 where Hð2Þ ðk þ 2lÞ=q; ks ¼ x2 =t2s ; ts ¼ l=q; n ðÞ denotes the second kind of Hankel function of order n; kp ¼ x =tp ; tp ¼ kp, ks denote the complex wave numbers for P and S wave of the lining structure; tp and ts denote the compressional wave velocity and the shear wave velocity; dn, en, mn, nn are arbitrary coefficient to be determined by the boundary conditions. Considering the fundamental field equations in the polar coordinate (r, h), the displacements and the stresses of the lining are written as (Pao and Mow, 1973)

@ u 1 @w þ @r r @h 1 @ u @w ¼  r @h @r

urII ¼

ð26aÞ

uhII

ð26bÞ

rrrII rrhII

"  # @ 2 u @ 1 @w ¼ k 2 r 2 u þ 2l þ @r 2 @r r @h " " #  # 1 @2u 1 @u 1 @2w @ 1 @w þl 2 ¼ 2l  r  2 r @r @h r @h r @h2 @r r @r

where subscript II designates the functions in the lining.

ð26cÞ ð26dÞ

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4. Boundary conditions The presence of a concrete pipeline made of a linear elastic material creates an impervious boundary along the periphery of the cavity. Thus the boundary conditions use a completely impermeable case. For impermeable case, the relative displacement between the solid skeleton and the pore fluid vanishes. At the conjunctive surface of the poroelastic medium and the lining, the continuity conditions are

urrI ¼ urrII

ð27aÞ

uhhI ¼ uhhII

ð27bÞ

rrrI ¼ rrrII rrhI ¼ rrhII

ð27dÞ

ð27cÞ

For impermeable boundary condition, the relative displacement between the solid skeleton and the pore fluid is zero. From Eq. (16), we obtain

wrI ¼ 0

ð28Þ

At inner surface of the lining, the stress free conditions are

rrrII ¼ 0 rrhII ¼ 0

ð29aÞ ð29bÞ

Eqs. (27)–(29) form a set of infinite algebraic equations for determining the coefficients an, bn, cn, dn, en, mn, nn. It should be pointed out that the above equations are all in infinite sums. Therefore, the system of equations must be solved by truncating the infinite terms into the finite terms. For the number of terms in series solution N takes 12, the stresses and the pore pressures have a good convergence. Therefore, the number of terms truncated from the infinite series N is takes as 12 for each part of numerical results. 5. Numerical results The dynamic stress concentration factor r* is defined as the ratio of the hoop stress to maximum amplitude of the incident stress at the same point

r ¼

rhh 2 ; r0 ¼ Re½ðk þ 2lÞkf u0  r0

ð30Þ

For the case of impermeable condition, the pore pressure concentration factor is defined as the ratio of the pore pressure on the boundary of cavity to maximum amplitude of the pore pressure at the same point

pf ¼

pf ; pf 0

2

pf 0 ¼ ReðAf kf u0 Þ

ð31Þ

In order to demonstrate the performance of present methodology, a typical poroelastic problem with known numerical solution is discussed first. Consider a circular pipeline buried in an infinite poroelastic saturated medium. The parameters of the poroelastic medium are as follows: l = 1.0  107 Pa; m = 0.33; qs = 2700 kg/m3; qf = 1000 kg/m3; n = 0.3; a = 0.999; M = 1.0  108 Pa; g = 1.0  102 Pa s; kd = 1.0  106 m2. The parameters of the lining are as follows: q = 1700 kg/m3; m = 0.25; l = 0.33  107 Pa. The ratio of outer radius to inner radius of the lining v = a2/a1 = 1.1. The dimensionless wave

90 1.0

120

60

Present result Kattis (2003)

0.8 0.6

30

150

0.4

σ

*

0.2 0.0 180

0

0.2 0.4 0.6

330

210

0.8 1.0

240

300 270

Fig. 2. Comparison present result with Kattis et al. (2003).

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(a)

90 120

6.0

σ

1.5

χ =1.05 χ =1.1

5.6

120

60

χ =1.05 χ =1.1

1.0 0.5

30

150

5.2

90

(b)

60

0.0

30

150

-0.5

* 4.8

σ

4.4

* -1.0 -1.5

0

180

-2.0 180

4.4

0

-1.5

4.8

-1.0

5.2

-0.5

330

210

330

210

0.0

5.6

0.5

6.0

240

1.0

300

240

1.5

270

ω=5, Re(kf )a=0.2

300

ω=5, Re(kf )a=1.0

270

Fig. 3. Stresses versus polar angle for incident P1 wave in a pipeline: (a) Re(kf)a = 0.2 and (b) Re(kf)a = 1.0.

(a)

90 1.2

120

60

χ =1.05 χ =1.1

1.1

χ =1.05 χ =1.1

60

0.2 0.0

*

* 0.9

pf

pf

0.8 180

120

0.4

30

150

1.0

90

(b)

30

150

-0.2 -0.4 180 -0.4

0

0.9

0

-0.2

1.0

0.0

330

210

1.1

330

210

0.2 240

1.2

300

0.4

300 ω=5, Re(kf )a=1.0

240

ω=5, Re(kf )a=0.2

270

270

Fig. 4. Pore pressures versus polar angle for incident P1 wave in a pipeline: (a) Re(kf)a = 0.2 and (b) Re(kf)a = 1.0.

(a)

90 7.2

120

χ =1.05 χ =1.1

60

6.4

90

(b) 1.6

120

60

χ =1.05 χ =1.1

0.8 5.6

30

150

0.0

30

150

4.8

σ

*

-0.8 4.0 0

180 4.0

σ

* -1.6 180

0

-1.6 4.8 5.6

330

210

-0.8 0.0

330

210

6.4 7.2

240

300 270

ω=100, Re(kf )a=0.2

0.8 1.6

240

300

ω=100, Re(kf )a=1.0

270

Fig. 5. Stresses versus polar angle for incident P1 wave in a pipeline: (a) Re(kf)a = 0.2 and (b) Re(kf)a = 1.0.

number Re(kf)a = 1.0; the incident angle b = 0°. Fig. 2 shows a comparison of the present method with the boundary element method reported by Kattis et al. (2003). The agreement with the boundary element method solution is excellent. In the remainder of this paper, the dynamic response of lining with concrete and subjected to a seismic wave with different frequencies, dimensionless wave numbers and thicknesses of the lining are calculated as a numerical example. The

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X.-L. Zhou et al. / Mechanics Research Communications 36 (2009) 898–905 90

(a)

120

1.2

60

χ =1.05 χ =1.1

1.0

(b)

90 120

0.4

χ =1.05 χ =1.1

60

0.2

0.8

30

150

30

150

0.0 0.6

*

pf

-0.2

0.4 0.2

180

0

0.4

180

0

-0.2

0.6 0.8

*

p f -0.4 -0.4

330

210

1.0 1.2

0.0

330

210

0.2 240

300 270

ω=100, Re(kf )a=0.2

0.4

240

300

ω=100, Re(kf )a=1.0

270

Fig. 6. Pore pressures versus polar angle for incident P1 wave in a pipeline: (a) Re(kf)a = 0.2 and (b) Re(kf)a = 1.0.

material parameters for the poroelastic medium in this example assume the following values: qs = 2500 kg/m3; qf = 1000 kg/ m3; n = 0.3; l = 1.0  107 Pa; m = 0.25; a = 0.999; M = 1.0  108 Pa; g = 1.0  102 Pa s; kd = 1.0  107 m2. The parameters for the lining are as follows: q = 3000 kg/m3; m = 0.3; l = 0.2  108 Pa. The incident angle b = 0°. Figs. 3–6 show the distribution of the hoop stresses and the pore pressures versus the polar angle h for P1 incident harmonic wave. The different values of the frequencies x = 5 and 100; the dimensionless wave numbers Re(kf)a = 0.2 and 1.0; the thicknesses of the lining v = 1.05 and 1.1. Figs. 3 and 4 show that for x = 5, very large difference of the stresses and the pore pressures around lining is observed. For low dimensionless wave number (Re(kf)a = 0.2), the stresses and the pore pressures decrease while the thicknesses of lining increasing. However, for high dimensionless wave number (Re(kf)a = 1.0), there is only very small difference of the stresses and the pore pressures around the lining when increasing the thicknesses of the lining. The stresses and the pore pressures have a little decrease when increasing the thicknesses of the lining. Figs. 3 and 4 also indicate that the stresses and the pore pressures decrease when increasing the dimensionless wave numbers. From the above results in Figs. 3 and 4, it can be also observed that the most important parameters for the response of lining in a poroelastic medium is the dimensionless wave number. For low value of the wave number, the thickness of the lining is significant and decreases the maximum hoop stress, while for high value of the wave number, the thickness of the lining has small influence on the stress and the pore pressure. Figs. 5 and 6 show the distribution of hoop stresses and pore pressures versus different wave numbers and the thicknesses of lining, for x = 100. Same conclusions can be drawn as in the previous case. Moreover, as frequencies increasing, the stresses and pore pressures increase obviously, especially for low wave number case. 6. Conclusion An efficient and accurate semi-analytical method of solving harmonic wave scattering by a circular pipeline subjected to a seismic wave has been proposed in the paper. This method is based on Biot’s theory for saturated porous medium. Numerical results show that, frequencies, dimensionless wave numbers, and thicknesses of the lining have direct influences on the response of the lining. For a low value of the wave number, the thickness of the lining is significant in decreasing the maximum hoop stress, while for a high value of the wave number, the thickness of the lining has small influence on the stress and the pore pressure. As frequencies increasing, the stresses and pore pressures obviously increase, especially for the case of low wave number. Acknowledgements This research was sponsored by Shanghai Leading Academic Discipline Project (Project No.: B208) and Shanghai Science and Technology Action Plan (Project No.: 08DZ1205400). This research was also supported by Key Laboratory of Geotechnical Mechanics and Engineering of the Ministry of Water Resources (Project No.: G07-09) and the Foundation of Jiangxi Educational Committee (Project No.: GJJ09367). References Bachmat, Y., Bear, J., 1983. The dispersive flux in transport phenomena in porous media. Adv. Water Resour. 6, 169–174. Berryman, J.G., 1985. Scattering by a spherical imhomogeneity in a fluid-saturated porous medium. J. Math. Phys. 26, 1408–1427. Biot, M.A., 1956. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low frequency rang. J. Acoust. Soc. Am. 28, 168–178. Biot, M.A., 1962. Mechanics of deformation and acoustic propagation in porous media. J. Appl. Phys. 33, 1482–1498. Bourbie, T., Coussy, O., Zinszner, B.E., 1987. Acoustics of Porous Media. Gulf Publishing, Houston.

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