Modeling of wave reflection in gas hydrate-bearing sediments

Wave Motion 85 (2019) 67–83

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Modeling of wave reflection in gas hydrate-bearing sediments Haomiao Qiu a,b , Tangdai Xia a,b , Bingqi Yu a,b , Weiyun Chen c , a b c

∗

Research Center of Coastal and Urban Geotechnical Engineering, Zhejiang University, Hangzhou 310058, China MOE Key Laboratory of Soft Soils and Geoenvironmental Engineering, Zhejiang University, Hangzhou 310058, China Institute of Geotechnical Engineering, Nanjing Tech University, Nanjing 210009, China

highlights • The theoretical models of wave reflection in gas hydrate-bearing sediments, based on a Biot-type three-phase theory, are presented for the free open-pore and sealed-pore boundaries.

• The general solutions are compared with the degenerate solutions based on Biot theory by assuming the gas hydrate and sediment frame are weakly coupled or strongly coupled.

• The influences of the content and distribution pattern of gas hydrate on the reflection properties are analyzed numerically.

article

info

Article history: Received 26 March 2018 Received in revised form 4 September 2018 Accepted 20 November 2018 Available online 22 November 2018 Keywords: Gas hydrate-bearing sediments Biot-type three-phase theory Two-phase type model Reflection

a b s t r a c t The content and distribution pattern of the gas hydrate are the key factors affecting the free surface reflection of the gas hydrate-bearing sediment. In this study, a theoretical model of wave reflection in gas hydrate-bearing sediment is discussed for the incidence of P1wave and S1-wave to analyze the influences of the gas hydrate. The gas hydrate-bearing sediment is modeled by a Biot-type three-phase theory and assumed to be composed of sediment grains with connected pore occupied by the mixture of fluid and hydrate. The incoming wave is split into three reflected compressional waves and two reflected shear waves at the free surface of a gas hydrate-bearing sediment half-space. The analytical solutions for various reflected waves at open-pore and sealed-pore boundaries are obtained in terms of displacement potentials. Then, two-phase type models based on Biot theory are proposed by assuming the gas hydrate and sediment frame are weakly coupled or essentially consolidated to verify the rationality of the theoretical model. A third two-phase type model is given by assuming there is no gas hydrate in the pore. At last, the influences of the content and distribution pattern of gas hydrate, the frequency of the incident wave, and the permeability condition on the horizontal displacements and displacement ratios are analyzed respectively through numerical calculation. It is revealed that the solutions based on the Biot-type three-phase theory are very consistent with the solutions based on Biot theory if the gas hydrate and sediment frame are weakly coupled or essentially consolidated. Furthermore, the model proposed in this study can be used to analyze the influence of the distribution pattern of the hydrate. © 2018 Elsevier B.V. All rights reserved.

1. Introduction Gas hydrates are ice-like crystalline solids composed of water molecules surrounding individual gas molecules [1]. As non-traditional energy sources, hydrates have received widespread attraction due to their widespread distribution in ∗ Corresponding author. E-mail address: [email protected] (W. Chen). https://doi.org/10.1016/j.wavemoti.2018.11.003 0165-2125/© 2018 Elsevier B.V. All rights reserved.

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H. Qiu, T. Xia, B. Yu et al. / Wave Motion 85 (2019) 67–83

ocean sediments, deep lake sediments and permafrost [2,3]. Pore-space gas hydrate-bearing sediments are composed of sediment grains with connected pore occupied by the mixture of fluid and hydrate. The presence of hydrate dramatically changes the properties of wave propagation of sediments, which has been extensively investigated [4–6]. Current researches mainly focused on the propagation characteristics of bulk waves, especially the effects of gas hydrate content on phase velocity and attenuation of fast waves [1,7–12]. The reflection and transmission properties of waves at the surface of the gas hydrate-bearing sediments are of fundamental importance in seismology, energy exploration, and geotechnical engineering. However, there have been few systematic investigations on this issue until now. At present, there are mainly three theories, i.e., classical elasticity theory, Biot theory and Biot-type three-phase theory, being applied to investigate the reflection and transmission of waves at the interface of gas hydrate-bearing sediments. The problems based on classical elasticity theory are research foundations which are well developed and used in practical engineering after many researchers’ efforts [13–15]. In classical elasticity theory, the gas hydrate-bearing sediments are considered to be scale-independent continuous solids. The effective medium theory, or velocity model [7–9], is used to establish the relationships between bulk density, modulus (or wave velocity) and content of each component of the composite solid material. However, unlike classical elastic materials, many geological materials, especially energy-rich materials, contain connected pores. The presence of pores and filler in the pores significantly alters the acoustic properties of the materials. The classical elastic model, based on the equivalent theory, cannot describe the pore morphology, permeability, fluid–solid coupling and other characteristics accurately. In order to describe the effects of pore and pore fluid, the two-phase porous medium theory is adopted to describe the acoustic properties of gas hydrate-bearing sediments. Biot [16,17] was perhaps the first one who established a two-phase theory describing wave propagation in uniform-pore fluid-saturated isotropic homogeneous porous elastic media. The theory successfully predicted the existence of slow compressional wave, which had been observed by Plona [18] and Berryman [19]. The gas hydrate-bearing sediments are generally considered to be three phase media consisting of sediment particles, hydrates, and pore fluids. When using Biot theory, the hydrate phase is either combined with fluid phase to form a new fluid phase, or combined with sediment grains phase to form a new solid phase, depending on the distributions of gas hydrate. However, this degradation can only describe the two extremes of hydrate being suspended in pores or hydrate completely as sediment frame. In some cases, Biot theory is no longer applicable in describing wave propagation in gas hydrate-bearing sediments. To this end, a Biot-type three-phase theory was improved for unconsolidated sands by Lee and Waite [1] using well log compressional and shear wave velocity data from the Mallik 5L-38 permafrost gas hydrate research well in Canada. The Biot-type three-phase theory was firstly derived by Leclaire et al. [20] describing the propagation and attenuation of the bulk waves in frozen porous media such as frozen soils and permafrost. Unlike the assumption of full saturation in the traditional Biot theory, this theory assumes that one solid phase acts as the solid frame and the other solid phase coexists with the fluid in the pore. Due to the existence of the solid in the pore, besides two types of compressional waves and one type of shear wave, which appear in fluid saturated porous elastic media, an additional compressional wave and an additional shear wave emerge. Subsequently, the existence of two types of shear waves was verified through the experiment by Leclaire et al. [21]. Considering the interaction between pore solid and substrate solid, the applicability of the theory to gas hydrate-bearing sediments and muddy sandstones was discussed by Carcione and Seriani [22], Carcione et al. [23], and Carcione et al. [24]. Considering the effects of hydrate distribution and hydrate content, Lee and Waite [1] improved the estimation of model moduli, which can well describe the propagation speed of waves in pore-scale gas hydrate-bearing sediments. However, Lee’s work focused on the speed of wave propagation and did not describe the wave attenuation well. Guerin and Goldberg [12] found that the friction between hydrates and sediments, coupled with an absence of inertial coupling between gas hydrates and the pore fluid, are the dominant modes of energy dissipation for wave motion in gas hydrate-bearing sediments. The problem of the wave reflection at the half-space surface is of fundamental importance and has attracted great attentions among researchers in different fields. The surface reflection and energy partitioning of incident bulk waves in classical elasticity media were investigated systematically by Knopoff et al. [25] and Trifunac [26,27]. Then Gourgiotis et al. [28] researched the reflection waves in half-spaces of microstructured materials governed by dipolar gradient elasticity. Singh et al. [29] studied the reflection of plane wave in a micropolar thermoelastic solid half-space with diffusion. Based on Biot theory, Deresiewicz and Rice [30], Tajuddin and Hussaini [31] discussed the effect of boundaries on the reflection of incident P- and SV-waves in a poroelastic half-space saturated by a viscous fluid. The effects of the dry-frame moduli on reflection of plane P- and SV-waves in a poroelastic half-space saturated with inviscid or viscous fluid were analyzed by Lin et al. [32] and Rjoub [33]. To the best of the authors’ knowledge, there is hardly any systemic and thorough research on the reflection of plane waves at the free boundary of a half-space which is based on the Biot-type three-phase theory. Therefore, the present paper launches on this problem. Two permeability conditions, i.e., both the open-pore and the sealed-pore boundaries are considered. Twophase models based on Biot theory are used to verify the feasibility of the proposed theoretical models. The paper is organized as follows. The wave equations for gas hydrate-bearing sediments are briefly presented in Section 2. Then, in Section 3, the reflection models are established at free boundary for an incidence of P1-wave and S1-wave. Two-phase models are given in Section 4. Numerical analysis is carried out in Section 5. Finally, a set of conclusions are given in Section 6.

H. Qiu, T. Xia, B. Yu et al. / Wave Motion 85 (2019) 67–83

69

Fig. 1. Pore-scale distributions of gas hydrate (gray) and sediment grains (black).

2. Wave propagation in gas hydrate-bearing sediment In general, the gas hydrate-bearing sediment is assumed to be a continuum consisting of sediment grains with connected pores occupied by the mixture of water and gas hydrate. The indices ‘‘s’’, ‘‘w’’, and ‘‘h’’ are used to represent the three phases in gas hydrate-bearing sediment, i.e., sediment grains phase, pore water phase, and pore gas hydrate phase, respectively. Therefore, the volume fraction can be expressed as

φs = 1 − φ,

φw = sr φ,

φh = (1 − sr ) φ,

(1)

where φ is the porosity; sr is the water saturation. Apparently, one key difference between the fully water-saturated sediment and gas hydrate-bearing sediment is the existence of hydrate in the pores. The content and distribution patterns of hydrate will significantly affect the propagation properties of waves in the sediment. Four pore-scale gas hydrate distribution patterns are given by Dvorkin et al. [4] and Waite et al. [6], as shown in Fig. 1. In Fig. 1(C) and (D), the propagation of waves in sediment can be modeled by fully water-saturated porous elastic theory (Biot theory) and are not in the scope of this work. However, for the distribution patterns of hydrate in Fig. 1(A) and (B), the hydrate formed in the pore is assumed to have no direct contact or partially in contact with sediment grain. In these cases, the traditional Biot theory may not be applicable. As an alternative, a Biot-type three-phase theory is adopted in this study, and the isotropic stress–strain relationships considering the coupling between sediment grains and hydrate are given by Carcione et al. [23]:

( ) σijs = K1 θ s + C12 θ w + C13 θ h δij + 2µ1 dsij + µ13 dhij ,

(2a)

σ = C12 θ + K2 θ + C23 θ , ( ) σijh = C13 θ s + C23 θ w + K3 θ h δij + 2µ3 dhij + µ13 dsij ,

(2b)

w

s

w

h

(2c)

where σijs and σijh are the stress tensors; σ w is the average stress of pore water; θ α (α = s, w, h) is the volume strain of the α phase; dαij is the deviator strain. When the microstructure is considered, elastic moduli are not only related to the properties and volume fraction of each phase, but also related to the pore morphology of hydrate as well as the sediment stiffening due to consolidation. The expressions for the nine elastic moduli could be expressed as [1,23]: K1 = [(1 − c1 ) φs ]2 Kav + Ksm , C12 = (1 − c1 ) φs φw Kav ,

K3 = [(1 − c3 ) φh ]2 Kav + Khm ,

C13 = (1 − c1 ) (1 − c3 ) φs φh Kav ,

µ1 = [(1 − g1 ) φs ] µav + µsm , 2

2 K2 = φw Kav ,

(3a)

C23 = (1 − c3 ) φw φh Kav ,

µ13 = (1 − g1 ) (1 − g3 ) φs φh µav ,

(3b)

µ3 = [(1 − g3 ) φh ] µav + µhm , 2

(3c)

in which c1 = Ksm /φs Ks ,

g1 = µsm /φs µs ,

c3 = Khm /φh Kh ,

Kav = [(1 − c1 ) φs /Ks + φw /Kw + (1 − c3 ) φh /Kh ]

−1

,

g3 = µhm /φh µh ;

µav = [(1 − g1 ) φs /µs + φw /iωηw + (1 − g3 ) φh /µh ]−1 .

Ks , Kw and Kh are the bulk moduli of each phase, respectively; µs and µh are the shear moduli of sediment grains phase and gas hydrate phase, respectively. Ksm and µsm are the bulk and shear moduli of sediment frame; Khm and µhm are the bulk and shear moduli of hydrate frame; ηw is the dynamic viscosity of the pore water; ω is the angle frequency; i is the imaginary unit. Introducing the Pride consolidation parameter α and contact parameter ε to respectively account for the sediment stiffening due to consolidation and the supporting effect of the gas hydrate on the sediment frame, the expressions for the frame moduli could be expressed as [1]:

(1 − φw − εφh ) Ks (1 − φw − εφh ) µs , µsm = , 1 + α (φw + εφh ) 1 + αγ (φw + εφh ) φh µh φ h Kh = , µhm = , 1 + α (1 − φh ) 1 + αγ (1 − φh )

Ksm =

(4a)

Khm

(4b)

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H. Qiu, T. Xia, B. Yu et al. / Wave Motion 85 (2019) 67–83

with

γ = (1 + 2α) /(1 + α) . In absence of body forces, the three coupled equations for the motion of sediment grains, pore water, and hydrate, can be expressed as [1,12,23]:

) ) ( ( ¨ hi + b12 u˙ si − u˙ w + b13 u˙ si − u˙ hi = σijs,j , ρ11 u¨ si + ρ12 u¨ w i i + ρ13 u ) ) ( ( ˙ hi = σ,wi , ˙ si + b23 u˙ w ¨ hi + b12 u˙ w ρ12 u¨ si + ρ22 u¨ w i −u i −u i + ρ23 u ) ) ( ( ¨ hi + b13 u˙ hi − u˙ si + b23 u˙ hi − u˙ w = σijh,j . ρ13 u¨ si + ρ23 u¨ w i i + ρ33 u

(5a) (5b) (5c)

α

where ui (α = s, w, h) are the displacement vectors; the overlaying dots indicate time derivatives. Considering the interaction between gas hydrate and sediment, the expressions of six dynamics mass coefficients ρij can be expressed as [12,23]

ρ11 = a13 φs ρs + (a12 − 1) φw ρw + (a31 − 1) φh ρh ,

(6a)

ρ22 = (a12 + a23 − 1) φw ρw φs φw Kav ,

(6b)

ρ33 = (a13 − 1) φs ρs + (a23 − 1) φw ρw + a31 φh ρh ,

(6c)

ρ12 = − (a12 − 1) φw ρw ,

(6d)

ρ13 = − (a13 − 1) φs ρs − (a31 − 1) φh ρh ,

(6e)

ρ23 = − (a23 − 1) φw ρw ,

(6f)

where ρs , ρw and ρh are the true density of each phase, respectively; αij is the inertial drag parameter of the phase i flowing through the phase j, also referred to as tortuosity, and can be found in [12,23]. The dissipative coefficients b12 and b23 are [20] 2 b12 = ηw φw /κs ,

2 b23 = ηw φw /κh ,

(7)

where ηw is the dynamic viscosity of the pore water; κs and κh are the effective permeability of the sediment and of the hydrate matrices, calculated from the sediment and hydrate frame permeabilities κs0 and κh0 by Kozeny–Carman relationships [20]:

κs = κs0 s3r ,

κh = κh0 (φ/φh )2 (φw /φs )3 .

(8)

The friction between hydrate and sediment is the dominant mode of energy dissipation and then b13 can be expressed as [12] b13 = b013 (φh φs )2 , where b013

(9)

is a viscous reference value that is independent of volume fractions, i.e., b013

is only related to the contact properties of the sediment grains and hydrates. With the aid of the usual Helmholtz resolution of a vector [16,20,23], the three displacement vectors can be conveniently written in the following forms: uαi = ∇ϕ α + ∇ × ψα , α

∇ · ψα = 0,

(10)

α

where ϕ and ψ (α = s, w, h) are the scalar and vector potential functions of α phase, respectively. Plugging Eq. (10) into Eq. (5), two sets of equations, one corresponding to compressional wave potentials and the other corresponding to shear wave potentials, can be obtained as [20,23] [ρ] {φ ¨ } + [b] {φ˙ } − [R] [∆] {φ} = 0,

(11a)

¨ } + [b] {ψ ˙ } − [µ] [∆] {ψ} = 0, [ρ] {ψ

(11b)

in which

[ ρ11 [ρ] = ρ12 ρ13 [ [R] =

ρ12 ρ22 ρ23

] ρ13 ρ23 , ρ33

K1 + 4µ1 /3 C12 C13 + 2µ13 /3

C12 K2 C23

b12 + b13 −b12 −b13

[ [b] =

C13 + 2µ13 /3 C23 , K3 + 4µ3 /3

]

⎡ 2 ⎤ ] ∇ 0 0 −b13 −b23 , [∆] = ⎣ 0 ∇ 2 0 ⎦ , b13 + b23 0 0 ∇2 ⎡ ⎤ ⎡ s ⎤ ] ϕs ψ µ1 0 µ13 0 0 0 , {φ} = ⎣ ϕ w ⎦ , {ψ} = ⎣ ψw ⎦ . [µ] = µ13 0 µ3 ϕh ψh

−b12 b12 + b23 −b23 [

The time-harmonic plane-wave solution of Eq. (11) are assumed to be

{ s w h} { s w s h s } [ ( )] ϕ , ϕ , ϕ = AP , δP AP , δP AP exp i ωt − k p · r ,

(12a)

H. Qiu, T. Xia, B. Yu et al. / Wave Motion 85 (2019) 67–83

71

Fig. 2. Reflected waves generated by incident P1-wave (or S1-wave) at free boundary.

{ s w h} { s w s h s } ψ , ψ , ψ = BS , δS BS , δS BS exp [i (ωt − k s · r )] ,

(12b)

where kp and ks are complex wave vectors of the compressional and shear waves, respectively; r is the position vector. The bulk waves are homogeneous waves, and the propagation vectors and attenuation vectors are pointing in same directions. Therefore, the following relationships can be derived as l2p = k 2p ,

l2s = k 2s ,

(13)

where lp and ls are the wave numbers, which are moduli of complex wave vectors. Substituting Eq. (12) into Eq. (11), the linear system of equations for controlling amplitude components of compressional and shear waves can be obtained as T Asp

Aw p

[

[

Q Bss

]T

= 0,

(14a)

] h T

= 0,

(14b)

Ahp

Bw s

Bs

in which T = ω2 ρ − iωb − l2p R,

Q = ω2 ρ − iωb − l2s µ.

Since Eq. (14) has nonzero solutions, the characteristic equations for the compressional and shear waves can be derived as det (T) = 0,

det (Q) = 0.

(15)

Compared with the work of Leclaire [20], the derivation process from Eq. (12) to Eq. (15) has been partially modified according to the author’s habits. Therefore, compared with the characteristic equation obtained by Leclaire [20], both sides of the equal sign of Eq. (15) are multiplied by ω2 . It is suggested from Eq. (15) that there exist three compressional waves (herein referred to as P1-wave, P2-wave, and P3-wave, respectively) and two shear waves (S1-wave and S2-wave) in gas hydrate-bearing sediment. The phase velocity of P1-wave and S1-wave is fast while their attenuation is small, which are consistent with the compressional and shear waves in classical elasticity media. The P2-wave, P3-wave, and S2-waves are slow waves and are strongly attenuated. These five bulk waves are dispersive, which means that the phase velocity is related to the frequency. Substituting the complex wave numbers obtained from Eq. (15) into Eq. (14), the relations between the amplitudes of different phases in Eq. (12) for different bulk waves can be obtained as

δPiw = (t21 t13 − t11 t23 ) /(t12 t23 − t22 t13 ) ,

δPih = (t21 t12 − t11 t22 ) /(t13 t22 − t23 t12 ) ,

δ = (q21 q13 − q11 q23 ) /(q12 q23 − q22 q13 ) , w Sj

δSjh = (q21 q12 − q11 q22 ) /(q13 q22 − q23 q12 ) ,

(16a) (16b)

where i = 1, 2, and 3; j = 1 and 2; tij and qij are the elements in the matrices T and Q. 3. Modeling of wave reflection at free boundary The aim of the present work is to establish a wave reflection model in gas hydrate-bearing sediment, as shown in Fig. 2. A plane P1-wave or S1-wave with an angular frequency ω and an incident angle θ0 propagates in gas hydrate-bearing sediment half-space, and then arrives at the free surface. As a consequence, five reflected waves (i.e. P1, P2, P3, S1 and S2-waves) are generated. In the two-dimensional xz-plane (P-SV system), we take ∂/∂ y ≡ 0. Thus the shear wave potential function vector in Eq. (7) can be expressed as follows:

ψα = 0

(

ψα

0 ,

)

(17)

and the displacements can be expressed as uαx = ∂ϕ α /∂ x − ∂ψ α /∂ z ,

uαz = ∂ϕ α /∂ z + ∂ψ α /∂ x.

(18)

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3.1. Boundary conditions The boundary conditions should be explicit for the reflection problems at the free boundary. The free boundary conditions of gas hydrate-bearing sediments are proposed based on the free boundary conditions in the Biot theory suggested by Tajuddin [34], as well as Tajuddin and Hussaini [31]. Firstly, considering the coupling effects between hydrate and sediment, it is reasonable to assume that the displacement between the two solid phases is coordinated when the amplitude of the incident wave is small. Therefore, the boundary conditions can be expressed as usz = uhz ,

usx = uhx .

(19)

Then, it is obvious that the normal stress and tangential stress at the free boundary are zero, i.e.

σzs + σ w + σzh = 0,

σxzs + σxzh = 0.

(20)

There is another condition related to the permeability at the boundary, which can be expressed as

σ w = 0 (for open-pore boundary),

(21a)

∂σ /∂ z = 0 (for sealed-pore boundary).

(21b)

w

It is noteworthy that if the coupling between hydrate and sediment frame is not considered, i.e. there is no direct contact between the two solid phases, the boundary condition is more reasonable in the following form:

σzh = 0,

σxzh = 0.

(22)

It is found in the calculation that there is almost no difference between the two boundary conditions when using the same parameters of the gas hydrate-bearing sediments. For the purpose of comparison, we adopt the form of Eq. (19) in this paper. 3.2. Incident P1-wave For the incidence of P1-wave, the displacement potential of the incident wave can be expressed as

{ s w h} { s w s h s } ϕin , ϕin , ϕin = Ain , δP1 Ain , δP Ain exp [i (ωt − lP1 sin θ0 x + lP1 cos θ0 z )] ,

(23)

The potential functions for the reflected waves are 3 { s w h} ∑ { s w s h s } ϕrP , ϕrP , ϕrP = ArPi , δPi ArPi , δPi ArPi exp [i (ωt − lxPi x − lzPi z )] ,

(24a)

i=1 2 { s } ∑ { s w s h s } [ ( )] ψrS , ψrSw , ψrSh = BrSj , δSj BrSj , δSj BrSj exp i ωt − lxSj x − lzSj z ,

(24b)

j=1

where i = 1, 2 and 3; j = 1 and 2; lxPi and lxSj are the components of the wave numbers in the x-axis for reflected waves. And lzPi = p.v.

√

l2Pi − l2xPi ,

lzSj = p.v.

√

l2Sj − l2xSj ,

(25)

are the components of the wave numbers in the z-axis; p.v. represents the principal value of the square root of a complex number. Since the components of the wave numbers of incident and reflected waves are equal in the x-axis (Snell law), the following relationships can be obtained as lP1 sin θ0 = lxPi = lxSj .

(26)

By combining the potential functions and boundary conditions, the following equations of the potential amplitude relations between the incident P1-wave and reflected waves in matrix form can be obtained: Mt AsrP1 , AsrP2 , AsrP3 , BsrS1 , BsrS2

]T

˜ AsrP1 , AsrP2 , AsrP3 , BsrS1 , BsrS2 M

]

[

[ t

= Asin Nt (for open-pore boundary), = Asin˜ Nt (for sealed-pore boundary),

(27a) (27b) Asin

where the corresponding elements in the matrices are given in Appendix in detail. Since the amplitude is assumed to be known, the amplitude of reflected waves can be solved. The reflection coefficients for each waves Rj (j = rP1, rP2, rP3, rS1 and rS2) can be defined as the ratios of the corresponding displacement amplitude. For the incidence of P1-wave, it can be expressed as Rj = Asj lj /Asin lP1 .

(28)

H. Qiu, T. Xia, B. Yu et al. / Wave Motion 85 (2019) 67–83

73

Furthermore, the normalized surface displacement at the free surface can be obtained as

⎛ uz |z =0 = ⎝−Asin lz +

3 ∑

ArPi lzPi +

2 ∑

i=1

⎛ ux |z =0 = ⎝Asin lx +

3 ∑

⎞/ Asin lP1 ,

(

BrSj lx ⎠

)

(29a)

j=1

ArPi lx −

i=1

2 ∑

⎞/ Asin lP1 .

)

(

BrSj lzSj ⎠

(29b)

j=1

3.3. Incident S1-wave For the incidence of S1-wave, the displacement potential of the incident wave can be expressed as

} } { { s h s w s Bin exp [i (ωt − lS1 cos θ0 x + lS1 sin θ0 z )] , Bin , δS1 ψin , ψinw , ψinh = Bsin , δS1

(30)

Through similar mathematical derivation, the following equations of the amplitude coefficients of the reflected waves in matrix form can be derived: Rt AsrP1 , AsrP2 , AsrP3 , BsrS1 , BsrS2

]T

= Bsin Pt (for open-pore boundary), [ ] T ˜ Rt AsrP1 , AsrP2 , AsrP3 , BsrS1 , BsrS2 = Bsin˜ Pt (for sealed-pore boundary),

[

(31a) (31b)

The elements of matrices are given in Appendix in detail. Supposing the amplitude of the incident wave is known, the amplitude of the reflected waves for S1-wave incidence can be obtained by solving Eq. (28), and reflection coefficients Rj can be expressed as Rj = Asj lj /Bsin lS1 .

(32)

Furthermore, using Eq. (15), the amplitudes of vertical and horizontal displacements at free boundary can be solved as

⎛ uz |z =0 = ⎝Bsin lx +

3 ∑

ArPi lzPi +

i=1

⎛ ux |z =0 = ⎝Bsin lz +

3 ∑

2 ∑

⎞/ Bsin lS1 ,

(33a)

Bsin lS1 .

(33b)

(

BrSj lx ⎠

)

j=1

ArPi lx −

i=1

2 ∑

⎞/ )

(

BrSj lzSj ⎠

j=1

4. Model verifications Although the gas hydrate-bearing sediment is composed of three phases, it can be regarded as the fully water-saturated sediment in some special cases. In order to verify the rationality of the reflection model based on the Biot-type three-phase theory, reflection models based on the Biot theory are discussed and the general solutions are compared with the degenerate solutions in this section. 4.1. Modeling of wave reflection based on the Biot theory The governing equations of the compressional and shear waves in fully water-saturated sediment in terms of displacement potentials are given as follows [16,32]: [ρ] {φ ¨ } + [b] {φ˙ } − [R] [∆] {φ} = 0,

(34a)

¨ } + [b] {ψ ˙ } − [µ] [∆] {ψ} = 0. [ρ] {ψ

(34b)

in which

[ [ρ] =

[ [µ] =

ρ11 ρ12 N 0

ρ12 ρ22 ] 0 , 0

]

,

[ [b] =

[ {φ} =

ϕs ϕw

b −b

]

−b b

,

]

,

[∆] =

[ {ψ} =

ψs ψw

]

[

∇2

0

0

∇2

]

,

[ [R] =

P Q

Q R

]

,

,

where ρ12 = − (a12 − 1) φρf , ρ11 = (1 − φ) ρs − ρ12 , and ρ22 = φρf − ρ12 are the dynamic mass coefficients, b is the coefficient of dissipation, and are given as, b = ηf φ 2 /κf . Constant ρs and ρf are the densities of the solid material and pore fluid; ηf is the absolute viscosity of the fluid; κf is the intrinsic permeability of frame. The four elastic moduli are given as [32] N = µsm ,

(35a)

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H. Qiu, T. Xia, B. Yu et al. / Wave Motion 85 (2019) 67–83

4 (1 − φ − Ksm /Ks )2 Ks + N , 3 (1 − φ − Ksm /Ks ) + φ Ks /Kf φ (1 − φ − Ksm /Ks ) , Q = (1 − φ − Ksm /Ks ) + φ Ks /Kf φ2 , R= (1 − φ − Ksm /Ks ) + φ Ks /Kf P = Ksm +

(35b) (35c) (35d)

Through similar mathematical derivation, the characteristic equations for the compressional and shear waves can be derived as det ˜ T = 0,

det ˜ Q = 0,

( )

( )

(36)

in which

˜ µ. T = ω2˜ ρ − iω˜ b − l2p˜ R, ˜ Q = ω2˜ ρ − iω˜ b − l2s˜ Thus, there exist two compressional waves (herein referred to as P1-wave and P2-wave, respectively) and one shear wave (S1-wave) in fully water-saturated sediment. The relations between the amplitudes of various bulk waves in different phases can be obtained as

˜ δPiw = −˜ t11 /˜ t12 , ˜ δSw = −˜ q11 /˜ q12 .

(37)

The amplitude coefficients of reflected waves in matrix form can be reduced to: Mb AsrP1 , AsrP2 , BsrS

[

]T

[ b

]T

,

AsrP2

b

[

AsrP1

,

AsrP2

b

[

AsrP1

,

AsrP1

˜ M R

˜ R

,

BsrS

T BsrS T AsrP2 BsrS

,

]

,

]

= Asin Nb (Open-pore boundary for the incidence of P1-wave), (Sealed-pore boundary for the incidence of P1-wave),

(38a)

=

Asin Nb

=

Bsin Pb

(Open-pore boundary for the incidence of S1-wave),

(38c)

=

Bsin Pb

(Sealed-pore boundary for the incidence of S1-wave).

(38d)

˜

˜

(38b)

Through the same way, the reflection coefficients Rj and the amplitudes of vertical and horizontal displacement can be solved. 4.2. Special cases Three special cases that can be degenerated into Biot theory are discussed in this section, and method of determining the material constants is given in detail. Case 1: As shown in Fig. 1(A), when the gas hydrate formed in the pore floats in the water and does not directly contact with the sediment frame, the gas hydrate and sediment frame are weakly coupled. At this point, in addition to the Biot-type threephase theory, the Biot theory can also approximately describe the propagation of waves in gas hydrate-bearing sediment. Hypothetically, hydrate is embedded in the pore water and the deformation is coordinated with water. Water and hydrate in the pore can be regarded as a new fluid, and the three-phase medium can be degenerated into a fully fluid-saturated medium. ′ In this case, according to Eqs. (4a) and (8), the bulk modulus Ksm , shear modulus µ′sm and effective intrinsic permeability κf′ of sediment frame are as follows: ′ Ksm = (1 − φ) Ks /(1 + αφ) ,

µ′sm = (1 − φ) µs /(1 + αγ φ) ,

κf′ = κs0 s3r .

(39)

The mixing density and bulk modulus of two-phase fluid can be expressed as [35]

ρf = sr ρw + (1 − sr ) ρh ,

Kf = [sr /Kf + (1 − sr ) /Kh ]−1 .

(40)

Case 2: As shown in Fig. 1(B), the hydrate formed in the pore may partially in contact with sediment frame. Considering the extreme case, the hydrate is considered to have the greatest contact with the sediment frame, i.e., the gas hydrate and sediment frame are essentially consolidated. At this point, the hydrate can be considered as part of the sediment frame, and the three-phase medium can be degenerated into a fully fluid-saturated medium (hereinafter referred to as model B). The volume fraction of pore water φw in Biot-type three-phase theory becomes the porosity in Model B, and the fluid in ′′ the pore is water. Similarly, according to Eqs. (4a) and (8), the bulk modulus Ksm , shear modulus µ′′sm and effective intrinsic permeability κf′′ of sediment frame are as follows: ′′ Ksm = (1 − φw ) Ks /(1 + αφw ) ,

µ′′sm = (1 − φw ) µs /(1 + αγ φw ) ,

κf′′ = κs0 s3r .

(41)

Case 3: The last special case corresponds to the absence of hydrate formation in the pores. Neglecting the presence of hydrate i.e., sr = 1, the material constants ρi3 , bi3 , Ri3 , and µi3 will vanish, and the Biot-type three-phase theory can be ′′′ degraded into classical two-phase Biot theory. From Eqs. (4a) and (8), the bulk modulus Ksm , shear modulus µ′′′ sm and effective intrinsic permeability κf′′′ of sediment frame are as follows: ′′′ Ksm = (1 − φ) Ks /(1 + αφ) ,

µ′′′ sm = (1 − φ) µs /(1 + αγ φ) ,

κf′′′ = κs0 .

(42)

H. Qiu, T. Xia, B. Yu et al. / Wave Motion 85 (2019) 67–83

75

Table 1 Physical parameters of the gas hydrate-bearing sediment. Parameters

Symbol

Model 1

Porosity, % Consolidation degree, Grain density, kg/m3 Water density, kg/m3 Hydrate density, kg/m3 Grain bulk modulus, GPa Grain shear modulus, GPa Pore fluid bulk modulus, GPa Hydrate bulk modulus, GPa Hydrate shear modulus, GPa Sediment permeability, m2 Hydrate permeability, m2 Water viscosity, Pa s Grain/hydrate coefficient of friction, kg/(m3 s)

φ α ρs ρw ρh

35 25 2700 1000 900 38 44 2.25 7.9 3.3 5.0 × 10−11 1.0 × 10−5 1.8 × 10−3 2.2 × 108

Ks

µs Kw Kh

µh κs0 κh0 ηw

b013

Fig. 3. Amplitude ratios and phase shifts in case 1 and 2 with an incident P1-wave: solid line for three-phase type model; dash line for two-phase type model.

4.3. Model comparison For the validation purpose, the models based on the Biot-type three-phase theory (hereinafter referred to as three-phase type model) and Biot theory (hereinafter referred to as two-phase type model) are compared in special cases. For case 1, the contact parameter is taken to be 1, and for case 2, the contact parameter is taken to be 0. The frequency of incident wave is taken to be 10 Hz and the water saturation is taken to be 60% in this section. The values of other parameters could refer to Guerin and Goldberg [12] and Lee and Waite [1], as listed in Table 1. As shown in Fig. 3, the amplitude ratio |Rj | and phase shift argRj are functions of incident angle for an incident P1wave. Although, the incoming P1-wave in Biot-type three-phase theory is split into five waves (P1, S1, P2, P3 and S2) at the boundary instead of the usual two waves (P1 and S1) in classical elasticity theory [15,26], or three waves (P1, S1, and P2) in Biot theory [32,33], the amplitude ratios of reflected P3-wave and S2-wave are significantly smaller than the other reflected waves. Therefore, the amplitude ratios and phase shifts of P3-wave and S2-wave are omitted in this section. Case 1 (red line) shows the results when the gas hydrate and sediment frame are weakly coupled. It is observed that, there is almost no difference between the two models in Fig. 3(a) and (b), whereas in Fig. 3(c) the amplitude ratio |RP2 | of two-phase type model is greater than that of three-phase type model. Case 2 (blue line) shows the results when the gas hydrate and

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H. Qiu, T. Xia, B. Yu et al. / Wave Motion 85 (2019) 67–83

Fig. 4. Normalized surface displacements in case 1 and 2 with an incident P1-wave: solid line for three-phase type model; dash line for two-phase type model.

sediment frame are essentially consolidated. Except that the amplitude ratio |RP1 | of the three-phase type model is slightly smaller than that of the two-phase type model, the results of other waves are very consistent. As depicted in Fig. 3(d) and (e), unlike the reflection model based on the classical elasticity theory, the phase angles argRP1 and argRS1 of the two models approach but not equal to 180◦ . In the two cases, the incident angle has the same effect on the amplitude ratios and phase angles of reflected waves, and the results of the calculation are similar when the angle of incidence is the same. Fig. 4 presents the results for the normalized horizontal and vertical components of surface displacements as a function of incident angle for an incident P1-wave. As shown in the figure, the vertical component of the surface displacement caused by the incident waves and the reflected waves is twice the displacement caused by the incident wave at the normal incidence in all cases and theories. That is to say, the amplification effect of the free boundary on the vertical incident P1 wave is applicable for all cases. It can also be seen from the figure that the vertical displacement decreases, the horizontal displacement first increases to reach the extreme value and then decreases, and the displacement ratio (ux /uz ) increases, as the incident angle increases. The displacements of the two models are also very consistent in all figures in the two cases. Fig. 5 depicts the amplitude ratio and phase shift for an incident S1-wave. Two critical angles can be seen in all curves. The first one is due to the phase velocity of the S1-wave being less than that of the P1-wave. It is approximately 21◦ in case 1 and 27◦ in case 2. The second one is 45◦ for all curves. At this angle, the amplitude ratio |RP1 | equals to 0. In other words, the total reflection of the S-wave occurs at the incident angle of 45◦ . This finding is consistent with the results in classical elasticity theory [15,26]. Below the first critical angle, results of the two models are nearly coincident, and the phase angle argRP1 is close to 0, and the phase angle argRS1 is close to π . The phenomenon that the amplitude ratio |RP2 | of the two-phase type model is greater than that of the three-phase type model also occurs for an incident S1-wave. The normalized horizontal and vertical components of surface displacements and the displacement ratio are illustrated in Fig. 6. The horizontal component of the surface displacement caused by the incident S1-waves and the reflected waves is twice the displacement caused by the incident S1-wave at the normal incidence in all cases and theories, however, the vertical components has a value of zero. The vertical components of displacement reach a minimum value, and the horizontal components and the amplitude ratios have a maximum value around the first critical angle in all curves. It can be seen from the figure that the results of the two models are very close, except for the vicinity of the first critical angle. In all the above analysis, except for the amplitude ratio |RP2 | in Case 1, the calculation results of the two models are very close, validating the rationality of the proposed model. The deviations of P2-wave may be because the shear resistance of the hydrate is neglected when the pore water and hydrate are equivalent to a new fluid in case 1. Unlike the two-phase model, which can only consider the extreme distribution patterns of the hydrate, the biggest advantage of the proposed model is that the contact parameters are introduced to characterize the effect of the degree of contact between the hydrate and the sediment frame. 5. Numerical results and discussions In general, in addition to the physical properties, the content and coupling effects with the sediment frame of the gas hydrate are the most influential factors for the reflection properties in gas hydrate-bearing sediment. In this section, the numerical examples are conducted to analyze the nature of dependence of the horizontal displacement and the displacement ratio for incident P1-wave and S1-wave on the angle of incidence, the liquid saturation, the contact parameter, the frequency of the incident wave, and permeability condition, respectively. The consolidation parameter α is taken to be 25, and the porosity is 0.35. The values of the physical parameters are summarized in Table 1. 5.1. Effects of incident angle on reflected waves The amplitude ratios |Rj | of the proposed model as a function of the incident angle are plotted in Fig. 7 to investigate the transformation of wave modes at free boundary before the parameter sensitivity study. The contact parameter is taken to

H. Qiu, T. Xia, B. Yu et al. / Wave Motion 85 (2019) 67–83

77

Fig. 5. Amplitude ratios and phase shifts in case 1 and 2 with an incident S1-wave: solid line for three-phase type model; dash line for two-phase type model.

Fig. 6. Normalized surface displacements in case 1 and 2 with an incident S1-wave: solid line for three-phase type model; dash line for two-phase type model.

0.12 (a value given by Lee and Waite [1] for the Mallik 5L-38 permafrost gas hydrate research well, western Canada), the frequency of the incident wave is taken to be 10 Hz, and the water saturation is assumed to be 60% in this example. Fig. 7(a)– (c) depict the results of incident P1-wave. When the incident P1-wave strikes the boundary perpendicularly, i.e., the incent angle equals zero, the amplitude ratio |RP1 | is close to one, the amplitude ratios |RS1 | and |RS2 | are zero, and the amplitude ratios |RP2 | and |RP3 | are close to zero. |RP1 | decreases with an increase of incident angle before reaching the minimum values, and have a reverse tendency thereafter. |RS1 | increases with an increase of incident angle before reaching the maximum value, then decrease thereafter. When the incident angle is 90◦ , i.e., the grazing incidence, only the reflected P1-wave exists. Fig. 7(d)–(f) depict the results of incident S1-wave. When the incident angle equals zero, the amplitude ratio |RS1 | is close to one, the amplitude ratios |RP1 |, |RP2 | and |RP3 | are zero, and the amplitude ratio |RS2 | is close to zero. When the incident angle equals 45◦ and 90◦ , only the reflected S1-wave exists. 5.2. Effects of water saturation and contact parameter on displacements To investigate the effects of the content and coupling effects of the gas hydrate on the behavior of the proposed model, the horizontal displacements and displacement ratios at open-pore boundary are plotted in Figs. 8–10. The frequency of the incident wave is taken to be 10 Hz, and the water saturation is assumed to be 20%, 40%, 60%, 80% and 100%, respectively.

78

H. Qiu, T. Xia, B. Yu et al. / Wave Motion 85 (2019) 67–83

Fig. 7. Effects of incident angle on amplitude ratios of reflected waves: a–c for an incident P1-wave; d–f for an incident S1-wave.

Fig. 8(a)–(c) illustrate the results for the horizontal displacements at surface as a function of incident angle for an incident P1-wave. It is worth noting that when the water saturation is 100%, there is no gas hydrate in the pore, and the reflection model turns into Case 3, as shown in the black curve. The contact parameter is taken to be 0, 0.12 and 1 for the purposes of comparison. It is observed that the water saturation has different influences on horizontal displacement with different contact parameters. Lower water saturation causes larger horizontal displacements when the hydrate and sediment are essentially consolidated (ε = 0), whereas the opposite trend appears when they are weakly coupled (ε = 1). Similar to Fig. 8(a), the horizontal displacement increases as the hydrate content increases in Fig. 8(b) (ε = 0.12). The displacement ratios ux / uz are depicted in Fig. 8(d)–(f). Five different cases of water saturation are also considered. As the vertical displacement is insensitive to water saturation and contact parameters, the effect of contact parameter and water saturation on the amplitude ratios is consistent with the effect of horizontal displacement. It is also worth noting that, the amplitude ratio is 1, i.e., the horizontal displacement equals to the vertical displacement around θ0 = 70◦ in the black line. As the water saturation increases, the corresponding incident angle decreases in Fig. 8(d)–(e), but increases in Fig. 8(f). Similarly, the horizontal surface displacement and displacement ratio are depicted for an incident S1-wave in Fig. 9. The water saturation also has different effects on horizontal displacements with different contact parameters. As the water saturation decreases, the first critical angle increases with ε = 0 and 0.12, but decreases with ε = 1. The displacements are almost independent of water saturation with small incident angles and the influences increase near the critical angle. Additionally, the horizontal displacement is far greater than the vertical displacement below the first critical angle, and as the water saturation decreases, the displacement ratio decreases with ε = 0 and 0.12, but increases with ε = 1. Fig. 10 shows the results for the horizontal displacements and displacement ratios as a function of contact parameter for an incident P1-wave (θ0 = 70◦ ) and incident S1-waves (θ0 = 15◦ and 60◦ ). The other material parameters remain to be constants, and the black solid curve still represents the results of Case3 in this example. As observed on Fig. 10(a) and (d), for an incident P1-wave, the horizontal displacements and displacement ratios decrease with the increases of the contact parameter, and the water saturation is higher, the scope of decrease is bigger. There is a contact parameter that causes the effect of water saturation to have an opposite trend, as evidenced by the results of Fig. 8. It can be seen from Fig. 10(b) and (e), for an incident S1-wave with θ0 = 15◦ (less than the first critical angle), the horizontal displacements and displacement ratios increase with the increases of the contact parameter, and the water saturation is higher, the scope of increase is bigger. The critical contact parameter that causes the opposite effect of water saturation still exists before the critical angle. In fact, the difference in reflection properties with different contact parameters can be approximated by classical elasticity theory. The reflection properties based on the classical elasticity theory mainly depend on the undrained Poisson’s ratio of the gas hydrate-bearing sediment under the condition of low frequency. When the contact parameter equals to 1, the undrained bulk modulus increases, but the shear modulus remains constant with the increasing hydrate content. Therefore, the undrained Poisson’s ratio of the medium is increased. When the contact parameter is less than 1, the undrained bulk

H. Qiu, T. Xia, B. Yu et al. / Wave Motion 85 (2019) 67–83

79

Fig. 8. Effects of water saturation on horizontal displacements and displacement ratios for an incident P1-wave: a–c horizontal displacements ux ; d–f displacement ratios ux /uz .

modulus and shear modulus increase with the increasing hydrate content. Therefore, the undrained Poisson’s ratio of the medium may decrease or increase. Therefore, the influence of hydrate content is significantly influenced by the pattern of hydrate distribution. 5.3. Effects of frequency and permeability condition on displacements Low-frequency and permeable boundary condition are used in all of the above examples. To investigate the effects of the frequency and permeability condition, the horizontal displacements and displacement ratios are plotted as a function of frequency in logarithmic coordinates at open-pore and sealed-pore boundary in Fig. 11 for an incident P1-wave (θ0 = 70◦ ) and incident S1-waves (θ0 = 15◦ and 60◦ ). The contact parameter is taken to 0.12, the water saturation is assumed to be 60%, and the other material parameters remain to be constants in this example. It can be seen that results are independent of boundary permeable conditions at low frequency. As the frequency increases, the effects of the permeable condition increase. It can be seen from Fig. 11(a) and (d), as the increase of frequency, the horizontal displacement decreases at openpore boundary and increases at sealed-pore boundary, whereas the displacement ratios increase at both open-pore and sealed-pore boundaries. The horizontal displacement at open-pore boundary is less that at sealed-pore boundary, whereas the displacement ratio shows the opposite phenomenon. From Fig. 11(b) and (e), for an incident S1-wave with θ0 = 15◦ (less than the first critical angle), both the horizontal displacements and displacement ratios decrease with the increase of frequency at both open-pore and sealed-pore boundaries. It can be also found that both the horizontal displacements and displacement ratios at open-pore boundary is less that at sealed-pore boundary. From Fig. 11(c) and (f), for an incident S1-wave with θ0 = 60◦ , the horizontal displacements and displacement ratios show the same trend as the frequency changes. 6. Conclusions The theoretical models of wave reflection in gas hydrate-bearing sediments, based on a Biot-type three-phase theory, are presented at free open-pore and sealed-pore boundaries. The amplitude ratio, phase shift and normalized surface displacements based on the Biot-type three-phase theory are compared with the results based on Biot theory in special cases to verify the feasibility of the proposed theoretical models. The transformation of wave modes at free boundary for the incidence of both P1-wave and S1-wave with different incident angles is investigated, then the effects of the incident angle, the liquid saturation, the contact parameter, the frequency of the incident wave, and permeability condition on the horizontal displacement and displacement ratios are analyzed for the incidence of both P1-wave and S1-wave. Based on the previous discussion, the following points are of notice:

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H. Qiu, T. Xia, B. Yu et al. / Wave Motion 85 (2019) 67–83

Fig. 9. Effects of water saturation on horizontal displacements and displacement ratios for an incident S1-wave: a–c horizontal displacements ux ; d–f displacement ratios ux /uz .

Fig. 10. Effects of contact parameter on horizontal displacements and displacement ratios for an incident P1- and S1-wave: a–c horizontal displacements ux ; d–f displacement ratios ux /uz .

(i) Although in most cases there are five reflected waves in gas hydrate-bearing sediments based on Biot-type three-phase theory, some special cases may occur at particular incident angles. For example, for an incident of P1-wave, no reflected

H. Qiu, T. Xia, B. Yu et al. / Wave Motion 85 (2019) 67–83

81

Fig. 11. Effects of c frequency and permeability condition on horizontal displacements and displacement ratios for an incident P1- and S1-wave: a–c horizontal displacements ux ; d–f displacement ratios ux / uz .

shear wave is generated at the normal incidence, and only a reflected P1-wave appears at the grazing incidence. Moreover, no reflected compressional wave is generated at the normal incidence, and only a reflected S1-wave appears at the grazing incidence or the incident angle equals 45◦ for an incident of S1-wave. The amplitudes ratios of reflected P1-wave and S1wave are much greater than the other reflected waves and have the same trends of variation with those based on Biot theory and classical elasticity theory. (ii) When the gas hydrate and sediment frame are weakly coupled or essentially consolidated, calculation results based on Biot-type three-phase theory are very consistent with Biot theory. (iii) The horizontal displacements and displacement ratios are independent of boundary permeable conditions at low frequency. (iv) The presence of hydrate in the pore and the supporting effect on the sediment frame has significantly influences on the horizontal displacements and displacement ratios. The horizontal displacements and displacement ratios increase with the increasing gas hydrate content when the hydrate strongly support sediment frame. However, displacements decrease with the increasing gas hydrate content when the hydrate has no support for sediment frame. Acknowledgments This work was supported by the National Natural Science Foundation of China [grant numbers 51378463, 41502285, 41877243]; and the Natural Science Foundation of Jiangsu Province (grant number BK20150952). Appendix. The explicit expressions of the elements The elements in Mt and Nt in Eq. (27a), for the incidence of plane P1-wave at open-pore boundary based on the Biot-type three-phase theory, are given as follows: h mt11 = 1 − δP1 lzP1 ,

h mt12 = 1 − δP2 lzP2 ,

h mt21 = 1 − δP1 lxP1 ,

h mt22 = 1 − δP2 lxP2 ,

(

(

)

(

)

(

h mt31 = D + E δP1 lxP1 lzP1 ,

(

)

h mt13 = 1 − δP3 lzP3 ,

)

(

h mt14 = 1 − δS1 lxS1 ,

(

)

h mt15 = 1 − δS2 lxS2 ,

(

h mt32 = D + E δP2 lxP2 lzP2 ,

)

)

(

)

h mt33 = D + E δP3 lxP3 lzP3 ,

(

)

−l2zS1 + l2xS1 , ) ( ) 2 ( ) 2 w h h = 0.5D + 0.5E δ −l2zS2 + l2xS2 , mt41 = A + BδP1 + C δP1 lP1 + D + E δP1 lzP1 ( ) ( ) ( ) ( ) 2 w 2 h 2 t w h 2 h = A + BδP2 + C δ lP2 + D + E δP2 lzP2 , m43 = A + BδP3 + C δP3 lP3 + D + E δP3 lzP3 ,

h mt34 = 0.5D + 0.5E δS1

)(

mt35 mt42

)(

( (

h S2 h P2

)

(

)

h h h mt23 = 1 − δP3 lxP3 , mt24 = δS1 − 1 lzS1 , mt25 = δS2 − 1 lzS2 ,

)

(

)

(

)

82

H. Qiu, T. Xia, B. Yu et al. / Wave Motion 85 (2019) 67–83 h lxS1 lzS1 , mt44 = D + E δS1

)

(

h w l2P1 , + C23 δP1 mt51 = C12 + K2 δP1

h lxS2 lzS2 , m145 = D + E δS2

)

(

)

(

h w l2P2 + C23 δP2 mt52 = C12 + K2 δP2

)

(

h w l2P3 , + C23 δP3 mt53 = C12 + K2 δP3

)

(

nt2 = −mt21 ,

nt1 = mt11 ,

mt54 = mt55 = 0.

nt3 = mt31 ,

nt4 = −mt41 ,

nt5 = −mt51 .

in which A = K1 + C12 + C13 − 2µ1 /3 − µ13 /3, D = 2µ1 + µ13 ,

B = C12 + K2 + C23 ,

C = C13 + C23 + K3 − µ13 /3 − 2µ3 /3,

E = µ13 + 2µ3 .

˜ t and ˜ The elements in M Nt in Eq. (27b), for the incidence of plane P1-wave at sealed-pore boundary based on the Biot-type three-phase theory, are given as follows: ) 2 ( h w ˜t51 = C12 + K2 δP1 lP1 lP1 , + C23 δP1 (i = 1, 2, 3, 4 j = 1, 2, 3, 4, 5) , m ) 2 ( ) 2 ( t h w t h w ˜t51 . ˜t55 = 0, ˜ ˜54 = m ˜53 = C12 + K2 δP3 + C23 δP3 lP3 lzP3 , m nt5 = m = C12 + K2 δP2 + C23 δP2 lP2 lzP2 , m ˜ nti = nti ,

˜tij = mtij , m ˜t52 m

The elements in Rt and Pt in Eq. (31a), for the incidence of plane S1-wave at open-pore boundary based on the Biot-type three-phase theory, are given as follows: rijt = mtij ,

t , (i = 1, 2, 3, 4, 5 j = 1, 2, 3, 4, 5) , pt1 = r14

r pr2 = −r24 ,

r pr3 = −r34 ,

r pr4 = r44 ,

pr5 = 0.

The elements in ˜ Rt and ˜ Pt in Eq. (31b), for the incidence of plane S1-wave at sealed-pore boundary based on the Biot-type three-phase theory, are given as follows:

˜ ˜tij , rijt = m

t t t t pt1 = −˜ r14 , ˜ pt2 = ˜ r24 , ˜ pt3 = −˜ r34 , ˜ pt4 = ˜ r44 , ˜ pt5 = 0. (i = 1, 2, 3, 4, 5 j = 1, 2, 3, 4, 5) , ˜

The elements in Mb and Nb in Eq. (38a), for the incidence of plane P1-wave at open-pore boundary based on the Biot theory, are given as follows: mb11 = 2NlxP1 lzP1 , mb12 = 2NlxP2 lzP2

w 2 mb13 = N −l2zS + l2xS , mb21 = (P + R − 2N ) l2P1 + (R + Q ) δP1 lP1 + 2Nl2zP1 ,

(

)

w 2 mb22 = (P + R − 2N ) l2P2 + (R + Q ) δP2 lP2 + 2Nl2zP2 ,

mb23 = 2NlxS lzS ,

w mb31 = R + Q δP1 l2P1 ,

(

)

w mb32 = R + Q δP2 l2P2 ,

(

)

mb33 = 0. nb1 = mb11 , nb2 = −mb21 ,

nb3 = −mb31 .

˜ b and ˜ The elements in M Nb in Eq. (38b), for the incidence of plane P1-wave at sealed-pore boundary based on the Biot theory, are given as follows: ˜bij = mbij , m

˜ nbi = nbi ,

(i = 1, 2 j = 1, 2, 3) ,

w ˜b31 = R + Q δP1 m l2P1 lzP1 ,

(

)

w ˜b32 = Q + Q δP2 m l2P2 lzP2 ,

(

)

˜b33 = 0. m

˜ ˜b31 . nb3 = −m The elements in Rb and Pb in Eq. (38c), for the incidence of plane S1-wave at open-pore boundary based on the Biot theory, are given as follows: rijb = mbij ,

b b , pb2 = r23 , (i = 1, 2, 3 j = 1, 2, 3) , pb1 = −r13

pb3 = 0.

The elements in ˜ Rb and ˜ Pb in Eq. (38d), for the incidence of plane S1-wave at open-pore boundary based on the Biot theory, are given as follows:

˜ ˜tij , rijt = m

t t pt1 = −˜ r13 , ˜ pt2 = ˜ r23 , ˜ pb5 = 0. (i = 1, 2, 3 j = 1, 2, 3) , ˜

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