reflection and transmission of plane waves at a water–porous sediment interface with a...

Transp Porous MedDOI 10.1007/s11242-014-0286-7

Reflection and Transmission of Plane Waves at aWater–Porous Sediment Interface with aDouble-Porosity Substrate

Dan-Dan Lyu · Jin-Ting Wang · Feng Jin ·Chu-Han Zhang

Received: 17 June 2013 / Accepted: 4 February 2014© Springer Science+Business Media Dordrecht 2014

Abstract This paper investigates the wave propagation at the interface between the oceanand the ocean floor. The ocean floor is assumed to be composed of covered porous sedi-ment with an underlying double-porosity substrate. For this purpose, plane wave reflectionand transmission in the coupled water–porous sediment–double-porosity substrate system areanalytically solved in terms of displacement potentials. Using numerical examples, the effectsof the material properties of the underlying double-porosity substrate on the reflection coef-ficients are discussed in detail. Variations in pore and fracture fluid, fracture volume fraction,and permeability coefficients are considered. In addition, two cases of boundary conditionsat the porous sediment–double-porosity substrate interface, i.e., sealed-pore boundary andopen-pore boundary, are compared in the numerical calculations. Results show that mater-ial property variations in the double-porosity substrate may significantly affect the reflectedwave in the overlying water if the sandwiched sediment depth is less than the critical value.

Keywords Plane wave · Reflection and transmission · Ocean floor · Porous sediment ·Double-porosity substrate

1 Introduction

Investigating wave propagation at the water–ocean floor interface is important in classifyingthe material properties of the ocean floor. Studies on wave reflection and transmission atthe ocean–sediment interface over the last three decades could be divided into two maincategories in terms of the ocean floor model.

In the first category, the ocean floor was modeled as homogeneous semi-infinite porousmedia. Stoll and Kan (1981), Wu et al. (1990), Santos et al. (1992), Albert (1993), Dennemanet al. (2002), Sharma (2004), Madeo and Gavrilyuk (2010), and other researchers investigated

D.-D. Lyu · J.-T. Wang (B) · F. Jin · C.-H. ZhangState Key Laboratory of Hydroscience and Engineering,Tsinghua University, Beijing 100084, People’s Republic of Chinae-mail: [email protected]

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the refraction and reflection of acoustic waves at a flat water–sediment interface by modelingthe sediment as porous media based on the Biot theory. Williams et al. (2001) and Yang etal. (2001) studied wave scattering at a rough fluid–poroelastic seafloor interface. Ohkawaet al. (2005) studied the reflection and transmission coefficients of a fluid–sandy seabedconsidering volume scattering. Cui and Wang (2003) analyzed the effect of squirt flow on thereflection and refraction of elastic waves at a fluid–fluid-saturated poroelastic solid interfacebased on the Biot–squirt flow model.

In the second category, a number of studies analyzed the reflection and transmissionof acoustic waves at the interface of water and layered seabed sediments (fluid or solid)with an underlying semi-infinite solid substrate. Hawker and Foreman (1978) studied theplane wave reflection coefficient of an acoustic wave impinging upon an arbitrary number ofinhomogeneous (fluid) sediment layers overlying a semi-infinite homogeneous solid substrate(basement). Kuo (1992) developed an acoustic wave scattering model for three homogeneouslayers. These layers comprised a thin solid sediment layer sandwiched by semi-infinite waterand solid basalt media. Hovem and Kristensen (1992) discussed the reflection loss of planeacoustic waves in shallow water with a relatively soft sediment bottom layer over a hardsolid half-space. Ainslie (1996) analyzed the exact solutions for plane wave reflection andtransmission in a layered fluid sediment sandwiched between a top uniform fluid and a bottomuniform solid substrate. Wang et al. (2013) investigated wave propagation at the ocean–porous sediment interface with an underlying solid substrate. Their results have showed thatthe porous sediment layer has a significant effect on the reflected wave in the overlying water.When the sediment layer exceeds a critical depth, the underlying solid substrate is negligible.

Among the literatures in the second category, the ocean floor was separated by layeredsediments and an underlying elastic solid substrate. However, some limitations occur whenthe heterogeneity of the underlying substrate is taken into account, such as the fractures andcracks. It has been confirmed that there exist two kinds of porosities in fractured or crackedrocks, i.e., matrix porosity and fractures or cracks. Matrix with a very low permeability anda relatively high porosity serves as the fluid reservoir, whereas fractures or cracks with alow porosity and a high permeability serve as the main path of fluid transport. Barenblatt etal. (1960) pioneered in proposing a double-porosity model to investigate fluid transport infissured rocks. A number of researchers investigated the properties of double-porosity mediafrom various aspects based on Barenblatt’s approach. Warren and Root (1963) developed animproved model that allows for coupling between rock deformation and fluid flow. Tuncay andCorapcioglu (1996a,b) used the volume-averaging technique to investigate wave propagationin fractured porous media. Berryman and Wang (1995, 2000) derived the phenomenologicalequations for the poroelastic behavior of double-porosity media and identified the relevantcoefficients in these liner equations. Dai et al. investigated the reflection and transmission ofelastic waves at the interface between an elastic solid and a double-porosity medium (Dai etal. 2006a), a fluid–saturated porous solid and a double-porosity solid (Dai et al. 2006b), andwater and a double-porosity medium (Dai and Kuang 2008).

Based on the abovementioned researches, it can be concluded that it is better to modelocean floor as a marine sediment layer and an underlying fractured or cracked rock substrate.However, to the best of our knowledge, no investigation on the reflection and transmission ofplane waves at the water–ocean floor interface by separating the ocean floor as a sediment layerand an underlying double-porosity substrate has been reported. This problem is meaningfuland essential for exploring oil, gas, or water contained in the fissured rocks of the ocean floor.

In this study, the reflection and transmission of plane waves at the water–ocean floorinterface are investigated, where the ocean floor is assumed to be composed of coveredporous sediment and an underlying double-porosity substrate. Using numerical examples,

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Reflection and Transmission of Plane Waves

Fig. 1 Physical model of thewater–sediment–substrate systemwith a plane wave incident fromthe water

the effects of the properties of the double-porosity substrate, such as fluid property, volumefraction of fracture, and permeability on the reflected waves, are discussed. In addition, theinfluence of the boundary conditions at the interface of the porous sediment and the underlyingdouble-porosity substrate is analyzed.

2 Governing Equations

The analyzed system is composed of the overlying water, the sandwiched porous sedimentlayer, and the underlying double-porosity substrate, as shown in Fig. 1. The overlying wateris assumed as an ideal compressible fluid. The sediment is modeled as linear porous mediausing the Biot theory and the underlying substrate as double-porosity media according toBerryman and Wang (1995, 2000). Using a Cartesian system with the x-axis rightward andz-axis downward, z = 0 and z = h represent the interfaces separating the overlying water,the porous sediment, and the underlying double-porosity substrate, respectively, where h isthe depth of the porous sediment layer. These two interfaces are assumed flat in the followinginvestigations.

2.1 Double-Porosity Substrate

By generalizing Biot’s theory, Berryman and Wang (2000) developed the equations of motionfor double-porosity media. Omitting the fluid–fluid coupling coefficients between the matrixpore and the fracture, the motion equations are as follows:

⎛⎜⎝ρ11 ρ12 ρ13

ρ12 ρ22 0

ρ13 0 ρ33

⎞⎟⎠

⎛⎜⎝

u

U(1)

U(2)

⎞⎟⎠ +

⎛⎜⎝

b12 + b13 −b12 −b13

−b12 b12 0

−b13 0 b13

⎞⎟⎠

⎛⎜⎝

u

U(1)

U(2)

⎞⎟⎠

=⎛⎜⎝

G∇2u + (Ku + 1

3 G) ∇e + Ku∇

(B(1)ζ (1) + B(2)ζ (2)

)

v(1)ϕ(1)∇ (A12e − A22ζ

(1) − A23ζ(2)

)

v(2)ϕ(2)∇ (A13e − A23ζ

(1) − A33ζ(2)

)

⎞⎟⎠ (1)

where ∇ and ∇2 are the gradient and Laplace operators, respectively. u, U(1), and U(2)

are the displacement vectors of the solid, the matrix pore fluid, and the fracture pore fluid,

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respectively; The over dot symbol denotes the differential with respect to time; e = ∇·u is theoverall volume strain, ζ (1) = −v(1)ϕ(1)∇ · (U(1) − u) and ζ (2) = −v(2)ϕ(2)∇ · (U(2) − u)are the increments of fluid content in the matrix and the fracture, respectively; v(1) andv(2) = 1−v(1) are the volume fractions of the matrix and the fracture pore, respectively; ϕ(1)

and ϕ(2) (=unity) are the matrix and fracture porosities, respectively; ∇,∇·, and ∇2 denotethe gradient, divergence, and Laplace operators, respectively; G is the shear modulus; Ku =

K1−3K(β(1)B(1)+β(2)B(2)) is the undrained bulk modulus for the double-porosity media;β(1)

and β(2) are the poroelastic expansion coefficients; α is the Biot–Willis parameter; K isthe drained frame bulk modulus of the whole and Ks is the unjacketed bulk modulus ofthe whole; B(1) and B(2) are the pore pressure buildup coefficients for the matrix and thefractures, respectively; ρ11, ρ12, ρ13, ρ22, and ρ33 are the mass coefficients. b12 and b13 arethe viscosity coupling coefficients; A12, A13, A22, A23 and A33 are the coefficients; and thedetail expressions for the coefficients (ρ11, ρ12, ρ13, ρ22, and ρ33; b12 and b13; A12, A13,A22, A23, and A33) can be found in Berryman and Wang (2000) and Dai et al. (2006a).

Based on the Helmholtz theorem, the three displacement vectors u, U(1), and U(2) can becast in the form

u = ∇ϕs + ∇ × ψs, (2)

U(1) = ∇ϕm + ∇ × ψm, (3)

U(2) = ∇ϕf + ∇ × ψf , (4)

where ∇× is the operator; ϕs and ψs are the scalar and vector potential functions of the solidphase, respectively; ϕm and ψm are the scalar and vector potential functions of the fluid inthe matrix pores, respectively; and ϕf and ψf are the scalar and vector potential functions ofthe fluid in the fractures, respectively.

The constitutive equations for double-porosity media are of the form

σi i =(

Ku − 2

3G

)e − Ku(B

(1)ζ (1) + B(2)ζ (2))+ 2Geii , (5)

σi j = 2Gei j , (6)

−p(1) = A12e − A22ζ(1) − A23ζ

(2), (7)

−p(2) = A13e − A23ζ(1) − A33ζ

(2), (8)

where σi j is the component of the total stress on an element of volume attached to the skeletonframe; ei j = (ui, j + u j,i )/2 is the component of the solid strain; and p(1) and p(2) are thefluid pressures in the matrix and the fracture, respectively.

2.2 Porous Sediment

Based on the Biot theory, the motion equations of linear isotropic porous media can beexpressed in terms of displacement vectors of the solid skeleton and pore fluid as follows(Stoll and Kan 1981):

μ∇2u + (H − μ)∇e − C∇ζ = ρu − ρf w, (9)

C∇e − M∇ζ = ρf u − mw − η/kw, (10)

where w = n(u − U), e = ∇ · u, and ζ = ∇ · w; u and U are the displacement vectors ofthe solid skeleton and the pore fluid, respectively; ρ = (1 − n)ρs + nρf is the total massdensity; and ρs and ρf are the solid skeleton density and the pore fluid density, respectively.

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n, η, and k are the porosity, fluid viscosity, and Darcy’s permeability coefficient, respectively.m = cρf /n, where c is the experimental determined parameter. The constant μ is the shearmodulus of the skeletal frame, and the constants H , C , and M characterize the volumetricresponse of the frame.

The displacement vector u depends on the scalar and vector displacement potentials, i.e.,ϕs and ψs, as follows:

u = ∇ϕs + ∇ × ψs (11)

and w depends on ϕf and ψf similarly as follows:

w = ∇ϕf + ∇ × ψf , (12)

where ∇× is the curl operator.The constitutive equations for the isotropic porous sediment are of the form

σi i = He − 2u(e j j + ekk)− Cζ, (13)

σi j = 2Gei j , (14)

pf

= Mζ − Ce, (15)

where σi j is the component of the total stress on an element of volume attached to the skeletonframe. pf is the pore fluid pressure, and ei j is the strain of the skeleton.

The constants H , C , and M are defined as follows (Biot 1956a,b; Stoll and Kan 1981):

H = [(Kr − Kb)

2/(D − Kb)] + Kb + 4μ/3, (16)

C = [Kr (Kr − Kb)]/(D − Kb), (17)

M = K 2r /(D − Kb), (18)

where D = Kr[1+n(Kr/Kf −1)]. Kr and Kf are the bulk modulus of the individual sedimentgrains and the bulk modulus of the pore water, respectively. Kb andμ, which are the bulk andshear moduli of the particle assemblage, respectively, are complex to account for the variousforms of energy dissipation that occurs at the grain contacts. At the high frequency range,the viscosity coefficient η should be replaced by a dynamic complex value ηF(κ). Detaileddescription of F(κ) is given in Biot (1956b) and Wang et al. (2013).

2.3 Ideal Compressible Water

The motion equation of ideal compressible fluid is given by

Kw∇∇ · U′ = ρwU′, (19)

where Kw is the bulk modulus, ρw is the fluid density, and U′ is the displacement vector.Ideal compressible fluid can only transmit a compressible wave, and the displacement

vector may be expressed as

U′ = ∇ϕw, (20)

where ϕw is the scalar displacement potential of the ideal fluid.The dynamic pressure pw in the fluid has the following form:

pw (x, z, t) = −ρw∂2φw

∂t2 , (21)

where t is the time variable.

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3 Boundary Conditions

3.1 Interface Between the Ideal Fluid and the Porous Media (z = 0)

The following equations defining the boundary conditions are obtained based on the consti-tutive equations for ideal compressible water and porous media (Stoll and Kan 1981) and thedisplacement potentials in Sect. 2.

(1) The compatibility between the normal pore fluid movement in and out of the skeletalframe and the normal fluid displacement is given by

U ′z = (1 − n)uz + nU z = uz − wz, (22)

where U ′z, uz,U z , and wz are the z components of U′, u, U, and w, respectively.

(2) The equilibrium of the total normal traction on the porous sediment and on the fluidpressure is given by

− pw = σ zz, (23)

where σ zz is the normal component of the total stress on an element of the porous sediment.

(3) The equilibrium between the ideal fluid pressure pw and the pore fluid pressure pf isgiven by

pw = pf . (24)

(4) The equilibrium of the tangential traction on the porous skeleton is given by

σ xz = 0. (25)

3.2 Interface Between the Porous Sediment and the Double-Porosity Substrate (z = h)

Similar to the boundary conditions used by Deresiewicz and Skalak (1963) for the two porousmedia and those used by Denneman et al. (2002) for the fluid–porous media, two kinds ofboundary conditions are adopted: (a) open-pore boundary and (b) sealed-pore boundary. Forthe open-pore boundary, the matrix pore and the fracture are both open at the boundary.For the sealed-pore boundary, the matrix pore and the fracture are all sealed. The accuratedefinition of the two boundary conditions can be found in the subsequent description.

(1) The continuity of the normal displacements of solid is given by

uz = uz . (26)

(2) The continuity of the tangential displacement of solid is given by

ux = ux . (27)

(3) The equilibrium of the normal traction on the skeletal frame is given by

σzz = σ zz . (28)

(4) The equilibrium of the tangential traction on the skeletal frame is given by

σxz = σ xz . (29)

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(5) The proportionality between the pressure drop in matrix across the interface and relativevelocities in double-porosity media is given by Deresiewicz and Skalak (1963)

pf − p(1) = T (1)v(1)ϕ(1)(U (1)z − uz), (30)

where p(1) is the fluid pressure in the matrix and T (1) is the coefficient for surface flowimpedance. The two limiting cases that are of special interest are T (1) = 0 and T (1) → ∞.

The open-pore case T (1) = 0 implies free flow of fluid in the matrix across the poroussediment–double-porosity substrate interface and leads to

pf = p(1). (31)

For the sealed-pore case T (1) → ∞, no fluid flow exists in the matrix across the poroussediment–double-porosity substrate interface. Substituting T (1) → ∞ in Eq. (30) leads to

U (1)z − uz = 0. (31’)

(6) The proportionality between the pressure drop in fracture across the interface and relativevelocities in double-porosity media can be given in this form

pf − p(2) = T (2)v(2)ϕ(2)(U (2)z − uz) (32)

Similarly, we can obtain

pf = p(2) (33)

for open-pore boundary and

U (2)z − uz = 0 (33’)

for sealed-pore boundary.(7) The continuity of fluid flow across the boundary is given by

wz = v(1)ϕ(1)(U (1)z − uz)+ ν(2)ϕ(2)(U (2)

z − uz). (34)

For sealed-pore boundary condition, Eq. (34) is reduced to

wz = 0. (34’)

4 Formulations of the System

As shown in Fig. 1, a plane compressible wave with an angular frequency ω and an angleof incidence θi crosses the fluid–porous sediment interface from the semi-infinite water. Thereflected P wave is generated in the region z < 0 based on Eq. (19) of the ideal compressiblefluid. The transmitted and reflected P1,P2, and S waves are generated in the region 0 <

z < h based on Eqs. (9) and (10) of the porous media. The transmitted P1,P2,P3, and Swaves are generated in the region z > h based on Eq. (1) of the double-porosity media.All these waves have equal wave numbers in the x direction using Snell’s law, and theirdisplacement potentials can be defined as follows (Stoll and Kan 1981; Berryman and Wang2000; Denneman et al. 2002; Wang et al. 2013):For the region z < 0

φw = Aiw exp[i (ωt − lz z − lx x)

] + Arw exp[i (ωt + lz z − lx x)

]. (35)

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For the region 0 < z < h

φs = Atp1exp

[i(ωt − lzp1 z − lx x

)] + Atp2exp


)]

+ Arp1exp

[i(ωt + lzp1 z − lx x

)] + Arp2exp


)], (36)

φf = Atp1δp1 exp


)] + Atp2δp2 exp


)]

+ Arp1δp1 exp


)] + Arp2δp2 exp


)], (37)

ψ s = Bt exp[i(ωt − lzsz − lx x

)] + Br exp[i(ωt + lzsz − lx x

)], (38)

ψ f = Btδs exp[i(ωt − lzsz − lx x

)] + Brδs exp[i(ωt + lzsz − lx x

)]. (39)

For the region z > h,

φs = Atp1exp


)] + Atp2exp


)]

+ Atp1exp


)], (40)

φm = Atp1δ(1)p1

exp[i(ωt − lzp1 z − lx x

)] + Atp2δ(1)p2


)]

+ Atp1δ(1)p3


)], (41)

φf = Atp1δ(2)p1


)] + Atp2δ(2)p2


)]

+ Atp1δ(2)p3


)], (42)

ψs = Bte exp[i (ωt − lzsz − lx x)

], (43)

ψm = Btsδ(1)s exp

[i (ωt − lzsz − lx x)

], (44)

ψf = Btsδ(2)s exp

[i (ωt − lzsz − lx x)

], (45)

where A and B denote the amplitude of the P and S waves, respectively. l denotes the wavenumber, and δ is the amplitude ratio of the solid skeleton potential to the pore fluid incrementpotential. δ(1) and δ(2) are the amplitude ratios of the fluid pore potential and the fracture poreto the solid potential, respectively. The overbar denotes the porous media, and the subscriptsi, r, and t denote the incidence, reflection, and transmission, respectively. The subscripts xand z denote the x and z components, respectively, and the subscripts p, s, p1, p2, and p3

denote the P, S, P1,P2, and P3 waves, respectively. The wave number components lz and lx

are expressed as

lz = lw cos θi , lx = lw sin θi , (46)

where lw = ω/Vw, Vw is the velocity of compressible wave in the overlying water.According to Deresiewicz and Rice (1962), δpi , and δs are given by

δpi = Hg2i − ρω2

Cg2i − ρfω2

, i = 1, 2 (47)

δs = ρω2 − μg2s

ρfω2 , (48)

where

g21,2 = − (

Hρfω2/n + Mρω2 − 2Cρfω

2 − i HωηF (κ)/k)

2(C2 − H M

)

∓√(

Hρfω2/n+Mρω2−2Cρfω2−i HωηF (κ)/k)2−4

(C2−HM

) (ρ2

f ω4−ρρfω4/n+iρω3ηF (κ)/k

)

2(C2−HM

)

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and

g2s = ρρfω

4 − ρ2f ω

4 − iρω3ηF (κ)/k

μ(ρfω2/k − iωηF (κ)/k

) .

According to Dai et al. (2006b), δ(1)pi , δ(2)pi , δ

(1)s , and δ(2)s are cast as follows:

δ(1)pi = t21t13 − t11t23

t12t23 − t22t13, at l pi = lw, i = 1, 2, 3 (49)

δ(2)pi = t21t12 − t11t22

t13t22 − t23t12, at l pi = lw, i = 1, 2, 3 (50)

δ(1)s = −q12

q22, at ls = ls (51)

δ(2)s = −q13

q33, at ls = ls (52)

where ti j and qi j are the elements in the characteristic equations for the compressional andshear waves, respectively (Dai et al. 2006b).

Combining the displacement potential Eqs. (35)–(45) and the boundary conditionEqs. (22)–(34), the following formula for the unknown constants Arw, Atp1

, Atp2, Bt , Arp1

,Arp2

, Br , Atp1, Atp2

, Atp3, and Bts can be obtained as follows:

[a][

Arw Atp1Atp2

Bt Arp1Arp2

Br Atp1Atp2

Atp3Bts

]T = Aiw [b] (53)

where Aiw is the known constant, [a] is a 11 × 11 matrix, and [b] is a column matrix with 11elements. The elements of the matrices [a] and [b] can be found in Appendix.

5 Numerical Example

The wave reflection and transmission of a coupled water–porous sediment–double-porositysubstrate system (Fig. 1) subjected to a compressible plane wave incident from the semi-infinite water are investigated using the formulations derived in Sect. 4. This study focuseson the effect of the underlying double-porosity substrate on the reflected wave of the over-lying water. Various properties of the double-porosity media are considered. Based on theconclusions of Deresiewicz and Skalak (1963) and the practice of other researchers (Denne-man et al. 2002; Stoll and Kan 1981; Dai et al. 2006a,b), the matrix [a] in Eq. (53) is alwaysnon-singular. In each case, Eq. (53) is solved by precisely calculating the inverse matricesof [a]. The amplitude ratio of the potential for the reflected wave to the potential for theincident wave Arw/Aiw is plotted to illustrate the effects of the double-porosity substrate onthe reflection coefficients in the overlying water.

5.1 Material Properties

Based on References (Stoll 1977; Stoll and Kan 1981; Williams et al. 2001; Berrymanand Wang 1995; Dai et al. 2006b), the material properties of water, porous sediment, anddouble-porosity substrate are listed in Table 1. The property variations of the double-porositysubstrate are considered. Except in special notation, the open-pore boundary condition isadopted at the sediment–double-porosity media interface, and the material parameters arefixed to Kf = 2.3 × 109 Pa, v(2) = 0.011, k(11) = 10−16 m2, k(22) = 10−11 m2, and ρf =1,000 kg/m3.

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Table 1 Material properties of water, porous sediment, and double-porosity substrate

Water

Kw = 2 × 109 Pa, ρw = 1,000 kg/m3

Porous sediment

Kr = 3.6 × 1010 Pa, Kf = 2.0 × 109 Pa, ρs = 2,650 kg/m3, ρf = 1,000 kg/m3,μ = (1 + 0.048i)× 26.1 MPa,

Kb = (1 + 0.048i)× 43.6 MPa, n = 0.47, c = 1.25, η = 10−3 Pa s, k = 10−10 m2, α = 3.9 × 10−5 m

Double-porosity substrate

K = 8 × 109 Pa, Ks = 5.45 × 1010 Pa, K (1) = 1.7 × 1010 Pa, K (1)s = 5.55 × 1010 Pa,Kf = 2.3 × 109 and 1.3 × 109 Pa,

v = 0.25, η = 10−3 Pa s, v(2) = 0.011, 0.005 and 0.001, ϕ(1) = 0.0011, B(1) = 0.992, α = 0.85,

k(11) = 10−14, 10−16, and 10−18 m2, k(22) = 10−11, 10−12, 10−14 m2, ρs = 3,000 kg/m3,ρf = 1,000 and 800kg/m3

5.2 Effect of the Porous Sediment Depths

As mentioned in Sect. 1, Wang et al. (2013) investigated wave propagation at the ocean–porous sediment interface with an underlying solid substrate. Their results have shown thatwhen the sediment layer exceeds a critical depth, the underlying solid substrate is negligible.Similarly, the effect of the porous sediment depths with an underlying double-porosity sub-strate is investigated in this section. Figure 2 shows the variation of the reflection coefficientArw/Aiw of the overlying water at different sediment depths h as a function of the incidentangle θi for the two excitation frequencies f (= ω/2π) = 100 Hz and 1 kHz, respectively.The reflection coefficient Arw/Aiw obviously varies with depth change in the porous medialayer.

In the case of h = 0 m, the analysis system eliminates the porous media and representsthe wave reflection and transmission at the interface separating the semi-infinite water anda double-porosity substrate (Dai and Kuang 2008). Wang et al. (2013) have indicated that acritical angle exists when the reflected wave is parallel to the interface, and Arw/Aiw is inunity when the porous sediment is absent (h = 0 m) at the critical angle. However, no criticalangle is observed in this study, as shown in Fig. 2. This is attributed to the attenuation effectsin the double-porosity substrate (Stoll and Kan 1981).

As the sediment depth thickens, the variations in Arw/Aiw with θi are apparently differentfrom that in the case of h = 0 m for both f = 100 Hz and 1 kHz. In the case of f = 100 Hz,Arw/Aiw at sediment depths of 100 and 200 m at a larger θi (e.g., θi > 70◦) approachesthat in the semi-infinite porous sediment case. Meanwhile, Arw/Aiw at a smaller θi remainsto oscillate around the value in the semi-infinite porous sediment case. The same trend isobserved in the case of f = 1 kHz (Fig. 2b) but with a higher rate of convergence.

Moreover, the reflection coefficient Arw/Aiw at a sediment depth of 500 m becomesclearly identical with that in the semi-infinite porous sediment case when f = 100 Hz. Thecorresponding sediment depth is 22 m for f = 1 kHz, which results from high attenuation ofthe porous sediment when the excitation frequency is high. That is to say, the sediment layerexists a critical depth as well when the substrate is modeled by the double-porosity media,and the critical depth values are 500 m for f = 100 Hz and 22 m for f = 1 kHz, which areconsistent with the conclusions of Wang et al. (2013).

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Fig. 2 Variation of reflection coefficient Arw/Aiw with θi for different depths h, a f = 100 Hz and bf = 1 kHz

5.3 Effect of Fluid Properties in the Double-Porosity Substrate

Distinguishing the type of fluid stored in the double-porosity substrate by investigatingreflected waves in overlying water is important. This section adopts two kinds of pore fluid forthe double-porosity substrate to discuss its influence on the reflected wave in the underlyingwater. This pore fluid is characterized by different densities and bulk moduli. The follow-ing conditions are assumed: Kf = 2.3 × 109 Pa and ρf = 1,000 kg/m3 for fluid I andKf = 1.3 × 109 Pa and ρf = 800 kg/m3 for fluid II.

Figure 3 shows the reflection coefficient Arw/Aiw in the overlying water varying withfrequency at two incident angles θi = 0◦ and 60◦ when the sediment depth is h = 5 m andthe pore and fracture fluids in the double-porosity substrate are fluids I and II, respectively. Itcan be observed that the difference in Arw/Aiw between the two kinds of fluid is not significantwhen the frequency is lower than 50 Hz at the two incident angles. Figure 3b shows that theeffect of fluid properties is even negligible at the incident angle of 60◦ when f < 50 Hz.

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Fig. 3 Variation of reflection coefficient Arw/Aiw with frequency at h = 5 m with fluids I and II, a θi = 0◦and b θi = 60◦

In the frequency range of 50–1,000 Hz, the trends of oscillation and decay of Arw/Aiw areapproximately the same, and a clear difference in the amplitude can be observed. This resultis attributed to the resistance difference in the two kinds of fluid. When f > 1,000 Hz, thereflection coefficient Arw/Aiw with the two kinds of fluid becomes identical at the givenincident angles. That is to say, the reflection coefficient Arw/Aiw in the overlying water isonly sensitive to the properties of fluids within a certain range of frequencies (50–1,000 Hz).

5.4 Effect of Permeability of the Double-Porosity Substrate

This section analyzes viscous loss in the double-porosity substrate on the wave attenuationin the overlying water. Viscous loss is attributable to the viscous flow of the pore fluid andthe fracture fluid with respect to the solid frame, which is related to the permeability ofthe matrix and the fracture in the double-porosity substrate. Therefore, the variations of thepermeability coefficient k(11) and k(22) of the double-porosity substrate are considered.

123


Fig. 4 Variation of reflection coefficient Arw/Aiw with incident angle θi at h = 5 m for k = 10−14, 10−16,and 10−18 m2, a f = 100 Hz and b f = 10 kHz

5.4.1 Effect of Permeability of the Matrix in the Double-Porosity Substrate

The permeability coefficient k(11) of the double-porosity substrate is assumed as 10−14,10−16, and 10−18 m2. Figure 4 shows the reflection coefficient Arw/Aiw of the overlying wateras a function of the incident angle θi at h = 5 m for f = 100 Hz and 10 kHz. Figure 4 shows thatthe permeability coefficient k(11) has little effect on the reflection coefficient Arw/Aiw, whichis nearly identical to different permeability coefficients k(11) when f = 100 Hz and 10 kHz.A slight difference occurs within the incident angle range of 20◦–60◦ when f = 100 Hz.

5.4.2 Effect of Permeability of the Fracture in the Double-Porosity Substrate

The permeability coefficient k(22) is taken to be 10−11, 10−12, and 10−14 m2. Figure 5 showsthe variation of the reflection coefficient Arw/Aiw with the incident angle θi at h = 5 m forf = 10 Hz, 100 Hz, 1 kHz, and 10 kHz. The effect of the permeability coefficient k(22) on

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D.-D. Lyu et al.

Fig. 5 Variation of reflection coefficient Arw/Aiw with incident angle θi at h = 5 m for k(22) =10−11, 10−12, and 10−14 m2, a f = 10 Hz; b f = 100 Hz; c f = 1 kHz; and d f = 10 kHz

Arw/Aiw is insignificant at high frequency (10 kHz). However, Arw/Aiw demonstrates cleardifferences among the three different k(22) values in the case of f = 1 kHz, 100 Hz, and10 Hz. The reflection coefficient Arw/Aiw in the case of k = 10−12 m2 is largely differentfrom that of k = 10−11 m2. In the case of f = 100 Hz, larger permeability leads to smallerArw/Aiw when the incident angle exceeds 20◦, and the oscillation amplitude becomes largerwith larger permeability in the case of f = 1 kHz when θi > 20◦. Compared with theresults in Sect. 5.4.1, the permeability of the fractures in the double-porosity substrate clearlyaffects Arw/Aiw more significantly than the permeability of the matrix. This phenomenonis primarily attributed to the very small permeability of the matrix compared with that ofthe fracture. Thus, the permeability of the fracture dominates fluid transport in the double-porosity substrate.

5.5 Effect of Volume Fraction of Fracture v(2) in the Double-Porosity Substrate

In this section, the effect of volume fraction of fracture v(2) in double-porosity substrate isinvestigated. Theoretically, the variation of the volume fraction of fracture v(2) will lead tochange in other properties of the double-porosity substrate. However, the other propertiesare assumed to remain unchanged because of the lack of relationship between the volumefraction of fracture and the other properties for double-porosity media.

Taking the volume fraction of fracture v(2) as 0.011, 0.005, and 0.001 and assuming that thegeneral porosity ϕ = 0.0121 remains constant, Figure 6 shows the variation in the reflection

123


Fig. 6 Variation of reflection coefficient Arw/Aiw with incident angle θi at h = 5 m with v(2) = 0.011, 0.005,and 0.001, a f = 10 Hz; b f = 100 Hz; c f = 1 kHz; and d f = 10 kHz

coefficient Arw/Aiw with the incident angle θi at h = 5 m with v(2) = 0.011, 0.005, and0.001 for f = 10 Hz, 100 Hz, 1 kHz, and 10 kHz, respectively. Clear differences are observedfor different volume fractions of fracture especially at low frequencies, e.g., f = 10 Hz and100 Hz. The difference is reduced when the frequency of the plane wave increases and isgradually diminished as f approaches 10 kHz. Moreover, the reflection coefficient Arw/Aiw

with v(2) = 0.001 tends to be larger than that with v(2) = 0.011 and 0.005 at most incidentangles. This is primarily because smaller volume fraction of fracture results in less viscousloss in fluids. Based on the aforementioned phenomenon, we can conclude that the effect ofvolume fraction of fracture v(2) in the double-porosity substrate on the reflection coefficientArw/Aiw is significant, especially in a low frequency range.

5.6 Effect of Boundary Condition at the Porous Sediment–Double-Porosity Interface

In the previous sections, the open-pore boundary condition is assumed to investigate thevariations of reflection coefficient Arw/Aiw. In this section, the effect of boundary condi-tion at the porous sediment–double-porosity interface is analyzed. The reflection coefficientArw/Aiw as a function of sediment depth at θi = 0◦ for open-pore and sealed-pore boundaryconditions with f = 100 Hz and 1 kHz, respectively, is plotted to compare the two boundaryconditions (Fig. 7).

Figure 7 shows that the difference in Arw/Aiw with open-pore and sealed-pore boundaryconditions is gradually reduced as the sediment depth increases and eventually disappears.

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D.-D. Lyu et al.

Fig. 7 Variation of reflection coefficient Arw/Aiw with sediment depth at θi = 0◦ for two cases of boundaryconditions: (i) open-pore boundary and (ii) sealed-pore boundary, a f = 100 Hz and b f = 1 kHz

This phenomenon is not difficult to understand because when the sediment layer exceedsthe critical depth, i.e., h > 500 m and h > 22 m for f = 100 Hz and 1 kHz, respectively,the double-porosity substrate is negligible, as discussed in Sect. 5.2. Otherwise, the bound-ary condition has a significant effect on Arw/Aiw in the overlying water. The amplitude ofreflection coefficient Arw/Aiw with open-pore boundary condition is notably larger than thatwith sealed-pore boundary condition if the sandwiched sediment depth is less than the criticalvalue. This phenomenon is mainly attributed to the flow impedance at the sealed-pore bound-ary. Meanwhile, the boundary condition at the porous sediment–double-porosity substrateinterface has significant effects on the wave reflection and transmission at the interface ofthe overlying water and porous sediment if the double-porosity substrate is not negligible.

6 Conclusions

The reflection and transmission of plane waves at the interface between the water and theporous sediment with an underlying double-porosity substrate were investigated in the cur-rent study. Analytical solutions were derived based on the displacement potential functionsfor compressible waves incident from the semi-infinite water to the interface. Numericalcalculations were performed for both the open-pore boundary and sealed-pore boundary todiscuss in detail the effects of the double-porosity substrate properties, which include fluidproperty, volume fraction of fracture, and permeability on the reflected waves. Based on thenumerical results, the conclusions may be summarized as follows:

(1) The attenuation effect in the double-porosity substrate reduces Arw/Aiw in the overlyingwater at the critical angle in the absence of porous sediment. When the sediment layerexceeds a critical depth, the underlying double-porosity substrate is negligible, which issimilar to the conclusion in Wang et al. (2013).

(2) The effect of fluid properties in the double-porosity substrate on Arw/Aiw depends onexcitation frequency. Changes in the amplitude of Arw/Aiw due to the variation of fluidproperties can be only observed within a certain frequency range.

(3) Changes in the permeability of the fracture in double-porosity substrate have a significanteffect on Arw/Aiw, whereas the permeability of the matrix only slightly affects thereflection coefficient Arw/Aiw.

(4) Different volume fractions of fracture v(2) in the double-porosity substrate result in aclear difference in the reflection coefficient Arw/Aiw in the low frequency range.

123


(5) Compared with the open-pore boundary condition, the sealed-pore boundary conditionreduces the amplitude of the reflection coefficient Arw/Aiw in the overlying water if thesandwiched sediment depth is less than the critical value.

The abovementioned findings show that the reflected wave in the overlying water is sensitiveto the double-porosity substrate with the variation of its material properties if the sandwichedsediment depth is less than the critical value. Therefore, feedback on ocean floor behaviorbased on wave reflection signals is possible.

Acknowledgments The present research was supported by the National Natural Science Foundation of China(No. 51179093), the National Basic Research Program of China (No. 2011CB013602), and the Program forNew Century Excellent Talents in University (No. NCET-10-0531). The support is gratefully acknowledged.

Appendix

The elements of the matrix [a] in Eq. (53) are given by

a11 = lz, a12 = (1 − δp1

)lzp1 , a13 = (

1 − δp2

)lzp2 , a14 = (

1 − δs)

lx ,

a15 = − (1 − δp1

)lzp1 , a16 = − (

1 − δp2

)lzp2 , a17 = (

1 − δs)

lx , a18 = 0,

a19 = 0, a1,10 = 0, a1,11 = 0; a21 = ω2ρw,

a22 = − (H − δp1 C

) (l2zp1

+ l2x

)+ 2μl2

x , a23 = − (H − δp2 C

) (l2zp2

+ l2x

)+ 2μl2

x ,

a24 = −2Glxlzs, a25 = − (H − δp1 C

) (l2zp1

+ l2x

)+ 2μl2

x ,

a26 = − (H − δp2 C

) (l2zp2

+ l2x

)+ 2Gl2

x , a27 = 2μlx lzs,

a28 = 0, a29 = 0, a2,10 = 0, a2,11 = 0; a31 = ω2ρw,

a32 = (−C + δp1 M) (

l2zp1

+ l2x

), a33 = (−C + δp2 M

) (l2zp2

+ l2x

),

a34 = 0, a35 = (−C + δp1 M) (

l2zp1

+ l2x

), a36 = (−C + δp2 M

) (l2zp2

+ l2x

),

a37 = 0, a38 = 0, a39 = 0, a3,10 = 0, a3,11 = 0; a41 = 0,

a42 = 2lx lzp1 , a43 = 2lx lzp2 , a44 = −l2zs + l2

x , a45 = −2lx lzp1 , a46 = −2lx lzp2 ,

a47 = −l2zs + l2

x , a48 = 0, a49 = 0, a4,10 = 0, a4,11 = 0; a51 = 0,

a52 = −lzp1 exp(−ilzp1 h

), a53 = −lzp2 exp

(−ilzp2 h), a54 = −lx exp

(−ilzsh),

a55 = lzp1 exp(ilzp1 h

), a56 = lzp2 exp

(ilzp2 h

), a57 = −lx exp

(ilzsh

),

a58 = lzp1 exp(−ilzp1 h

), a59 = lzp2 exp

(−ilzp2 h), a5,10 = lzp3 exp

(−ilzp3 h),

a5,11 = lx exp(−ilzsh); a61 = 0, a62 = −lx exp(−ilzp1 h

), a63 = −lx exp

(−ilzp2 h),

a64 = lzs exp(−ilzsh

), a65 = −lx exp

(ilzp1 h

), a66 = −lx exp

(ilzp2 h

),

a67 = −lzs exp(ilzsh

), a68 = lx exp

(−ilzp1 h), a69 = lx exp

(−ilzp2 h),

a6,10 = lx exp(−ilzp3 h

), a6,11 = −lzs exp(−ilzsh); a71 = 0,

a72 =[− (

H − δp1 C) (

l2zp1

+ l2x

)+ 2Gl2

x

]exp

(−ilzp1 h),

a73 =[− (

H − δp2 C) (

l2zp2

+ l2x

)+ 2Gl2

x

]exp

(−ilzp2 h),

123

D.-D. Lyu et al.

a74 = −2Glxlzs exp(−ilzsh),

a75 =[− (

H − δp1 C) (

l2zp1

+ l2x

)+ 2Gl2

x

]exp

(ilzp1 h

),

a76 =[− (

H − δp2 C) (

l2zp2

+ l2x

)+ 2Gl2

x

]exp

(ilzp2 h

),

a77 = 2Glxlzs exp(ilzsh),

a78 ={(Ku − 2

3G)(l2

zp1+ l2

x )+ 2Gl2zp1

−Ku

[B(1)v(1)ϕ(1)(1 − δ(1)p1

)+ B(2)v(2)ϕ(2)(1 − δ(2)p1)](l2

zp1+ l2

x )

}exp

(−ilzp1 h),

a79 ={(Ku − 2

3G)(l2

zp2+ l2

x )+ 2Gl2zp2

−Ku

[B(1)v(1)ϕ(1)(1 − δ(1)p2

)+ B(2)v(2)ϕ(2)(1 − δ(2)p2)](l2

zp2+ l2

x )

}exp

(−ilzp2 h),

a7,10 ={(Ku − 2

3G)(l2

zp3+ l2

x )+ 2Gl2zp3

−Ku

[B(1)v(1)ϕ(1)(1 − δ(1)p3

)+ B(2)v(2)ϕ(2)(1 − δ(2)p3)](l2

zp3+ l2

x )

}exp

(−ilzp3 h),

a7,11 = 2Glxlzs exp(−ilzsh); a81 = 0, a82 = 2Glxlzp1 exp(−ilzp1 h

),

a83 = 2Glxlzp2 exp(−ilzp2 h

), a84 = G

(−l

2zs + l2

x

)exp

(−ilzsh),

a85 = −2Glxlzp1 exp(ilzp1 h

), a86 = −2Glxlzp2 exp

(ilzp2 h

),

a87 = G(−l

2zs + l2

x

)exp

(ilzsh

), a88 = −2Glxlzp1 exp

(−ilzp1 h),

a89 = −2Glxlzp2 exp(−ilzp2 h

),

a8,10 = −2Glxlzp3 exp(−ilzp3 h

), a8,11 = G(l2

zs − l2x ) exp(−ilzsh);

a91 = 0, a92 = (C − δp1 M

) (l2zp1

+ l2x

)exp

(−ilzp1 h),

a93 = (C − δp2 M

) (l2zp2

+ l2x

)exp

(−ilzp2 h),

a94 = 0, a95 = (C − δp1 M

) (l2zp1

+ l2x

)exp

(ilzp1 h

),

a96 = (C − δp2 M

) (l2zp2

+ l2x

)exp

(ilzp2 h

), a97 = 0,

a98 =[−A13 + A23v

(1)ϕ(1)(1 − δ(1)p1)+ A33v

(2)ϕ(2)(1 − δ(2)p1)](l2

x + l2zp1) exp(−ilzp1 h),

a99 =[−A13 + A23v

(1)ϕ(1)(1 − δ(1)p2)+ A33v

(2)ϕ(2)(1 − δ(2)p2)](l2


a9,10 =[−A13 + A23v

(1)ϕ(1)(1 − δ(1)p3)+ A33v

(2)ϕ(2)(1 − δ(2)p3)](l2


a9,11 = 0 (if the open-pore boundary condition is applied) or

a′91 = a′

92 = a′93 = a′

94 = a′95 = a′

96 = a′97 = 0, a′

98 = lzp1(1 − δ(2)p1) exp(−ilzp1 h),

a′99 = lzp2(1 − δ(2)p2

) exp(−ilzp2 h), a′9,10 = lzp3(1 − δ(2)p3

) exp(−ilzp3 h),

a′9,11 = lx (1 − δ(2)s ) exp(−ilzp1 h) exp(−ilzsh)

(if the sealed-pore boundary condition is applied);

123


a10,1 = 0, a10,2 = (C − δp1 M

) (l2zp1

+ l2x

)exp

(−ilzp1 h),

a10,3 = (C − δp2 M

) (l2zp2

+ l2x

)exp

(−ilzp2 h), a10,4 = 0,

a10,5 = (C − δp1 M

) (l2zp1

+ l2x

)exp

(ilzp1 h

),

a10,6 = (C − δp2 M

) (l2zp2

+ l2x

)exp

(ilzp2 h

), a10,7 = 0,

a10,8 =[−A12 + A22v

(1)ϕ(1)(1−δ(1)p1)+ A23v

(2)ϕ(2)(1−δ(2)p1)](l2


a10,9 =[−A12 + A22v

(1)ϕ(1)(1−δ(1)p2)+ A23v

(2)ϕ(2)(1−δ(2)p2)](l2


a10,10 =[−A12 + A22v

(1)ϕ(1)(1−δ(1)p3)+ A23v

(2)ϕ(2)(1−δ(2)p3)](l2


a10,11 = 0 (if the open-pore boundary condition is applied) or

a′10,1 = a′

10,2 = a′10,3 = a′

10,4 = a′10,5 = a′

10,6 = a′10,7 = 0,

a′10,8 = lzp1(1 − δ(1)p1

) exp(−ilzp1 h), a′10,9 = lzp2(1 − δ(1)p2

) exp(−ilzp2 h),

a′10,10 = lzp3(1 − δ(1)p3

) exp(−ilzp3 h),

a′10,11 = lx (1 − δ(1)s ) exp(−ilzsh) (if the sealed-pore boundary condition is applied);a11,1 = 0,

a11,2 = −lzp1δp1 exp(−ilzp1 h),

a11,3 = −lzp2δp2 exp(−ilzp2 h),

a11,4 = −lxδs exp(−ilzsh),

a11,5 = lzp1δp1 exp(ilzp1 h),

a11,6 = lzp2δp2 exp(ilzp2 h),

a11,7 = −lxδs exp(ilzsh),

a11,8 =[v(1)ϕ(1)(δ

(1)p1

− 1)+ v(2)ϕ(2)(δ(2)p1

− 1)]

lzp1 exp(−ilzp1 h),

a11,9 =[v(1)ϕ(1)(δ

(1)p2

− 1)+ v(2)ϕ(2)(δ(2)p2

− 1)]


a11,10 =[v(1)ϕ(1)(δ

(1)p3

− 1)+ v(2)ϕ(2)(δ(2)p3

− 1)]


a11,11 =[v(1)ϕ(1)(δ(1)s − 1)+ v(2)ϕ(2)(δ(2)s − 1)

]lzs exp(−ilzsh)

(if the open-pore boundary condition is applied) or

a′11,1 = 0, a′

11,2 = −lzp1δp1 exp(−ilzp1 h), a′11,3 = −lzp2δp2 exp(−ilzp2 h),

a′11,4 = −lxδs exp(−ilzsh),

a′11,5 = lzp1δp1 exp(ilzp1 h), a′

11,6 = lzp2δp2 exp(ilzp2 h), a′11,7 = −lxδs exp(ilzsh),

a′11,8 = 0, a′

11,9 = 0, a′11,10 = 0, a′

11,11 = 0

(if the sealed-pore boundary condition is applied).

The elements of the matrix [b] in Eq. (53) are given by

b1 = lz, b2 = −ω2ρw, b3 = −ω2ρw,

b4 = b5 = b6 = b7 = b8 = b9 = b10 = b11 = 0.

123

D.-D. Lyu et al.

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