Inhomogeneous wave propagation in partially saturated soils

Wave Motion 93 (2020) 102470

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Inhomogeneous wave propagation in partially saturated soils ∗

M.S. Barak a , M. Kumar b , , M. Kumari a , A. Singh a a b

Department of Mathematics, Indira Gandhi University, Meerpur, Rewari, Haryana, 122503, India Department of Mathematics, Dr. B R Ambedkar Govt. College, Dabwali, Sirsa, Haryana, 125104, India

article

info

Article history: Received 11 October 2018 Received in revised form 9 September 2019 Accepted 18 November 2019 Available online 26 November 2019 Keywords: Inhomogeneous wave Phase velocity Attenuation coefficient Partially saturated Dispersive

a b s t r a c t In the present study, inhomogeneous plane harmonic waves propagation in dissipative partially saturated soils are investigated. The analytical model for the dissipative partially saturated soils is solved in terms of Christoffel equations. These Christoffel equations yields the existence of four wave (three longitudinal and one shear) modes in partially saturated soils. Christoffel equations are further solved to determine the complex velocities and polarizations of four wave modes. Inhomogeneous propagation is considered through a particular specification of complex slowness vector. A finite non-dimensional inhomogeneity parameter is considered to represent the inhomogeneous nature of these four waves. Impact of tortuosity parameter on the movement of pore fluids is considered. Hence, the considered model is capable of describing the wave behavior at high as well as mid and low frequencies. Numerical example is considered to study the effects of inhomogeneity parameter, saturation of water, porosity, permeability, viscosity of fluid phase and wave frequency on the velocity and attenuation of four waves. It is observed that all the waves are dispersive in nature (i.e., frequency dependent). © 2019 Elsevier B.V. All rights reserved.

1. Introduction Partially saturated soils mainly consist of three phases (i.e., solid, liquid and gas). In the recent years, a considerable increase in theoretical and experimental studies is observed to understand the behavior of unsaturated soils. In the present scenario, unsaturated soil mechanics is named as young science. Several researchers derived the effective stress equations for unsaturated soils. In 1970s, Fredlund and his coworkers developed the fundamental theories for partially saturated soils. Fredlund and Morgenstern [1] derived the constitutive relations for volume change in unsaturated soils. A general formulation for one-dimensional consolidation in which the air and water phases are assumed to be continuous, was presented at almost the same time by Fredlund and Hasan [2] and Lloret and Alonso [3]. This formulation is based on two continuity equations, one for the water phase and other for the air phase, which have to be solved simultaneously to give water and air pressures at any time. In the method developed by Fredlund and Hasan, the constitutive relations proposed by Fredlund and Morgenstern [1] were incorporated. The three-dimensional consolidation problem was studied by Dakshanamurthy and Fredlund [4] using an uncoupled approach. Conte [5] developed the model to analyze the coupled and uncoupled consolidation in unsaturated soils under plane-strain conditions. He derived the differential equations governing the coupled and uncoupled consolidation, based on the approach proposed by Fredlund and his coworkers, and then he solved these equations. Lo and Sposito [6] studied the acoustic waves in unsaturated soils. They studied the effect of hysteresis on the phase velocity and attenuation coefficients of acoustic waves. They found that the effect ∗ Corresponding author. E-mail addresses: [email protected] (M.S. Barak), [email protected] (M. Kumar), [email protected] (M. Kumari), [email protected] (A. Singh). https://doi.org/10.1016/j.wavemoti.2019.102470 0165-2125/© 2019 Elsevier B.V. All rights reserved.

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M.S. Barak, M. Kumar, M. Kumari et al. / Wave Motion 93 (2020) 102470

of hysteresis becomes more apparent as the wave excitation frequency increases. Lo et al. [7] studied the poroelastic theory of consolidation in unsaturated soils. Whalley et al. [8] calculated the velocities of shear waves in two unsaturated soils (loamy sand/sandy clay loam) at various matric potentials and confining pressures. They exposed that how the velocity of shear wave is connected with the void ratio, matric potential and net stress. Albers [9] studied the linear elastic wave propagation in unsaturated sands, silts, loams and clays. She investigated the phase velocities and attenuations of some unconsolidated soils and compared to the acoustic behavior of sandstone filled by a water–air mixture. Albers [10] investigated the influence of hysteretic behavior of the capillary pressure on the wave propagation in partially saturated soils. Conte et al. [11] derived the analytical expressions of shear and dilatational wave velocities (VS and VP) in terms of some physical and constitutive parameters of unsaturated soils. They did not study these wave characteristics numerically, only studied analytically. Ghasemzadeh and Abounouri [12] presented the model for propagation of waves in partially saturated soils. In this model, the tortuous path of each fluid phase is considered. Therefore, the inertial coupling mechanism which exists between the fluid and the solid particles in higher frequencies is accounted. They studied the propagation of compressional and shear waves in partially saturated soils by using the tradition potential method. They found that four (three longitudinal and one shear) waves exist in such medium. In the potential method, elastodynamic equation can be decomposed into two standard wave equations: a scalar wave equation for scalar potential and a vector wave equation for vector potential. But, due to the presence of viscosity in pore fluids, the medium become dissipative. Attenuated waves propagate in such a dissipative medium. A general representation of an attenuated wave is defined through its inhomogeneous propagation, i.e., different directions for propagation and attenuation. Therefore, in the present study we discuss the propagation of inhomogeneous time-harmonic plane waves in dissipative media. Contrary to the traditionally potentials method, we looks directly for the solution of elastodynamic equation in terms of the displacement vector. The elastodynamic equations reduced to Christoffel equations by using the suitable ansatz solution for a plane harmonic wave. The equations of motion developed by Ghasemzadeh and Abounouri [12] are solved for the propagation of harmonic plane waves. The solution is obtained in the form of Christoffel equations, which are solved further to calculate the complex velocities and polarizations of three longitudinal waves and one transverse wave. For any of these four attenuated waves, a general inhomogeneous propagation is considered through a particular specification of complex slowness vector. Inhomogeneity of an attenuated wave is represented through a finite non-dimensional parameter. The effects of various subsurface hydrological properties and wave frequency are investigated graphically on the propagation characteristics of four waves, with the help of MATLAB software. Hence, in this contribution, a different, more systematic, approach is used to investigate the propagation of inhomogeneous plane waves in the partially saturated soils. 2. Basic equations The mathematical model proposed by Ghasemzadeh and Abounouri [12] has been considered as reference work, wherein equations of motion for partially saturated soils containing two immiscible fluids (say, a gas and water) are derived. The equations of motion, in the absence of body force, are given by

σij,j = ρs (1 − φ )u¨ i + ρw φw v¨ i + ρa φa w ¨ i, φw pw,i = (τw − 1)φw ρw (u¨ i − v¨ i ) + bw (u˙ i − v˙ i ) − φw ρw v¨ i , φa pa,i = (τa − 1)φa ρa (u¨ i − w ¨ i ) + ba (u˙ i − w ˙ i ) − φa ρa w ¨ i,

(1)

where ui , vi and wi represent the solid, gas and water particles displacement components respectively. The indices s, w and a represent the three phases of composite medium such as solid, water and air, respectively. Else, the variable indices in the tensors can take the values 1, 2, 3. The partial time derivative is represented by a dot over variable. The τ ’s are used to define effective tortuosity, ρ ’s are material densities of the different phases. φw (= φ Sw ) and φa (= φ (1 − Sw )) represent the volume fractions of water and air phases, respectively. Sw is the saturation of water and φ is the soil porosity. The parameters bw and ba represent the inertial coupling of two fluid phases with solid are given by bj =

φj2 ηj kj

,

j = w, a,

where ηw and ηa represent the viscosity of water and air phases, respectively. kw and ka are the effective permeability of water and air phases, respectively. Following Smeulders [13], the fluid momentum equations to incorporate the inertial coupling terms caused by the tortuosity of fluid phases to describe the pore-water and pore-air movement through the soil are given by

φw ρw v¨ = −φw ▽ pw + (τw − 1)φw ρw (u¨ − v¨ ) + bw (u˙ − v˙ ) φa ρa w ¨ = −φa ▽ pa + (τa − 1)φa ρa (u¨ − w ¨ ) + ba (u˙ − w ˙) where τw and τa are effective tortuosity of water and air phases. The stress–strain relations for the composite medium are given by

σij = (Huk,k + φw [χ L + (1 − χ )C ]vk,k + φa [χ C + (1 − χ )N ]wk,k )δij + G(ui,j + uj,i ),

M.S. Barak, M. Kumar, M. Kumari et al. / Wave Motion 93 (2020) 102470

3

pw = (Wuk,k − φw Lvk,k − φa C wk,k )δij , pa = (Muk,k − φw C vk,k − φa N wk,k )δij ,

(2)

where δij is Kronecker symbol. G is the shear modulus, χ is effective stress parameter. The other elastic coefficients appearing in the above equations are given in Appendix A.1. Equations of motion, in terms of the displacement components are written as follows.

ζ uj,ij + ς vj,ij + ξ wj,ij + Gui,jj − (τw − 1)φw ρw (u¨ i − v¨ i ) − (τa − 1)φa ρa (u¨ i − w ¨ i) −bw (u˙ i − v˙ i ) − ba (u˙ i − w ˙ i ) = ρs (1 − φ )u¨ i , −φw Wuj,ij + φw2 Lvj,ij + φw φa C wj,ij + (τw − 1)φw ρw (u¨ i − v¨ i ) + bw (u˙ i − v˙ i ) = φw ρw v¨ i , −φa Muj,ij + φw φa vj,ij + φa2 N wj,ij + (τa − 1)φa ρa (u¨ i − w ¨ i ) + ba (u˙ i − w ˙ i ) = φa ρa w ¨ i,

(3)

where

ζ = H + G + W φw + M φa ,

ς = φw [χ L + (1 − χ )C ] − (φw )2 L − φw φa C ,

ξ = φa [χ C + (1 − χ )N ] − (φa )2 N − φw φa C . 3. Harmonic plane waves The displacement components to study the propagation of plane harmonic wave in the medium are given by (ui , vi , wi ) = (Ai , Bi , Ci ) exp {iω(pj xj − t)}, (i = 1, 2, 3),

(4)

where the vectors (A1 , A2 , A3 ), (B1 , B2 , B3 ) and (C1 , C2 , C3 ) are defined as the polarizations of the solid, water and gas particles in the composite medium respectively. The slowness vector (p1 , p2 , p3 ) = N/V represents the propagation/attenuation of a wave through a unit vector N = (N1 , N2 , N3 ) and the velocity V . By substituting (4) in (3), we get nine homogeneous equations, given by

[ζ Ni Nj + (G − {⟨ρw ⟩ + ⟨ρa ⟩ − φw ρw − φa ρa + ρs (1 − φ )}V 2 )δij ]Aj +[ς Ni Nj + V 2 {⟨ρw ⟩ − ρw φw }δij ]Bj + [ξ Ni Nj + V 2 {⟨ρa ⟩ − ρa φa }δij ]Cj = 0,

(5)

[−W φw Ni Nj + V {⟨ρw ⟩ − φw ρw }δij ]Aj + [L(φw ) Ni Nj − V ⟨ρw ⟩δij ]Bj + [C φw φa Ni Nj ]Cj = 0,

(6)

[−M φa Ni Nj + V 2 {⟨ρa ⟩ − φa ρa }δij ]Aj + [C φw φa Ni Nj ]Bj + [N(φa )2 Ni Nj − V 2 ⟨ρa ⟩δij ]Cj = 0,

(7)

2

where

⟨ρa ⟩ = τa ρa φa −

2

ι ba , ω

⟨ρw ⟩ = τw ρw φw −

2

ι bw . ω

The Eqs. (6) and (7) are solved into two relations, given by Bi = Γij Aj ,

Γ =

Ci = ∆ij Aj ,

∆=

b0 a0 c0 a0

(I − NT N) + (I − NT N) +

b0 V 4 + b1 V 2 + b2 a0 V 4 + a1 V 2 + a2 c0 V 4 + c1 V 2 + c2 a0 V 4 + a1 V 2 + a2

NT N,

(8)

NT N,

(9)

where, I is the identity matrix and NT identify the transpose of the row-matrix N = (N1 , N2 , N3 ). The above relations interrelate the polarizations (displacements) of solid particles with water and gas particles in the medium. The polarization vector A defines the polarization of solid particles in the aggregate. Polarizations of the water and gas particles are calculated from the relations (8) and (9), respectively. The constants appearing in the above equations are given in Appendix A.2. Substituting (8) and (9) in (5), we get the following relations Dij Aj = 0,

D = a3 (I − NT N) + b3 NT N,

(10)

The constants appearing in the above equation are given in Appendix A.3. The system of equation (10) represent the Christoffel equations. Non-trivial solution of this system is ensured by a cubic equation (in V 2 ) d3 V 6 + d2 V 4 + d1 V 2 + d0 = 0,

(11)

where,

⎛

⎞

⎡

d3 a0 ⎜ d 2 ⎟ ⎢ a1 ⎝ d ⎠=⎣ a 1 2 d0 0

b0 b1 b2 0

⎤

⎡

( ) c0 0 ρ0 c1 ⎥ ⎢ a0 −⟨ρw ⟩ − φw ρw −⎣ c2 ⎦ a1 −⟨ρa ⟩ − φa ρa 0 a2

0 b0 b1 b2

⎤

( ) 0 ζ +G c0 ⎥ ς , c1 ⎦ ξ c2

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M.S. Barak, M. Kumar, M. Kumari et al. / Wave Motion 93 (2020) 102470

and a linear equation (in V 2 ), given by G−

( ) b0 c0 b0 c0 ρ0 − ⟨ρw ⟩ − ⟨ρa ⟩ + φw ρw + φa ρa V 2 = 0. a0

a0

a0

(12)

a0

The three roots of the cubic equation (11) explains the existence of three longitudinal waves (i.e., polarization vector is parallel to propagation vector) propagating with velocities (Vj , j = 1, 2, 3). For convenience, the three longitudinal waves identified with velocities V1 , V2 and V3 are termed as P1 , P2 and P3 wave, respectively. In an analogs manner, the single root of Eq. (12), explains the existence of lone transverse wave (i.e., polarization ( b

c

vector is normal to propagation vector) propagating with velocity V4 , given by the relation V42 = G/ ρ0 − ⟨ρw ⟩ a0 − ⟨ρa ⟩ a0 0

0

)

+φw ρw ba0 + φa ρa ac0 . This lone transverse wave is identified as S wave. 0

0

3.1. Inhomogeneous waves The general plane waves propagating in a dissipative medium are inhomogeneous waves [14]. The propagation of an attenuated wave is defined with complex slowness vector. The real and imaginary parts of complex slowness vector are termed as propagation vector and attenuation vector, respectively. In general, the inhomogeneity of an attenuating wave is represented through the difference in the directions of its propagation vector and attenuation vector. A finite, nondimensional parameter δ [15] is used to define the inhomogeneous propagation of attenuated waves. Following Sharma and Kumar [16], in terms of inhomogeneity parameter (δ ), the complex slowness vector p of an attenuated wave is written as follows. p=

1

v

ˆ ], [(1 + ıβ )nˆ + ıδ m

(13)

ˆ and an orthogonal unit vector m ˆ identifies the propagation–attenuation plane. For where the propagation direction n δ = 0, the attenuated wave is considered to be propagating as homogeneous wave. This implies that inhomogeneous propagation of the attenuated wave is represented through the deviation of δ from zero. Hence, the magnitude of inhomogeneity parameter δ is considered as the strength of the inhomogeneous wave. For the given value of propagation ˆ and an orthogonal unit vector m ˆ and inhomogeneity parameter δ ∈ (0, 1), using the relations p = N/V , direction n NNT = 1 and λ = V 2 , we obtain |λ|2 v 2 = −2β , ℑ(λ)

ℜ(λ) β= + ℑ(λ)

√ (

ℜ(λ) 2 ) + 1 − δ2 , ℑ(λ)

ˆ + ıδ m ˆ (1 + ıβ )n . N= √ 2 2 1 − β − δ + ı2β

(14)

The attenuation coefficient in terms of inhomogeneity parameter (δ ) is defined as

√ α = ω β 2 + δ 2 /v.

(15)

4. Numerical example The derivations for velocities involve a large number of parameters. Then, to study the dependence of the propagation velocity and attenuation on various properties, a particular numerical example is considered. A sand containing an air– water mixture is chosen for numerical model of partially saturated soils. The values for the elastic constants are given by ρs = 2650 kgm−3 , ρa = 1.1 kgm−3 , ρw = 1000 kgm−3 , G = 50 Mpa, Ka = 0.11 Mpa, Kw = 2.25 Gpa, ηa = 18 × 10−6 pa s, ηw = 1 × 10−3 pa s, ν = 0.3, φ = 0.3, a = 10, m = 1, n = 2, λ = 2.2, Sr = 0.05, ψr = 2000 kPa, k0 = 10−11 m2 . 4.1. Velocity and attenuation Fig. 1 shows the effect of wave frequency on the phase velocities (vj , j = 1, 2, 3, 4) and attenuation coefficients (αj , j = 1, 2, 3, 4) of P1 , P2 P3 and S waves with δ ∈ [0, 1). It is clearly visible that the velocities of all the waves may reduced up to one-half with the change in propagation from homogeneous (i.e., δ = 0) to nearly-evanescent (i.e., δ = 1). The velocities (attenuation coefficients) of all the waves decreases (increases) with the increase of inhomogeneity parameter. The velocities and attenuation coefficients of all waves are increasing with increase of wave frequency. However, increase of frequency may have little impact on the velocities (v1 , v4 ) of two faster waves. But the slower waves (i.e., P2 , P3 ) propagate faster at high frequency. Finally, it is observed that all the waves are dispersive in nature (i.e., frequency dependent) owing to relative movement of the fluids particles respect to solid particles, which causes viscous dissipation of energy. Fig. 2 displays the effect of liquid saturation on the phase velocities and attenuation coefficients of four waves (P1 , P2 , P3 , S). The exact value Sw = 0 is taken in case of no water/all gas. However, for case of no water/all gas, the exact value Sw = 0 was resulting in some singularity in numerical calculation. Hence this case is represented with Sw = 0.01. A significant influence of liquid saturation is observed on slower waves (i.e., P1 , P2 ) velocities and attenuation coefficients. However, little bit impact of liquid saturation is noticed on the velocities and attenuation coefficients of two faster waves (P1 , S). Fig. 3 illustrates the variations of phase velocities and attenuation coefficients

M.S. Barak, M. Kumar, M. Kumari et al. / Wave Motion 93 (2020) 102470

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Fig. 1. Phase velocities (vj , j = 1, 2, 3, 4) and attenuation coefficients (αj , j = 1, 2, 3, 4) of P1 , P2 , P3 , S waves respectively; variations with inhomogeneity parameter (δ ) and wave frequency (ω); Sw = 0.8, φ = 0.4, ψ = 10 000, ν = 0.3, ηw = 1 × 10−3 Pa s, k0 = 10−11 m2 .

with liquid saturation (Sw ) for three different values of wave frequency (ω). A significant impact of wave frequency is observed on the velocities and attenuation coefficients of two slower waves. However, the impact of frequency on the velocities and attenuation coefficients of two faster waves is seems beyond Sw = 0.4. Phase velocities of all the waves increases with the increase of wave frequency. A peak in the attenuation coefficient of P3 wave is observed around Sw = 0.05. The attenuation coefficients of P1 , P2 , S waves are very much sensitive to any change in wave frequency

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M.S. Barak, M. Kumar, M. Kumari et al. / Wave Motion 93 (2020) 102470

Fig. 2. Phase velocities (vj , j = 1, 2, 3, 4) and attenuation coefficients (αj , j = 1, 2, 3, 4) of P1 , P2 , P3 , S waves respectively; variations with inhomogeneity parameter (δ ) and water saturation (Sw ); ω = 1kHz, φ = 0.4, ψ = 10 000, ν = 0.3, ηw = 1 × 10−3 Pa s, k0 = 10−11 m2 .

for Sw ∈ (0, 1). The phase velocity of P2 wave strengthens a lot with the increase of wave frequency for Sw ∈ (0, 0.5). Fig. 4 displays the variations of phase velocities and attenuation coefficients of P1 , P2 , P3 , S waves with inhomogeneity parameter (δ ) for three different values of porosity (φ ). It is observed that with the increase of porosity the phase velocities (attenuation coefficients) of P1 , P3 , S increases (decreases) for δ ∈ (0, 1). Quite opposite, the phase velocity (attenuation coefficient) of P2 wave decreases (increases) with the increase of porosity. Finally, a significant impact of porosity is

M.S. Barak, M. Kumar, M. Kumari et al. / Wave Motion 93 (2020) 102470

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Fig. 3. Phase velocities (vj , j = 1, 2, 3, 4) and attenuation coefficients (αj , j = 1, 2, 3, 4) of P1 , P2 , P3 , S waves respectively; variations with water saturation (Sw ) and wave frequency (ω); δ = 0.1, φ = 0.4, ψ = 1000, ν = 0.3, ηw = 1 × 10−3 Pa s, k0 = 10−11 m2 .

visible on phase velocities of P1 , P2 , P3 , S waves. For three different values of water viscosity ηw = 1, 3, 5 mPa.s, the variations of phase velocities and attenuation coefficients of P1 , P2 , P3 and S waves with δ ∈ (0, 1) are shown in Fig. 5. It is clearly visible from the figure that the phase velocities of P1 , P2 , P3 and S waves increase with the increase of water viscosity (ηw ). A significant increase in the phase velocities of slower waves (P2 , P3 ) is observed with the increase of water viscosity. However, almost no impact of water viscosity is observed on the attenuation coefficients of two faster waves

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M.S. Barak, M. Kumar, M. Kumari et al. / Wave Motion 93 (2020) 102470

Fig. 4. Phase velocities (vj , j = 1, 2, 3, 4) and attenuation coefficients (αj , j = 1, 2, 3, 4) of P1 , P2 , P3 , S waves respectively; variations with inhomogeneity parameter (δ ) and porosity (φ ); ω = 1kHz, Sw = 0.8, ψ = 10 000, ν = 0.3, ηw = 1 × 10−3 Pa s, k0 = 10−11 m2 .

(P1 , S). Fig. 6 illustrates the variations of phase velocities and attenuation coefficients with inhomogeneity parameter (δ ) for three different values of permeability (k0 ). It is clearly visible from the plot that the phase velocities of all the waves increase with the increase of permeability. A significant increase (decrease) in the phase velocities (attenuation coefficients) of slower waves (P2 , P3 ) is observed with the increase of intrinsic permeability. However, almost no impact of permeability is observed on the attenuation coefficients of two faster waves (P1 , S). The characteristics of P1 and S

M.S. Barak, M. Kumar, M. Kumari et al. / Wave Motion 93 (2020) 102470

9

Fig. 5. Phase velocities (vj , j = 1, 2, 3, 4) and attenuation coefficients (αj , j = 1, 2, 3, 4) of P1 , P2 , P3 , S waves respectively; variations with inhomogeneity parameter (δ ) and water viscosity (ηw ); ω = 1kHz, Sw = 0.8, ψ = 10 000, ν = 0.3, φ = 0.4, k0 = 10−11 m2 .

waves are driven mainly by the skeleton. P2 wave is due to the interaction between the solid and fluid phases and P3 wave is associated with capillary pressure fluctuations and having a phase speed dependent on the slope of the relationship between capillary pressure and wetting fluid saturation. The S and P1 waves are driven mainly by the skeleton, therefore,

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M.S. Barak, M. Kumar, M. Kumari et al. / Wave Motion 93 (2020) 102470

Fig. 6. Phase velocities (vj , j = 1, 2, 3, 4) and attenuation coefficients (αj , j = 1, 2, 3, 4) of P1 , P2 , P3 , S waves respectively; variations with inhomogeneity parameter (δ ) and permeability (k0 ); ω = 1kHz, Sw = 0.9, ψ = 10 000, ν = 0.3, φ = 0.4, ηw = 1 × 10−3 pa s.

it is obvious that mass density of the solid is important factor responsible for the any change in the wave characteristics (propagation/attenuation) of these waves. Whereas, wave characteristics of P2 and P3 waves mainly depend on the first pore fluid and capillary pressure, respectively. Therefore, P1 and S waves are less insensitive then the slower P2 and P3 to several parameters.

M.S. Barak, M. Kumar, M. Kumari et al. / Wave Motion 93 (2020) 102470

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5. Conclusions In the recent years, the study of propagation characteristics (velocity/attenuation) in partially saturated rocks has received much attention of researchers. The velocity and attenuation coefficients of seismic waves depend on the various properties of soils such as elastic moduli, viscosity, porosity and permittivity etc. Hence, the identification of correct velocity and attenuation coefficient of seismic wave is essential to predict and understand the characteristics of the soils. Some interesting consequences of present study are explained as follows.

• Phase velocities of all the waves may reduced up to one-half with the change in propagation from homogeneous (i.e., δ = 0) to nearly-evanescent (i.e., δ = 1). All the waves are dispersive in nature (i.e., frequency dependent) owing to relative movement of the fluid particles respect to solid particles, which causes viscous dissipation of energy. The phase velocity of P2 wave strengthens a lot with the increase of wave frequency for Sw ∈ (0, 0.5). • The increase of porosity may increases the phase velocities of P1 , P3 , S waves but reduce the attenuation coefficients of P1 , S waves. However, it certainly decrease/increase the phase velocity/attenuation coefficient of P2 wave. • The increase of water viscosity/permeability may increase the phase velocities of all the waves but attenuation coefficients of P1 , S waves are unaffected. • Effect of wave frequency, liquid saturation, porosity, water viscosity, permeability and inhomogeneity parameter on the phase velocities and attenuation coefficients of slower waves (i.e., P2 , P3 ) seems to be significant. This wave propagation study is an attempt to explore the role of subsurface features (wave frequency, saturation of water, inhomogeneity parameter, and viscosity of water and porosity of pore space) in the propagation of seismic waves. The advancement in knowing the structure of the earth consists of more accurate determination of seismic velocities and attenuation. The measurements of seismic wave velocities and seismic attenuation are the common tools, which are used to analyze this subsurface fluid movement. The characterization of residual stresses in the reservoirs, earth masses and composite materials could be possible through analyzing the velocity variations in them. The expressions derived for the propagation characteristics (velocity/attenuation) further, may be used to prepare a platform for studying the surface waves and to study the reflection/refraction/scattering process at the boundaries of partially saturated soils. This study may be helpful to the prospecting seismologists working for improved oil recovery. Appendix A.1. H = 2as G − χ W − (1 − χ )M , L=

Kw [φ

s a m1

+

Ka (ms1 mw 2

−

W =−

ms2 mw 1)

D s w s w A = φa mw 1 + Ka (m1 m2 − m2 m1 ),

]

,

Kw A D

N =

+ φw L + φa C , Ka [φ

s w m1

+

M=−

Kw (ms1 mw 2

−

Ka B D

+ φw C + φa N ,

ms2 mw 1)

D s w B = φw ma1 + Kw (ms1 mw 2 − m2 m1 ),

]

,

C =

s w Kw Ka (ms1 mw 2 − m2 m1 )

D

,

s w s D = (ms1 mw 2 − m2 m1 )(φa Kw + φw Ka ) + φa φw m1 ,

where as (= ν/(1 − 2ν )) is related to Poisson’s ratio (ν ) of soil skeleton, Kw bulk modulus of water, Ka bulk modulus of air, ms1 is the coefficient of volume change with respect to net normal stress, ms2 is the coefficient of volume change w with respect to matric suction, mw 1 is the coefficient of water volume change with respect to net normal stress, m2 is a s a s w w the coefficient of water volume change with respect to matric suction. m1 (= m1 − m1 ) and m2 (= m2 − m2 ) are the coefficients of air volume change with respect to net normal stress and metric suction, respectively. The effective stress s parameter χ can be derived by the relation χ = ms2 /ms1 . Cosentini [17] showed that mw 1 = m2 . The model parameters φw and φa are given by [18]

φ w = C (ψ )

[ w

m2 = −

θs (ln(e +

ψ ( )n ))m

]

ma1 = (1 − χ )ms1 ,

C (ψ ) = 1 −

106

)

[(

θs

r

ψr

φa = φ − φw ,

a

1 ( ψ+ψ )

ln(1 +

,

ψ

(ln(e + ( a )n ))m

+

ma2 = χ ms1 − mw 2,

ln(1 + 1− ln(1 +

ψ ψr 106

ψr

)

ln(1 + ln(1 +

)( −mn

)

ψ ψr 106

ψr

) )

,

ms1 =

3(1 − 2ν ) 2(1 + ν )G

θs ψ

(ln(e + ( a )n ))m+1

)(

,

ψ n−1 ean + ψ n

)]

,

where ψ (kPa) is soil metric suction, ψr (kPa) is residual soil suction, θs is volumetric water content at saturation, e is natural log base, a (kPa)is a soil parameter related to the air entry value of the soil, n is a soil parameter related to the rate of desaturation, and m is a soil parameter which is related to the residual water content of the soil. The effective tortuosity of water (τw ) and air (τa ) phases, and tortuosity of porous medium τ are given by

τw =

τ (1 − Sr ) , Sw − Sr

τa =

τ (1 − Sr − Se ) , 1 − Sw − Se

( ) 1 τ = 1 − 0.5 1 − φ

12

M.S. Barak, M. Kumar, M. Kumari et al. / Wave Motion 93 (2020) 102470

where Se is equilibrium saturation of air phase. Assuming the equilibrium saturation Se = 0, Brooks and Corey [19] derived the effective permeability of water (kw ) and air (ka ) phases are as follows:

( kw = k0

Sw − Sr

)(2+3λ)/λ

1 − Sr

(

,

ka = k0

1−

Sw − Sr

)2 (

( 1−

1 − Sr

Sw − Sr

)(2+λ)/λ )

1 − Sr

,

where k0 is the intrinsic permeability of medium and λ is pore-size distribution index. A.2. a0 = ⟨ρa ⟩⟨ρw ⟩,

2 L⟨ρa ⟩ + φa2 N ⟨ρw ⟩), a1 = −(φw

b0 = ⟨ρa ⟩(⟨ρw ⟩ − φw ρw ), b2 = φa2 φw (NW − MC ),

2 , a2 = (LN − C 2 )φa2 φw

b1 = φa φw C (⟨ρa ⟩ − φa ρa ) − φw W ⟨ρa ⟩ − φa2 N(⟨ρw ⟩ − φw ρw ), c0 = ⟨ρw ⟩(⟨ρa ⟩ − φa ρa ),

2 L(⟨ρa ⟩ − φa ρa ), c1 = φa φw C (⟨ρw ⟩ − φw ρw ) − φa M ⟨ρw ⟩ − φw

2 φa (ML − WC ). c2 = φw

A.3. a3 = G − (ρ0 − ⟨ρw ⟩

b0 a0

− ⟨ρa ⟩

c0 a0

+ φw ρw

b0 a0

+ φa ρa

b3 = ζ + G − ρ0 V 2 + [ς + V 2 {⟨ρw ⟩ − φw ρw }][

c0 a0

)V 2

b0 V 4 + b1 V 2 + b2 a0 V 4 + a1 V 2 + a2

] + [ξ + V 2 {⟨ρa ⟩ − φa ρa }][

c0 V 4 + c1 V 2 + c2 a0 V 4 + a1 V 2 + a2

],

ρ0 = ⟨ρw ⟩ + ⟨ρa ⟩ − φw ρw − φa ρa + ρs (1 − φ ). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

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