1 Introduction

As we know,every kind of waves contains the information about the physical properties of the propagation medium. Therefore,the analysis of wave propagation is not only of great significance in theoretical study but also of distinctive practical value in many engineering fields such as geotechnical engineering,oil exploration,earthquake engineering,and vibration test^{[1, 2]}. To achieve that,it is essential to study the relation between the parameters about the physical properties of the medium and the waves propagating in it. Porous media,such as saturated and unsaturated media,are very common in reality. In the past years,the effects of the parameters on the waves in porous media have been widely studied.

(i) The waves propagating in porous media have been widely studied in the cases of considering or neglecting the compressibility of the solid grain^{[3, 4, 5, 6, 7]}. However,those studies were conducted separately,and no comparison has ever been made between the results in those two different cases. Consequently,we have no idea about the effects on the results in the case of neglecting the compressibility of solid grain. Therefore,it is necessary to make assessment about the effects of the compressibility of solid grain.

(ii) The frequency effects have been widely investigated on the wave propagation velocity and attenuation^{[8, 9, 10, 11, 12]}. Beskos et al.^[13] conducted a numerical analysis about the frequency effects on the velocities and attenuations of bulk waves in saturated rocks with different combinations of porosities and permeabilities. Kim et al.^[14] and Sharma^[15] numerically illuminated how the velocities and attenuations of the bulk waves in dissipative porous media were affected by a combined parameter,namely,the product of frequency and fluid viscosity. Thereinto,Kim et al.^[14] compared the results in the cases with different values of porosity,bulk modulus of solid skeleton,and Poisson’s ratio. Lu et al.^[11] showed the frequency effects on the propagation characteristics of the bulk waves under several different values of fluid viscosity. Liu and De Boer^[8],Zhang et al.^[16],and Chen et al.^[17] numerically analyzed the effects of frequency and saturation on the velocities and attenuations of the surface waves in the cases of several different values of permeability,Poisson’s ratio,and coupling-damping coefficient (a combined parameter of porosity,fluid viscosity,and permeability). To sum up,apart from frequency,the analysis about how the wave propagation velocity and attenuation vary with the continuous change of each single parameter has seldom been made.

(iii) Among the available references,only Yang et al.^[18] once numerically analyzed the variations of the propagation velocities and attenuations of the bulk waves in saturated soils with continuous changes in the frequency,porosity,permeability,and fluid viscosity,respectively. However,in the numerical calculation,the relation between the porosity and the moduli of the porous media (porosity-moduli relation for short) was not considered. Instead,when the porosity effects on the wave propagationcharacteristics were analyzed,the bulk modulus and the shear modulus of the solid skeleton were assumed to be constants when the porosity increased. For instance,the bulk modulus and the shear modulus of the solid skeleton in the case of n = 0.05 (n,the notation of porosity) were the same as those in the case of n = 0.9. Obviously,that kind of processing is unreasonable,and does not agree with the fact. Besides,the degeneration of the numerical results and the effect of Poisson’s ratio were not discussed.

Based on the deficiencies mentioned above,the present paper further analyzes the effects of frequency on the propagation characteristics of the bulk waves in a saturated porous medium. The numerical results in the cases of considering and neglecting the compressibility of the solid grain are compared. Then,the obtained results are compared with those of Yang et al.^[18]. The effects of the porosity and Poisson’s ratio on the wave propagation velocity and attenuation are studied with the consideration of the porosity-moduli relation,and the degeneration of the numerical results is also discussed. Conclusions are given in the end.

2 Basic theoretical framework

2.1 Governing equations

In Biot’s theory^[19],the dynamic governing equations for wave propagation in saturated porous media can be expressed in terms of the displacements as follows:

where e = ∇u,and ε = ∇U,in which u and U are the displacement vectors for the solid phase and the fluid phase,respectively. b = n²η/κ_d,where n is the porosity of the porous medium. η is the fluid viscosity,and κd is the intrinsic permeability (with the dimension L²). ρ11 and ρ22 represent the mass coefficients of the solid phase and the fluid phase,respectively. ρ₁₂ represents the mass coupling coefficient between the solid phase and the fluid phase. nρ_f = ρ₁₂ + ρ₂₂,and (1 − n)ρ_s = ρ₁₁+ρ₁₂,where ρs and ρf represent the density of solid grain and fluid,respectively. A,R,Q,and N are the Biot elastic coefficients,and they can be expressed as follows: where λ and μ are the Lame constants of the solid skeleton,respectively. α and M are the Biot coefficients representing the compressibility of the fluid and solid grain,respectively.

The expressions of A,R,and Q are usually different in the literature^{[20, 21, 22]}. However, they are essentially consistent after substituting the relations among the parameters mentioned below.

When the compressibility of the solid grain is considered,the relation between the two Biot coefficients,α and M,and the bulk moduli of the porous media can be expressed as follows:

where K_s,K_b,and K_f are the bulk moduli of the solid grain,the solid skeleton,and the fluid,respectively.

When the compressibility of solid grain is neglected,i.e.,K_s → ∞,Eq. (3) can be rewritten as follows:

As we know,for the elastic solid skeleton,the stress-strain relation is

From Eq. (5),we can derive and ε_v is the volumetric strain expressed defined by

According to the elastic theory,we have

where E and ν are the elastic modulus and Poisson’s ratio of the solid skeleton,respectively.

Combining Eqs. (6)-(8),we have

For a porous medium whose constituents are ascertained,the bulk moduli of the solid grain and fluid,i.e.,K_s and K_f,are constants,while the bulk modulus of the solid skeleton,K_b,will decrease with the increase in the porosity. Ignoring the porosity effect on the relevant Poisson ratio,Luo and Stevens^[23] described the relation between K_b and the porosity as follows: As we can see from Eqs. (9)-(12),for a porous medium whose constituents are ascertained,the bulk modulus of the solid grain,K_s,is invariable. However,the bulk modulus and the Lame constants of the solid skeleton,i.e.,K_b,λ,and N,will vary when the porosity and Poisson ratio change.

2.2 Theoretical solution

According to the Helmholtz decomposition theorem,the displacement vectors u and U can be decomposed into

where φ_s and φ_f are the scalar potentials of the solid displacement and fluid displacement, respectively. ψ_s and ψ_f are the vector potentials satisfying

Substituting the divergence and the curl operator into Eq. (1),respectively,we can obtain the governing equations for the compressional wave and the shear wave propagating in the saturated porous media as follows^[24]:

where

The general form of the solution for the plane bulk waves can be assumed as follows:

where i is the imaginary unit,i= . ω is the angular frequency defined by

where f is the frequency. A_s,A_f,B_s,and Bf are the amplitudes of the wave motion. r is the position vector. k_p and k_s are the wave number vectors for the compressional wave and the shear wave,respectively.

In this paper,we only study the bulk waves in a two-dimensional plane,i.e.,the xz-plane of the Cartesian coordinate system. Therefore,

Substituting Eq. (16) into Eq. (14),we can obtain

where v_p is the theoretical expression for the velocity of the compressional wave defined by

and

To guarantee the existence of nonzero solutions for A_s and A_f in Eq. (18),the following condition must be satisfied:

Simplifying Eq. (19),we obtain From Eq. (20),we can draw the conclusion that there exist two kinds of compressional waves in a saturated porous medium,which are known as the fast compressional wave,P₁,and the slow compressional wave,P₂. Furthermore,we assume that the theoretical solutions of Eq. (20) are v_p1 and v_p2. Therefore,the corresponding complex wave numbers are where v_p1 and v_p2 are the theoretical expressions for the wave velocities of the P₁ wave and the P₂ wave,respectively,and both of them are complex numbers. Here,for the purpose of discussion,we define vp1 and vp2 as complex velocities.

Take the absolute value of the imaginary part of k_pi as the attenuation coefficient. Then, the attenuations of the P₁ wave and the P₂ wave can be defined by

where Im represents the imaginary part of a complex number.

When the propagation characteristics of the waves in porous media are studied,some researchers^{[8, 25, 26]} take the real part of the complex velocity,i.e.,Re(v_pi),as the wave propagation velocity. However,it is more reasonable and more widely accepted to take the phase velocity as the wave propagation velocity^{[10, 15, 16, 27]},i.e.,

Substituting Eq. (17) into Eq. (15),we can obtain

To guarantee the existence of nonzero solutions for B_s and B_f in Eq. (24),the following condition must be satisfied: Simplifying Eq. (25),we obtain From Eq. (26),we can infer that there exists one shear wave,i.e.,the S wave,in a saturated porous medium,and obtain the complex velocity v_s. The complex wave number for the S wave is Similarly,the attenuation of the S wave,δ_s,is defined by and the wave propagation velocity is

3 Numerical results and discussion

Since the numerical results about the effects of permeability and fluid viscosity on the bulk waves in the case of considering the porosity-moduli relation are consistent with those obtained by Yang et al.^[18] in which the porosity-moduli relation is neglected,we focus on analyzing the effects of frequency,porosity,and Poisson’s ratio in the following.

The common parameters for the numerical calculation are as follows:

3.1 Effects of frequency

Figure 1 depicts the effects of frequency on the propagation velocities and attenuations of the bulk waves in a saturated porous medium in the cases of considering and neglecting the compressibility of solid grain,respectively.

As we can see from Fig. 1,for each bulk wave,the variation tendency of the velocity is consistent with that of the attenuation. The P₁ wave,known as the fast compressional wave, propagates the fastest,while attenuates the most slowly. The P₂ wave,known as the slow compressional wave,attenuates the fastest.

Biot^[19] introduced a characteristic frequency as follows:

Substituting the values of the relevant parameters into Eq. (30),the characteristic frequency of the porous medium considered here is f_c = 2 547 Hz.

According to the numerical results and the position of f_c shown in Fig. 1,the whole frequency band can be divided into three parts,i.e.,low frequency band,medium frequency band,and high frequency band. Specifically,,the medium frequency band refers to a certain frequency range around f_c,and the left and right sides of the medium frequency band are the low frequency band and the high frequency band,respectively.

Based on the above analysis,more details regarding the frequency effects are found as follows.

(i) For the P₁ wave and the S wave,both the velocities and attenuations almost keep constant in the low frequency band and the high frequency band,while grow rapidly with the increase in the frequency in the medium frequency band. It is also worth pointing out that the attenuations of the P₁ wave and the S wave are nearly zero in the low frequency band.

(ii) For the P₂ wave,unlike the P₁ wave and the S wave whose variation curves can be distinctly divided into three different segments,when the frequency increases,the velocity and attenuation grow at first,and then almost keep invariable when the frequency is beyond f_c.

From Fig. 1,we can also compare the results in the cases of considering and neglecting the compressibility of solid grain.

In the case of considering the compressibility of solid grain,substituting the values of the relevant parameters into Eq. (3),we can obtain the Biot coefficient α,i.e.,α = 1/3. According to Eq. (4),the case when α = 1 corresponds to the case of neglecting the compressibility of solid grain.

Figure 1 also shows that whether considering the compressibility of solid grain or not,there is no effect on the variation tendency of the curves.

For the P₁ wave,the velocity in the case when α = 1 is greater than that in the case when α = 1/3,and the two curves are almost parallel to each other (see Fig. 1(a)). Instead, the attenuation in the case when α = 1/3 is greater than that in the case when α = 1. The discrepancy between them is quite tiny in the low frequency band,and then gradually enlarges in the medium frequency band. When the frequency increases to the high frequency band,the discrepancy keeps constant.

For the P₂ wave,the velocity corresponding to α = 1/3 is slightly greater than that corresponding to α = 1,while the attenuation is the opposite. Besides,within the characteristic frequency,the difference between the two velocity curves is tiny,which is also the case for the attenuation curves.

For the S wave,the curves when α = 1/3 and α = 1 almost overlap each other,which means that the compressibility of solid grain has little effect on the velocity and attenuation of the S wave.

3.2 Effect of porosity

As mentioned before,compared with Yang et al.^[18],this paper considers the porosity-moduli relation of the porous media. Based on the methods used by this paper and Yang et al.^[18],the porosity effects on the propagation velocities and attenuations of bulk waves are illustrated in Fig. 2,and the range of the porosity discussed here is from 0.001 to 0.999.

As shown in Fig. 2,there exist obvious changes in the velocities and attenuations as the porosity varies,and the numerical results obtained by the methods of this paper and Yang et al.^[18] are remarkably different.

For the P₁ wave,based on the theory of this paper,as the porosity increases from 0.001 to 0.999,the velocity decreases,while the attenuation increases gently at first,and then decreases. Instead,in the case of Yang et al.^[18],the velocity grows with the increase in the porosity,while the attenuation grows sharply at the beginning,and then decreases.

For the P₂ wave,the velocities of both the two cases almost vary in the same route,decreasing with the increase in the porosity. However,the starting points and the ending points of the two curves are different. So far as attenuation is concerned,in the case of this paper,it grows slightly at first,and then increases sharply when the porosity approaches 1.0. However,in the case of Yang et al.^[18],the attenuation grows sharply at the very beginning,and then almost keeps invariable.

For the S wave,both the velocities and the attenuations corresponding to the cases of this paper and Yang et al.^[18] vary in the opposite way. Specifically,in the case of this paper,the velocity decreases with the increase in the porosity,and the attenuation grows gently at first, and then increase sharply when the porosity gets close to 1.0.

In the above analysis,we have compared the numerical results of this paper and Yang et al.^[18]. In the following,through a degeneration analysis,we will further discuss the rationality of the results obtained by the two cases.

When n = 0,the saturated porous medium degenerates into an elastic solid,where only two kinds of bulk waves exist,i.e.,the compressional wave and the shear wave,and the slowcompressional wave,P2,disappears. As we know,the bulk waves in an elastic solid are of no attenuation. Just as shown in Figs. 2(b),2(d),and 2(f),the attenuations of both this paper and Yang et al.^[18] are nearly zero when n is 0.001.

In the case when n = 0,according to Eqs. (7)-(12),the relevant parameters of the elastic solid are

Then,the velocities of the compressional wave and the shear wave in the elastic solid are

In the above numerical calculation,when n = 0.001,the results of this paper are

which are very close to the velocities of the bulk waves in the elastic solid. However,the results of Yang et al.^[18] are

which deviate considerably from the results in the elastic solid.

When n = 1,the saturated porous medium degenerates into a pure fluid. The fluid is assumed to be unable to sustain the shear strain in Biot’s theory. Therefore,only a kind of bulk wave,namely,the compressional wave,exists in the fluid,and the P₂ wave and the S wave disappear. Consequently,when the porosity approaches 1.0,the attenuation of the P₁ wave is supposed to decrease to nearly zero,while the attenuations of the P2 wave and the S wave are supposed to grow rapidly. As shown in Figs. 2(b),2(d),and 2(f),when n gets close to 1.0,the attenuations of the P₁ wave in the cases of both this paper and Yang et al.^[18] decrease close to zero,and the attenuations of the P₂ wave and the S wave in the case of this paper exhibit a rapid increase. However,the attenuations of the P₂ wave and the S wave in the case of Yang et al.^[18] have no obvious changes when n approaches 1.0,and are much smaller than the results of this paper.

In the case of n = 1,combined with Eqs. (7)-(12),the corresponding parameters of the fluid are

Then,the velocity of the compressional wave in the fluid is

According to the numerical calculation,when n = 0.999,the velocity of the P1 wave of this paper is

which is quite close to the result in Eq. (35),while in Yang et al.^[18],

As we can infer from the above degeneration analysis,although both this paper and Yang et al.^[18] are based on Biot’s theory,this paper considers the porosity-moduli relation of the porous media in the parametric studies. Therefore,the numerical results of this paper are more rational than those of Yang et al.^[18].

3.3 Effects of Poisson’s ratio

Figure 3 shows the effects of Poisson’s ratio ν on the propagation velocities and attenuations of the bulk waves. The values of ν considered here are from 0.01 to 0.498.

From Figs. 3(a),3(c),and 3(e),we can see that the velocities of the P₁ wave and the S wave almost decrease linearly when ν increases. While for the P₂ wave,the velocity decreases quite slightly at the beginning,and then decreases sharply when ν gets close to 0.5. Instead,as shown in Figs. 3(b),3(b),and 3(f),the attenuations of the P₂ wave and the S wave grow gently with the increase in ν at first,and then increase sharply when ν approaches 0.5. For the P₁ wave, the attenuation increases at first,then decreases,and has a rapid increase when ν approaches 0.5. Besides,the numerical results shown in Fig. 3 also verify the former conclusion that the P₂ wave attenuates the fastest. The attenuation of the P₂ wave has orders of magnitude larger than those of the P₁ wave and the S wave.

4 Conclusions

In this paper,considering the porosity-moduli relation of the porous medium,we study the effects of frequency,porosity,and Poisson’s ratio on the propagation characteristics of bulk waves in saturated porous media,and compare the obtained results with those of Yang et al.^[18]. Conclusions can be drawn as follows.

(i) According to the characteristic frequency,f_c,the whole frequency band can be naturally divided into three parts,i.e.,,low frequency band,medium frequency band,and high frequency band. For the P₁ wave and the S wave,both the velocities and attenuations almost keep invariable in the low frequency band and the high frequency band,while grow sharply in the medium frequency band. For the P₂ wave,the velocity and attenuation grow with the increase in the frequency within f_c,and then keep constant when the frequency is beyond f_c.

(ii) The compressibility of the solid grain has a distinctive effect on the P₁ wave,and a tiny effect on the P₂ wave,while no effect on the S wave.

(iii) For a saturated porous medium whose constituents are ascertained,in the case of considering the porosity-moduli relation,the velocity of each bulk wave decreases as n increases from 0.001 to 0.999,and the attenuation of the P₁ wave increases gradually firstly,then decreases to nearly zero,while the attenuations of the P₂ wave and the S wave grow slightly at first,then increase sharply when n gets close to 1.0.

(iv) In contrast with the case of neglecting the porosity-moduli relation,the numerical results in the case of considering the porosity-moduli relation can be degenerated into the results of elastic solid and pure fluid,thereby verifying the rationality of this paper. The porosity-moduli relation cannot be neglected when the porosity effects on the wave propagation characteristics are studied.

(v) Poisson’s ratio ν has obvious effects on the propagation characteristics of the bulk waves in saturated porous media. Generally speaking,as ν increases,the velocities of the P₁ wave, the P₂ wave,and the S wave decrease all the way,while the attenuations vary gently at the beginning,and then grow rapidly when ν approaches 0.5.