Elsevier

Wave Motion

Volume 61, March 2016, Pages 73-82
Wave Motion

Rayleigh waves in orthotropic fluid-saturated porous media

https://doi.org/10.1016/j.wavemoti.2015.10.007Get rights and content

Highlights

  • The propagation of Rayleigh waves in orthotropic non-viscous fluid-saturated porous media is investigated.

  • Stroh’s formalism is obtained from the basic equations in matrix form.

  • The secular equation of the wave in explicit form is derived using the method of polarization vector.

  • It is not a complex equation as the one derived previously. It is a real equation.

Abstract

In this paper, we are interested in the propagation of Rayleigh waves in orthotropic fluid-saturated porous media. This problem was investigated by Liu and Liu (2004). The authors have derived the secular equation of the wave but that secular equation is still in implicit form. The main aim of this paper is to derive explicit secular equation of the wave. By employing the method of polarization vector, the secular equations of Rayleigh waves in explicit form is obtained. This equation recovers the dispersion equation of Rayleigh waves propagating in pure orthotropic elastic half-spaces. Remarkably, the secular equation obtained is not a complex equation as the one derived by Liu and Liu, it is a really real equation.

Introduction

Elastic surface waves, discovered by Rayleigh  [1] more than 120 years ago for compressible isotropic elastic solids, have been studied extensively and exploited in a wide range of applications in seismology, acoustics, geophysics, telecommunications industry and materials science, for example. It would not be far-fetched to say that Rayleigh’s study of surface waves upon an elastic half-space has had fundamental and far-reaching effects upon modern life and many things that we take for granted today, stretching from mobile phones through to the study of earthquakes, as addressed by Adams et al.  [2]. A huge number of investigations have been devoted to this topic. As written in  [3], one of the biggest scientific search engines Google.Scholar returns more than a million links for request “Rayleigh waves”.

For the Rayleigh waves, their dispersion equations in explicit form are very significant in practical applications. They can be used for solving the direct (forward) problems: studying effects of material parameters on the wave velocity, and especially for the inverse problems: determining material parameters from the measured values of the wave speed. Thus, the secular equations in explicit form are always the main purpose of any investigation related to Rayleigh waves.

Since most of the geological material can be treated as some kind of fluid-saturated porous media, the study of elastic wave propagation and attenuation through a fluid-saturated porous medium has significant practical meaning in many fields such as earthquake engineering, soil dynamics, geophysics and hydrology.

Deresiewicz  [4] first researched the Rayleigh waves in porous media based on Biots theory. Johns  [5] studied the Rayleigh waves in isotropic saturated soil, but only one kind of longitudinal waves was considered in his characteristic equations and thus the equations obtained by him are not complete. Tajuddin  [6] considered two kinds of the longitudinal waves and got the characteristic equations for isotropic fluid-saturated porous media. Hirai  [7] analyzed the Rayleigh waves in isotropic layered fluid-saturated porous media by finite element methods, and the effects of the dynamic permeability on the propagation characteristic of Rayleigh waves are discussed. Sharma and Gogna  [8] studied the Rayleigh waves in transversely isotropic porous media but the dissipation is ignored. Liu and de Boer  [9] studied the Rayleigh waves in isotropic fluid-saturated porous media using mixture theory. Kumar  [10] analyzed the influence of the heterogeneous base on the propagation of Rayleigh waves in isotropic fluid-saturated porous layer. Liu and Liu  [11], [12] discussed the fluid viscous effects on the propagation characteristic of Rayleigh waves in transversely isotropic fluid-saturated porous media. All investigations mentioned above are mainly focused on the isotropic and transversely isotropic fluid-saturated porous media. However, according to the research results of multi-scale analysis, geological materials with random cracks/joints distribution, such as rocks, can be classified as some kinds of transversely isotropic or orthotropic materials after statistical treatment  [13]. The propagation of Rayleigh waves in non-viscous-fluid-saturated orthotropic porous media was studied recently by Lui and Lui  [14]. The authors have derived the secular equation of the Rayleigh wave. However, that secular equation is still in implicit form.

The main purpose of this paper is to derive the secular equation in explicit form of Rayleigh waves propagating in a non-viscous-fluid-saturated orthotropic porous half-space. Firstly, the basic equations are written in matrix form. Then, from this matrix equation we arrive immediately Stroh’s formalism  [15]. Secondly, based on Stroh’s formalism, the secular equation in explicit form is derived using the method of polarization vector. It is shown that the secular equation obtained recovers the one of Rayleigh waves propagating in orthotropic elastic half-spaces. Remarkably, the dispersion equation obtained by Lui and Lui  [14] is really complex equation, while the one derived in this paper is only real equation.

Section snippets

Basic equations in matrix form

In this section, we recall the basic equations governing the motion of the non-viscous-fluid saturated porous media. These equations are then rewritten in matrix form. A non-viscous-fluid saturated porous medium is understood as follows: (i) The porous medium is elastic. (ii) The fluid in pores is non-viscous, i.e. the viscosity of the fluid is neglected. This assumption leads to the absence of dissipation (see Section 3 in Ref.  [16]). (iii) The medium is isothermal. Following Biot  [17], the

Rayleigh waves. Stroh’s formalism

Now we consider a two-dimensional problem so that the motion is independent of the variable x2 and u20. In particular, we consider a Rayleigh wave traveling with the velocity c(>0), the wave number k(>0), in the x1-direction and decaying in the x3-direction. Then, the Rayleigh wave fields are sought in the following form: u1=U1(y)eik(x1ct),u3=U3(y)eik(x1ct),u20,w3=W(y)eik(x1ct),σ31=ikT1(y)eik(x1ct),σ33=ikT3(y)eik(x1ct),σ320,p=ikP(y)eik(x1ct), where y=kx3. Substituting (21) into Eq.

Explicit secular equation

From the free-traction condition (27) we have ξ(0)=[U1(0)U3(0)W(0)000]T. Substituting (31) into Eq. (30) ​yields U¯(0)TK(n)U(0)=0,n=2,1,1,2,3, where U(0)=[U1(0)U2(0)W(0)]T, n=2,1,1,2,3. Suppose U1(0)0, then the vector U(0) can be written as U(0)=U1(0)[1αβ]T, where α=U2(0)/U1(0), β=W(0)/U1(0) are complex number, α=αˆ1+iαˆ2, β=β1+iβ2, αˆ1, αˆ2, β1, β2 are real. Introducing the expression of U(0) into Eq. (32) and taking into account the symmetry of matrix K(n) we have [1α¯β¯][K11(n)K12(n)K13(

Implicit secular equation

Eliminating u from Eq. (22) we obtain an equation for t, namely r1tir2tr3t=0,0y<+,where r1=K1,r2=N1K1+K1N1T,r3=N1K1N1TN2. According to (26), (27), t(+)=t(0)=0. One can see that the general solution of (46) satisfying the decay condition t(+)=0 ​is t=γ1eis1yt(1)+γ2eis2yt(2)+γ3eis3yt(3), where γ1,γ1,γ3 are constants to be determined, s1,s2,s3 are (distinct from each other) three roots with positive imaginary parts of the characteristic equation: det|s2r1sr2+r3|=0det|NsI|=0, that

Dimensionless secular equation

It will be convenient in use if we have in hand the dimensionless version of the secular equation (42). Multiplying two sides of Eq. (42) by M4ϕ6/C556 and after some manipulations we arrive at the equation: 4Dˆ1Dˆ2Dˆ32=0,where Dˆ1=dˆ1(Kˆ23Kˆ33(3)ϕKˆ33Kˆ23(3))+rx[Kˆ11(Kˆ33(3)+Kˆ23(3))Kˆ11(3)(Kˆ23+ϕKˆ33)],Dˆ2=ϕ[dˆ1(Kˆ22Kˆ23(3)Kˆ23Kˆ22(3))+dˆ2(Kˆ23Kˆ11(3)Kˆ11Kˆ23(3))+rx(Kˆ22Kˆ11(3)Kˆ11Kˆ22(3))],Dˆ3=dˆ1(ϕ2Kˆ33Kˆ22(3)Kˆ22Kˆ33(3))+dˆ2(Kˆ11Kˆ33(3)ϕKˆ33Kˆ11(3))+rx(Kˆ22Kˆ11(3)ϕKˆ

Conclusions

In this paper, we consider the propagation of Rayleigh waves in non-viscous fluid-saturated orthotropic porous media. The secular equation of the wave is derived using the method of polarization vector. This secular equation obtained is totally explicit and is an algebraical equation of 12th order. Moreover, it is a real equation, not a complex equation as the one obtained previously. The secular equation obtained recovers the one for pure elastic solids. Some numerical examples are considered

Acknowledgments

The work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 107.02-2014.04 and the ICTP-IAEA Sandwich Training Educational Programme.

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