Effect of stresses on wave propagation in fluid-saturated porous media

https://doi.org/10.1016/j.ijengsci.2021.103519Get rights and content

Abstract

The mechanism of the effect of stresses on wave propagation in a fluid-saturated porous media has not been well understood. The goal of this paper is to fill this gap. First we formulated the general equations of motion in a homogeneously pre-stressed fluid-saturated medium and use them to derive explicit expressions of velocity dependence on stresses. Equations for fast and slow longitudinal waves allow substantial simplifications. The feasibility of fast longitudinal wave simplification is verified by comparison with experimental data available in literature.

Introduction

Present work focuses on the effect of stresses on wave propagation in fluid saturated porous media. Effect of saturation on the overall mechanical properties of a porous material is discussed in literature starting from classical works of Gassmann (1951a, 1951b). Review of the available results on the effect of fluid saturation on overall material properties of porous materials was done by Sevostianov (2020). The main applications involve petroleum geophysics, underwater acoustics, exploration seismology, soil engineering, biomechanics (Cowin & Cardoso, 2011; Ramırez-Torres et al., 2017; Vilchevskaya, Levin, Seyedkavoosi & Sevostianov, 2019; Cieszko& Kubik., 2020; Hodaei, Maghoul & Popplewell, 2020; Ju, Gong, Chang & Sun, 2020; Zhou, Cui & Sevostianov, 2020). Physical properties of fluid saturated materials are very sensitive to stresses. Kanaun (2019) considered effect of pressure of injected fluid on cavities in poroelastic media under quasistatic conditions. Guo, Vardakis, Chou and Ventikos (2020) used poroelastic model to account for stress effect on fluid transport in biological systems. Lei, Liao, Patil and Zhao (2021) proposed an analytical model for stress dependent permeability of saturated porous media focusing on permeability and extend of saturation. Effect of stresses on the elastic wave propagation in a fluid-saturated porous media attracted attention of researchers from 1980s. Pao, Sachse and Fukuoka (1984) considered change of elastic wave velocity caused by the static stress and introduced the term acoustoelasticity for such problems. Grinfeld and Norris (1996) extended the acoustoelastic theory applicable to fluid-saturated porous media. Ba, Carcione, Cao, Yao and Du (2013) accounted for nonlinear term in static strain. Wang and Tian (2014, 2017) gave the equations for small dynamic fields superimposed on a static deformation of a fluid-saturated porous media based on the finite deformation theory of the continuum and Biot theory. Fu and Fu (2018) used the poroacoustoelasticity theory developed by Ba et al. (2013) and a dual-porosity model to account for compliant pore by using a semiempirical equation. Qu, Liu, Cui and Lv (2018) initially estimated some third-order elastic moduli of dry and saturated rock. Winkler and McGowan (2004) measured three third-order elastic constants in several dry and water-saturated rocks, and showed that classical acoustoelastic theory for an elastic medium did not fully describe the stress dependence of velocities in water-saturated rock. They explained the difficulties with correct prediction of the wave speeds in a water saturated rock by neglecting effects of dispersion.

The acoustoelasticity of elastic solid for infinitesimal dynamic fields superimposed on static deformation has been adopted and used in geophysics (Liu & Sinha, 2003; Liu, Zhou, Cui & Kundu, 2019; Vernik, 2016) and non-destructive evaluation of defects (Abderahmane, Emery & Daniel, 2021; Chaki & Bourse, 2009). To the best of our knowledge, however, the general equations for small dynamic fields superimposed on static deformation of isotropic fluid-saturated porous media including viscous and dispersion effects have not been given in literature and, due to that, acoustoelasticity has not been used for simulation of the wave propagation in fluid-saturated porous media. As a result, acoustic logging is not used to detect in-situ stress in fluid saturated porous media, in contrast with the homogeneous solid materials (Lei, Bikash, Sinha & Sanders, 2012; Liu et al., 2019; Schmitt, Currie & Zhang, 2012). The main reason for this lack is complexity of the mathematical description that involves seven elastic constants.

In the present work, we derive motion equations for a statistically homogeneous pre-deformed fluid-saturated porous medium accounting for viscous and dispersion effects. These equations yield explicit relations between wave speeds and stresses. Possible simplified approximations for the general theory, convenient for applications, are analyzed and compared with experimental data available in literature.

Section snippets

Equations of motion for pre-stressed fluid-saturated porous media

In this Section, we derive non-linear Biot equations for finite deformations. The six energy equations can be written in Lagrangian description in the following form (Biot, 1962)ddt(Tu˙i)+Du˙i=ddXjΘui,j,(i=1,2,3)ddt(Tw˙i)+Dw˙i=ddXjΘwi,j,(i=1,2,3)where Θis the potential of poroelastic medium, D is the viscous dissipation and T is kinetic energy; ui are the displacements of solid skeleton and wi represent the flow of the fluid relative to the solid: wi=φ(Uiui), with Ui staying for

Dependence of the wave speeds on applied stress

Two plane sinusoidal waves are represented byud(X,t)=u0eiω(n·XVt)wd(X,t)=w0eiω(n·XVt) n is a unit normal to the wave front, u0 and w0 are wave displacements, respectively; X is the position vector, ω is the angular frequency, t is time and V is the wave phase velocity in the direction n

Substitution Eqs. (13) and (14) into Eqs. (11) and (12) yields(Zikρ0V2δik)uk0+(Nikρf0V2δik)wk0=0,(i=1,2,3)(Nkiρf0δikV2)uk0+(Mkiρ˜V2δik)wk0=0,(i=1,2,3)where Zik=Aijklnlnj,Nik=Bijnknj,Nki=Bjkninj and Mki=Bnink

Simplification for longitudinal waves

In order to analyze explicitly the poro-acoustoelastic effect of stresses on fast and slow longitudinal waves, we need to simplify Eq. (29). Geertsma and Smith, (1961) showed that most porous materials|4(ρρ˜ρf2)(HM(Mα)2)|<<|(ρM+Hρ˜2ρfMα)|2

Assuming that the changes ΔAijkl induced by the applied stress are small, this condition can be written as.|4(ρρ˜ρf2)(A3333BB332)|<<|(ρB+A3333ρ˜2ρfB33)|2

Neglecting small quantities according to Eqs. (32), (29) can be written asvP12ρB+A3333ρ˜2ρfB33(ρρ˜ρ

Comparison with the experiment for the fast longitudinal waves

In this section, we compare predictions of the simplified Eq. (35) with experimental data of Qu et al. (2018) and Fu and Fu (2018) and discuss whether part of the third-order elastic moduli can be ignored. The poroelastic constants used in calculations are given in Table 1. The frequencies of longitudinal and transverse waves are 0.5 MHz, 0.25 MHz Qu et al., 2018) and 0.6 MHz and 0.3 MHz (Fu & Fu, 2018). The six third-order elastic moduli( ν1,ν2,ν3,γ2,γ3 and γ ) are obtained by using least

Conclusions

In the present paper, we analyzed dependence of wave speeds in a poroelastic media on stresses. We first derived the general equations of motion and Christoffel equation for a pre-stressed poroelastic medium and obtained the explicit formulas for the fast and slow longitudinal waves, and transverse waves s functions of stresses. Explicit expressions for longitudinal wave speeds involve seven third-order elastic constants. For fast and slow longitudinal waves we proposed simplified equations

Declaration of Competing Interest

None

Acknowledgments

Financial supports of the National Natural Science Foundation of China grant #42074139 and NSF grant # 2011220 are gratefully acknowledged.

References (33)

  • I. Sevostianov et al.

    Connection between electrical conductivity and diffusion coefficient of a conductive porous material filled with electrolyte

    International Journal of Engineering Science

    (2017)
  • E. Vilchevskaya et al.

    Replacement relations for a viscoelastic material containing multiple inhomogeneities

    International Journal of Engineering Science

    (2019)
  • H.Q. Wang et al.

    Acoustoelastic theory for fluid-saturated porous media

    Acta Mechanica Solida Sinica

    (2014)
  • H.Q. Wang et al.

    Incremental algorithm for acoustoelastic theory of large static pre-deformed fluid-saturated porous media

    Progress in Computational Fluid Dynamics

    (2017)
  • J. Zhou et al.

    Effect of saturation on the elastic properties and anisotropy of cortical bone

    International Journal of Engineering Science

    (2020)
  • A. Abderahmane et al.

    Effects of multiaxial pre-stress on lamb and shear horizontal guided waves

    The Journal of the Acoustical Society of America

    (2021)
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