Effect of stresses on wave propagation in fluid-saturated porous media
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)where is the potential of poroelastic medium, D is the viscous dissipation and is kinetic energy; are the displacements of solid skeleton and represent the flow of the fluid relative to the solid: , with staying for
Dependence of the wave speeds on applied stress
Two plane sinusoidal waves are represented by is a unit normal to the wave front, and are wave displacements, respectively; is the position vector, is the angular frequency, is time and is the wave phase velocity in the direction
Substitution Eqs. (13) and (14) into Eqs. (11) and (12) yieldswhere and
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
Assuming that the changes induced by the applied stress are small, this condition can be written as.
Neglecting small quantities according to Eqs. (32), (29) can be written as
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( and ) 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)
- et al.
Guided ultrasonic waves for non-destructive monitoring of the stress levels in prestressed steel strands
Ultrasonics
(2009) - et al.
Equations and fundamental characteristics of transverse waves propagating in fluid-saturated porous materials
International Journal of Engineering Science
(2020) - et al.
A multiple-network poroelastic model for biological systems and application to subject-specific modeling of cerebral fluid transport
International Journal of Engineering Science
(2020) - et al.
An overview of the acoustic studies of bone-like porous materials, and the effect of transverse acoustic waves
International Journal of Engineering Science
(2020) - et al.
Effects of pore characteristics on water-oil two-phase displacement in non-homogeneous pore structures: A pore-scale lattice Boltzmann model considering various fluid density ratios
International Journal of Engineering Science
(2020) Cavities and cracks subjected to pressure of injected fluid in poroelastic media
International Journal of Engineering Science
(2019)- et al.
A new permeability model for argillaceous porous media under stress dependence with clay swelling
International Journal of Engineering Science
(2021) - et al.
The influence of anisotropic growth and geometry on the stress of solid tumors
International Journal of Engineering Science
(2017) - et al.
Crustal stress determination from boreholes and rock cores: Fundamental principles
Tectonophysics
(2012) Gassmann equation and replacement relations in micromechanics: A review
International Journal of Engineering Science
(2020)
Connection between electrical conductivity and diffusion coefficient of a conductive porous material filled with electrolyte
International Journal of Engineering Science
Replacement relations for a viscoelastic material containing multiple inhomogeneities
International Journal of Engineering Science
Acoustoelastic theory for fluid-saturated porous media
Acta Mechanica Solida Sinica
Incremental algorithm for acoustoelastic theory of large static pre-deformed fluid-saturated porous media
Progress in Computational Fluid Dynamics
Effect of saturation on the elastic properties and anisotropy of cortical bone
International Journal of Engineering Science
Effects of multiaxial pre-stress on lamb and shear horizontal guided waves
The Journal of the Acoustical Society of America
Cited by (16)
Acoustoelastic DZ-MT model for stress-dependent elastic moduli of fractured rocks
2024, International Journal of Rock Mechanics and Mining SciencesEllipticity of gradient poroelasticity
2023, International Journal of Engineering ScienceA review of advanced materials, structures and deformation modes for adaptive energy dissipation and structural crashworthiness
2022, Thin-Walled StructuresCitation Excerpt :The incompressible nature of the oil also resulted in brittle failure and leakage of the ferrofluid, this was also observed by Caglayan et al. [384] for MRF-filled PU foam sandwich panels. An investigation considering hypervelocity (> 1 km/s) impact conditions by Liu et al. [387] revealed the relationship between stress wave propagation speed on compressive strength and pore pressure, and a need for further investigations to characterize the response when the internal pore pressure exceeds 35 MPa. These concepts are equally applicable to any porous structure or material, particularly foams, and are correspondingly applicable to virtually any foam-filled structure proposed in the open literature (and discussed for selected cases in Sections 4.3, 4.2 and 3.1.2) such as automotive and rail bumper frames, building supports and various protective sandwich panels, among others.
Higher-order elastic constitutive relation: Micro mechanism and application to acoustoelasticity
2022, Computer Physics CommunicationsCitation Excerpt :However, the tangent moduli depend on the strain state [11–13], which is not suitable for describing wave propagation, especially acoustoelastic effect. The relation between stress and propagation velocity is described by acoustoelastic coefficient, which is composed of SOECs and TOECs [3,4]. If the tangent modulus method is used to establish the acoustoelastic constitutive model, the acoustoelastic coefficients will be composed of the tangent modulus and depend on the strain state.
Influence of external clamping pressure on nanoscopic mechanical deformation and catalyst utilization of quaternion PtC catalyst layers for PEMFCs
2022, Renewable EnergyCitation Excerpt :The thickness of ionomer layers is not constant and varies randomly as shown in Fig. 1(b). The interconnectivity ratios of PtC, ionomer, and pore phases are determined by the 3D percolation theory coupled with the cluster labeling processes [41–43]. Finally, the ECUF is calculated as a ratio of active catalyst sites to the total catalyst NPs using TPBs.
Reasonable value range of damage stress during rock brittle failure under compression
2024, Geomechanics and Geophysics for Geo-Energy and Geo-Resources