Wave propagation in a pre-stressed anisotropic generalized thermoelastic medium
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB 2010
Received: 21 February 2009
Accepted: 17 December 2009
Published: 17 June 2010
Four attenuated waves propagate in a pre-stressed anisotropic generalized thermoelastic medium. The propagation phenomenon in this medium is explained through two systems. One of them, relating the temperature variation in the medium to the particle displacement, is free from the explicit effect of pre-stress. The other system defines Christoffel equations for the medium. These equations are modified with a matrix, which involves phase direction and pre-stress components. A propagation-attenuation plane is defined for given directions of propagation and attenuation of plane harmonic waves. A finite non-dimensional parameter defines the inhomogeneity strength of an attenuated wave. A complex vector is defined to calculate complex velocities of the four waves from the complex roots of a quartic equation. The complex slowness vector of the attenuated wave in the medium is resolved to calculate its propagation (phase) velocity, quality factor and angle of attenuation. Numerical example is considered to study the propagation characteristics of each of the four attenuated waves in the pre-stressed medium. The presence of anisotropic symmetries and anelasticity are also considered in the medium. Effect of pre-stress is analyzed on the propagation characteristics of each of the four attenuated waves.
Key wordsInitial stress thermoelastic inhomogeneous waves attenuation
The crustal rocks are always subjected to stresses. The slow process of creep inside the earth creates a differential stress environment (Hanks and Raleigh, 1980; McGarr, 1980) in the crust, which is responsible for the preferential alignments in the Earth, ranging from mineral orientations, grains, or microcracks to sedimentary folds or regional fractures. The difference between confined tectonic stress and pore-fluid pressure conducts the flow of fluid to a reservoir through the connected cracks. For the presence of pre-stress or aligned cracks, an elastic medium behaves anisotropic to wave propagation. Hence, the almost universal presence of anisotropy is observed in many types of rocks at many depths and in many geological and tectonic environments. Two recent studies (Prikazchikov and Rogerson, 2003; Sharma, 2005) contribute to the understanding of wave propagation characteristics of anisotropic materials under initial stress. A latest book by Carcione (2007) explains the importance of anisotropy for wave propagation studies in real materials.
Temperature variations play a significant role in the modification of cracks and the flow of fluid (Paulsson et al., 1994). These modifications in microcracks are responsible for the dynamism around geothermal reservoirs and the sedimentary basins. Stixrude and Lithgow-Bertelloni (2005) have argued the merit of fundamental thermodynamic relations as the basis for the description of thermoelastic behaviour of in-situ minerals. Theory of thermoelasticity is used to understand such dynamical systems that involve interactions between mechanical work and thermal changes. Few generalised theories of thermoelasticity have been defined with the introduction of relaxation in temperature field. The theory with one relaxation time (Lord and Shulman, 1967) is termed as LS theory and another with two relaxation times (Green and Lindsay, 1972) is termed as GL theory. Using these modified theories, a large number of problems have been studied on the propagation of plane waves in generalized thermoelastic media (El-Karamany et al., 2002; Sharma et al., 2003). Sharma et al. (2000) studied plane harmonic waves in orthotropic thermoelastic materials. In a recent study (Sharma, 2006), the author considered the general anisotropy in thermoelastic medium and derived a mathematical model to calculate the complex velocities of four waves in the medium. Correspondence was, also, established between the generalised theory of thermoelasticity and the homogenization based (u, p) theory of poroelasticity.
The attenuation of waves in thermoelastic medium comes from the memory effects allowed to heat conduction. However, a more realistic scheme is defined with memory effects allowed for all the constitutive properties of thermoelastic coupling. Giorgi et al. (2001) studied a linear theory for thermoviscoelastic materials but with thermal behaviour represented through the heat conduction equation. Generalizations of El-Karamany et al. (2002) and El-Karamany and Ezzat (2002) described and established relaxation effects in mechanical properties as well as thermo-mechanical coupling in thermoviscoelastic media. Analogous to the correspondence principle in classical elasticity, the complex values of appropriate constitutive quantities may define the time-harmonic material dissipation in thermoviscoelastic medium (Caviglia and Morro, 2005).
The present study considers the propagation and attenuation of inhomogeneous waves, in a pre-stressed anisotropic generalized thermoelastic (hereafter, referred as SAGT) medium. In a similar study (Sharma, 2005), the author studied the propagation of homogeneous waves in anisotropic poroelastic medium in the presence of initial stress. But, for the use of Biot’s theory of poroelasticity (Biot, 1956) in this paper, the present work can not be obtained through the poroelastic-thermoelastic correspondence (Sharma, 2006). Moreover, the inhomogeneous propagation of attenuated waves is considered in the present work. The complex slowness vector of an inhomogeneous wave is constructed with its propagation direction and inhomogeneity strength. Numerical examples are computed to analyze the propagation characteristics of the attenuated waves in the elastic/viscoelastic media of different anisotropies. Anisotropic variations in propagation characteristics are also analysed for an isotropic medium pre-stressed with hydrostatic pressure.
2. Definition of the Problem
The propagation of plane harmonic waves may be explained through two systems of equations. One of them relates the wave-induced temperature in the medium to the displacement of its particles. The other system provides modified Christoffel equation for the medium. The Christoffel equations are solved into a quartic equation. The four roots of this equation define the complex velocities of the four attenuated waves in SAGT medium.
The inhomogeneity of an attenuated wave in the medium is represented through an inhomogeneity parameter (δ). For a wave, of given inhomogeneity (value of δ), propagating along a given direction , bisection method is used to calculate a parameter (β) to represent homogeneous attenuation. The parameter β is used, further, to calculate the phase velocity, quality factor and attenuation angle of the wave considered.
The numerical examples are considered to compute the propagation characteristics (velocity, attenuation) of each of the four inhomogeneous waves in SAGT medium. The presence of anisotropic symmetries and anelasticity in the medium is considered in the numerical models.
3. SAGT (pre-stressed anisotropic generalized thermoelastic) Medium
3.1 Basic equations
3.2 Plane harmonic waves
In Eq. (8), the matrices Y and Σ N represent the effects of thermoelastic coupling and pre-stress, respectively, on the propagation of elastic waves. With the substitution of Σ N = 0, this equation governs the anisotropic propagation of thermoelastic wave (Sharma, 2006). 3.3 Four attenuated waves
3.4 Inhomogeneous plane waves
In a dissipative medium, an attenuated wave may be homogeneous or inhomogeneous. An angle between propagation vector and attenuation vector, in general, represents the inhomogeneous character of a plane wave. The phase vector N, in real space, yields same direction for propagation vector and attenuation vector and, hence, represents the homogeneous waves. So, an inhomogeneous wave may be represented, only, with N as a complex (dual) vector.
4. Numerical Examples
Propagation of an attenuated wave is characterised by its phase velocity (υ) and quality factor of attenuation (Q). In general, the inhomogeneity of an attenuating wave is represented through the difference in the directions of its propagation vector and attenuation vector. In other words, an angle (γ) between equi-amplitude plane and equi-phase plane of an attenuating plane wave represent its inhomogeneous character. To avoid forbidden directions (Krebes and Le, 1994) for γ, an inhomogeneity of an attenuated wave is defined with the non-dimensional inhomogeneity parameter (δ). The relation (15) relates the angle γ to the parameter δ. Hence, phase velocity (υ), quality factor (Q) and attenuation angle (γ) are the main propagation characteristics of an inhomogeneous wave. These characteristics are the functions of δ and, in anisotropic media, these are the functions of propagation direction also. Variations of these characteristics with propagation direction explain the anisotropic behaviour of inhomogeneous waves. The value of δ defines a general attenuated wave varying from homogenous propagation (δ = 0) to evanescent wave (β = 0). The arbitrary anisotropy always provides a liberty to consider any realistic anisotropy with symmetries.
For general direction (θ, ϕ), in three-dimensional space, the propagation direction is defined by (sinθ cosϕ, sinθ sinϕ, cosθ). The vertical plane ϕ = 0.34π is the fixed propagation-attenuation plane for numerical computation. In this plane, the propagation direction is considered with θ, varying from 0 to 90° and the orthogonal vector is considered along θ + π/2. The three values of δ(= 0.01, 0.1, 0.4) are used to represent the variations in inhomogeneity strength of a wave. The smallest value (0.01) of δ represents a nearly homogeneous (or weakly inhomogeneous) wave. It was noted that the thermoelasticity theory (LS or GL) has only a negligible effect on the propagation characteristics of inhomogeneous waves. Hence, the numerical results are exhibited, only, for GL theory (i.e., m = 2). Three main anisotropies (triclinic, monoclinic, orthotropic) are considered in thermoelasticity as well as initial-stress.
The anisotropy in the medium has a significant effect on the propagation of qS2-wave. The behavior of anisotropic variations seem to be quite sensitive to the larger (between 0.1 to 0.4) values of δ. Similar to qS1-wave, the value of γ is around 90°.
5. Concluding Remarks
The slower is an elastic wave, more it is affected by the presence of anisotropic symmetry. The presence of anisotropic symmetries have a very significant effect on the variations of velocity with propagation direction. The negligible effect of anisotropic symmetry on the velocities of qP and qT waves may indicate that the quasi-longitudinal waves are less sensitive to the presence of anisotropic symmetry.
The attenuations represented by weakly inhomogeneous (i.e., δ = 0.01) faster waves are negligible. This implies that the large attenuation in a dissipative medium may be explained only with strongly inhomogeneous waves. The values of γ near 90° imply the near-evanescent character of inhomogeneous waves in the dissipative medium.
The qT-waves experience the largest attenuation among all the four waves in the medium. The angle (γ) between propagation and attenuation direction of qT-wave is much away from 90° and varies a lot with the value of δ.For δ = 0.01, the significant value of Q−1 and zero value of γ implies that attenuated qT-wave propagates as a homogeneous wave. That means, homogeneous qT-waves may be able to explain large attenuation also, contrary to the three faster waves in the medium.
Viscoelasticity in the medium may not have any effect on the phase velocities of any of the four waves. However, the attenuation angles (γ) and coefficients (Q−1) of three faster waves increase with ∊. The exception is qT-wave, which is not affected by the anelastic nature of the medium.
The propagation of quasi-thermal (qT) wave may not be affected with the presence of pre-stress in SAGT medium. The other three waves may not be changing their velocities and attenuation angle with the presence of pre-stress. However, pre-stress may affect the attenuation of these waves but only for few propagation directions. This implies that, to affect the velocities of the waves, the pre-stress values should be much greater than that assumed in the numerical model.
The effect of anisotropy induced by hydrostatic pre-stress in an isotropic thermoelastic medium is observed only on two split-shear (qS1, qS2) waves. The isotropic propagation behavior of other two (qP, qT) waves is not affected by the presence this stress-induced anisotropy.
Seismic waves generated in Earth’s interior provide images that help us to better understand the pattern of mantle convection that drives plate motions. Anisotropy and dissipation, which also influence seismic-wave propagation, may be characterized better to extract additional information on flow directions, temperature variations and the presence of partial melting (Romanowicz, 2008).
The boundary between the core and mantle is one of the most enigmatic regions of Earth’s interior. It holds the key to understand a host of geophysical phenomena—including the formation of plumes in the mantle, interactions between core and mantle, and the ultimate fate of subducting slabs of crust that are driven into the interior by tectonic forces. Investigations of this region largely depend on interpreting the behaviour of seismic waves, which have shown that it is highly complex (Duffy, 2004)
The MgSiO3 perovskite, generally accepted to be the major component of the lower mantle. It is found to be highly anisotropic in all portions of the lower mantle and the nature of anisotropy changes significantly with depth. Wentzcovitch et al. (1998) calculated anisotropy of seismic wave velocities as a function of pressure (depth). Anisotropy at the topmost lower mantle can be attributed to the preferred orientation of perovskite.
Knowledge of the elastic properties of the dominant (Mg, Fe, Al)(Si, Al)O3 perovskite phase of the Earth’s lower mantle, including the pressure and temperature dependence of the bulk and shear moduli, is critical for analyses of its chemical composition and thermal regime including the significance of the lateral variations of seismic wave speeds (Jackson, 1998; Deschamps and Trampert, 2004; Mattern et al., 2005).
The prediction of pre-drill overpressure is required for the monitoring of hydrocarbon production in the boreholes (Sayers et al., 2002).
The extent of fracturing in a region of a borehole, which is a vital factor in the extraction of oil and geothermal heat. The information on fracture distribution (from velocity inversion) is used to estimate the anisotropic permeability of the fracture rock system (Gibson and Toksoz, 1990).
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