Thermoelasticity is important in seismic propagation due to the effects related to wave attenuation and velocity dispersion We have applied a novel finite-difference (FD) solver of the Lor

Simulation of thermoelastic waves based on the Lord-Shulman theory

We establish a generalization of the thermoelasticity wave equation to the porous case, including the Lord–Shulman (LS) and Green–Lindsay (GL) theories that involve a set of relaxation times (
 $$\tau _i, \ i = 1, \ldots , 4$$
 ). The dynamical equations predict four propagation modes, namely, a fast P wave, a Biot slow wave, a thermal wave, and a shear wave. The plane-wave analysis shows that the GL theory predicts a higher attenuation of the fast P wave, and consequently a higher velocity dispersion than the LS theory if $$\tau _1 = \tau _2 > \tau _3$$
 , whereas both models predict the same anelasticity for $$\tau _1 = \tau _2 = \tau _3$$
 . We also propose a generalization of the LS theory by applying two different Maxwell–Vernotte–Cattaneo relaxation times related to the temperature increment (
 $$\tau _3$$
 ) and solid/fluid strain components (
 $$\tau _4$$
 ), respectively. The generalization predicts positive quality factors when $$\tau _4 \ge \tau _3$$
 , and increasing $$\tau _4$$
 further enhances the attenuation. The wavefields are computed with a direct meshing algorithm using the Fourier pseudospectral method to calculate the spatial derivatives and a first-order explicit Crank–Nicolson time-stepping method. The propagation illustrated with snapshots and waveforms at low and high frequencies is in agreement with the dispersion analysis. The study can be useful for a comprehensive understanding of wave propagation in high-temperature high-pressure fields.

Generalized Thermo-poroelasticity Equations and Wave Simulation

The classical Zener model of thermoelasticity can be represented by a mechanical (or viscoelastic) model based on two springs and a dashpot, commonly called standard-linear solid, whose parameters ...

Thermoelasticity and P-wave simulation based on the Cole-Cole model

In this study, a singular boundary method (SBM) formulation is developed for the transient coupled dynamic thermoelastic problems. Before applying the SBM, a method based on Helmholtz decomposition and eigenvalue analysis is presented to derive the fundamental solutions for the governing equations in the frequency-domain, and the source singularity issue associated with the fundamental solutions is overcome with newly proposed analytical formulas. Because of the semi-analytical property of the fundamental solutions, the SBM avoids the domain discretization. Finally, the frequency-domain SBM solutions are converted to the time-domain solutions via the exponential window method (EWM) with the Hanning window. The accuracy and convergence of the present SBM-EWM is tested to several numerical examples with different loads and geometries. 

A singular boundary method for transient coupled dynamic thermoelastic analysis

We study the reflection and transmission coefficients of plane waves incident at an interface between two isotropic thermoelastic half spaces and compare them with those of the elastic case. The models include the classical-Biot (B) and extended Lord-Shulman (LS) theories, and predict reflected and transmitted fast-compressional (P), thermal (T) and shear (S) waves. The coefficients are formulated in terms of incidence and inhomogeneity angles, medium properties and potential functions. We consider different incident wave types and inhomogeneity angles to analyze the magnitude, phase and energy ratio of the plane waves, and perform a comparison with the isothermal (elastic) theory. The thermoelastic and elastic models predict different energy partitions between the P and S modes, satisfying the conservation of energy. The LS model exhibits higher T-wave thermal attenuation with increasing inhomogeneity angle at high frequencies, accordingly predicting more interference energy. The angle affects the energy partitions, particularly at the critical angle and near grazing incidence for an incident P wave, which satisfies the conservation of energy. Beyond the critical angle, the energy flux perpendicular to the interface of the isothermal model vanishes, while it is significant in the thermoelastic case. The T-wave magnitudes increase when the thermal conductivity (relaxation time) increases.

/pdf/reflection-and-transmission-of-inhomogeneous-plane-waves-in-kmj79h7w.pdf

Reflection and Transmission of Inhomogeneous Plane Waves in Thermoelastic Media

Pand SV-wave scattering by thermally-constrained saturated cracks couples elastic deformation with temperature. We estimate the scattering attenuation and velocity dispersion by aligned fluid-saturated cracks with finite thickness, which are sparsely and randomly embedded in an isothermal and isotropic elastic medium. By incorporating thermoelastic effects into the representation theorem with the non-interaction Foldy approximation, we formulate the dispersion and attenuation of P and SV waves induced by displacement discontinuities across aligned cracks constrained by a thermoelastic boundary condition. The frequency-dependent response as a function of temperature is calculated for various Pand SV-wave incidence angles. The resulting scattering attenuation and dispersion are compared with those of the conventional aligned-crack model. The examples show that the dissipation increases with increasing temperature, but the scattering is inhibited at a given temperature. The P wave shows a strong sensitivity to temperature along the direction normal to the cracks, whereas the SV wave has an opposite behavior. This theory has the potential to assess the

Thermoelastic Dispersion and Attenuation of P and SV Wave Scattering by Aligned Fluid-Saturated Cracks of Finite Thickness in an Isothermal Elastic Medium

High temperature affects the seismic properties of cracked and faulted reservoirs and can be an indicator for their detection. To this purpose, we study the wave-induced thermal flux (WITF) and develop two exact solutions for the scattering of compressional waves by a circular crack filled with a compressible fluid, where the approach is based on thermally permeable and impermeable boundary conditions. We obtain the phase velocity and attenuation as a function of frequency, which show that there are two loss mechanisms, i.e., thermoelastic dissipation at low frequencies and elastic scattering at high frequencies. Basically, when the crack size is comparable to the thermal and elastic wavelengths, there is substantial dispersion and attenuation (anelasticity) in the WITF and scattering frequency ranges, respectively. This means that the spatial inhomogeneity scale for inducing WITF is much smaller than that of scattering and the two mechanisms can be discriminated. The dependence of the compressional-wave velocity and attenuation on the compressibility and thermal expansion of the crack-filling fluid is different depending on the thermal diffusion rates at the crack interface. The anelasticity is much higher in the fully permeable case. This model has the potential to evaluate thermoelastic properties and heterogeneity at different scales from seismic responses.

Wave-induced thermal flux and scattering of P-waves in a medium with aligned circular cracks

Abstract The theory of wave propagation in non-isothermal porous rocks has been introduced in geophysics in recent years by combining the single-phase-lag (SPL) Lord-Shulman (LS) model of heat conduction with Biot’s poroelasticity theory. However, the theory of SPL thermoporoelasticity is inadequate in describing the lagging behavior of thermal relaxation for wave dissipation due to fluid and heat flow effects. We address this problem by incorporating a dual-phase-lags (DPL) model of heat conduction into thermoporoelasticity, utilizing analytical solutions and numerical simulations. The DPL model involves two lagging times: the (macroscopic) heat-flux lagging time τq from the LS model and an additionally introduced lagging time τT of temperature gradient that characterizes the fluid phase. A plane-wave analysis predicts four propagation waves, namely, fast P, slow P, thermal (T), and shear (S). We calculate wavefield snapshots by using a finite-difference solver for the DPL thermoelastic equations and provide further insight into the physics of two lagging times for porous rocks. The simulations show that the DPL model induces higher thermal attenuation and larger velocity dispersion compared to the SPL model, especially at high frequencies. The influence of fluids is crucial for wave propagation within thermoporoelastic media.

Simulation of wave propagation in thermoporoelastic media with dual-phase-lag heat conduction

We have developed a hypersingular boundary integral equation (BIE) approach to simulate wave scattering by inclusions such as karstic cavities and gypsum salts. The solution of conventional...

Evaluation of the hypersingular boundary integral equation for acoustic wave-propagation simulations

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