On the Green function of the Lord–Shulman thermoelasticity equations

2019 ◽  
Vol 220 (1) ◽  
pp. 393-403 ◽  
Author(s):  
Zhi-Wei Wang ◽  
Li-Yun Fu ◽  
Jia Wei ◽  
Wanting Hou ◽  
Jing Ba ◽  
...  
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Parabolic Type ◽  
Second Order ◽  
P Wave ◽  
Thermal Mode ◽  
S Wave ◽  
Order Tensor ◽  
P Waves ◽  
Thermal Conductivities

SUMMARY Thermoelasticity extends the classical elastic theory by coupling the fields of particle displacement and temperature. The classical theory of thermoelasticity, based on a parabolic-type heat-conduction equation, is characteristic of an unphysical behaviour of thermoelastic waves with discontinuities and infinite velocities as a function of frequency. A better physical system of equations incorporates a relaxation term into the heat equation; the equations predict three propagation modes, namely, a fast P wave (E wave), a slow thermal P wave (T wave), and a shear wave (S wave). We formulate a second-order tensor Green's function based on the Fourier transform of the thermodynamic equations. It is the displacement–temperature solution to a point (elastic or heat) source. The snapshots, obtained with the derived second-order tensor Green's function, show that the elastic and thermal P modes are dispersive and lossy, which is confirmed by a plane-wave analysis. These modes have similar characteristics of the fast and slow P waves of poroelasticity. Particularly, the thermal mode is diffusive at low thermal conductivities and becomes wave-like for high thermal conductivities.

Green's function of the Lord–Shulman thermo-poroelasticity theory

2020 ◽  
Vol 221 (3) ◽  
pp. 1765-1776 ◽  
Author(s):  
Jia Wei ◽  
Li-Yun Fu ◽  
Zhi-Wei Wang ◽  
Jing Ba ◽  
José M Carcione
Keyword(s):  
Plane Wave ◽  
Green’S Function ◽  
Green's Function ◽  
Heterogeneous Media ◽  
Numerical Algorithms ◽  
Heat Diffusion ◽  
P Wave ◽  
Wave Analysis ◽  
S Wave ◽  
S Waves

SUMMARY The Lord–Shulman thermoelasticity theory combined with Biot equations of poroelasticity, describes wave dissipation due to fluid and heat flow. This theory avoids an unphysical behaviour of the thermoelastic waves present in the classical theory based on a parabolic heat equation, that is infinite velocity. A plane-wave analysis predicts four propagation modes: the classical P and S waves and two slow waves, namely, the Biot and thermal modes. We obtain the frequency-domain Green's function in homogeneous media as the displacements-temperature solution of the thermo-poroelasticity equations. The numerical examples validate the presence of the wave modes predicted by the plane-wave analysis. The S wave is not affected by heat diffusion, whereas the P wave shows an anelastic behaviour, and the slow modes present a diffusive behaviour depending on the viscosity, frequency and thermoelasticity properties. In heterogeneous media, the P wave undergoes mesoscopic attenuation through energy conversion to the slow modes. The Green's function is useful to study the physics in thermoelastic media and test numerical algorithms.

Pseudo-elastic PP and PSSP modeling in stratified media

Geophysics ◽  
2021 ◽  
pp. 1-56
Author(s):  
Lasse Amundsen ◽  
Bjørn Ursin
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Fourier Transforms ◽  
P Wave ◽  
P Waves ◽  
Simple Modification ◽  
Time Space ◽  
P Domain ◽  
Band Limited ◽  
Time Band

Many modeling techniques have been developed to find the acoustic and elastic responses of a stack of plane layers to a plane spectral wave. For an elastic medium bounded above by an acoustic half-space, the acoustic wave propagator matrix modeling method can be modified to model pseudoelastic PP arrivals and PSSP arrivals. PP arrivals propagate as pure longitudinal (P) waves in the layers, whereas PSSP arrivals propagate as shear (S) waves in the elastic part of the model. A simple modification of the pseudoelastic PP response modeling scheme allows modeling of primary P reflections. A primary reflection event involves just one reflection in the plane stratified model and thus excludes internal multiples. The propagator modeling scheme is formulated in the frequency-horizontal slowness domain. By applying inverse Fourier transforms over the frequency and horizontal wavenumbers, where the wavenumber is the horizontal slowness divided by the frequency, modeled seismograms are computed and displayed in the time-space domain. By applying an inverse Fourier transform over the frequency for selected horizontal slowness components, the computed seismograms can be shown in the intercept time-horizontal slowness ( τ- p) domain. When the source wavelet is unity for frequencies of interest, the τ- p domain seismograms become plane-wave Green’s function seismograms. The p-traces of the Green’s function primary P-wave seismograms accumulate with increasing time band-limited step functions weighted by reflection strengths.

Simulation of thermoelastic waves based on the Lord-Shulman theory

Geophysics ◽  
2021 ◽  
Vol 86 (3) ◽  
pp. T155-T164
Author(s):  
Wanting Hou ◽  
Li-Yun Fu ◽  
José M. Carcione ◽  
Zhiwei Wang ◽  
Jia Wei
Keyword(s):  
Thermal Conductivity ◽  
Expansion Coefficient ◽  
Velocity Dispersion ◽  
Wave Attenuation ◽  
Splitting Method ◽  
Grazing Incidence ◽  
P Wave ◽  
Staggered Grid ◽  
Thermal Mode ◽  
P Waves

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 Lord-Shulman thermoelasticity equations to compute synthetic seismograms that include the effects of the thermal properties (expansion coefficient, thermal conductivity, and specific heat) compared with the classic forward-modeling codes. We use a time splitting method because the presence of a slow quasistatic mode (the thermal mode) makes the differential equations stiff and unstable for explicit time-stepping methods. The spatial derivatives are computed with a rotated staggered-grid FD method, and an unsplit convolutional perfectly matched layer is used to absorb the waves at the boundaries, with an optimal performance at the grazing incidence. The stability condition of the modeling algorithm is examined. The numerical experiments illustrate the effects of the thermoelasticity properties on the attenuation of the fast P-wave (or E-wave) and the slow thermal P-wave (or T-wave). These propagation modes have characteristics similar to the fast and slow P-waves of poroelasticity, respectively. The thermal expansion coefficient has a significant effect on the velocity dispersion and attenuation of the elastic waves, and the thermal conductivity affects the relaxation time of the thermal diffusion process, with the T mode becoming wave-like at high thermal conductivities and high frequencies.

Numerical simulation of azimuthal acoustic logging in a borehole penetrating a rock formation boundary

Geophysics ◽  
2016 ◽  
Vol 81 (3) ◽  
pp. D283-D291 ◽  
Author(s):  
Peng Liu ◽  
Wenxiao Qiao ◽  
Xiaohua Che ◽  
Xiaodong Ju ◽  
Junqiang Lu ◽  
...  
Keyword(s):  
Domain Model ◽  
P Wave ◽  
S Wave ◽  
Angle Variation ◽  
P Waves ◽  
Acoustic Logging ◽  
S Waves ◽  
Rock Formation ◽  
Logging Tool ◽  
Difference Time

We have developed a new 3D acoustic logging tool (3DAC). To examine the azimuthal resolution of 3DAC, we have evaluated a 3D finite-difference time-domain model to simulate a case in which the borehole penetrated a rock formation boundary when the tool worked at the azimuthal-transmitting-azimuthal-receiving mode. The results indicated that there were two types of P-waves with different slowness in waveforms: the P-wave of the harder rock (P1) and the P-wave of the softer rock (P2). The P1-wave can be observed in each azimuthal receiver, but the P2-wave appears only in the azimuthal receivers toward the softer rock. When these two types of rock are both fast formations, two types of S-waves also exist, and they have better azimuthal sensitivity compared with P-waves. The S-wave of the harder rock (S1) appears only in receivers toward the harder rock, and the S-wave of the softer rock (S2) appears only in receivers toward the softer rock. A model was simulated in which the boundary between shale and sand penetrated the borehole but not the borehole axis. The P-wave of shale and the S-wave of sand are azimuthally sensitive to the azimuth angle variation of two formations. In addition, waveforms obtained from 3DAC working at the monopole-transmitting-azimuthal-receiving mode indicate that the corresponding P-waves and S-waves are azimuthally sensitive, too. Finally, we have developed a field example of 3DAC to support our simulation results: The azimuthal variation of the P-wave slowness was observed and can thus be used to reflect the azimuthal heterogeneity of formations.

scholarly journals Green’s Function Estimates for Time-Fractional Evolution Equations

2019 ◽  
Vol 3 (2) ◽  
pp. 36
Author(s):  
Ifan Johnston ◽  
Vassili Kolokoltsov
Keyword(s):  
Green’S Function ◽  
Elliptic Operator ◽  
Green's Function ◽  
Green's Functions ◽  
Evolution Equations ◽  
Green’S Functions ◽  
Variable Coefficients ◽  
Second Order ◽  
Fractional Evolution Equations ◽  
Order Elliptic Operator

We look at estimates for the Green’s function of time-fractional evolution equations of the form D 0 + * ν u = L u , where D 0 + * ν is a Caputo-type time-fractional derivative, depending on a Lévy kernel ν with variable coefficients, which is comparable to y - 1 - β for β ∈ ( 0 , 1 ) , and L is an operator acting on the spatial variable. First, we obtain global two-sided estimates for the Green’s function of D 0 β u = L u in the case that L is a second order elliptic operator in divergence form. Secondly, we obtain global upper bounds for the Green’s function of D 0 β u = Ψ ( - i ∇ ) u where Ψ is a pseudo-differential operator with constant coefficients that is homogeneous of order α . Thirdly, we obtain local two-sided estimates for the Green’s function of D 0 β u = L u where L is a more general non-degenerate second order elliptic operator. Finally we look at the case of stable-like operator, extending the second result from a constant coefficient to variable coefficients. In each case, we also estimate the spatial derivatives of the Green’s functions. To obtain these bounds we use a particular form of the Mittag-Leffler functions, which allow us to use directly known estimates for the Green’s functions associated with L and Ψ , as well as estimates for stable densities. These estimates then allow us to estimate the solutions to a wide class of problems of the form D 0 ( ν , t ) u = L u , where D ( ν , t ) is a Caputo-type operator with variable coefficients.

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