Simulation of wave propagation in linear thermoelastic media

Geophysics ◽  
2019 ◽  
Vol 84 (1) ◽  
pp. T1-T11 ◽  
Author(s):  
José M. Carcione ◽  
Zhi-Wei Wang ◽  
Wenchang Ling ◽  
Ettore Salusti ◽  
Jing Ba ◽  
...  
Keyword(s):  
Thermal Conductivity ◽  
Wave Propagation ◽  
Grid Method ◽  
P Wave ◽  
Thermal Mode ◽  
S Wave ◽  
Splitting Algorithm ◽  
Low Frequencies ◽  
First Order ◽  
Time Stepping

We have developed a numerical algorithm for simulation of wave propagation in linear thermoelastic media, based on a generalized Fourier law of heat transport in analogy with a Maxwell model of viscoelasticity. The wavefield is computed by using a grid method based on the Fourier differential operator and two time-integration algorithms to cross-check solutions. Because the presence of a slow quasistatic mode (the thermal mode) makes the differential equations stiff and unstable for explicit time-stepping methods, first, a second-order time-splitting algorithm solves the unstable part analytically and a Runge-Kutta method the regular equations. Alternatively, a first-order explicit Crank-Nicolson algorithm yields more stable solutions for low values of the thermal conductivity. These time-stepping methods are second- and first-order accurate, respectively. The Fourier differential provides spectral accuracy in the calculation of the spatial derivatives. The model predicts three propagation modes, namely, a fast compressional or (elastic) P-wave, a slow thermal P diffusion/wave (the T-wave), having similar characteristics to the fast and slow P-waves of poroelasticity, respectively, and an S-wave. The thermal mode is diffusive for low values of the thermal conductivity and wave-like for high values of this property. Three velocities define the wavefront of the fast P-wave, i.e., the isothermal velocity in the uncoupled case, the adiabatic velocity at low frequencies, and a higher velocity at high frequencies.

Canonical analytical solutions of wave-induced thermoelastic attenuation

2020 ◽  
Vol 221 (2) ◽  
pp. 835-842 ◽  
Author(s):  
José M Carcione ◽  
Davide Gei ◽  
Juan E Santos ◽  
Li-Yun Fu ◽  
Jing Ba
Keyword(s):  
P Wave ◽  
Layered System ◽  
Thermal Mode ◽  
S Wave ◽  
Low Frequencies ◽  
High Frequencies ◽  
Cylindrical Cavities ◽  
Relaxation Curves ◽  
Wave Induced ◽  
Flow Attenuation

SUMMARY Thermoelastic attenuation is similar to wave-induced fluid-flow attenuation (mesoscopic loss) due to conversion of the fast P wave to the slow (Biot) P mode. In the thermoelastic case, the P- and S-wave energies are lost because of thermal diffusion. The thermal mode is diffusive at low frequencies and wave-like at high frequencies, in the same manner as the Biot slow mode. Therefore, at low frequencies, that is, neglecting the inertial terms, a mathematical analogy can be established between the diffusion equations in poroelasticity and thermoelasticity. We study thermoelastic dissipation for spherical and cylindrical cavities (or pores) in 2-D and 3-D, respectively, and a finely layered system, where, in the latter case, only the Grüneisen ratio is allowed to vary. The results show typical quality-factor relaxation curves similar to Zener peaks. There is no dissipation when the radius of the pores tends to zero and the layers have the same properties. Although idealized, these canonical solutions are useful to study the physics of thermoelasticity and test numerical algorithm codes that simulate thermoelastic dissipation.

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.

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.

scholarly journals A new acoustic assumption for orthorhombic media

2020 ◽  
Vol 223 (2) ◽  
pp. 1118-1129
Author(s):  
Mohammad Mahdi Abedi ◽  
Alexey Stovas
Keyword(s):  
Wave Propagation ◽  
Conventional Approach ◽  
P Wave ◽  
Model Parameters ◽  
Governing Equations ◽  
S Wave ◽  
S Waves ◽  
Exploration Seismology ◽  
And Inversion ◽  
Approximate Equations

SUMMARY In exploration seismology, the acquisition, processing and inversion of P-wave data is a routine. However, in orthorhombic anisotropic media, the governing equations that describe the P-wave propagation are coupled with two S waves that are considered as redundant noise. The main approach to free the P-wave signal from the S-wave noise is the acoustic assumption on the wave propagation. The conventional acoustic assumption for orthorhombic media zeros out the S-wave velocities along three orthogonal axes, but leaves significant S-wave artefacts in all other directions. The new acoustic assumption that we propose mitigates the S-wave artefacts by zeroing out their velocities along the three orthogonal symmetry planes of orthorhombic media. Similar to the conventional approach, our method reduces the number of required model parameters from nine to six. As numerical experiments on multiple orthorhombic models show, the accuracy of the new acoustic assumption also compares well to the conventional approach. On the other hand, while the conventional acoustic assumption simplifies the governing equations, the new acoustic assumption further complicates them—an issue that emphasizes the necessity of simple approximate equations. Accordingly, we also propose simpler rational approximate phase-velocity and eikonal equations for the new acoustic orthorhombic media. We show a simple ray tracing example and find out that the proposed approximate equations are still highly accurate.

PROPELLER NOISE FROM A LIGHT AIRCRAFT FOR LOW-FREQUENCY MEASUREMENTS OF THE SPEED OF SOUND IN A MARINE SEDIMENT

2002 ◽  
Vol 10 (04) ◽  
pp. 445-464 ◽  
Author(s):  
MICHAEL J. BUCKINGHAM ◽  
ERIC M. GIDDENS ◽  
FERNANDO SIMONET ◽  
THOMAS R. HAHN
Keyword(s):  
Water Column ◽  
Speed Of Sound ◽  
Low Frequency ◽  
Deep Ocean ◽  
Fine Sand ◽  
P Wave ◽  
Difference Frequency ◽  
S Wave ◽  
Low Frequencies ◽  
La Jolla

The sound from a light aircraft in flight is generated primarily by the propeller, which produces a sequence of harmonics in the frequency band between about 80 Hz and 1 kHz. Such an airborne sound source has potential in underwater acoustics applications, including inversion procedures for determining the wave properties of marine sediments. A series of experiments has recently been performed off the coast of La Jolla, California, in which a light aircraft was flown over a sensor station located in a shallow (approximately 15 m deep) ocean channel. The sound from the aircraft was monitored with a microphone above the sea surface, a vertical array of eight hydrophones in the water column, and two sensors, a hydrophone and a bender intended for detecting shear waves, buried 75 cm deep in the very-fine-sand sediment. The propeller harmonics were detected on all the sensors, although the s-wave was masked by the p-wave on the buried bender. Significant Doppler shifts of the order of 17%, were observed on the microphone as the aircraft approached and departed from the sensor station. Doppler shifting was also evident in the hydrophone data from the water column and the sediment, but to a lesser extent than in the atmosphere. The magnitude of the Doppler shift depends on the local speed of sound in the medium in which the sensor is located. A technique is described in which the Doppler difference frequency between aircraft approach and departure is used to determine the speed of sound at low-frequencies (80 Hz to 1 kHz) in each of the three environments, the atmosphere, the ocean and the sediment. Several experimental results are presented, including the speed of sound in the very fine sand sediment at a nominal frequency of 600 Hz, which was found from the Doppler difference frequency of the seventh propeller harmonic to be 1617 m/s.

Wavefield simulation and data-acquisition-scheme analysis for LWD acoustic tools in very slow formations

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. E59-E68 ◽  
Author(s):  
Hua Wang ◽  
Guo Tao
Keyword(s):  
Wave Velocity ◽  
Cutoff Frequency ◽  
P Wave ◽  
Flexural Wave ◽  
S Wave ◽  
Low Frequencies ◽  
Wave Velocities ◽  
Dipole System ◽  
P Wave Velocity ◽  
Wavefield Simulation

Propagating wavefields from monopole, dipole, and quadrupole acoustic logging-while-drilling (LWD) tools in very slow formations have been studied using the discrete wavenumber integration method. These studies examine the responses of monopole and dipole systems at different source frequencies in a very slow surrounding formation, and the responses of a quadrupole system operating at a low source frequency in a slow formation with different S-wave velocities. Analyses are conducted of coherence-velocity/slowness relationships (semblance spectra) in the time domain and of the dispersion characteristics of these waveform signals from acoustic LWD array receivers. These analyses demonstrate that, if the acoustic LWD tool is centralized properly and is operating at low frequencies (below 3 kHz), a monopole system can measure P-wave velocity by means of a “leaky” P-wave for very slow formations. Also, for very slow formations a dipole system can measure the P-wave velocity via a leaky P-wave and can measure the S-wave velocity from a formation flexural wave. With a quadrupole system, however, the lower frequency limit (cutoff frequency) of the drill-collar interference wave would decrease to 5 kHz and might no longer be neglected if the surrounding formation becomes a very slow formation, with S-wave velocities at approximately 500 m/s.

scholarly journals The effects of tool eccentricity on individual P and S head waves in monopole acoustic LWD

2021 ◽  
Vol 18 (1) ◽  
pp. 74-84
Author(s):  
Yunjia Ji ◽  
Xiao He ◽  
Hao Chen ◽  
Xiuming Wang
Keyword(s):  
P Wave ◽  
Branch Points ◽  
Wave Excitation ◽  
Integration Technique ◽  
S Wave ◽  
Excitation Spectra ◽  
Low Frequencies ◽  
S Waves ◽  
Head Waves ◽  
Logging While Drilling

Abstract Velocities of P and S waves are main goals of downhole acoustic logging. In this work, we study the effects of an off-center acoustic tool on formation P and S head waves in monopole logging while drilling (LWD), which will be helpful for accurate interpretation of recorded logs. We first develop an analytic method to solve the wavefields of this asymmetric LWD model. Then using a branch-cut integration technique, we evaluate the contributions of branch points associated with P and S waves, and further investigate the effects of tool eccentricity on their characteristics of excitation, attenuation and waveforms. The analyses reveal that the variation of the excitation and attenuation of both P and S head waves with eccentricity depends on frequencies and receiver azimuths strongly. Besides, new resonance peaks appear in excitation spectra due to influences of poles of multipole modes near branch points when the monopole tool is off-center. According to semblance results of individual compressional and shear waveforms, extracted velocities are not affected by tool eccentricity in both fast and slow formations. In fast formations, spectra analyses indicate that S-wave excitation is more sensitive to tool eccentricity than P-wave. Moreover, resonance peaks in P-wave excitation spectra increase with the increasing eccentricity in all directions. In slow formations, off-center tools almost have no influence on both P and S waves at low frequencies, which suggests that the effects of tool eccentricity can be reduced by adjusting the source's operating frequency.

The effects of pore‐fluid salinity on ultrasonic wave propagation in sandstones

Geophysics ◽  
1998 ◽  
Vol 63 (3) ◽  
pp. 928-934 ◽  
Author(s):  
Simon M. Jones ◽  
Clive McCann ◽  
Timothy R. Astin ◽  
Jeremy Sothcott
Keyword(s):  
Wave Propagation ◽  
Ultrasonic Wave ◽  
Wave Attenuation ◽  
Clay Content ◽  
Seismic Wave Propagation ◽  
Ultrasonic Pulse ◽  
Pore Fluid ◽  
P Wave ◽  
S Wave ◽  
Shear Moduli

Petrophysical interpretation of increasingly refined seismic data from subsurface formations requires a more fundamental understanding of seismic wave propagation in sedimentary rocks. We consider the variation of ultrasonic wave velocity and attenuation in sandstones with pore‐fluid salinity and show that wave propagation is modified in proportion to the clay content of the rock and the salinity of the pore fluid. Using an ultrasonic pulse reflection technique (590–890 kHz), we have measured the P-wave and S-wave velocities and attenuations of 15 saturated sandstones with variable effective pressure (5–60 MPa) and pore‐fluid salinity (0.0–3.4 M). In clean sandstones, there was close agreement between experimental and Biot model values of [Formula: see text], but they diverged progressively in rocks containing more than 5% clay. However, this effect is small: [Formula: see text] changed by only 0.6% per molar change in salinity for a rock with a clay content of 29%. The variation of [Formula: see text] with brine molarity exhibited Biot behavior in some samples but not in others; there was no obvious relationship with clay content. P-wave attenuation was independent of pore‐fluid salinity, while S-wave attenuation was weakly dependent. The velocity data suggest the frame bulk and shear moduli of sandstones are altered by changes in the pore‐fluid salinity. One possible mechanism is the formation damage caused by clay swelling and migration of fines in low‐molarity electrolytes. The absence of variation between the attenuation in water‐saturated and brine‐saturated samples indicates the attenuation mechanism is relatively unaffected by changes in the frame moduli.

Automatic Classification of Impact-Echo Spectra II

2011 ◽  
pp. 199-205
Author(s):  
Addisson Salazar ◽  
Arturo Serrano
Keyword(s):  
Neural Networks ◽  
Wave Propagation ◽  
Propagation Velocity ◽  
Sampling Frequency ◽  
P Wave ◽  
Material Surface ◽  
S Wave ◽  
Wave Propagation Velocity ◽  
Impact Echo

We study the application of artificial neural networks (ANNs) to the classification of spectra from impact-echo signals. In this paper we focus on analyses from experiments. Simulation results are covered in paper I. Impact-echo is a procedure from Non-Destructive Evaluation where a material is excited by a hammer impact which produces a response from the material microstructure. This response is sensed by a set of transducers located on material surface. Measured signals contain backscattering from grain microstructure and information of flaws in the material inspected (Sansalone & Street, 1997). The physical phenomenon of impact-echo corresponds to wave propagation in solids. When a disturbance (stress or displacement) is applied suddenly at a point on the surface of a solid, such as by impact, the disturbance propagates through the solid as three different types of stress waves: a P-wave, an S-wave, and an R-wave. The P-wave is associated with the propagation of normal stress and the S-wave is associated with shear stress, both of them propagate into the solid along spherical wave fronts. In addition, a surface wave, or Rayleigh wave (R-wave) travels throughout a circular wave front along the material surface (Carino, 2001). After a transient period where the first waves arrive, wave propagation becomes stationary in resonant modes of the material that vary depending on the defects inside the material. In defective materials propagated waves have to surround the defects and their energy decreases, and multiple reflections and diffraction with the defect borders become reflected waves (Sansalone, Carino, & Hsu, 1998). Depending on the observation time and the sampling frequency used in the experiments we may be interested in analyzing the transient or the stationary stage of the wave propagation in impact- echo tests. Usually with high resolution in time, analyzes of wave propagation velocity can give useful information, for instance, to build a tomography of a material inspected from different locations. Considering the sampling frequency that we used in the experiments (100 kHz), a feature extracted from the signal as the wave propagation velocity is not accurate enough to discern between homogeneous and different kind of defective materials. The data set for this research consists of sonic and ultrasonic impact-echo signal (1-27 kHz) spectra obtained from 84 parallelepiped-shape (7x5x22cm. width, height and length) lab specimens of aluminium alloy series 2000. These spectra, along with a categorization of the quality of materials among homogeneous, one-defect and multiple-defect classes were used to develop supervised neural network classifiers. We show that neural networks yield good classifications (

Approximation of pure acoustic seismic wave propagation in TTI media

Geophysics ◽  
2011 ◽  
Vol 76 (5) ◽  
pp. WB97-WB107 ◽  
Author(s):  
Chunlei Chu ◽  
Brian K. Macy ◽  
Phil D. Anno
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Elastic Wave ◽  
Wave Equations ◽  
Computational Cost ◽  
P Wave ◽  
Anisotropy Axis ◽  
S Wave ◽  
Time Migration ◽  
Tti Media

Pseudoacoustic anisotropic wave equations are simplified elastic wave equations obtained by setting the S-wave velocity to zero along the anisotropy axis of symmetry. These pseudoacoustic wave equations greatly reduce the computational cost of modeling and imaging compared to the full elastic wave equation while preserving P-wave kinematics very well. For this reason, they are widely used in reverse time migration (RTM) to account for anisotropic effects. One fundamental shortcoming of this pseudoacoustic approximation is that it only prevents S-wave propagation along the symmetry axis and not in other directions. This problem leads to the presence of unwanted S-waves in P-wave simulation results and brings artifacts into P-wave RTM images. More significantly, the pseudoacoustic wave equations become unstable for anisotropy parameters [Formula: see text] and for heterogeneous models with highly varying dip and azimuth angles in tilted transversely isotropic (TTI) media. Pure acoustic anisotropic wave equations completely decouple the P-wave response from the elastic wavefield and naturally solve all the above-mentioned problems of the pseudoacoustic wave equations without significantly increasing the computational cost. In this work, we propose new pure acoustic TTI wave equations and compare them with the conventional coupled pseudoacoustic wave equations. Our equations can be directly solved using either the finite-difference method or the pseudospectral method. We give two approaches to derive these equations. One employs Taylor series expansion to approximate the pseudodifferential operator in the decoupled P-wave equation, and the other uses isotropic and elliptically anisotropic dispersion relations to reduce the temporal frequency order of the P-SV dispersion equation. We use several numerical examples to demonstrate that the newly derived pure acoustic wave equations produce highly accurate P-wave results, very close to results produced by coupled pseudoacoustic wave equations, but completely free from S-wave artifacts and instabilities.

Sign in / Sign up

Export Citation Format

Share Document