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 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.

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.

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.

Seismic wave attenuation and modulus dispersion in sandstones

Geophysics ◽  
2016 ◽  
Vol 81 (3) ◽  
pp. D211-D231 ◽  
Author(s):  
James W. Spencer ◽  
Jacob Shine
Keyword(s):  
Wave Attenuation ◽  
High Viscosity ◽  
P Wave ◽  
S Wave ◽  
Local Flow ◽  
Shear Moduli ◽  
Low Viscosity ◽  
Confining Pressures ◽  
Wave Induced ◽  
Dispersion And Attenuation

We have conducted laboratory experiments over the 1–200 Hz band to examine the effects of viscosity and permeability on modulus dispersion and attenuation in sandstones and also to examine the effects of partial gas or oil saturation on velocities and attenuations. Our results have indicated that bulk modulus values with low-viscosity fluids are close to the values predicted using Gassmann’s first equation, but, with increasing frequency and viscosity, the bulk and shear moduli progressively deviate from the values predicted by Gassmann’s equations. The shear moduli increase up to 1 GPa (or approximately 10%) with high-viscosity fluids. The P- and S-wave attenuations ([Formula: see text] and [Formula: see text]) and modulus dispersion with different fluids are indicative of stress relaxations that to the first order are scaling with frequency times viscosity. By fitting Cole-Cole distributions to the scaled modulus and attenuation data, we have found that there are similar P-wave, shear and bulk relaxations, and attenuation peaks in each of the five sandstones studied. The modulus defects range from 11% to 15% in Berea sandstone to 16% to 26% in the other sandstones, but these would be reduced at higher confining pressures. The relaxations shift to lower frequencies as the viscosity increased, but they do not show the dependence on permeability predicted by mesoscopic wave-induced fluid flow (WIFF) theories. Results from other experiments having patchy saturation with liquid [Formula: see text] and high-modulus fluids are consistent with mesoscopic WIFF theories. We have concluded that the modulus dispersion and attenuations ([Formula: see text] and [Formula: see text]) in saturated sandstones are caused by a pore-scale, local-flow mechanism operating near grain contacts.

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.

Analysis of wave-induced pore pressure changes recorded during the 1980 Mammoth Lakes, California, earthquake sequence

1984 ◽  
Vol 74 (4) ◽  
pp. 1395-1407
Author(s):  
Gerald M. Mavko ◽  
Ed Harp
Keyword(s):  
Pore Pressure ◽  
Vertical Surface ◽  
P Wave ◽  
Earthquake Sequence ◽  
Surface Velocity ◽  
Elastic Model ◽  
S Wave ◽  
Porous Saturated Medium ◽  
Wave Induced ◽  
Mammoth Lakes

Abstract Acceleration and wave-induced pore pressure were recorded in a saturated sand during the 1980 Mammoth Lakes, California, earthquake sequence. For the largest event recorded, the pore pressure was observed to be proportional to vertical surface acceleration during the P-wave arrivals and proportional to horizontal surface velocity during the S-wave arrivals. The results can be quantitatively explained with a linear elastic model of a porous saturated medium, such that pore pressure depends on dilatation and is independent of shear strain. A slight frequency dependence in the ratio of pore pressure to dilatation indicates local fluid flow on the scale of individual pores. The good agreement between observations and theory indicates that the deformation was primarily linear, even though maximum shear strains were close to the typical thresholds for liquefaction.

A theoretical study of acoustic S-wave and P-wave velocity logging with conventional and dipole sources in soft formations

Geophysics ◽  
1988 ◽  
Vol 53 (10) ◽  
pp. 1334-1342 ◽  
Author(s):  
Graham A. Winbow
Keyword(s):  
Wave Velocity ◽  
Water Waves ◽  
Fluid Velocity ◽  
Soft Materials ◽  
P Wave ◽  
Stoneley Wave ◽  
Drilling Mud ◽  
S Wave ◽  
Low Frequencies ◽  
P Wave Velocity

This paper is focused on the special features of the wavetrains recorded by conventional and dipole sonic logging tools in soft formations defined to be those whose shar velocity is less than the sound velocity of drilling mud. Such formations are commonn in the Gulf Coast, the Canadian Arctic, the Bass Strait of Australia, and many other region. A conventional logging tool operating at normal frequencies [Formula: see text] records P waves, water waves, and Stoneley waves in soft formations. A dipole tool records modal waves and water waves at frequencies of order 15 kHz, but produces almost pure S-wave first arrivals at low frequencies [Formula: see text] since at 1 kHz, a mode which we refer to as a “dipole Stoneley wave” is efficiently excited. For very soft materials such as clays, where the formation P-wave velocity can be less than the fluid velocity, the formation P velocity can be logged by operating a conventional sonic tool at low frequencies [Formula: see text] so as to excite a leaky mode traveling at very close to the formation P-wave velocity. Water waves are not important for high‐velocity formations where they arrive at the trailing edge of the modal part of the wavetrain. However, in soft formalions they form a prominent part of the wavetrain at normal logging frequencies [Formula: see text] and disappear at low frequencies [Formula: see text]. Water waves are carried by leaky modes.

A simple method of predicting S-wave velocity

Geophysics ◽  
2006 ◽  
Vol 71 (6) ◽  
pp. F161-F164 ◽  
Author(s):  
Myung W. Lee
Keyword(s):  
Wave Velocity ◽  
P Wave ◽  
Unconsolidated Sediments ◽  
S Wave ◽  
Amplitude Analysis ◽  
Simple Method ◽  
Low Frequencies ◽  
Wave Velocities ◽  
Aspect Ratios ◽  
P Wave Velocity

Prediction of shear-wave velocity plays an important role in seismic modeling, amplitude analysis with offset, and other exploration applications. This paper presents a method for predicting S-wave velocity from the P-wave velocity on the basis of the moduli of dry rock. Elastic velocities of water-saturated sediments at low frequencies can be predicted from the moduli of dry rock by using Gassmann’s equation; hence, if the moduli of dry rock can be estimated from P-wave velocities, then S-wave velocities easily can be predicted from the moduli. Dry rock bulk modulus can be related to the shear modulus through a compaction constant. The numerical results indicate that the predicted S-wave velocities for consolidated and unconsolidated sediments agree well with measured velocities if differential pressure is greater than approximately [Formula: see text]. An advantage of this method is that there are no adjustable parameters to be chosen, such as the pore-aspect ratios required in some other methods. The predicted S-wave velocity depends only on the measured P-wave velocity and porosity.

Statistical analysis of irregular wave-guide influences on regional seismic discriminants in China

1998 ◽  
Vol 88 (1) ◽  
pp. 74-88 ◽  
Author(s):  
Guangwei Fan ◽  
Thorne Lay
Keyword(s):  
Wave Energy ◽  
P Wave ◽  
Western China ◽  
The Tibetan Plateau ◽  
Large Scatter ◽  
S Wave ◽  
Low Frequencies ◽  
Short Period ◽  
Propagation Effects ◽  
Regional Phases

Abstract Short-period regional phases play an important role in identifying low-magnitude seismic events in the context of monitoring the Comprehensive Test Ban Treaty. Amplitude ratios of regional phases comprised mainly of P-wave energy (Pn, Pg) to those comprised mainly of S-wave energy (Sn, Lg) effectively discriminate between explosions and earthquakes in many regions, particularly at frequencies higher than 3 Hz. At lower frequencies, discrimination is usually poor due to large scatter that causes overlapping of event populations. Scatter in regional discriminant measures such as Pg/Lg ratios is caused by both source and propagation effects, and reducing the scatter imparted by the latter is essential to improving the discriminant performance when events do not share identical paths. Regional phases experience distance-dependent amplitude variations due to effects such as critical angle amplification, geometric spreading, and attenuation. Discriminant measures are usually corrected for empirically determined distance trends for a given region, but large scatter persists after such corrections. This study seeks to develop more sophisticated empirical corrections for path properties in order to further reduce the scatter in regional discriminant measures caused by propagation effects. Broadband seismic waveforms recorded at station WMQ, in western China, demonstrate that regional Pg/Lg ratios show significant distance dependence for frequencies less than 6 Hz. However, variations in crustal structure cause additional path-specific amplitude fluctuations that are not accounted for by regionally averaged distance corrections. Blockage of Lg phases on paths traversing the margins of the Tibetan Plateau is one such effect. Regression analysis demonstrates that Pg/Lg ratios measured at WMQ display significant correlations with path-specific properties such as mean elevation, topographic roughness, basement depth, and crustal thickness. Multiple regressions using optimal combinations of parameters yield corrections that reduce variance in Pg/Lg measurements for frequencies less than 3 Hz by a factor of 2 or more relative to standard distance corrections. This should systematically enhance the performance of the Pg/Lg discriminant at low frequencies. The method presented here can be used for all regions and all short-period regional discriminants. It is likely that the extraordinary crustal heterogeneity in western China represents an extreme case of path-dependent effects.

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