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 Local and regional P-wave spectral attenuation model for Iran

2020 ◽  
Vol 224 (1) ◽  
pp. 241-256
Author(s):  
Ehsan Moradian Bajestani ◽  
Anooshiravan Ansari ◽  
Ehsan Karkooti
Keyword(s):  
Wave Attenuation ◽  
Vertical Component ◽  
P Wave ◽  
P Waves ◽  
Attenuation Model ◽  
Hypocentral Distance ◽  
Anelastic Attenuation ◽  
Curve Versus ◽  
Path Coverage ◽  
Spectral Attenuation

SUMMARY A robust frequency-dependent local and regional P-wave attenuation model is estimated for continental paths in the Iranian Plateau. In order to calculate the average attenuation parameters, 46 337 vertical-component waveforms related to 9267 earthquakes, which are recorded at the Iranian Seismological Center (IRSC) stations, have been selected in the distance range 10–1000 km. The majority of the event's magnitudes are less than 4.5. This collection of records provides high spatial ray path coverage. Results indicate that the shape of attenuation P-wave curve versus distance is not uniform and has three distinct sections with hinges at 90 and 175 km. A trilinear model for attenuation of P-wave amplitude in the frequency range 1–10 Hz is proposed in this study. Fourier spectral amplitudes are found to decay as R−1.2 (where R is hypocentral distance), corresponding to geometric spreading within 90 km from the source. There is a section from 90 to 175 km, where the attenuation is described as R0.8, and the attenuation is described well beyond 175 km by R−1.3. Moreover, the average quality factor for Pg and Pn waves (QPg and QPn), related to anelastic attenuation is obtained as Qpg= (54.2 ± 2.6)f(1.0096±0.07) and Qpn= (306.8 ± 7.4)f (0.51±0.05). There is a good agreement between the results of the model and observations. Also, the attenuation model shows compatibility with the recent regional studies. From the results it turns out that the amplitude of P waves attenuates more rapidly in comparison with the global models in local distances.

scholarly journals Seismic wave attenuation and dispersion resulting from wave-induced flow in porous rocks — A review

Geophysics ◽  
2010 ◽  
Vol 75 (5) ◽  
pp. 75A147-75A164 ◽  
Author(s):  
Tobias M. Müller ◽  
Boris Gurevich ◽  
Maxim Lebedev
Keyword(s):  
Fluid Flow ◽  
Velocity Dispersion ◽  
Wave Attenuation ◽  
Field Measurements ◽  
Theoretical Models ◽  
Seismic Attenuation ◽  
P Wave ◽  
Induced Flow ◽  
Wave Induced ◽  
Attenuation And Dispersion

One major cause of elastic wave attenuation in heterogeneous porous media is wave-induced flow of the pore fluid between heterogeneities of various scales. It is believed that for frequencies below [Formula: see text], the most important cause is the wave-induced flow between mesoscopic inhomogeneities, which are large compared with the typical individual pore size but small compared to the wavelength. Various laboratory experiments in some natural porous materials provide evidence for the presence of centimeter-scale mesoscopic heterogeneities. Laboratory and field measurements of seismic attenuation in fluid-saturated rocks provide indications of the role of the wave-induced flow. Signatures of wave-induced flow include the frequency and saturation dependence of P-wave attenuation and its associated velocity dispersion, frequency-dependent shear-wave splitting, and attenuation anisotropy. During the last four decades, numerous models for attenuation and velocity dispersion from wave-induced flow have been developed with varying degrees of rigor and complexity. These models can be categorized roughly into three groups ac-cording to their underlying theoretical framework. The first group of models is based on Biot’s theory of poroelasticity. The second group is based on elastodynamic theory where local fluid flow is incorporated through an additional hydrodynamic equation. Another group of models is derived using the theory of viscoelasticity. Though all models predict attenuation and velocity dispersion typical for a relaxation process, there exist differences that can be related to the type of disorder (periodic, random, space dimension) and to the way the local flow is incorporated. The differences manifest themselves in different asymptotic scaling laws for attenuation and in different expressions for characteristic frequencies. In recent years, some theoretical models of wave-induced fluid flow have been validated numerically, using finite-difference, finite-element, and reflectivity algorithms applied to Biot’s equations of poroelasticity. Application of theoretical models to real seismic data requires further studies using broadband laboratory and field measurements of attenuation and dispersion for different rocks as well as development of more robust methods for estimating dissipation attributes from field data.

P-wave seismic attenuation by slow-wave diffusion: Effects of inhomogeneous rock properties

Geophysics ◽  
2006 ◽  
Vol 71 (3) ◽  
pp. O1-O8 ◽  
Author(s):  
José M. Carcione ◽  
Stefano Picotti
Keyword(s):  
Fluid Flow ◽  
Wave Attenuation ◽  
Fluid Pressure ◽  
Seismic Attenuation ◽  
P Wave ◽  
Rock Properties ◽  
Stratified Medium ◽  
P Waves ◽  
Low Frequencies ◽  
Attenuation Mechanism

Recent research has established that the dominant P-wave attenuation mechanism in reservoir rocks at seismic frequencies is because of wave-induced fluid flow (mesoscopic loss). The P-wave induces a fluid-pressure difference at mesoscopic-scale inhomogeneities (larger than the pore size but smaller than the wavelength, typically tens of centimeters) and generates fluid flow and slow (diffusion) Biot waves (continuity of pore pressure is achieved by energy conversion to slow P-waves, which diffuse away from the interfaces). In this context, we consider a periodically stratified medium and investigate the amount of attenuation (and velocity dispersion) caused by different types of heterogeneities in the rock properties, namely, porosity, grain and frame moduli, permeability, and fluid properties. The most effective loss mechanisms result from porosity variations and partial saturation, where one of the fluids is very stiff and the other is very compliant, such as, a highly permeable sandstone at shallow depths, saturated with small amounts of gas (around 10% saturation) and water. Grain- and frame-moduli variations are the next cause of attenuation. The relaxation peak moves towards low frequencies as the (background) permeability decreases and the viscosity and thickness of the layers increase. The analysis indicates in which cases the seismic band is in the relaxed regime, and therefore, when the Gassmann equation can yield a good approximation to the wave velocity.

Poroelastic model to relate seismic wave attenuation and dispersion to permeability anisotropy

Geophysics ◽  
2000 ◽  
Vol 65 (1) ◽  
pp. 202-210 ◽  
Author(s):  
Jorge O. Parra
Keyword(s):  
Seismic Waves ◽  
Velocity Dispersion ◽  
Wave Attenuation ◽  
Test Site ◽  
Compressional Wave ◽  
P Wave ◽  
Low Velocity ◽  
Stiffness Constant ◽  
Principal Axes ◽  
Permeability Anisotropy

A transversely isotropic model with a horizontal axis of symmetry, based on the Biot and squirt‐flow mechanisms, predicts seismic waves in poroelastic media. The model estimates velocity dispersion and attenuation of waves propagating in the frequency range of crosswell and high‐resolution reverse vertical seismic profiling (VSP) (250–1250 Hz) for vertical permeability values much greater than horizontal permeability parameters. The model assumes the principal axes of the stiffness constant tensor are aligned with the axes of the permeability and squirt‐flow tensors. In addition, the unified Biot and squirt‐flow mechanism (BISQ) model is adapted to simulate cracks in permeable media. Under these conditions, the model simulations demonstrate that the preferential direction of fluid flow in a reservoir containing fluid‐filled cracks can be determined by analyzing the phase velocity and attenuation of seismic waves propagating at different azimuth and incident angles. As a result, the fast compressional wave can be related to permeability anisotropy in a reservoir. The model results demonstrate that for a fast quasi-P-wave propagating perpendicular to fluid‐filled cracks, the attenuation is greater than when the wave propagates parallel to the plane of the crack. Theoretical predictions and velocity dispersion of inter‐well seismic waves in the Kankakee Limestone Formation at the Buckhorn test site (Illinois) demonstrate that the permeable rock matrix surrounding a low‐velocity heterogeneity contains vertical cracks.

Seismic modeling of low-frequency “shadows” beneath gas reservoirs

Geophysics ◽  
2014 ◽  
Vol 79 (6) ◽  
pp. D417-D423 ◽  
Author(s):  
Elmira Chabyshova ◽  
Gennady Goloshubin
Keyword(s):  
Wave Propagation ◽  
Arrival Time ◽  
Wave Attenuation ◽  
Low Frequency ◽  
P Wave ◽  
Wave Reflections ◽  
Gas Reservoirs ◽  
P Waves ◽  
Seismic Frequencies ◽  
Time Variations

P-wave amplitude anomalies below reservoir zones can be used as hydrocarbon markers. Some of those anomalies are considerably delayed relatively to the reflections from the reservoir zone. High P-wave attenuation and velocity dispersion of the observed P-waves cannot justify such delays. The hypothesis that these amplitude anomalies are caused by wave propagation through a layered permeable gaseous reservoir is evaluated. The wave propagation through highly interbedded reservoirs suggest an anomalous amount of mode conversions between fast and slow P-waves. The converted P-waves, which propagated even a short distance as slow P-waves, should be significantly delayed and attenuated comparatively, with the fast P-wave reflections. The amplitudes and arrival time variations of conventional and converted P-wave reflections at low seismic frequencies were evaluated by means of an asymptotic analysis. The calculations confirmed that the amplitude anomalies due to converted P-waves are noticeably delayed in time relatively to fast P-wave reflections. However, the amplitudes of the modeled converted P-waves were much lower than the amplitude anomalies observed from exploration cases.

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.

Implications of a composite source model and seismic-wave attenuation for the observed simplicity of small earthquakes and reported duration of earthquake initiation phase

1998 ◽  
Vol 88 (5) ◽  
pp. 1171-1181
Author(s):  
S. K. Singh ◽  
M. Ordaz ◽  
T. Mikumo ◽  
J. Pacheco ◽  
C. Valdés ◽  
...  
Keyword(s):  
Seismic Waves ◽  
Wave Attenuation ◽  
Source Model ◽  
P Wave ◽  
Bright Spot ◽  
Power Law Distribution ◽  
P Waves ◽  
Velocity Amplitude ◽  
Initiation Phase ◽  
Composite Source Model

Abstract An examination of P waves recorded on near-source, velocity seismograms generally shows that most small earthquakes (Mw < 2 to 3) are simple. On the other hand, larger earthquakes (Mw ≧ 4) are most often complex. The simplicity of the seismograms of Mw < 2 to 3 events may reflect the simplicity of the source (and, hence, may imply that smaller and larger earthquakes are not self-similar) or may be a consequence of attenuation of seismic waves. To test whether the attenuation is the cause, we generated synthetic P-wave seismograms from a composite circular source model in which subevent rupture areas are assumed to follow a power-law distribution. The rupture of an event is assumed to initiate at a random point on the fault and to propagate with a uniform speed. As the rupture front reaches the center of a subevent patch (all of which are circular), a P pulse is radiated that is calculated from the kinematic source model of Sato and Hirasawa (1973). Synthetic P-wave seismograms, which are all complex, are then convolved with an attenuation operator for different values of t*. The results show that the observed simplicity of small events (Mw < 2 to 3) may be entirely explained by attenuation if t* ≧ 0.02 sec. The composite source model predicts that the average time delay between the initiation of the rupture and the rupture of the largest patch, τ, scales as M01/3, such that log τ = (1/3) log M0 − 8.462. This relation is very similar to that reported by Umeda et al. (1996) between M0 and the observed time difference between the initiation of the rupture and the rupture of the “bright spot.” It roughly agrees with the relation between M0 and the duration of the initiation phase reported by Ellsworth and Beroza (1995) and Beroza and Ellsworth (1996). The relation also fits surprisingly well the data on duration of slow initial phase, tsip, and M0, reported by Iio (1995). One possible explanation of this agreement may be that the composite source model, which is essentially the “cascade” model, successfully captures the evolution of the earthquake source process and that the rupture initiation and the abrupt increase in the velocity amplitude observed on seismograms by previous researchers roughly corresponds to the rupture of the first subevent and the breaking of the largest subevent in the composite source model.

Ultrasonic attenuation in Glenn Pool rocks, northeastern Oklahoma

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 465-478 ◽  
Author(s):  
Andrew P. Shatilo ◽  
Carl Sondergeld ◽  
Chandra S. Rai
Keyword(s):  
Phase Velocity ◽  
Reservoir Characterization ◽  
Velocity Dispersion ◽  
Wave Attenuation ◽  
Ultrasonic Attenuation ◽  
Clay Content ◽  
P Wave ◽  
Phase Velocity Dispersion ◽  
Q Model ◽  
Water Saturated

Ultrasonic P-wave attenuation and phase velocity dispersion have been estimated for 29 samples of sandstones and 13 samples of shales from a Glenn Pool oil reservoir using a pulse transmission technique. The measurements were performed under effective pressures from atmospheric to 15 MPa. There is a strong correlation between attenuation coefficient and phase velocity dispersion. Even though the observed attenuation may deviate from a “constant Q” model, it generally agrees with a minimum‐phase prediction. Attenuation in the water‐saturated sandstones increases with porosity and permeability. We found no correlation between the attenuation and clay content within the sandstone subset. Attenuation in the shales is much less than that in the sandstones. This difference may be used in reservoir characterization.

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