Theory and modeling of constant-Q P- and S-waves using fractional time derivatives

Geophysics ◽  
2009 ◽  
Vol 74 (1) ◽  
pp. T1-T11 ◽  
Author(s):  
José M. Carcione
Keyword(s):  
Heterogeneous Media ◽  
Fourier Method ◽  
P Wave ◽  
Difference Operators ◽  
S Wave ◽  
Central Difference ◽  
S Waves ◽  
The Time Domain ◽  
Derivatives Of ◽  
Modeling Algorithm

I have developed and solved the constant-[Formula: see text] model for the attenuation of P- and S-waves in the time domain using a new modeling algorithm based on fractional derivatives. The model requires time derivatives of order [Formula: see text] applied to the strain components, where [Formula: see text] and [Formula: see text], with [Formula: see text] the P-wave or S-wave quality factor. The derivatives are computed with the Grünwald-Letnikov and central-difference fractional approximations, which are extensions of the standard finite-difference operators for derivatives of integer order. The modeling uses the Fourier method to compute the spatial derivatives, and therefore can handle complex geometries and general material-property variability. I verified the results by comparison with the 2D analytical solution obtained for wave propagation in homogeneous Pierre Shale. Moreover, the modeling algorithm was used to compute synthetic seismograms in heterogeneous media corresponding to a crosswell seismic experiment.

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.

Processing, correlating, and interpreting converted shear waves from borehole data in southern Alberta

Geophysics ◽  
1990 ◽  
Vol 55 (6) ◽  
pp. 660-669 ◽  
Author(s):  
W. T. Geis ◽  
R. R. Stewart ◽  
M. J. Jones ◽  
P. E. Katapodis
Keyword(s):  
P Wave ◽  
Equation Modeling ◽  
S Wave ◽  
Borehole Data ◽  
P Waves ◽  
S Waves ◽  
Sonic Log ◽  
Lake Formation ◽  
Southern Alberta ◽  
The Time Domain

Borehole measurements coupled with phase information from Zoeppritz equation modeling has assisted in accurate correlation between a VSP converted S-wave section and both the surface and VSP P-wave sections from southern Alberta. For the most part, both the character and polarities of the sections agree; however, there are some differences. Some reflections are stronger and more distinct on the S-wave section than on the P-wave section. Spectral analysis of the time‐domain upgoing P-wave and S-wave energy shows that the frequency content of the S-waves is comparable to the P-waves. Thus, the slower velocity S-waves have a shorter wavelenght and provide better vertical resolution of some interfaces. Other upgoing S-wave modes can interfere with the P‐SV mode and contribute to the differences between the P- and S-wave sections. The match between P-wave and S-wave velocities ([Formula: see text] and [Formula: see text]), determined from VSP traveltime inversion and the full‐waveform sonic log, is best in the Paleozoic carbonate section; there is some discrepancy in Cretaceous sandstone intervals. A basal salt unit in the Paleozoic Beaverhill Lake formation has a VSP‐determined [Formula: see text] ratio of 1.97, suggesting that salt can be distinguished from carbonates using both P-wave and S-wave velocity information in this region.

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.

Scalar reverse‐time depth migration of prestack elastic seismic data

Geophysics ◽  
2001 ◽  
Vol 66 (5) ◽  
pp. 1519-1527 ◽  
Author(s):  
Robert Sun ◽  
George A. McMechan
Keyword(s):  
Seismic Waves ◽  
Velocity Model ◽  
P Wave ◽  
Common Source ◽  
Reverse Time ◽  
S Wave ◽  
Wave Source ◽  
Reflected Waves ◽  
S Waves ◽  
Elastic Data

Reflected P‐to‐P and P‐to‐S converted seismic waves in a two‐component elastic common‐source gather generated with a P‐wave source in a two‐dimensional model can be imaged by two independent scalar reverse‐time depth migrations. The inputs to migration are pure P‐ and S‐waves that are extracted by divergence and curl calculations during (shallow) extrapolation of the elastic data recorded at the earth’s surface. For both P‐to‐P and P‐to‐S converted reflected waves, the imaging time at each point is the P‐wave traveltime from the source to that point. The extracted P‐wave is reverse‐time extrapolated and imaged with a P‐velocity model, using a finite difference solution of the scalar wave equation. The extracted S‐wave is reverse‐time extrapolated and imaged similarly, but with an S‐velocity model. Converted S‐wave data requires a polarity correction prior to migration to ensure constructive interference between data from adjacent sources. Synthetic examples show that the algorithm gives satisfactory results for laterally inhomogeneous models.

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.

Comparison of P‐ and S‐wave velocities and Q’S from VSP and sonic log data

Geophysics ◽  
1994 ◽  
Vol 59 (10) ◽  
pp. 1512-1529 ◽  
Author(s):  
Gopa S. De ◽  
Donald F. Winterstein ◽  
Mark A. Meadows
Keyword(s):  
Velocity Dispersion ◽  
P Wave ◽  
S Wave ◽  
S Waves ◽  
Wave Velocities ◽  
Sonic Log ◽  
Transit Times ◽  
Wave Slope ◽  
Northern Alberta ◽  
Q Values

We compared P‐ and S‐wave velocities and quality factors (Q’S) from vertical seismic profiling (VSP) and sonic log measurements in five wells, three from the southwest San Joaquin Basin of California, one from near Laredo, Texas, and one from northern Alberta. Our purpose was to investigate the bias between sonic log and VSP velocities and to examine to what degree this bias might be a consequence of dispersion. VSPs and sonic logs were recorded in the same well in every case. Subsurface formations were predominantly clastic. The bias found was that VSP transit times were greater than sonic log times, consistent with normal dispersion. For the San Joaquin wells, differences in S‐wave transit times averaged 1–2 percent, while differences in P‐wave transit times averaged 6–7 percent. For the Alberta well, the situation was reversed, with differences in S‐wave transit times being about 6 percent, while those for P‐waves were 2.5 percent. For the Texas well, the differences averaged about 4 percent for both P‐ and S‐waves. Drift‐curve slopes for S‐waves tended to be low where the P‐wave slopes were high and vice versa. S‐wave drift‐curve slopes in the shallow California wells were 5–10 μs/ft (16–33 μs/m) and the P‐wave slopes were 15–30 μs/ft (49–98 μs/m). The S‐wave slope in sandstones in the northern Alberta well was up to 50 μs/ft (164 μs/m), while the P‐wave slope was about 5 μs/ft (16 μs/m). In the northern Alberta well the slopes for both P‐ and S‐waves flattened in the carbonate. In the Texas well, both P‐ and S‐wave drifts were comparable. We calculated (Q’s) from a velocity dispersion formula and from spectral ratios. When the two Q’s agreed, we concluded that velocity dispersion resulted solely from absorption. These Q estimation methods were reliable only for Q values smaller than 20. We found that, even with data of generally outstanding quality, Q values determined by standard methods can have large uncertainties, and negative Q’s may be common.

Velocity anisotropy in shale determined from crosshole seismic and vertical seismic profile data

Geophysics ◽  
1990 ◽  
Vol 55 (4) ◽  
pp. 470-479 ◽  
Author(s):  
D. F. Winterstein ◽  
B. N. P. Paulsson
Keyword(s):  
Seismic Profile ◽  
P Wave ◽  
Isotropic Model ◽  
S Wave ◽  
Vertical Seismic Profile ◽  
Near Surface ◽  
S Waves ◽  
Velocity Anisotropy ◽  
Wave Velocities ◽  
Late Tertiary

Crosshole and vertical seismic profile (VST) data made possible accurate characterization of the elastic properties, including noticeable velocity anisotropy, of a near‐surface late Tertiary shale formation. Shear‐wave splitting was obvious in both crosshole and VSP data. In crosshole data, two orthologonally polarrized shear (S) waves arrived 19 ms in the uppermost 246 ft (75 m). Vertically traveling S waves of the VSP separated about 10 ms in the uppermost 300 ft (90 m) but remained at nearly constant separation below that level. A transversely isotropic model, which incorporates a rapid increase in S-wave velocities with depth but slow increase in P-wave velocities, closely fits the data over most of the measured interval. Elastic constants of the transvesely isotropic model show spherical P- and [Formula: see text]wave velocity surfaces but an ellipsoidal [Formula: see text]wave surface with a ratio of major to minor axes of 1.15. The magnitude of this S-wave anisotropy is consistent with and lends credence to S-wave anisotropy magnitudes deduced less directly from data of many sedimentary basins.

scholarly journals Travel-Time Inversion Method of Converted Shear Waves Using RayInvr Algorithm

2021 ◽  
Vol 11 (8) ◽  
pp. 3571
Author(s):  
Genggeng Wen ◽  
Kuiyuan Wan ◽  
Shaohong Xia ◽  
Huilong Xu ◽  
Chaoyan Fan ◽  
...  
Keyword(s):  
Travel Time ◽  
Wave Velocity ◽  
P Wave ◽  
Inversion Method ◽  
Time Inversion ◽  
Ocean Bottom Seismometer ◽  
S Wave ◽  
S Waves ◽  
Inversion Model ◽  
Travel Time Inversion

The detailed studies of converted S-waves recorded on the Ocean Bottom Seismometer (OBS) can provide evidence for constraining lithology and geophysical properties. However, the research of converted S-waves remains a weakness, especially the S-waves’ inversion. In this study, we applied a travel-time inversion method of converted S-waves to obtain the crustal S-wave velocity along the profile NS5. The velocities of the crust are determined by the following four aspects: (1) modelling the P-wave velocity, (2) constrained sediments Vp/Vs ratios and S-wave velocity using PPS phases, (3) the correction of PSS phases’ travel-time, and (4) appropriate parameters and initial model are selected for inversion. Our results show that the vs. and Vp/Vs of the crust are 3.0–4.4 km/s and 1.71–1.80, respectively. The inversion model has a similar trend in velocity and Vp/Vs ratios with the forward model, due to a small difference with ∆Vs of 0.1 km/s and ∆Vp/Vs of 0.03 between two models. In addition, the high-resolution inversion model has revealed many details of the crustal structures, including magma conduits, which further supports our method as feasible.

Crustal structures across the young and oblique North-eastern Gulf of Aden margin

2020 ◽  
Author(s):  
Louise Watremez ◽  
Sylvie Leroy ◽  
Elia d'Acremont ◽  
Stéphane Rouzo
Keyword(s):  
Seismic Waves ◽  
Seismic Refraction ◽  
P Wave ◽  
Gulf Of Aden ◽  
S Wave ◽  
Acoustic Basement ◽  
S Waves ◽  
P Wave Velocity ◽  
North Eastern ◽  
Lower Crustal

<p>The Gulf of Aden is a young and active oceanic basin, which separates the south-eastern margin of the Arabian Plate from the Somali Plate. The rifting leading to the formation of the north-eastern Gulf of Aden passive margin started ca. 34 Ma ago when the oceanic spreading in this area initiated at least 17.6 Ma ago. The opening direction (N26°E) is oblique to the mean orientation of the Gulf (N75°E), leading to a strong structural segmentation.</p><p>The Encens cruise (2006) allowed for the acquisition of a large seismic refraction dataset with profiles across (6 lines) and along (3 lines) the margin, between the Alula-Fartak and Socotra-Hadbeen fracture zones, which define a first order segment of the Gulf. P-wave velocity modelling already allowed us to image the crustal thinning and the structures, from continental to oceanic domains, along some of the profiles. A lower crustal intermediate body is observed in the Ashawq-Salalah segment, at the base of the transitional and oceanic crusts. The nature of this intermediate body is most probably mafic, linked to a post-rift thermal anomaly. The thin (1-2 km) sediment layer in the study area allows for a clear conversion of P-waves to S-waves at the top basement. Thus, most seismic refraction records show very clear S-wave arrivals.</p><p>In this study, we use both P-wave and S-wave arrivals to delineate the crustal structures and segmentation along and across the margin and add insight into the nature of the rocks below the acoustic basement. P-wave velocity modelling allows for the delineation of the structure variations across and along the margin. The velocity models are used as a base for the S-wave modelling, through the definition of Poisson’s ratios in the different areas of the models. Picking and modelling of S-wave arrivals allow us to identify two families of converted waves: (1) seismic waves converted at the basement interface on the way up, just before arriving to the OBS and (2) seismic waves converted at the basement on the way down, which travelled into the deep structures as S-waves. The first set of arrivals allows for the estimation the S-wave velocities (Poisson’s ratio) in the sediments, showing that the sediments in this area are unconsolidated and water saturated. The second set of arrivals gives us constraints on the S-wave velocities below the acoustic basement. This allows for an improved mapping of the transitional and oceanic domains and the confirmation of the mafic nature of the lower crustal intermediate body.</p>

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.

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