scholarly journals An acoustic eikonal equation for attenuating orthorhombic media

Geophysics ◽  
2017 ◽  
Vol 82 (4) ◽  
pp. WA67-WA81 ◽  
Author(s):  
Qi Hao ◽  
Tariq Alkhalifah
Keyword(s):  
Eikonal Equation ◽  
P Wave ◽  
S Wave ◽  
Seismic Wave Velocity ◽  
P Waves ◽  
Azimuthal Variation ◽  
Wave Parameters ◽  
Complex Valued ◽  
Complex Eikonal ◽  
Wave Complex

Attenuating orthorhombic models are often used to describe the azimuthal variation of the seismic wave velocity and attenuation in finely layered hydrocarbon reservoirs with vertical fractures. In addition to the P-wave related medium parameters, S-wave parameters are also present in the complex eikonal equation needed to describe the P-wave complex-valued traveltime in an attenuating orthorhombic medium, which increases the complexity of using the P-wave traveltime to invert for the medium parameters in practice. We have used the acoustic assumption to derive an acoustic eikonal equation that approximately governs the complex-valued traveltime of P-waves in an attenuating orthorhombic medium. For a homogeneous attenuating orthorhombic media, we solve the eikonal equation using a combination of the perturbation method and Shanks transform. For a horizontal attenuating orthorhombic layer, the real and imaginary parts of the complex-valued reflection traveltime have nonhyperbolic behaviors in terms of the source-receiver offset. Similar to the roles of normal moveout (NMO) velocity and anellipticity, the attenuation NMO velocity and the attenuation anellipticity characterize the variation of the imaginary part of the complex-valued reflection traveltime around zero source-receiver offset.

scholarly journals An acoustic eikonal equation for attenuating transversely isotropic media with a vertical symmetry axis

Geophysics ◽  
2017 ◽  
Vol 82 (1) ◽  
pp. C9-C20 ◽  
Author(s):  
Qi Hao ◽  
Tariq Alkhalifah
Keyword(s):  
Eikonal Equation ◽  
Wave Attenuation ◽  
Transversely Isotropic ◽  
S Wave ◽  
P Waves ◽  
The Earth ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Vti Media ◽  
Complex Valued

Seismic-wave attenuation is an important component of describing wave propagation. Certain regions, such as gas clouds inside the earth, exert highly localized attenuation. In fact, the anisotropic nature of the earth induces anisotropic attenuation because the quasi P-wave dispersion effect should be profound along the symmetry direction. We have developed a 2D acoustic eikonal equation governing the complex-valued traveltime of quasi P-waves in attenuating, transversely isotropic media with a vertical-symmetry axis (VTI). This equation is derived under the assumption that the complex-valued traveltime of quasi P-waves in attenuating VTI media are independent of the S-wave velocity parameter [Formula: see text] in Thomsen’s notation and the S-wave attenuation coefficient [Formula: see text] in Zhu and Tsvankin’s notation. We combine perturbation theory and Shanks transform to develop practical approximations to the acoustic attenuating eikonal equation, capable of admitting an analytical description of the attenuation in homogeneous media. For a horizontal-attenuating VTI layer, we also derive the nonhyperbolic approximations for the real and imaginary parts of the complex-valued reflection traveltime. These equations reveal that (1) the quasi SV-wave velocity and the corresponding quasi SV-wave attenuation coefficient given as part of Thomsen-type notation barely affect the ray velocity and ray attenuation of quasi P-waves in attenuating VTI media; (2) combining the perturbation method and Shanks transform provides an accurate analytic eikonal solution for homogeneous attenuating VTI media; (3) for a horizontal attenuating VTI layer with weak attenuation, the real part of the complex-valued reflection traveltime may still be described by the existing nonhyperbolic approximations developed for nonattenuating VTI media, and the imaginary part of the complex-valued reflection traveltime still has the shape of nonhyperbolic curves. In addition, we have evaluated the possible extension of the proposed eikonal equation to realistic attenuating media, an alternative perturbation solution to the proposed eikonal equation, and the feasibility of applying the proposed nonhyperbolic equation for the imaginary part of the complex-valued traveltime to invert for interval attenuation parameters.

scholarly journals New acoustic approximation for transversely isotropic media with a vertical symmetry axis

Geophysics ◽  
2019 ◽  
Vol 85 (1) ◽  
pp. C1-C12 ◽  
Author(s):  
Shibo Xu ◽  
Alexey Stovas ◽  
Tariq Alkhalifah ◽  
Hitoshi Mikada
Keyword(s):  
Wave Propagation ◽  
Seismic Data ◽  
Eikonal Equation ◽  
Symmetry Axis ◽  
Transversely Isotropic ◽  
P Wave ◽  
Seismic Data Processing ◽  
S Wave ◽  
P Waves ◽  
Acoustic Approximation

Seismic data processing in the elastic anisotropic model is complicated due to multiparameter dependency. Approximations to the P-wave kinematics are necessary for practical purposes. The acoustic approximation for P-waves in a transversely isotropic medium with a vertical symmetry axis (VTI) simplifies the description of wave propagation in elastic media, and as a result, it is widely adopted in seismic data processing and analysis. However, finite-difference implementations of that approximation are plagued with S-wave artifacts. Specifically, the resulting wavefield also includes artificial diamond-shaped S-waves resulting in a redundant signal for many applications that require pure P-wave data. To derive a totally S-wave-free acoustic approximation, we have developed a new acoustic approximation for pure P-waves that is totally free of S-wave artifacts in the homogeneous VTI model. To keep the S-wave velocity equal to zero, we formulate the vertical S-wave velocity to be a function of the model parameters, rather than setting it to zero. Then, the corresponding P-wave phase and group velocities for the new acoustic approximation are derived. For this new acoustic approximation, the kinematics is described by a new eikonal equation for pure P-wave propagation, which defines the new vertical slowness for the P-waves. The corresponding perturbation-based approximation for our new eikonal equation is used to compare the new equation with the original acoustic eikonal. The accuracy of our new P-wave acoustic approximation is tested on numerical examples for homogeneous and multilayered VTI models. We find that the accuracy of our new acoustic approximation is as good as the original one for the phase velocity, group velocity, and the kinematic parameters such as vertical slowness, traveltime, and relative geometric spreading. Therefore, the S-wave-free acoustic approximation could be further applied in seismic processing that requires pure P-wave data.

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.

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 Phased array compaction cell for measurement of the transversely isotropic elastic properties of compacting sediments

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. WA113-WA123 ◽  
Author(s):  
Kurt T. Nihei ◽  
Seiji Nakagawa ◽  
Frederic Reverdy ◽  
Larry R. Myer ◽  
Luca Duranti ◽  
...  
Keyword(s):  
Elastic Properties ◽  
Phased Array ◽  
Transversely Isotropic ◽  
P Wave ◽  
S Wave ◽  
P Waves ◽  
Array Measurements ◽  
Experimental Apparatus ◽  
Marine Sediment Sample ◽  
Plane P Waves

Sediments undergoing compaction typically exhibit transversely isotropic (TI) elastic properties. We present a new experimental apparatus, the phased array compaction cell, for measuring the TI elastic properties of clay-rich sediments during compaction. This apparatus uses matched sets of P- and S-wave ultrasonic transducers located along the sides of the sample and an ultrasonic P-wave phased array source, together with a miniature P-wave receiver on the top and bottom ends of the sample. The phased array measurements are used to form plane P-waves that provide estimates of the phase velocities over a range of angles. From these measurements, the five TI elastic constants can be recovered as the sediment is compacted, without the need for sample unloading, recoring, or reorienting. This paper provides descriptions of the apparatus, the data processing, and an application demonstrating recovery of the evolving TI properties of a compacting marine sediment sample.

scholarly journals Estimation of Geodynamic Properties using Seismic Techniques at a Steel Rolling Factory, Northwestern Gulf of Suez, Egypt

2020 ◽  
Vol 8 (6) ◽  
pp. 1785-1794
Keyword(s):  
Wave Velocity ◽  
Seismic Waves ◽  
Seismic Refraction ◽  
Industrial Area ◽  
P Wave ◽  
Gulf Of Suez ◽  
S Wave ◽  
P Waves ◽  
Steel Rolling ◽  
P Wave Velocity

The objective of the current investigations is to estimate the dynamic geotechnical properties necessary for evaluating the conditions of the subsurface in order to make better decisions for economic and safe designs of the proposed structures at a Steel Rolling Factory, Ataqa Industrial Area, Northwestern Gulf of Suez, Egypt. To achieve this purpose, four seismic refraction profiles were conducted to measure the velocity of primary seismic waves (P-waves) and four profiles were conducted using Multichannel Analysis of Surface Waves (MASW) technique in the same locations of refraction profiles to measure the velocity of shear waves (S-waves). SeisImager/2D Software Package was used in the analysis of the measured data. Data processing and interpretation reflect that the subsurface section in the study area consists of two layers, the first layer is a thin surface layer ranges in thickness from 1 to 4 meters with P-wave velocity ranges from 924 m/s to 1247 m/s and S-wave velocity ranges from 530 m/s to 745 m/s. The second layer has a P-wave velocity ranges from 1277 m/s to 1573 m/s and the S-wave velocity ranges from 684 m/s to 853 m/s. Geotechnical parameters were calculated for both layers. Since elastic moduli such as Poisson’s ratio, shear modulus, Young’s modulus, and bulk’s modulus were calculated. Competence scales such as material index, stress ratio, concentration index, and density gradient were calculated also. In addition, the ultimate and allowable bearing capacities

Neutron Resonance Parameters of 90Zr below 300 KeV

1977 ◽  
Vol 30 (4) ◽  
pp. 379 ◽  
Author(s):  
AR de L Musgrove ◽  
JA Harvey ◽  
WM Good
Keyword(s):  
Linear Accelerator ◽  
P Wave ◽  
Neutron Resonance ◽  
S Wave ◽  
Electron Linear Accelerator ◽  
Resonance Parameters ◽  
Oak Ridge ◽  
Wave Parameters ◽  
Transmission Data ◽  
Bombarding Energy

Transmission data taken at both the 80 and 200 m stations of the Oak Ridge Electron Linear Accelerator have provided resonance parameters for 90Zr to 300 keV bombarding energy. The average s-wave parameters obtained were: <D> = 8�6�1�6keY and So = (0�S4�0�14)x10-4 , while the average parameters for the two p-wave sequences were: for P1/2, <D> = 8�6 � 1 . 6 keY and SI = (3�9�1�0)x 10-4 ; for P3/2, <D> =o4�3�0�S keY and SI = (4�3�0�8)x 10-4 ?

Circularly polarized shear waves used for VSP measurements

Geophysics ◽  
1992 ◽  
Vol 57 (4) ◽  
pp. 643-646
Author(s):  
Hans A. K. Edelmann
Keyword(s):  
Shear Wave ◽  
Velocity Ratio ◽  
Shear Waves ◽  
Signal Amplitude ◽  
P Wave ◽  
Polarization Analysis ◽  
S Wave ◽  
P Waves ◽  
Additional Processing ◽  
Wave Recording

If shear waves are to be recorded, all other types of waves (including P waves) have to be regarded as noise. All data processing applied later is limited in its success, not so much by the character of the signal, but by the character of the noise superimposed on the signal. Therefore an optimum method for simultaneous P‐ and S‐wave recording does not exist per se. All efforts made in the field that help to enhance the relatively weak S‐wave signal enhance the possibility of a more detailed interpretation such as polarization analysis. In the course of shear‐wave investigations over a period of more than ten years, simultaneous P‐ and SV‐wave recording has yielded fairly good results for velocity ratio determination, but has never produced satisfying results for polarization analysis because of the interfering P‐wave events. When generating pure SH‐waves, however, P‐wave arrival amplitudes in a shot record can, under favorable conditions, be kept well below the SH‐wave amplitude (−40 dB). Through additional processing, a ratio of P‐ to SH‐signal amplitude of −60 dB can be reached. The improvement achieved by making separate shear‐wave recordings, obviously, must be weighed against the additional costs caused by these recordings.

Attenuation anisotropy in the linear-slip model: Interpretation of physical modeling data

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. WB165-WB176 ◽  
Author(s):  
Tatiana Chichinina ◽  
Irina Obolentseva ◽  
Leonid Gik ◽  
Boris Bobrov ◽  
Gerardo Ronquillo-Jarillo
Keyword(s):  
Symmetry Axis ◽  
Wave Attenuation ◽  
Joint Inversion ◽  
P Wave ◽  
Slip Model ◽  
S Wave ◽  
The Real ◽  
Complex Valued ◽  
Stack Model ◽  
Linear Slip Model

This study attempts to validate a mathematical formalism of introducing attenuation into Schoenberg’s linear slip model. This formalism is based on replacing the real-valued weaknesses by complex-valued ones. During an ultrasonic experiment, performed at a central frequency of [Formula: see text] on a plate-stack model with [Formula: see text]-thick Plexiglas™ plates, the velocity and attenuation (inverse of the quality factor [Formula: see text]) of P-, SH-, and SV-waves are measured in directions from 25° to 90° with the symmetry axis for dry and oil-saturated models and loading uniaxial pressures of 2 and [Formula: see text]. The velocity and attenuation data are fitted by the derived theoretical functions. The values of the real and imaginary parts of the complex-valued weaknesses are estimated. Thereal parts of the weaknesses, which have a clear physical meaning (they affect the weakening of the material), are three times larger for the dry model than for the oil-saturated one. The imaginary parts of the weaknesses are responsible for attenuation; their values are an order of magnitude smaller than the real parts. The derived expressions for angle-dependent velocities and attenuations can be used to distinguish between dry and oil-saturated fractures. In particular, the P-wave attenuation function in the symmetry-axis direction (normal to fracture planes) is different in dry and saturated media. The experiment shows that the plate-stack model is inhomogeneous because of the nonuniform pressure distribution, which degrades the experimental results and creates difficulties in the inversion for the complex-valued weaknesses — particularly in joint inversion of P- and S-wave data.

scholarly journals Numeric Simulation of Acoustic-Logging of Cave Formations

Energies ◽  
2020 ◽  
Vol 13 (15) ◽  
pp. 3908
Author(s):  
Fanghui Xu ◽  
Zhuwen Wang
Keyword(s):  
Excitation Frequency ◽  
Scattered Wave ◽  
P Wave ◽  
Borehole Wall ◽  
Stoneley Wave ◽  
S Wave ◽  
P Waves ◽  
Acoustic Logging ◽  
Source Distance ◽  
First Arrival

The finite difference (FD) method of monopole source is used to simulate the response of full-wave acoustic-logging in cave formations. The effect of the cave in the formation of borehole full-waves was studied. The results show that the radius of cave is not only linearly related to the first arrival of the compressional wave (P-wave), but also to the energy of the shear wave (S-wave). The converted S (S–S wave) and P-waves (S–P wave) are formed when the S-wave encounters the cave. If the source distance is small, the S–S and S–P waves are not separated, and the attenuation of the S-wave is not large, due to superposition of the converted waves. The S–P wave has been separated from the S-wave when the source distance is large, so the attenuation of the S-wave increases. The amplitude of the P and S–waves changes most when the distance of the cave to the borehole wall reaches a certain value; this value is related to the excitation frequency. The amplitude of the Stoneley wave (ST wave) varies directly with the radius of cave. If the radius of the cave is large, the energy of ST wave is weak. The scattered wave is determined by the radius and position of the cave. The investigation depth of a monopole source is limited. When the distance of the cave to the borehole wall exceeds the maximum investigation depth, the borehole acoustic wave is little affected by the cave. In actual logging, the development of the cave can be evaluated by using the first arrival of the P-wave and the energy of the S and ST waves.

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