Weak anisotropy-attenuation parameters

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. WB203-WB213 ◽  
Author(s):  
Václav Vavryčuk
Keyword(s):  
Quality Factor ◽  
Propagation Velocity ◽  
Elastic Anisotropy ◽  
Polarization Vector ◽  
Linear Functions ◽  
Weak Anisotropy ◽  
Simple Modification ◽  
Velocity Anisotropy ◽  
Slowness Vector ◽  
Lower Accuracy

Velocity anisotropy and attenuation in weakly anisotropic and weakly attenuating structures can be treated uniformly using weak anisotropy-attenuation (WAA) parameters. The WAA parameters are constructed in a way analogous to weak anisotropy (WA) parameters designed for weak elastic anisotropy. The WAA parameters generalize WA parameters by incorporating attenuation effects. They can be represented alternatively by one set of complex values or by two sets of real values. Assuming high-frequency waves and using the first-order perturbation theory, all basic wave quantities such as the slowness vector, the polarization vector, propagation velocity, attenuation, and the quality factor are linear functions of WAA parameters. Numerical modeling shows that perturbation equations have different accuracy for different wave quantities. The propagation velocity usually is calculated with high accuracy. However, the attenuation and quality factor can be reproduced with appreciably lower accuracy. This happens mostly when the strength of velocity anisotropy is higher than 10% and attenuation is moderate or weak [Formula: see text]. In this case, the errors of the attenuation or [Formula: see text]-factor can attain values comparable to the strength of anisotropy or even higher. A simple modification of the equations by including some higher-order perturbations improves accuracy by three to four times.

Reliability of the slowness and slowness-polarization methods for anisotropy estimation in VTI media from 3C walkaway VSP data

Geophysics ◽  
2013 ◽  
Vol 78 (5) ◽  
pp. WC93-WC102 ◽  
Author(s):  
Mehdi Asgharzadeh ◽  
Andrej Bóna ◽  
Roman Pevzner ◽  
Milovan Urosevic ◽  
Boris Gurevich
Keyword(s):  
Polarization Vector ◽  
Estimation Errors ◽  
Local Anisotropy ◽  
Weak Anisotropy ◽  
Polarization Measurements ◽  
Slowness Vector ◽  
Wide Range ◽  
Walkaway Vsp ◽  
Anisotropy Parameters ◽  
Vti Media

We studied the validity of qP-wave slowness and slowness-polarization methods for estimating local anisotropy parameters in transversely isotropic (TI) media by quantifying the estimation errors in a numerical exercise. We generated numerical slownesses and polarizations over two aperture ranges corresponding to a short offset walkaway vertical seismic profiling (VSP) and a long offset walkaway VSP for a range of TI models with vertical axis of symmetry (VTI). Synthetic data are equisampled over the phase angle range and contaminated with Gaussian noise. We inverted the data and compared the anisotropy parameters of the optimal model with the true model. We found that the selection of a proper methodology for VTI parameter estimation based on walkaway VSP measurements was mostly dependent on our ability to accurately estimate either horizontal components of qP-wave slowness vector or the polarization vector. With data contaminated with noise, methods that include the horizontal component of the slowness vector had greater accuracy than the methods that replace this information with polarization measurements. The estimations are particularly accurate when a wide range of propagation angle was available. For short offsets, only parameter [Formula: see text] could be reliably estimated. In the absence of long offsets, depending on the accuracy of polarization measurements, the method based on the weak anisotropy approximation for qP-wave velocity in VTI media or the method based on slowness and polarization vectors could be used to estimate [Formula: see text] and [Formula: see text]. If the horizontal components of the slowness vector were not available (a heterogeneous overburden), we used methods that were based on local measurements of the polarization vector. We found that, with accurate measurements of the polarization vector, the method based on exact relationship between vertical slowness and polarization dip could be used to estimate VTI parameters even for the cases in which the wide offset range was not available.

Velocity, attenuation, and quality factor in anisotropic viscoelastic media: A perturbation approach

Geophysics ◽  
2008 ◽  
Vol 73 (5) ◽  
pp. D63-D73 ◽  
Author(s):  
Václav Vavryčuk
Keyword(s):  
Elastic Medium ◽  
Order Approximation ◽  
Linear Functions ◽  
Perturbation Approach ◽  
Nonlinear Inversion ◽  
Slowness Surface ◽  
Simple Modification ◽  
First Order ◽  
Slowness Vector ◽  
Velocity Attenuation

Velocity, attenuation, and the quality [Formula: see text] factor of waves propagating in homogeneous media of arbitrary anisotropy and attenuation strength are calculated in high-frequency asymptotics using a stationary slowness vector, the vector evaluated at the stationary point of the slowness surface. This vector is generally complex-valued and inhomogeneous, meaning that the real and imaginary parts of the vector have different directions. The slowness vector can be determined by solving three coupled polynomial equations of the sixth order or by a nonlinear inversion. The procedure is simplified if perturbation theory is applied. The elastic medium is viewed as a background medium, and the attenuation effects are incorporated as perturbations. In the first-order approximation, the phase and ray velocities and their directions remain unchanged, being the same as those in the background elastic medium. The perturbation of the slowness vector is calculated by solving a system of three linear equations. The phase attenuation and phase [Formula: see text]-factor are linear functions of the perturbation of the slowness vector. Calculating the ray attenuation and ray [Formula: see text]-factor is even simpler than calculating the phase quantities because they are expressed in terms of perturbations of the medium without the need to evaluate the perturbation of the slowness vector. Numerical modeling indicates that the perturbations are highly accurate; the errors are less than 0.3% for a medium with a [Formula: see text]-factor of 20 or higher. The accuracy can be enhanced further by a simple modification of the first-order perturbation formulas.

Quartic moveout coefficient for a dipping azimuthally anisotropic layer

Geophysics ◽  
2004 ◽  
Vol 69 (3) ◽  
pp. 699-707 ◽  
Author(s):  
Andrés Pech ◽  
Ilya Tsvankin
Keyword(s):  
Reservoir Characterization ◽  
Fractured Reservoirs ◽  
Symmetry Plane ◽  
Exact Expression ◽  
Naturally Fractured Reservoirs ◽  
P Wave ◽  
Orthorhombic Symmetry ◽  
Fracture Density ◽  
Weak Anisotropy ◽  
Velocity Anisotropy

Interpretation and inversion of azimuthally varying nonhyperbolic reflection moveout requires accounting for both velocity anisotropy and subsurface structure. Here, our previously derived exact expression for the quartic moveout coefficient A4 is applied to P‐wave reflections from a dipping interface overlaid by a medium of orthorhombic symmetry. The weak‐anisotropy approximaton for the coefficient A4 in a homogeneous orthorhombic layer is controlled by the anellipticity parameters η(1), η(2), and η(3), which are responsible for time processing of P‐wave data. If the dip plane of the reflector coincides with the vertical symmetry plane [x1, x3], A4 on the dip line is proportional to the in‐plane anellipticity parameter η(2) and always changes sign for a dip of 30○. The quartic coefficient on the strike line is a function of all three η–parameters, but for mild dips it is mostly governed by η(1)—the parameter defined in the incidence plane [x2, x3]. Whereas the magnitude of the dip line A4 typically becomes small for dips exceeding 45○, the nonhyperbolic moveout on the strike line may remain significant even for subvertical reflectors. The character of the azimuthal variation of A4 depends on reflector dip and is quite sensitive to the signs and relative magnitudes of η(1), η(2), and η(3). The analytic results and numerical modeling show that the azimuthal pattern of the quartic coefficient can contain multiple lobes, with one or two azimuths of vanishing A4 between the dip and strike directions. The strong influence of the anellipticity parameters on the azimuthally varying coefficient A4 suggests that nonhyperbolic moveout recorded in wide‐azimuth surveys can help to constrain the anisotropic velocity field. Since for typical orthorhombic models that describe naturally fractured reservoirs the parameters η(1,2,3) are closely related to the fracture density and infill, the results of azimuthal nonhyperbolic moveout analysis can also be used in reservoir characterization.

Elastic velocity anisotropy in the presence of an anisotropic initial stress

1972 ◽  
Vol 62 (5) ◽  
pp. 1183-1193 ◽  
Author(s):  
F. A. Dahlen
Keyword(s):  
Initial Stress ◽  
Elastic Anisotropy ◽  
Body Wave ◽  
Homogeneous Medium ◽  
Body Waves ◽  
P Waves ◽  
S Waves ◽  
Velocity Anisotropy ◽  
First Order ◽  
Wave Velocities

Abstract The effect of a homogeneous anisotropic initial stress on the propagation of infinitesimal amplitude elastic body waves in a perfectly elastic, homogeneous medium is investigated. If the medium is inherently isotropic in the reference configuration and if the magnitude τ0 of the deviatoric part of the initial static stress is small compared to the rigidity μ of the medium, then the apparent body-wave velocities of P waves are unaffected by the initial stress to first order in τ0/μ. The apparent body-wave velocities of S waves are rendered anisotropic to first order, and this effect is described explicitly. It is concluded that the direct effect of an anisotropic initial stress cannot contribute appreciably to the observed velocity anisotropy of horizontally propagating P waves in the oceanic upper mantle. Those observations require an inherent elastic anisotropy of the oceanic uppermantle material.

Weak elastic anisotropy

Geophysics ◽  
1986 ◽  
Vol 51 (10) ◽  
pp. 1954-1966 ◽  
Author(s):  
Leon Thomsen
Keyword(s):  
Critical Parameter ◽  
Elastic Anisotropy ◽  
Horizontal Velocity ◽  
Elastic Media ◽  
Strong Anisotropy ◽  
Elastic Parameters ◽  
Weak Anisotropy ◽  
Anisotropic Parameter

Most bulk elastic media are weakly anisotropic. The equations governing weak anisotropy are much simpler than those governing strong anisotropy, and they are much easier to grasp intuitively. These equations indicate that a certain anisotropic parameter (denoted δ) controls most anisotropic phenomena of importance in exploration geophysics, some of which are nonnegligible even when the anisotropy is weak. The critical parameter δ is an awkward combination of elastic parameters, a combination which is totally independent of horizontal velocity and which may be either positive or negative in natural contexts.

Ray velocity and ray attenuation in homogeneous anisotropic viscoelastic media

Geophysics ◽  
2007 ◽  
Vol 72 (6) ◽  
pp. D119-D127 ◽  
Author(s):  
Václav Vavryčuk
Keyword(s):  
High Frequency ◽  
Sedimentary Rocks ◽  
Strong Anisotropy ◽  
Frequency Signal ◽  
High Frequency Signal ◽  
Velocity Anisotropy ◽  
Viscoelastic Media ◽  
Slowness Vector ◽  
Complex Valued ◽  
Unit Vectors

Asymptotic wave quantities such as ray velocity and ray attenuation are calculated in anisotropic viscoelastic media by using a stationary slowness vector. This vector generally is complex valued and inhomogeneous, and it predicts the complex energy velocity parallel to a ray. To compute the stationary slowness vector, one must find two independent, real-valued unit vectors that specify the directions of its real and imaginary parts. The slowness-vector inhomogeneity affects asymptotic wave quantities and complicates their computation. The critical quantities are attenuation and quality factor ([Formula: see text]-factor); these can vary significantly with the slowness-vector inhomogeneity. If the inhomogeneity is neglected, the attenuation and the [Formula: see text]-factor can be distorted distinctly by errors commensurate to the strength of the velocity anisotropy — as much as tens of percent for sedimentary rocks. The distortion applies to strongly as well as to weakly attenuative media. On the contrary, the ray velocity, which defines the wavefronts and physically corresponds to the energy velocity of a high-frequency signal propagating along a ray, is almost insensitive to the slowness-vector inhomogeneity. Hence, wavefronts can be calculated in a simplified way except for media with extremely strong anisotropy and attenuation.

Pyrolysis-induced P-wave velocity anisotropy in organic-rich shales

Geophysics ◽  
2014 ◽  
Vol 79 (2) ◽  
pp. D41-D53 ◽  
Author(s):  
Adam M. Allan ◽  
Tiziana Vanorio ◽  
Jeremy E. P. Dahl
Keyword(s):  
Wave Velocity ◽  
Elastic Anisotropy ◽  
Rock Physics ◽  
Confining Pressure ◽  
Source Rocks ◽  
Acoustic Properties ◽  
P Wave ◽  
Velocity Anisotropy ◽  
Geochemical Parameters ◽  
P Wave Velocity

The sources of elastic anisotropy in organic-rich shale and their relative contribution therein remain poorly understood in the rock-physics literature. Given the importance of organic-rich shale as source rocks and unconventional reservoirs, it is imperative that a thorough understanding of shale rock physics is developed. We made a first attempt at establishing cause-and-effect relationships between geochemical parameters and microstructure/rock physics as organic-rich shales thermally mature. To minimize auxiliary effects, e.g., mineralogical variations among samples, we studied the induced evolution of three pairs of vertical and horizontal shale plugs through dry pyrolysis experiments in lieu of traditional samples from a range of in situ thermal maturities. The sensitivity of P-wave velocity to pressure showed a significant increase post-pyrolysis indicating the development of considerable soft porosity, e.g., microcracks. Time-lapse, high-resolution backscattered electron-scanning electron microscope images complemented this analysis through the identification of extensive microcracking within and proximally to kerogen bodies. As a result of the extensive microcracking, the P-wave velocity anisotropy, as defined by the Thomsen parameter epsilon, increased by up to 0.60 at low confining pressures. Additionally, the degree of microcracking was shown to increase as a function of the hydrocarbon generative potential of each shale. At 50 MPa confining pressure, P-wave anisotropy values increased by 0.29–0.35 over those measured at the baseline — i.e., the immature window. The increase in anisotropy at high confining pressure may indicate a source of anisotropy in addition to microcracking — potentially clay mineralogical transformation or the development of intrinsic anisotropy in the organic matter through aromatization. Furthermore, the evolution of acoustic properties and microstructure upon further pyrolysis to the dry-gas window was shown to be negligible.

Microcrack‐induced versus intrinsic elastic anisotropy in mature HC‐source shales

Geophysics ◽  
1993 ◽  
Vol 58 (11) ◽  
pp. 1703-1706 ◽  
Author(s):  
Lev Vernik
Keyword(s):  
Elastic Anisotropy ◽  
Crack Density ◽  
Experimental Studies ◽  
Source Rocks ◽  
Hydrocarbon Generation ◽  
Clay Particles ◽  
Velocity Anisotropy ◽  
Vsp Data ◽  
Intrinsic Anisotropy

Recent experimental studies of ultrasonic velocity anisotropy in kerogen‐rich shales indicate that these rocks are characterized by a very strong anisotropic response related to a very fine, bedding‐parallel lamination of organic matter and preferred orientation of clay particles in the rock matrix (Vernik and Nur, 1992). This intrinsic anisotropy is further enhanced in thermally mature shales by bedding‐parallel microcracks caused by the processes of hydrocarbon generation (Vernik, paper in preparation). However, the potential of recognizing mature source‐rocks in situ using downhole sonic, crosshole seismic, or VSP data clearly depends on our ability to discriminate between these two major causes of elastic anisotropy, remove the effect of the intrinsic anisotropy, and estimate crack density and crack porosity from velocity measurements.

Complete phenomenon of reflection at the plane boundary of a dissipative anisotropic elastic medium

2020 ◽  
Vol 224 (2) ◽  
pp. 1015-1027
Author(s):  
M D Sharma ◽  
Suman Nain
Keyword(s):  
Elastic Medium ◽  
Propagation Velocity ◽  
Sagittal Plane ◽  
Amplitude Ratio ◽  
Plane Waves ◽  
Plane Boundary ◽  
Dual Vector ◽  
Inhomogeneous Waves ◽  
Slowness Vector ◽  
Anisotropic Elastic

SUMMARY A complex slowness vector governs the 3-D propagation of harmonic plane waves in a dissipative elastic medium with general anisotropy. In any sagittal plane, this dual vector is specified with phase direction, propagation velocity and coefficients for attenuation. A generalized reflection phenomenon is illustrated for incidence of inhomogeneous waves at the stress free boundary of the medium. Each reflected wave at the boundary is characterized by its propagation direction, propagation velocity, inhomogeneity, amplitude ratio, phase shift and energy flux. These propagation characteristics are exhibited graphically for a numerical example of anisotropic viscoelastic medium.

Polarization, phase velocity, and NMO velocity of qP-waves in arbitrary weakly anisotropic media

Geophysics ◽  
1998 ◽  
Vol 63 (5) ◽  
pp. 1754-1766 ◽  
Author(s):  
Ivan Pšenčk ◽  
Dirk Gajewski
Keyword(s):  
Phase Velocity ◽  
Polarization Vector ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Vertical Axis ◽  
Natural Generalization ◽  
Weak Anisotropy ◽  
Isotropic Media ◽  
Axis Of Symmetry ◽  
Approximate Formulas

We present approximate formulas for the qP-wave phase velocity, polarization vector, and normal moveout velocity in an arbitrary weakly anisotropic medium obtained with first‐order perturbation theory. All these quantities are expressed in terms of weak anisotropy (WA) parameters, which represent a natural generalization of parameters introduced by Thomsen. The formulas presented and the WA parameters have properties of Thomsen’s formulas and parameters: (1) the approximate equations are considerably simpler than exact equations for qP-waves, (2) the WA parameters are nondimensional quantities, and (3) in isotropic media, the WA parameters are zero and the corresponding equations reduce to equations for isotropic media. In contrast to Thomsen’s parameters, the WA parameters are related linearly to the density normalized elastic parameters. For the transversely isotropic media with vertical axis of symmetry, the equations presented and the WA parameters reduce to the equations and linearized parameters of Thomsen. The accuracy of the formulas presented is tested on two examples of anisotropic media with relatively strong anisotropy: on a transversely isotropic medium with the horizontal axis of symmetry and on a medium with triclinic anisotropy. Although anisotropy is rather strong, the approximate formulas presented yield satisfactory results.

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