SV‐wave velocities from P‐P and P‐SV data for transversely isotropic solids

Geophysics ◽  
1989 ◽  
Vol 54 (10) ◽  
pp. 1336-1338 ◽  
Author(s):  
Franklyn K. Levin
Keyword(s):  
Transversely Isotropic ◽  
Velocity Fields ◽  
P Wave ◽  
Square Root ◽  
Wave Velocities ◽  
Reflection Data ◽  
Good For ◽  
Transversely Isotropic Solids ◽  
Isotropic Solids

Tessmer and Behle (1988) have shown that for P‐SV reflections from subsurfaces consisting of beds of isotropic solids separated by horizontal interfaces, the stacking velocity found for P‐SV reflection data is approximately the square root of the product of the P‐wave and SV‐wave stacking velocities from that reflection, i.e., [Formula: see text]. Even though P‐SV traveltime curves are not hyperbolic, the approximation is surprisingly good for the velocity fields considered by Tessmer and Behle. Given P‐wave and P‐SV stacking velocities, a reasonably accurate SV‐wave stacking velocity could be computed.

On: “Weak elastic anisotropy,” by L. Thomsen (GEOPHYSICS, 52, 1954–1966, October, 1986).

Geophysics ◽  
1988 ◽  
Vol 53 (4) ◽  
pp. 558-559 ◽  
Author(s):  
Franklyn K. Levin
Keyword(s):  
Elastic Constants ◽  
Vertical Velocity ◽  
Elastic Waves ◽  
Elastic Anisotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
Sh Waves ◽  
The Individual ◽  
Transversely Isotropic Solids ◽  
Isotropic Solids

In a paper whose importance seems to have escaped notice, Thomsen (1) derived equations that give the moveout velocities of P, SV, and SH-waves when solids are weakly transversely isotropic and (2) tabulated experimentally determined elastic constants for a large number of rocks, crystals, and a few other solids. For rocks, one of the constants, delta, differed from zero by as much as 0.73 and −0.27. Delta is the fraction by which P-wave moveout velocity deviates from the vertical velocity [Thomsen’s equation (27a)]. Although some deltas indicated deviations from the vertical velocity smaller than 1 or 2 percent, most were larger and positive. Until the publication of Thomsen’s data, most of us concerned with elastic waves traveling in earth sections that act as transversely isotropic solids because the sections consist of thin beds had assumed the individual beds were isotropic solids, all with the same Poisson’s ratios. That assumption results in a zero value for delta and a moveout velocity equal to the vertical velocity. The validity of the assumption is now doubtful.

Seismic velocities in transversely isotropic media, II

Geophysics ◽  
1980 ◽  
Vol 45 (1) ◽  
pp. 3-17 ◽  
Author(s):  
Franklyn K. Levin
Keyword(s):  
Wave Velocity ◽  
Transversely Isotropic ◽  
P Wave ◽  
Seismic Velocities ◽  
Sh Wave ◽  
S Wave ◽  
Wave Velocities ◽  
P Wave Velocity ◽  
Surface Spread ◽  
Isotropic Solids

P‐wave, SV‐wave, and SH‐wave velocities are computed for transversely isotropic solids formed from two isotropic solids. The combinations are shale‐sandstone and shale‐limestone solids of an earlier paper (Levin, 1979), but one velocity of the nonshale component is allowed to vary over the range of Poisson’s ratios σ = 0 to σ = 0.45, i.e., from a rigid solid to a near‐liquid. When the S‐wave velocity of either the sandstone or limestone is varied, the ratio of horizontal P‐wave velocity to vertical P‐wave velocity goes through a maximum as σ increases and subsequently falls to values less than unity as σ approaches 0.5. The P‐wave velocity that would be found with a short surface spread also goes through a maximum and, at σ = 0.5, is less than the P‐wave velocity of either isotropic component. SV‐wave velocities found for data from a short spread are unreasonably large; SH‐wave velocities decrease monotonically as σ increases, but the ratio of horizontal SH‐wave velocity to vertical SH‐wave velocity goes through a minimum of unity.

Near-Tip Fields for Penny-Shaped Cracks in Magnetoelectroelastic Media

2006 ◽  
Vol 312 ◽  
pp. 41-46 ◽  
Author(s):  
Bao Lin Wang ◽  
Yiu Wing Mai
Keyword(s):  
Closed Form ◽  
Transversely Isotropic ◽  
Crack Plane ◽  
Axially Symmetric ◽  
Crack Configuration ◽  
Penny Shaped Crack ◽  
Electric And Magnetic Fields ◽  
Shaped Crack ◽  
Transversely Isotropic Solids ◽  
Isotropic Solids

This paper solves the penny-shaped crack configuration in transversely isotropic solids with coupled magneto-electro-elastic properties. The crack plane is coincident with the plane of symmetry such that the resulting elastic, electric and magnetic fields are axially symmetric. The mechanical, electrical and magnetical loads are considered separately. Closed-form expressions for the stresses, electric displacements, and magnetic inductions near the crack frontier are given.

Anisotropic geometrical-spreading correction for wide-azimuth P-wave reflections

Geophysics ◽  
2006 ◽  
Vol 71 (5) ◽  
pp. D161-D170 ◽  
Author(s):  
Xiaoxia Xu ◽  
Ilya Tsvankin
Keyword(s):  
Transversely Isotropic ◽  
Azimuthal Anisotropy ◽  
P Wave ◽  
Fractured Reservoir ◽  
Geometrical Spreading ◽  
Spreading Factor ◽  
Velocity Anisotropy ◽  
Reflection Data ◽  
Fitting Procedure ◽  
Avo Attributes

Compensation for geometrical spreading along a raypath is one of the key steps in AVO (amplitude-variation-with-offset) analysis, in particular, for wide-azimuth surveys. Here, we propose an efficient methodology to correct long-spread, wide-azimuth reflection data for geometrical spreading in stratified azimuthally anisotropic media. The P-wave geometrical-spreading factor is expressed through the reflection traveltime described by a nonhyperbolic moveout equation that has the same form as in VTI (transversely isotropic with a vertical symmetry axis) media. The adapted VTI equation is parameterized by the normal-moveout (NMO) ellipse and the azimuthally varying anellipticity parameter [Formula: see text]. To estimate the moveout parameters, we apply a 3D nonhyperbolic semblance algorithm of Vasconcelos and Tsvankin that operates simultaneously with traces at all offsets andazimuths. The estimated moveout parameters are used as the input in our geometrical-spreading computation. Numerical tests for models composed of orthorhombic layers with strong, depth-varying velocity anisotropy confirm the high accuracy of our travetime-fitting procedure and, therefore, of the geometrical-spreading correction. Because our algorithm is based entirely on the kinematics of reflection arrivals, it can be incorporated readily into the processing flow of azimuthal AVO analysis. In combination with the nonhyperbolic moveout inversion, we apply our method to wide-azimuth P-wave data collected at the Weyburn field in Canada. The geometrical-spreading factor for the reflection from the top of the fractured reservoir is clearly influenced by azimuthal anisotropy in the overburden, which should cause distortions in the azimuthal AVO attributes. This case study confirms that the azimuthal variation of the geometrical-spreading factor often is comparable to or exceeds that of the reflection coefficient.

Effective elastic properties of rocks with transversely isotropic background permeated by a set of vertical fracture cluster

Geophysics ◽  
2021 ◽  
pp. 1-78
Author(s):  
Da Shuai ◽  
Alexey Stovas ◽  
Jianxin Wei ◽  
Bangrang Di ◽  
Yang Zhao
Keyword(s):  
Standard Deviation ◽  
Elastic Wave ◽  
Significant Influence ◽  
Transversely Isotropic ◽  
Azimuth Angle ◽  
Effective Medium ◽  
P Wave ◽  
Wave Velocities ◽  
Anisotropy Parameters ◽  
Vertical Fractures

The linear slip theory is gradually being used to characterize seismic anisotropy. If the transversely isotropic medium embeds vertical fractures (VFTI medium), the effective medium becomes orthorhombic. The vertical fractures, in reality, may exist in any azimuth angle which leads the effective medium to be monoclinic. We apply the linear slip theory to create a monoclinic medium by only introducing three more physical meaning parameters: the fracture preferred azimuth angle, the fracture azimuth angle, and the angular standard deviation. First, we summarize the effective compliance of a rock as the sum of the background matrix compliance and the fracture excess compliance. Then, we apply the Bond transformation to rotate the fractures to be azimuth dependent, introduce a Gaussian function to describe the fractures' azimuth distribution assuming that the fractures are statistically distributed around the preferred azimuth angle, and average each fracture excess compliance over azimuth. The numerical examples investigate the influence of the fracture azimuth distribution domain and angular standard deviation on the effective stiffness coefficients, elastic wave velocities, and anisotropy parameters. Our results show that the fracture cluster parameters have a significant influence on the elastic wave velocities. The fracture azimuth distribution domain and angular standard deviation have a bigger influence on the orthorhombic anisotropy parameters in the ( x2, x3) plane than that in the ( x1, x3) plane. The fracture azimuth distribution domain and angular standard deviation have little influence on the monoclinic anisotropy parameters responsible for the P-wave NMO ellipse and have a significant influence on the monoclinic anisotropy parameters responsible for the S1- and S2-wave NMO ellipse. The effective monoclinic can be degenerated into the VFTI medium.

True bounds on Poisson's ratios for transversely isotropic solids

1992 ◽  
Vol 27 (1) ◽  
pp. 43-44 ◽  
Author(s):  
P S Theocaris ◽  
T P Philippidis
Keyword(s):  
Energy Density ◽  
Basic Principle ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Elastic Solid ◽  
Transversely Isotropic ◽  
Linear Elastic ◽  
Positive Strain ◽  
Transversely Isotropic Solids ◽  
Isotropic Solids

The basic principle of positive strain energy density of an anisotropic linear or non-linear elastic solid imposes bounds on the values of the stiffness and compliance tensor components. Although rational mathematical structuring of valid intervals for these components is possible and relatively simple, there are mathematical procedures less strictly followed by previous authors, which led to an overestimation of the bounds and misinterpretation of experimental results.

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