Focal mechanisms produced by shear faulting in weakly transversely isotropic crustal rocks

Geophysics ◽  
2006 ◽  
Vol 71 (5) ◽  
pp. D145-D151 ◽  
Author(s):  
Václav Vavryčuk
Keyword(s):  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Focal Mechanisms ◽  
Crustal Rocks ◽  
Linear Vector ◽  
Double Couple ◽  
Anisotropic Rocks ◽  
Anisotropy Parameters ◽  
Anisotropy Strength

Shear faulting in anisotropic rocks produces non-double-couple (non-DC) mechanisms. The non-DC mechanisms can comprise the isotropic (ISO) and compensated linear vector dipole (CLVD) components. The formulas for percentages of the ISO and CLVD are simplified under the assumption of weak transverse isotropy and can be expressed advantageously in terms of Thomsen’s anisotropy parameters. Shear faulting in crustal rocks with anisotropy strength of 10% can produce an ISO of up to 10% and a CLVD of up to 30%. Such values are significant and detectable in carefully determined focal mechanisms.

Analytic study of the effective parameters for determination of the NMO velocity function in transversely isotropic media

Geophysics ◽  
1997 ◽  
Vol 62 (6) ◽  
pp. 1855-1866 ◽  
Author(s):  
Jack K. Cohen
Keyword(s):  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
Velocity Function ◽  
Transversely Isotropic Media ◽  
Wave Phase Velocity ◽  
Isotropic Media ◽  
Anisotropy Parameters ◽  
Vti Media ◽  
Ray Parameter

In their studies of transversely isotropic media with a vertical symmetry axis (VTI media), Alkhalifah and Tsvankin observed that, to a high numerical accuracy, the normal moveout (NMO) velocity for dipping reflectors as a function of ray parameter p depends mainly on just two parameters, each of which can be determined from surface P‐wave observations. They substantiated this result by using the weak‐anisotropy approximation and exploited it to develop a time‐domain processing sequence that takes into account vertical transverse isotropy. In this study, the two‐parameter Alkhalifah‐Tsvankin result was further examined analytically. It was found that although there is (as these authors already observed) some dependence on the remaining parameters of the problem, this dependence is weak, especially in the practically important regimes of weak to moderately strong transverse isotropy and small ray parameter. In each of these regimes, an analytic solution is derived for the anisotropy parameter η required for time‐domain P‐wave imaging in VTI media. In the case of elliptical anisotropy (η = 0), NMO velocity expressed through p is fully controlled just by the zero‐dip NMO velocity—one of the Alkhalifah‐ Tsvankin parameters. The two‐parameter representation of NMO velocity also was shown to be exact in another limit—that of the zero shear‐wave vertical velociy. The analytic results derived here are based on new representations for both the P‐wave phase velocity and normal moveout velocity in terms of the ray parameter, with explicit expressions given for the cases of vanishing onaxis shear speed, weak to moderate transverse isotropy, and small to moderate ray parameter. Using these formulas, I have rederived and, in some cases, extended in a uniform manner various results of Tsvankin, Alkhalifah, and others. Examples include second‐order expansions in the anisotropy parameters for both the P‐wave phase‐velocity function and NMO‐velocity function, as well as expansions in powers of the ray parameter for both of these functions. I have checked these expansions against the corresponding exact functions for several choices of the anisotropy parameters.

A crack-induced stress approach to describe the tensile strength of transversely isotropic rocks

2002 ◽  
Vol 39 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Li Li ◽  
Michel Aubertin
Keyword(s):  
Tensile Strength ◽  
Transverse Isotropy ◽  
Local Stress ◽  
Transversely Isotropic ◽  
Bedding Plane ◽  
Tensile Failure ◽  
Induced Stress ◽  
Anisotropic Rocks ◽  
Indirect Tensile ◽  
Boundary Equations

Rocks are generally more or less anisotropic, depending on their structure at the scale of interest. In engineering applications, the magnitude of such anisotropy must often be determined for compressive as well as tensile loading conditions. In this paper, the authors present the results of an investigation on tensile failure of transversely isotropic rocks, based on Inglis' analytical solution for the stress at the boundary of an elliptical flaw. The strength of transversely isotropic rocks is assumed to be controlled by the maximum tensile local stress along the crack boundary. Equations are developed and compared with tensile test data taken from the literature. The results show that the proposed formulations represent well the direct and indirect tensile strength of anisotropic rocks as a function of bedding plane orientation. It is also shown that the proposed physical model correlates well with the results obtained from more empirical formulations.Key words: rock mechanics, anisotropy, transverse isotropy, tensile strength, Brazilian test, crack.

Seismic shear wave anisotropy of an anisotropic rock containing aligned cracks: theory and applications to experiment and field data

2019 ◽  
Vol 220 (1) ◽  
pp. 404-414
Author(s):  
Song Xu ◽  
Xiaoming Tang ◽  
Yuanda Su ◽  
Chunxi Zhuang
Keyword(s):  
Field Data ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Seismic Wave Propagation ◽  
Anisotropic Rock ◽  
Crustal Rocks ◽  
Acoustic Anisotropy ◽  
Seismic Shear ◽  
Realistic Situation ◽  
Anisotropy Measurement

SUMMARY Cracks universally exist in Earth's crustal rocks. Many rocks are intrinsically anisotropic, which, when coupled with crack-induced anisotropy, significantly affect seismic wave propagation through the rocks. Using the method of sphere equivalency of effective scattering, we have developed a technique to model the effective moduli of transversely isotropic (TI) media containing cracks. The modelling results show that the wave characteristics are significantly affected by the interaction of the two anisotropy mechanisms. To validate the validity and accuracy, the theory was applied to a recent experiment made with a vertical transverse isotropy (VTI) medium containing cracks and shows significantly better agreement with the data. For a more realistic situation, the new modelling was applied to interpret the borehole acoustic anisotropy measurement results from a fractured VTI formation, showing that the theory can adequately explain the anisotropic characteristics of the field data. With the validation and testing, the theoretical results advocated in this study can be used with confidence.

Moment tensors of double-couple microseismic sources in anisotropic formations

Geophysics ◽  
2019 ◽  
Vol 85 (1) ◽  
pp. KS1-KS11 ◽  
Author(s):  
Vladimir Grechka
Keyword(s):  
Hydraulic Fracturing ◽  
Transversely Isotropic ◽  
Focal Mechanisms ◽  
Weak Anisotropy ◽  
Moment Tensors ◽  
Numerical Examples ◽  
Double Couple ◽  
Microseismic Events ◽  
Anisotropy Coefficients

Moment tensors of kinematically double-couple microseismic events triggered in anisotropic formations are known to exhibit non-double-couple focal mechanisms. The weak anisotropy approximation of these mechanisms reveals the combinations of anisotropy coefficients of vertically transversely isotropic and orthorhombic focal regions responsible for the deviations of moment tensors from double couples. Numerical examples for models of typical unconventional shales indicate the non-double-couple components of moment tensors to be sufficiently large to cause misinterpretation of the nature of ruptures associated with hydraulic fracturing.

Seismic traveltime inversion for transverse isotropy

Geophysics ◽  
1990 ◽  
Vol 55 (2) ◽  
pp. 192-200 ◽  
Author(s):  
B. S. Byun ◽  
D. Corrigan
Keyword(s):  
Field Experiments ◽  
Seismic Anisotropy ◽  
Transverse Isotropy ◽  
Quantitative Measure ◽  
Transversely Isotropic ◽  
Velocity Anisotropy ◽  
Additional Information ◽  
Stiffness Coefficients ◽  
Anisotropy Parameters ◽  
Low Degrees

Quantitative measurements of seismic anisotropy can provide a valuable clue to the lithology and degree of stratification in sedimentary rocks with hydrocarbon potential. We present a practical technique for obtaining anisotropy parameters (i.e., five stiffness coefficients A, C, F, L, and M) from seismic traveltime measurements for horizontally layered, transversely isotropic media. The technique is based on the construction of ray‐velocity surfaces in terms of five measurement parameters. An iterative model‐based optimization scheme is then used to invert the traveltime parameters for the five stiffness coefficients in a layer‐stripping mode. Both model and field experiments are performed to demonstrate the feasibility of the method. The model experiment shows that inversion errors (especially in stiffness coefficients A, F, and M) increase with increasing number of layers. Despite these errors, the proposed method does provide a quantitative measure of velocity anisotropy as additional information that cannot be obtained readily from conventional methods. A field VSP data example shows the correlation between the anisotropy parameters and lithology: Chalk and shale exhibited high degrees of anisotropy, and sands showed low degrees of anisotropy.

scholarly journals Estimation of the anisotropy parameters of transversely isotropic shales with a tilted symmetry axis

2012 ◽  
Vol 190 (2) ◽  
pp. 1197-1203 ◽  
Author(s):  
Dariush Nadri ◽  
Joël Sarout ◽  
Andrej Bóna ◽  
David Dewhurst
Keyword(s):  
Symmetry Axis ◽  
Transversely Isotropic ◽  
Anisotropy Parameters

scholarly journals A new traveltime approximation for TI media

Geophysics ◽  
2012 ◽  
Vol 77 (4) ◽  
pp. C37-C42 ◽  
Author(s):  
Alexey Stovas ◽  
Tariq Alkhalifah
Keyword(s):  
Power Series ◽  
Eikonal Equation ◽  
Transversely Isotropic ◽  
Transversely Isotropic Medium ◽  
Efficient Tool ◽  
Trade Off ◽  
Background Medium ◽  
The Common ◽  
Anisotropy Parameters ◽  
Ti Media

In a transversely isotropic (TI) medium, the trade-off between inhomogeneity and anisotropy can dramatically reduce our capability to estimate anisotropy parameters. By expanding the TI eikonal equation in power series in terms of the aneliptic parameter [Formula: see text], we derive an efficient tool to estimate (scan) for [Formula: see text] in a generally inhomogeneous, elliptically anisotropic background medium. For a homogeneous-tilted transversely isotropic medium, we obtain an analytic nonhyperbolic moveout equation that is accurate for large offsets. In the common case where we do not have well information and it is necessary to resolve the vertical velocity, the background medium can be assumed isotropic, and the traveltime equations becomes simpler. In all cases, the accuracy of this new TI traveltime equation exceeds previously published formulations and demonstrates how [Formula: see text] is better resolved at small offsets when the tilt is large.

Wrinkling of a Fiber-Reinforced Membrane

2008 ◽  
Vol 76 (1) ◽  
Author(s):  
E. Shmoylova ◽  
A. Dorfmann
Keyword(s):  
Boundary Conditions ◽  
Energy Function ◽  
Mechanical Response ◽  
Transverse Isotropy ◽  
Base Material ◽  
Strain Energy Function ◽  
Transversely Isotropic ◽  
Incompressible Material ◽  
Fiber Reinforced ◽  
Relaxed Energy

In this paper we investigate the response of fiber-reinforced cylindrical membranes subject to axisymmetric deformations. The membrane is considered as an incompressible material, and the phenomenon of wrinkling is taken into account by means of the relaxed energy function. Two cases are considered: transversely isotropic membranes, characterized by one family of fibers oriented in one direction, and orthotropic membranes, characterized by two family of fibers oriented in orthogonal directions. The strain-energy function is considered as the sum of two terms: The first term is associated with the isotropic properties of the base material, and the second term is used to introduce transverse isotropy or orthotropy in the mechanical response. We determine the mechanical response of the membrane as a function of fiber orientations for given boundary conditions. The objective is to find possible fiber orientations that make the membrane as stiff as possible for the given boundary conditions. Specifically, it is shown that for transversely isotropic membranes a unique fiber orientation exists, which does not affect the mechanical response, i.e., the overall behavior is identical to a nonreinforced membrane.

A least squares method for earthquake mechanism determination using S-wave data

1964 ◽  
Vol 54 (6A) ◽  
pp. 2037-2047
Author(s):  
Agustin Udias
Keyword(s):  
Least Squares ◽  
Statistical Evaluation ◽  
Least Squares Method ◽  
Numerical Approach ◽  
Focal Mechanisms ◽  
Least Square ◽  
S Wave ◽  
Double Couple ◽  
Earthquake Mechanism

abstract In this paper a numerical approach to the determination of focal mechanisms based on the observation of the polarization of the S wave at N stations is presented. Least-square methods are developed for the determination of the orientation of the single and double couple sources. The methods allow a statistical evaluation of the data and of the accuracy of the solutions.

Transverse isotropy of the upper mantle in the vicinity of Pacific fracture zones

1969 ◽  
Vol 59 (1) ◽  
pp. 59-72
Author(s):  
Robert S. Crosson ◽  
Nikolas I. Christensen
Keyword(s):  
Upper Mantle ◽  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Seismic Wave Propagation ◽  
Fracture Zones ◽  
Mantle Material ◽  
Transversely Isotropic Media ◽  
Horizontal Symmetry ◽  
Isotropic Media

Abstract Several recent investigations suggest that portions of the Earth's upper mantle behave anisotropically to seismic wave propagation. Since several types of anisotropy can produce azimuthal variations in Pn velocities, it is of particular geophysical interest to provide a framework for the recognition of the form or forms of anisotropy most likely to be manifest in the upper mantle. In this paper upper mantle material is assumed to possess the elastic properties of transversely isotropic media. Equations are presented which relate azimuthal variations in Pn velocities to the direction and angle of tilt of the symmetry axis of a transversely isotropic upper mantle. It is shown that the velocity data of Raitt and Shor taken near the Mendocino and Molokai fracture zones can be adequately explained by the assumption of transverse isotropy with a nearly horizontal symmetry axis.

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