Weak elastic anisotropy and the tube wave

Geophysics ◽  
1993 ◽  
Vol 58 (8) ◽  
pp. 1091-1098 ◽  
Author(s):  
Andrew N. Norris ◽  
Bikash K. Sinha
Keyword(s):  
Shear Modulus ◽  
Elastic Moduli ◽  
Wave Speed ◽  
Elastic Anisotropy ◽  
Transversely Isotropic ◽  
Anisotropic Formation ◽  
Inversion Procedure ◽  
Anisotropy Parameters ◽  
Tube Wave ◽  
Simple Inversion

Tube‐wave speed in the presence of a weakly anisotropic formation can be expressed in terms of an effective shear modulus for an equivalent isotropic formation. When combined with expressions for the speeds of the SH‐ and quasi‐SV‐waves along the borehole axis, a simple inversion procedure can be obtained to determine three of the five elasticities of a transversely isotropic (TI) formation tilted at some known angle with respect to the borehole axis. Subsequently, a fourth combination of elastic moduli can be estimated from the expression for the qP‐wave speed along the borehole axis. The possibility of determining all five elasticities of a TI formation based on an assumed correlation between two anisotropy parameters is discussed.

scholarly journals Estimating a shear modulus of a transversely isotropic formation

Geophysics ◽  
1992 ◽  
Vol 57 (11) ◽  
pp. 1428-1434 ◽  
Author(s):  
K. J. Ellefsen ◽  
M. N. Toksöz ◽  
K. M. Tubman ◽  
C. H. Cheng
Keyword(s):  
Shear Modulus ◽  
Cost Function ◽  
Symmetry Axis ◽  
Synthetic Data ◽  
Clay Content ◽  
Transversely Isotropic ◽  
Optimization Methods ◽  
Current Estimate ◽  
Tube Wave ◽  
The Cost

We have developed a method that estimates a shear modulus [Formula: see text] of a transversely isotropic formation using the tube wave generated during acoustic logging. (The symmetry axis of the anisotropy is assumed to parallel the borehole.) The inversion, which is implemented in the frequency‐wavenumber domain, is based upon a cost function that has three terms: a measure of the misfit between the observed and predicted wavenumbers of the tube wave, a measure of the misfit between the current estimate for [Formula: see text] and the most‐likely value for [Formula: see text], and penalty functions that constrain the estimate to physically acceptable values. The largest contribution to the value of the cost function ordinarily comes from the first term, indicating that the estimate for [Formula: see text] depends mostly on the data. Because the cost function only has one minimum, it can be found using standard optimization methods. The minimum is well defined indicating that the estimate for [Formula: see text] is well resolved. Estimates for [Formula: see text] from synthetic data are almost always within 1 percent of their correct value. Estimates for [Formula: see text] from field data that were collected in a formation with a high clay content are typical of transversely isotropic rocks.

The elastic anisotropy of clay minerals

Geophysics ◽  
2016 ◽  
Vol 81 (5) ◽  
pp. C193-C203 ◽  
Author(s):  
Colin M. Sayers ◽  
Lennert D. den Boer
Keyword(s):  
Clay Minerals ◽  
Elastic Properties ◽  
Elastic Moduli ◽  
Elastic Anisotropy ◽  
Elastic Stiffness ◽  
Stiffness Tensor ◽  
Covalent Bonds ◽  
Best Fitting ◽  
Anisotropy Parameters ◽  
Elastic Stiffness Tensor

The layered structure of clay minerals produces large elastic anisotropy due to the presence of strong covalent bonds within layers and weaker electrostatic bonds in between. Technical difficulties associated with small grain size preclude experimental measurement of single-crystal elastic moduli. However, theoretical calculations of the complete elastic tensors of several clay minerals have been reported, using either first-principle calculations based on density functional theory or molecular dynamics. Because of the layered microstructure, the elastic stiffness tensor obtained from such calculations can be approximated to good accuracy as a transversely isotropic (TI) medium. The TI-equivalent elastic moduli of clay minerals indicate that Thomsen’s anisotropy parameters [Formula: see text] and [Formula: see text] are large and positive, whereas [Formula: see text] is small or negative. A least-squares inversion for the elastic properties of a best-fitting equivalent TI medium consisting of two isotropic layers to the elastic properties of clay minerals indicates that the shear modulus of the stiffest layer is considerably larger than the softest layer, consistent with the expected high compliance of the interlayer region in clay minerals. It is anticipated that the elastic anisotropy parameters derived from the best-fitting TI approximation to the elastic stiffness tensor of clay minerals will find applications in rock physics for seismic imaging, amplitude variation with offset analysis, and geomechanics.

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.

Simple inversion of time‐domain electromagnetic data

Geophysics ◽  
1984 ◽  
Vol 49 (7) ◽  
pp. 925-933 ◽  
Author(s):  
C. T. Barnett
Keyword(s):  
Time Domain ◽  
Eddy Currents ◽  
Early Time ◽  
Late Time ◽  
Current Filament ◽  
Equivalent Current ◽  
Time Domain Electromagnetic ◽  
Inversion Procedure ◽  
Current Filaments ◽  
Simple Inversion

The eddy currents induced in a thin confined conductor by a fixed‐loop time‐domain EM system can be represented by a single equivalent current filament. The equivalent current filament stays in the plane of the conductor at all times during the decay of the secondary field, but tends to migrate from a position of maximum primary field coupling at early time toward the center of the conductor at late time. This filament approximation is used in the design of a least‐squares inversion procedure which fits circular or rectangular current filaments to an observed eddy current distribution. The inversion procedure provides a rapid but precise means of estimating the position, size, and attitude of a conductor which has been detected by a time‐domain EM survey.

A design method for metamaterials: 3D transversely isotropic lattice structures with tunable auxeticity

2021 ◽  
Author(s):  
Xiang-Long Peng ◽  
Swantje Bargmann
Keyword(s):  
Elastic Anisotropy ◽  
Design Method ◽  
Transversely Isotropic ◽  
Longitudinal Direction ◽  
Lattice Structures ◽  
Structural Geometry ◽  
Young’S Moduli ◽  
Auxetic Structures ◽  
Material Space ◽  
Effective Young's Moduli

Abstract A method for designing 3D transversely isotropic auxetic lattice structures is proposed. Based on it, two new auxetic structures have been designed. Systematically, their effective elastic properties are investigated computationally and analytically in all loading directions. The effective Young's moduli and Poisson's ratios within the transverse plane and those along the longitudinal direction are widely tunable by tailoring the structural geometry. Both structures exhibit transverse and longitudinal auxeticities concurrently as well as separately. The proposed auxetic structures expand the existing auxetic material space in terms of elastic anisotropy.

Sensitivity of the Shear Wave Speed-Stress Relationship to Soft Tissue Material Properties and Fiber Alignment

2021 ◽  
Author(s):  
Jonathon Blank ◽  
Darryl Thelen ◽  
Matthew S. Allen ◽  
Joshua Roth
Keyword(s):  
Wave Propagation ◽  
Shear Modulus ◽  
Shear Wave ◽  
Wave Speed ◽  
Axial Stress ◽  
Beam Model ◽  
Fiber Alignment ◽  
Wave Speeds ◽  
Shear Wave Speed ◽  
Constitutive Properties

The use of shear wave propagation to noninvasively gauge material properties and loading in tendons and ligaments is a growing area of interest in biomechanics. Prior models and experiments suggest that shear wave speed primarily depends on the apparent shear modulus (i.e., shear modulus accounting for contributions from all constituents) at low loads, and then increases with axial stress when axially loaded. However, differences in the magnitudes of shear wave speeds between ligaments and tendons, which have different substructures, suggest that the tissue’s composition and fiber alignment may also affect shear wave propagation. Accordingly, the objectives of this study were to (1) characterize changes in the apparent shear modulus induced by variations in constitutive properties and fiber alignment, and (2) determine the sensitivity of the shear wave speed-stress relationship to variations in constitutive properties and fiber alignment. To enable systematic variations of both constitutive properties and fiber alignment, we developed a finite element model that represented an isotropic ground matrix with an embedded fiber distribution. Using this model, we performed dynamic simulations of shear wave propagation at axial strains from 0% to 10%. We characterized the shear wave speed-stress relationship using a simple linear regression between shear wave speed squared and axial stress, which is based on an analytical relationship derived from a tensioned beam model. We found that predicted shear wave speeds were both in-range with shear wave speeds in previous in vivo and ex vivo studies, and strongly correlated with the axial stress (R2 = 0.99). The slope of the squared shear wave speed-axial stress relationship was highly sensitive to changes in tissue density. Both the intercept of this relationship and the apparent shear modulus were sensitive to both the shear modulus of the ground matrix and the stiffness of the fibers’ toe-region when the fibers were less well-aligned to the loading direction. We also determined that the tensioned beam model overpredicted the axial tissue stress with increasing load when the model had less well-aligned fibers. This indicates that the shear wave speed increases likely in response to a load-dependent increase in the apparent shear modulus. Our findings suggest that researchers may need to consider both the material and structural properties (i.e., fiber alignment) of tendon and ligament when measuring shear wave speeds in pathological tissues or tissues with less well-aligned fibers.

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.

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