Changes in Crack Shape and Saturation in Laboratory-Induced Seismicity by Water Infiltration in the Transversely Isotropic Case with Vertical Cracks

Wave-mode separation can be achieved by projecting elastic wavefields onto mutually orthogonal polarization directions. In isotropic media, because the P-wave’s polarization vectors are consistent with wave vectors, the isotropic separation operators are represented by divergence and curl operators, which are easy to realize. In anisotropic media, polarization vectors deviate from wave vectors based on local anisotropic strength and separation operators lose their simplicity. Conventionally, anisotropic wave-mode separation is implemented either by direct filtering in the wavenumber domain or nonstationary filtering in the space domain, which are computationally expensive. Moreover, in conventional anisotropic separation, correcting for amplitude and phase changes of waveforms by applying separation operators is also more difficult than in an isotropic case. We have developed new operators for efficient wave-mode separation in vertical transversely isotropic (VTI) media. Our separation operators are constructed by local rotation of wave vectors to directions where the quasi-P (qP) wave is polarized. The deviation angles between the wave vectors and the qP-wave’s polarization vectors are explicitly estimated using the Poynting vectors. Obtaining polarization directions by rotating wave vectors yields separation operators in VTI media with the same forms as divergence and curl operators, except that the spatial derivatives are now rotated to implement wavefield projections in accurate polarization directions. The main increase in computational cost relative to isotropic separation operators is the estimation of the Poynting vectors, which is relatively small within elastic-wave extrapolation. As a result, applying the proposed operators is efficient. In the meantime, the waveforms corrections for divergence and curl operators can be directly extended for our new operators due to the similarities between these operators. By numerical exercises, we have determined that wave modes can be well-separated with small numerical cost using the present separation operators. The conservation of energy in wave-mode separation by applying waveform corrections was also verified.

Download Full-text

Effective Elastic Moduli of Ribbon-Reinforced Composites

Journal of Applied Mechanics ◽

10.1115/1.2888297 ◽

1990 ◽

Vol 57 (1) ◽

pp. 158-167 ◽

Cited By ~ 92

Author(s):

Y. H. Zhao ◽

G. J. Weng

Keyword(s):

Aspect Ratio ◽

Elastic Moduli ◽

Volume Fraction ◽

Transversely Isotropic ◽

Isotropic Case ◽

Effective Elastic Moduli ◽

Cross Sectional ◽

Isotropic Composite ◽

The Matrix ◽

The Hill

Based on the Eshelby-Mori-Tanaka theory the nine effective elastic constants of an orthotropic composite reinforced with monotonically aligned elliptic cylinders, and the five elastic moduli of a transversely isotropic composite reinforced with two-dimensional randomly-oriented elliptic cylinders, are derived. These moduli are given in terms of the cross-sectional aspect ratio and the volume fraction of the elliptic cylinders. When the aspect ratio approaches zero, the elliptic cylinders exist as thin ribbons, and these moduli are given in very simple, explicit forms as a function of volume fraction. It turns out that, in the transversely isotropic case, the effective elastic moduli of the composite coincide with Hill’s and Hashin’s upper bounds if ribbons are harder than the matrix, and coincide with their lower bounds if ribbons are softer. These results are in direct contrast to those of circular fibers. Since the width of the Hill-Hashin bounds can be very wide when the constituents have high modular ratios, this analysis suggests that the ribbon reinforcement is far more effective than the traditional fiber reinforcement.

Download Full-text

A recipe for practical full-waveform inversion in anisotropic media: An analytical parameter resolution study

Geophysics ◽

10.1190/geo2013-0366.1 ◽

2014 ◽

Vol 79 (3) ◽

pp. R91-R101 ◽

Cited By ~ 148

Author(s):

Tariq Alkhalifah ◽

René-Édouard Plessix

Keyword(s):

Waveform Inversion ◽

Transversely Isotropic ◽

Anisotropic Media ◽

Horizontal Velocity ◽

Isotropic Case ◽

Full Waveform Inversion ◽

Symmetry Direction ◽

Velocity Anisotropy ◽

Full Waveform ◽

Model Update

In multiparameter full-waveform inversion (FWI) and specifically one describing the anisotropic behavior of the medium, it is essential that we have an understanding of the parameter resolution possibilities and limits. Because the imaging kernel is at the heart of the inversion engine (the model update), we drew our development and choice of parameters from what we have experienced in imaging seismic data in anisotropic media. In representing the most common (first-order influence and gravity induced) acoustic anisotropy, specifically, a transversely isotropic medium with a vertical symmetry direction (VTI), with the [Formula: see text]-wave normal moveout velocity, anisotropy parameters [Formula: see text], and [Formula: see text], we obtained a perturbation radiation pattern that has limited trade-off between the parameters. Because [Formula: see text] is weakly resolvable from the kinematics of [Formula: see text]-wave propagation, we can use it to play the role that density plays in improving the data fit for an imperfect physical model that ignores the elastic nature of the earth. An FWI scheme that starts from diving waves would benefit from representing the acoustic VTI model with the [Formula: see text]-wave horizontal velocity, [Formula: see text], and [Formula: see text]. In this representation, the diving waves will help us first resolve the horizontal velocity and then reflections, if the nonlinearity is properly handled, could help us resolve [Formula: see text], and [Formula: see text] could help improve the amplitude fit (instead of the density). The model update wavenumber for acoustic anisotropic FWI is very similar to that for the isotropic case, which is mainly dependent on the scattering angle and frequency.

Download Full-text

Effect of anisotropy on Love wave propagation

Bulletin of the Seismological Society of America ◽

10.1785/bssa0580010259 ◽

1968 ◽

Vol 58 (1) ◽

pp. 259-266

Author(s):

Janardan G. Negi ◽

S. K. Upadhyay

Keyword(s):

Wave Propagation ◽

Frequency Dependence ◽

Characteristic Frequency ◽

Love Wave ◽

Transversely Isotropic ◽

Frequency Equation ◽

Isotropic Case ◽

Rigid Base ◽

Isotropic Layer ◽

Transversely Isotropic Layer

abstract A study on Love wave propagation in a transversely isotropic layer with stress free upper surface and underlying rigid base, is presented. The characteristic frequency equation is obtained and frequency dependence of the velocity parameters for different modes is analysed in detail. Several distinctive propagation phenomena which differ considerably from those in isotropic case are listed below:

Download Full-text

Elastic wave propagation in heterogeneous anisotropic media using the lumped finite‐element method

Geophysics ◽

10.1190/1.1468624 ◽

2002 ◽

Vol 67 (2) ◽

pp. 625-638 ◽

Cited By ~ 20

Author(s):

Jianfeng Zhang ◽

D. J. Verschuur

Keyword(s):

Wave Propagation ◽

Symmetry Axis ◽

Numerical Technique ◽

Computational Cost ◽

Vertical Plane ◽

Grid Cell ◽

Transversely Isotropic ◽

Isotropic Case ◽

Lamb’S Problem ◽

Anisotropic Structures

A numerical technique for wave‐propagation simulation in 2‐D heterogeneous anisotropic structures is presented. The scheme is flexible in incorporating arbitrary surface topography, inner openings, liquid/solid boundaries, and irregular interfaces, and it naturally satisfies the free‐surface conditions of complex geometrical boundaries. The algorithm, based on a discretization mesh of triangles and quadrilaterals, solves the problem using integral equilibrium around each node instead of satisfying elastodynamic differential equations at each node as in the finite‐difference method. This study is an extension of previous work for the elastic‐isotropic case. Besides accounting for anisotropy, a simplified quadrilateral grid cell with low computational cost is introduced. The transversely isotropic medium with a symmetry axis on the horizontal or vertical plane, as typically caused by a system of parallel cracks or fine layers, is discussed in detail. A 2‐D algorithm is presented that can handle the situation where the symmetry axis of the anisotropy does not lie in the 2‐D plane. The proposed scheme is successfully tested against an analytical solution for Lamb's problem with a symmetry axis normal to the surface and agrees well with a numerical solution of the reflectivity method for a plane‐layered model in the isotropic case. Computed radiation patterns show characteristics such as shear‐wave splitting and triplications of quasi‐SV wavefronts, as predicted by the theory. Examples of surface‐wave propagation in an anisotropic half‐space with a semicylindrical pit on the surface and mixed liquid/(anisotropic) solid model with an inclined liquid/solid interface are presented. Moreover, seismograms are modeled for dome‐layered and plane‐layered anisotropic structures.

Download Full-text

Analysis of Embedded and Surface Elliptical Flaws in Transversely Isotropic Bodies by the Finite Element Alternating Method

Journal of Applied Mechanics ◽

10.1115/1.2897204 ◽

1991 ◽

Vol 58 (2) ◽

pp. 435-443 ◽

Cited By ~ 5

Author(s):

H. Rajiyah ◽

S. N. Atluri

Keyword(s):

Finite Element ◽

Transversely Isotropic ◽

Generalized Solution ◽

Solution Procedure ◽

Isotropic Case ◽

The Finite Element Method ◽

Crack Face ◽

Elastic Symmetry ◽

Alternating Method ◽

Isotropic Bodies

The general analytical solution to the problem of a flat elliptical crack embedded in an infinite, transversely isotropic solid, oriented perpendicular to the axis of elastic symmetry, is derived along the lines of Vijayakumar and Atluri’s solution procedure for the isotropic case. The prior work of Kassir and Sih on this problem is limited to some constant and linear variations of normal and shear tractions on the crack face. The generalized solution is employed in the Schwarz-Neumann alternating method in conjunction with the finite element method. Such a method of analysis is shown to be an efficient way to evaluate the stress intensity factors along the flaw border.

Download Full-text

Scalar generalized‐screen algorithms in transversely isotropic media with a vertical symmetry axis

Geophysics ◽

10.1190/1.1487100 ◽

2001 ◽

Vol 66 (5) ◽

pp. 1538-1550 ◽

Cited By ~ 29

Author(s):

Jéro⁁me H. Le Rousseau ◽

Maarten V. de Hoop

Keyword(s):

Symmetry Axis ◽

Complex Structure ◽

Numerical Algorithms ◽

Transversely Isotropic ◽

Isotropic Case ◽

Polarization Angle ◽

Screen Method ◽

Transversely Isotropic Media ◽

Isotropic Media ◽

The One

The scalar generalized‐screen method in isotropic media is extended here to transversely isotropic media with a vertical symmetry axis (VTI). Although wave propagation in a transversely isotropic medium is essentially elastic, we use an equivalent “acoustic” system of equations for the qP‐waves which we prove to be accurate for both the dispersion relation and the polarization angle in the case of “mild” anisotropy. The enhanced accuracy of the generalized‐screen method as compared to the split‐step Fourier methods allows the extension to VTI media. The generalized‐screen expansion of the one‐way propagator follows closely the method used in the isotropic case. The medium is defined in terms of a background and a perturbation. The generalized‐screen expansion of the vertical slowness is based upon an expansion of the medium parameters simultaneously into magnitude and smoothness of variation. We cast the theory into numerical algorithms, and assess the accuracy of the generalized‐screen method in a particular VTI medium with complex structure (the BP Amoco Valhall model) in which multipathing is significant.

Download Full-text

Modelling transversely isotropic fiber-reinforced composites with unidirectional fibers and microstructure

Mathematics and Mechanics of Solids ◽

10.1177/1081286519838603 ◽

2019 ◽

Vol 24 (11) ◽

pp. 3444-3455 ◽

Cited By ~ 1

Author(s):

Ryan Barrage ◽

Stanislav Potapenko ◽

Maria Anna Polak

Keyword(s):

Elastic Moduli ◽

Transversely Isotropic ◽

Constitutive Law ◽

Fiber Reinforced Composites ◽

Isotropic Case ◽

Reinforced Composites ◽

Homogenization Procedure ◽

Isotropic Composite ◽

Unidirectional Fibers ◽

Micropolar Medium

This paper develops a micropolar constitutive model for a transversely isotropic composite material comprised of a polymer matrix and unidirectional fibers. The constitutive law follows Eringen’s model for a generally anisotropic micropolar medium and reduces the model to the transversely isotropic case. The model is then used to develop a boundary value problem with a homogenization procedure, with the goal of obtaining an analytical solution for the homogenized elastic moduli once solved. A variational principle is used to develop the boundary conditions.

Download Full-text