General problem of the theory of elasticity for a transversely isotropic cone

Time-lapse traveltime shifts of reflection events recorded above hydrocarbon reservoirs can be used to monitor production-related compaction and pore-pressure changes. Existing methodology, however, is limited to zero-offset rays and cannot be applied to traveltime shifts measured on prestack seismic data. We give an analytic 3D description of stress-related traveltime shifts for rays propagating along arbitrary trajectories in heterogeneous anisotropic media. The nonlinear theory of elasticity helps to express the velocity changes in and around the reservoir through the excess stresses associated with reservoir compaction. Because this stress-induced velocity field is both heterogeneous and anisotropic, it should be studied using prestack traveltimes or amplitudes. Then we obtain the traveltime shifts by first-order perturbation of traveltimes that accounts not only for the velocity changes but also for 3D deformation of reflectors. The resulting closed-form expression can be used efficiently for numerical modeling of traveltime shifts and, ultimately, for reconstructing the stress distribution around compacting reservoirs. The analytic results are applied to a 2D model of a compacting rectangular reservoir embedded in an initially homogeneous and isotropic medium. The computed velocity changes around the reservoir are caused primarily by deviatoric stresses and produce a transversely isotropic medium with a variable orientation of the symmetry axis and substantial values of the Thomsen parameters [Formula: see text] and [Formula: see text]. The offset dependence of the traveltime shifts should play a crucial role in estimating the anisotropy parameters and compaction-related deviatoric stress components.

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Love solutions in the linear inhomogeneous transversely isotropic theory of elasticity

International Applied Mechanics ◽

10.1007/s10778-010-0289-1 ◽

2010 ◽

Vol 46 (2) ◽

pp. 121-129 ◽

Cited By ~ 4

Author(s):

M. Yu. Kashtalyan ◽

J. J. Rushchitsky

Keyword(s):

Transversely Isotropic ◽

Theory Of Elasticity ◽

Isotropic Theory

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Approximate Elasticity Solution for Orthotopic Cylinder Under Hydrostatic Pressure and Band Loads

Journal of Applied Mechanics ◽

10.1115/1.3408416 ◽

1970 ◽

Vol 37 (1) ◽

pp. 101-108 ◽

Cited By ~ 10

Author(s):

A. P. Misovec ◽

J. Kempner

Keyword(s):

Approximate Solution ◽

Shell Theory ◽

Good Accuracy ◽

Three Dimensional ◽

Transversely Isotropic ◽

External Pressure ◽

Theory Of Elasticity ◽

Numerical Comparison ◽

Axial Loads ◽

Special Case

An approximate solution to the Navier equations of the three-dimensional theory of elasticity for an axisymmetric orthotopic circular cylinder subjected to internal and external pressure, axial loads, and closely spaced periodic radial loads is developed. Numerical comparison with the exact solution for the special case of a transversely isotropic cylinder subjected to periodic band loads shows that very good accuracy is obtainable. When the results of the approximate solution are compared with previously obtained results of a Flu¨gge-type shell solution of a ring-reinforced orthotropic cylinder, it is found that the shell theory gives fairly accurate representations of the deformations and stresses except in the neighborhood of discontinuous loads. The addition of transverse shear deformations does not improve the accuracy of the shell solution.

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Diagonalization of a three-dimensional system of equations in terms of displacements of the linear theory of elasticity of transversely isotropic media

Journal of Applied Mechanics and Technical Physics ◽

10.1134/s0021894413060126 ◽

2013 ◽

Vol 54 (6) ◽

pp. 971-988 ◽

Cited By ~ 2

Author(s):

N. I. Ostrosablin

Keyword(s):

Linear Theory ◽

Three Dimensional ◽

Transversely Isotropic ◽

Theory Of Elasticity ◽

Dimensional System ◽

System Of Equations ◽

Transversely Isotropic Media ◽

Isotropic Media ◽

Three Dimensional System

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Asymptotic Behavior of the Solution to the Axisymmetric Dynamical Problem of the Elasticity Theory for the Transversally Isotropic Spherical Layer of Small Thickness

UNIVERSITY NEWS. NORTH-CAUCASIAN REGION. NATURAL SCIENCES SERIES ◽

10.18522/1026-2237-2020-2-61-71 ◽

2020 ◽

pp. 61-71

Author(s):

M.F. Mehdiyev ◽

N.K. Akhmedov ◽

S.M. Yusubova

Keyword(s):

Dynamic Problem ◽

Transversely Isotropic ◽

Spherical Layer ◽

Theory Of Elasticity ◽

Dynamical Problem ◽

Algebraic Equations ◽

Elastic Band ◽

Small Thickness ◽

Homogeneous Solutions ◽

Linear Algebraic Equations

In this paper, we study the axisymmetric dynamic problem of the theory of elasticity for the transversely isotropic spherical layer of small thickness that does not contain any of the poles 0 and π. It is assumed that the lateral surface of the sphere is free of stresses, and boundary conditions are set on conical sections. Using the method of asymptotic integration of equations of the theory of elasticity, the dynamic problem of this theory is analyzed for the transversely isotropic spherical layer as the thin-walled parameter tends to zero. A possible form of wave formation in the transversely isotropic spherical layer has been studied depending on the frequency of the influencing forces. Homogeneous solutions are constructed and their classification is given. Asymptotic expansions of the homogeneous solutions are obtained, which make possible to calculate the stress-strain state for various values of the frequency of the influencing forces. It is shown that for the high-frequency oscillations in the first term of the asymptotics, the dispersion equation coincides with the well-known Rayleigh-Lamb equation for the elastic band. In the general case of loading on the sphere using the Hamilton variational principle, the boundary-value problem is reduced to the solving infinite systems of linear algebraic equations.

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Solution of the axisymmetric problem of the theory of elasticity for transversely isotropic bodies with the aid of generalized analytic functions

Journal of Applied Mathematics and Mechanics ◽

10.1016/0021-8928(74)90079-3 ◽

1974 ◽

Vol 38 (2) ◽

pp. 352-357 ◽

Cited By ~ 1

Author(s):

Iu.I. Solov'ev

Keyword(s):

Analytic Functions ◽

Axisymmetric Problem ◽

Transversely Isotropic ◽

Theory Of Elasticity ◽

Generalized Analytic Functions ◽

Isotropic Bodies

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Stability Loss in Thick Transversely Isotropic Cylindrical Shells Under Axial Compression

Journal of Applied Mechanics ◽

10.1115/1.2900822 ◽

1993 ◽

Vol 60 (2) ◽

pp. 506-513 ◽

Cited By ~ 31

Author(s):

G. A. Kardomateas

Keyword(s):

Axial Compression ◽

Critical Loads ◽

Shell Theory ◽

Three Dimensional ◽

Transversely Isotropic ◽

Stability Loss ◽

Theory Of Elasticity ◽

Dimensional Theory ◽

Perfect Agreement ◽

The Stability

The stability of equilibrium of a transversely isotropic thick cylindrical shell under axial compression is investigated. The problem is treated by making appropriate use of the three-dimensional theory of elasticity. The results are compared with the critical loads furnished by classical shell theories. For the isotropic material cases considered, the elasticity approach predicts a lower critical load than the shell theories, the percentage reduction being larger with increasing thickness. However, both the Flu¨gge and Danielson and Simmonds theories predict critical loads much closer to the elasticity value than the Donnell theory. Moreover, the values of n, m (number of circumferential waves and number of axial half-waves, respectively, at the critical point) for both the elasticity, and the Flu¨gge and the Danielson and Simmonds theories, show perfect agreement, unlike the Donnell shell theory.

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