A nonlinear elastic model for acoustic waves in a stressed medium is used to calculate tectonic stress‐induced changes in borehole flexural dispersions. Our theoretical analysis shows that a horizontal uniaxial stress in the formation causes a crossover in flexural dispersions for the radial polarization aligned parallel and normal to the stress direction. This crossover in flexural dispersions is caused by stress‐induced radial heterogeneities in acoustic wave velocities that are different in the two principal stress directions. Other sources of borehole flexural anisotropy caused by finely layered dipping beds, aligned fractures, or microstructures found in shales, exhibit neither such radial heterogeneities nor flexural dispersion crossovers. Consequently, a crossover in flexural dispersion can be used as an indicator of stress‐induced anistropy. In this situation, the fast shear polarization direction coincides with the far‐field uniaxial stress direction. The analysis also yields an expression for the largest shear stress parameter in terms of the fast and slow seismic shear‐wave velocities with shear polarization parallel and perpendicular to the far‐field stress direction.
S U M M A R Y A heterogeneous medium composed of inviscid fluid and solid constituents is pre-stressed, resulting in relative slip of material particles at the interfaces between the solid and the fluid. The standard theory of acoustoelasticity, which is concerned with small deformation superimposed upon large initial strain, is generalized here to include the effects of the interfacial slip. Difficulties arise from the possibility that the traction, viewed as a function of either the undeformed material (Lagrangian) coordinates or of the intermediate coordinates, is not necessarily continuous across the interface. It is shown that the problem is most easily considered in the intermediate coordinates, leading to a divergence formulation of the equations of small motion from which the interface conditions arise naturally. The theory is demonstrated for the problem of a fluid-filled borehole with a pressurized fluid and pre-strained solid. An explicit expression is found for the change in the speed of the tube wave, which is the quasi-static limit of the Stoneley wave mode.
An analysis is carried out of axisymmetric waves propagating along fluid-loaded cylindrical shells within the framework of linear elasticity and classical perfect-slip boundary conditions at the solid-fluid interface. Numerical solutions are obtained for various axisymmetric eigenmodes for a cylindrical shell in vacuum; a cylindrical shell surrounded by a liquid of infinite radial extent; a hypothetical liquid column with both the stress-free and displacementfree boundary conditions; a cylindrical shell with a liquid core; and a cylindrical shell immersed in an infinite liquid. Numerical results are obtained for both the radiating (leaky) and nonradiating eigenmodes of the system by a careful search of the complex eigenfrequencies of the associated boundary value problem. In particular, attenuation of leaky modes due to radiation of energy into the surrounding medium is expressed in terms of the imaginary part of the eigenfrequency. Computational results are presented for the dispersion curves as well as the displacement and stress amplitude component distributions along the radial direction for various propagating modes of the system. Practical benefits from such analyses are discussed.
Analyses of sonic logs in a horizontal well provide new information about mechanical properties of rocks, made possible by recent developments in our understanding of acoustic wave propagation in prestressed formations. Most sections of this horizontal well exhibit azimuthal shear isotropy, indicating isotropic stresses in the plane perpendicular to the well trajectory, leading to stable wellbore conditions. However, two sections show dipole dispersion crossovers that confirm the presence of stress‐induced shear anisotropy caused by a difference between the maximum and minimum stresses in the plane perpendicular to the well trajectory. The two dipole dispersions are obtained by processing the recorded waveforms by a modified matrix pencil algorithm. The fast‐shear direction is estimated from Alford rotation of the cross‐dipole waveforms. One section of the well exhibits the fast‐shear direction parallel to the overburden stress as the maximum stress direction, whereas the other section has the fast‐shear direction parallel to the horizontal stress that is larger than the overburden stress. The cause of this change in the fast‐shear direction is believed to be the well’s penetration into a 3-ft-thick bed with lower porosity and permeability and significantly higher elastic stiffnesses than those in the other part of the homogeneous, high‐permeability reservoir. A stiff bed is likely to have greater stresses in its plane than perpendicular to it, which would make the horizontal stresses greater than the vertical.
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