Sergio Kostek

Stress-induced azimuthal anisotropy in borehole flexural waves

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.

Axisymmetric wave propagation in fluid‐loaded cylindrical shells. I: Theory

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 displacement‐free 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...

Axisymmetric wave propagation in fluid-loaded cylindrical shells. II : Theory versus experiment

This paper discusses both theoretical and experimental aspects of axisymmetric wave propagation along fluid‐loaded cylindrical shells (excluding torsional modes). For a steel cylindrical shell with a fixed ratio of inner to outer radius, four different fluid configurations are considered: water inside and outside the steel shell; air inside and outside; water inside and air outside; and air inside and water outside. Calculations of the transient pressure response for the case of an axisymmetric ring source and a point receiver are made as a function of source–receiver separation. Experimentally, a PZT ring source and ring receiver are placed around a steel cylindrical shell with an outer radius of 9.53 mm and an inner radius of 7.94 mm. Waveforms are recorded for multiple source–receiver hydrophone spacings in the frequency band 50–240 kHz. Using a Prony’s method, the complex wave number as a function of frequency for each of the modes in the system is derived from both the theoretical and experimental wa...

The interaction of tube waves with borehole fractures; Part II, Analytical models

We solve, numerically, the equations of elastodynamics that govern the propagation of waves in a fluid‐filled borehole intersected by one or more fluid‐filled fractures, thus extending earlier work on formations presumed to be rigid. The model is axis symmetric, and it allows for arbitrary radial and vertical variations in the elastic properties of the formation. We have developed a novel gridding scheme that takes advantage of the assumed thinness of the fracture compared to any relevant wavelength; this technique obviates the necessity for a mesh within the fracture itself and thereby saves computational time and memory. We illustrate our technique by means of examples with single and double fractures with or without washouts, as well as examples with variable width fractures. We compute the frequency‐dependent reflection and transmission coefficients of tube waves by fractures from the time waveforms generated by the present method. Major conclusions of this work are (1) reflections of tube waves by fr...

Third‐order elastic constants for an inviscid fluid

In dealing with nonlinear problems involving fluids and solids, whether of a static or dynamic nature, a common description of the fields in terms of Eulerian or Lagrangian variables is desirable. Usually the former is used for fluids and the latter for solids. The choice of primitive variables also differs when dealing with fluids or solids. The material constants describing the constitutive behavior of these media will thus depend on the description adopted. In this paper, explicit relations are provided between third‐order elastic constants for an inviscid fluid and the more common coefficients, A and B, appearing in the Taylor expansion of the equation of state. The essential results are c111=−(5A+B), c112=−(A+B), and c123=A−B.

Stoneley and flexural modes in pressurized boreholes

The propagation of Stoneley and flexural waves in a fluid-filled borehole is adequately described by the linear equations of elasticity. However, when the borehole fluid is pressurized either due to the hydrostatic head at a given depth or with the aid of packers at the wellhead, both the fluid and the surrounding formation are subjected to biasing stresses. Under this situation, wave propagation along the borehole is described by the equations of motion for small dynamic fields superposed on a bias. The resulting formulation allows us to study the influence of a change in the fluid pressure on the Stoneley and flexural mode dispersion curves. Since the biasing stresses in the surrounding formation exhibit a radial decay away from the borehole, it is expedient to employ a perturbation technique to calculate changes in the borehole Stoneley and flexural wave dispersion curves as a function of hydrostatic pressure change in the fluid. A key advantage of this perturbation technique is that it separates contributions of the acoustoelastic effect due to the borehole fluid and that due to the formation. Insofar as the fluid nonlinear properties at a given pressure and temperature are known, the model provides a procedure for estimating the acoustoelastic coefficient of the formation for the borehole Stoneley and flexural wave velocities for a given change in the fluid pressure. The formation acoustoelastic coefficient can be expressed as a fractional change in the acoustic wave velocity caused by a unit change in the borehole pressure. Computational results show that acoustoelastic coefficients for the Stoneley and flexural waves are larger for formations with higher degree of nonlinearity which is typically associated with poorly consolidated rocks.

Tube waves and mandrel modes: Experiment and theory

The characteristics of tube waves in a borehole, with and without a solid cylindrical mandrel, which may be either elastic or poro-elastic, are compared. With an elastic mandrel, the tube waves are slower, more dispersive, and more sensitive to the formation shear modulus than without. Similarly extensional mode characteristics are compared in the presence of a formation. In the presence of an elastic formation the extensional mode is faster, more dispersive than without, and it is only slightly sensitive to the formation shear properties. These calculated characteristics are in excellent agreement with our measured data. Additionally, the characteristics of tube waves and extensional modes are studied in a borehole in the presence of a concentric liquid-saturated porous mandrel whose acoustic properties are calculated using the Biot theory. The coupling of the tube wave propagating in the annulus with the slow wave propagating in the porous mandrel introduces attenuation and additional dispersion to the ...

The equivalent force system of a monopole source in a fluid‐filled open borehole

One of the major contributors to the complexity of boundary-value problems pertaining to theoretical modeling of exploration elastodynamics is the presence of both vertical and horizontal discontinuities. It has been known for a long time that it is sometimes possible to replace certain boundaries by a system of images, provided the extra stresses and displacements induced by these images could indeed mimic the discontinuities caused by the said boundaries (e.g., Ben-Menahem and Singh, 1981). In this vein, we show that part of the field created by a monopole source acting on the axis of a fluid-filled open borehole surrounded by a homogeneous and isotropic formation, can be reconstructed with the aid of an equivalent force system (EFS) that mimics the geometrodynamic effects of the borehole. The advantages of the EFS are twofold. In the first place, it simplifies the physical setup and brings many seemingly different problems into a common denominator in a sense that they are reduced to fields of known basic force systems. Second, and this is not less important, much computer time is saved and numerical complexities are avoided.

Acoustic waves in pressurized boreholes: A finite difference formulation

A velocity-stress, finite difference formulation of acoustic waves in a fluid-filled, pressurized borehole yields synthetic waveforms for monopole or dipole sources before and after borehole pressurization. Processing of these waveforms using a variation of Pronys procedure isolates dominant arrivals in the full wave field. Differences between the slownesses of individual arrivals before and after pressurization provide stress-induced changes in propagation characteristics that are of importance in estimating mechanical properties of the formation. The borehole pressurization in an isotropic formation produces insignificant changes in the compressional head wave slownesses; and small changes in the shear head wave slownesses. The most significant changes occur in the Stoneley and flexural slownesses at relatively higher frequencies in the range of 5–10 kHz for a borehole of diameter 20.32 cm (8 inches). These differences in the Stoneley and flexural slowness dispersions for a known increase in the borehole pressure can be used to calculate the acoustoelastic coefficients of the formation. These coefficients are measures of nonlinear elastic parameters of the formation that are generally larger for poorly consolidated slow formations than those of tightly consolidated fast formations.

Nonlinear tube waves

The nonlinear characteristics of an acoustic tube wave propagating along the axis of a fluid-filled circular borehole in an elastic solid that is locally isotropic but whose properties may vary radially is considered. The analysis is carried out in the quasistatic limit. All terms through quadratic in the amplitude of the wave are considered and the amplitude of second-harmonic generation and the pressure dependence of the tube wave speed, dVr/d p, are expressed in terms of the fluid and formation nonlinear parameters. The results show that if there is no radial variation of the shear modulus of the solid then both the amplitude of second-harmonic generation and dVr/d p are independent of the third-order elastic constants of the solid and nearly equal to those of the fluid alone. If there is a radial variation of the shear modulus then the numerical calculations indicate that