scholarly journals Stoneley‐wave propagation in a fluid‐filled borehole with a vertical fracture

Geophysics ◽  
1991 ◽  
Vol 56 (4) ◽  
pp. 447-460 ◽  
Author(s):  
X. M. Tang ◽  
C. H. Cheng ◽  
M. N. Toksöz
Keyword(s):  
Boundary Condition ◽  
Wave Propagation ◽  
Wave Attenuation ◽  
Perturbation Technique ◽  
Quantitative Relationship ◽  
Acoustic Properties ◽  
Stoneley Wave ◽  
Fracture Aperture ◽  
Stoneley Waves ◽  
Vertical Fracture

The propagation of Stoneley waves in a fluid‐filled borehole with a vertical fracture is investigated both theoretically and experimentally. The borehole propagation excites fluid motion in the fracture and the resulting fluid flow at the fracture opening perturbs the fluid‐solid interface boundary condition at the borehole wall. By developing a boundary condition perturbation technique for the borehole situation, we studied the effect of this change in the boundary condition on the Stoneley propagation. Cases of both hard and soft formations have been investigated. The fracture has minimal effects on the Stoneley velocity, except in the very low frequency range in which the Stoneley velocity drastically decreases with decreasing frequency. Significant Stoneley‐wave attenuation is produced because of the energy dissipation into the fracture. The quantitative behavior of these effects depends not only on fracture aperture and borehole radius, but also on the acoustic properties of the formation and fluid. Ultrasonic experiments were performed to measure Stoneley propagation in laboratory fracture borehole models. Aluminum and lucite were used to simulate a hard and a soft formation, respectively. Array data for wave propagation were obtained and were processed using Prony’s method to give velocity and attenuation of Stoneley waves as a function of frequency. In both hard and soft formation cases, the experimental results agreed with the theoretical predictions. The important result of this study is that it presents a quantitative relationship between the Stoneley propagation and the fracture character in conjunction with formation and fluid properties. This relationship provides a method for estimating the characteristics of a vertical fracture by means of Stoneley wave measurements.

Simulation of borehole sonic waveforms in dipping, anisotropic, and invaded formations

Geophysics ◽  
2011 ◽  
Vol 76 (4) ◽  
pp. E127-E139 ◽  
Author(s):  
Robert K. Mallan ◽  
Carlos Torres-Verdín ◽  
Jun Ma
Keyword(s):  
Wave Propagation ◽  
Shear Modulus ◽  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Cutoff Frequency ◽  
Dispersion Curves ◽  
Flexural Wave ◽  
Stoneley Wave ◽  
Stoneley Waves ◽  
The Impact

A numerical simulation study has been made of borehole sonic measurements that examined shoulder-bed, anisotropy, and mud-filtrate invasion effects on frequency-dispersion curves of flexural and Stoneley waves. Numerical simulations were considered for a range of models for fast and slow formations. Computations are performed with a Cartesian 3D finite-difference time-domain code. Simulations show that presence of transverse isotropy (TI) alters the dispersion of flexural and Stoneley waves. In slow formations, the flexural wave becomes less dispersive when the shear modulus (c44) governing wave propagation parallel to the TI symmetry axis is lower than the shear modulus (c66) governing wave propagation normal to the TI symmetry axis; conversely, the flexural wave becomes more dispersive when c44 > c66. Dispersion decreases by as much as 30% at higher frequencies for the considered case where c44 < c66. Dispersion of Stoneley waves, on the other hand, increases in TI formations when c44 > c66 and decreases when c44 < c66. Dispersion increases by more than a factor of 2.5 at higher frequencies for the considered case where c44 < c66. Simulations also indicate that the impact of invasion on flexural and Stoneley dispersions can be altered by the presence of TI. For the case of a slow formation and TI, where c44 decreases from the isotropic value, separation between dispersion curves for cases with and without the presence of a fast invasion zone increases by as much as 33% for the flexural wave and by as much as a factor of 1.4 for the Stoneley wave. Lastly, presence of a shoulder bed intersecting the sonic tool at high dip angles can alter flexural dispersion significantly at low frequencies. For the considered case of a shoulder bed dipping at 80°, ambiguity in the flexural cutoff frequency might lead to shear-wave velocity errors of 8%–10%.

Complex roots of the Stoneley-wave equation

1972 ◽  
Vol 62 (1) ◽  
pp. 285-299
Author(s):  
Walter L. Pilant
Keyword(s):  
Wave Propagation ◽  
Fundamental Equation ◽  
Stoneley Wave ◽  
Physical Parameters ◽  
Branch Points ◽  
Stoneley Waves ◽  
Interface Waves ◽  
Material Parameters ◽  
Complex Roots ◽  
Wide Range

Abstract The equation governing elastic waves propagating along a solid-solid interface is found to have sixteen (16) independent roots on its eight (8) associated Riemann sheets. The range of existence (in terms of material parameters) for the real root corresponding to the propagation of Stoneley waves has long been known. It is found that outside this range there are two types of behavior. If the material of greater density has a velocity slightly greater than that of the material of lesser density, the unattenuated Stoneley waves make a transition to attenuated Interface waves, i.e., they leak energy away from the interface as they propagate along it. If the more dense material has a velocity more than about three times that of the less dense, then the Interface-wave root disappears and energy is propagated along the interface as Rayleigh waves. This Rayleigh-wave propagation is associated with a different root of the fundamental equation. On the other hand, if the material of greater density has a velocity much lower than that of the material of lower density (a case that is difficult to find physically), then no energy will be propagated along the interface at all. This result was unexpected. Some rather interesting behavior of the 16 roots was noted as the physical parameters were varied over a wide range. In addition to the normal collisions between pairs of roots, and between individual roots and branch points (with attendant Riemann sheet jumping), it was found that some roots go through the point at infinity and return with a change in sign. At least one unexpected case of a multiple root was found. Another case was noted in which a pair of complex roots change quadrants in the complex phase-velocity plane, leading to a discontinuity in root type. Finally, it was noted that, in a cyclic variation of the material parameters, it is possible to choose a path such that the roots, when followed individually, will not return to their original values. In fact, as many as five cycles in parameter space can be accomplished before the roots return. All this strange mathematical behavior seems to have no physical significance, but has been presented to increase understanding of the general behavior of the dispersion relations associated with elastic-wave propagation.

Stoneley wave attenuation and dispersion and the dynamic permeability correction

Geophysics ◽  
2019 ◽  
Vol 84 (4) ◽  
pp. WA1-WA10 ◽  
Author(s):  
Xiumei Zhang ◽  
Tobias M. Müller
Keyword(s):  
Hybrid Model ◽  
Wave Attenuation ◽  
Stoneley Wave ◽  
Relaxation Length ◽  
Permeability Model ◽  
P Waves ◽  
Stoneley Waves ◽  
Dynamic Permeability ◽  
Viscous Relaxation ◽  
Attenuation And Dispersion

Stoneley waves induce fluid pressure gradients in a permeable formation surrounding the borehole. These gradients are equilibrated through pressure diffusion, that is to say, slow P-waves in the context of Biot’s poroelasticity theory. Because slow P-waves are strongly sensitive to the formation permeability, the Stoneley-slow P-wave interaction can be used to retrieve the formation permeability from the attenuation and dispersion of Stoneley waves. The accuracy of this established technique in high-permeability formations deteriorates when slow P-waves are not pure diffusion waves; hence, the permeability dependence is more complicated. This effect on Stoneley waves is captured by applying the Johnson-Koplik-Dashen dynamic permeability model. Their model depends on a viscous relaxation length. However, in the estimation of formation permeability from Stoneley waves, this parameter is typically not measured but is estimated from an empirical equation, wherein material properties and microstructural descriptors are lumped together. When the so-calculated relaxation length is erroneous, the inverted formation permeability from the Stoneley wave is not correct either. To overcome this limitation and to provide a versatile alternative, the dynamic permeability problem is reformulated within the viscosity-extended Biot framework. Its physical basis is the conversion scattering in random media from slow P- to slow S-waves. The correlation length of this so-called stochastic dynamic permeability model can be derived from pore-scale images, and it also captures the effect of pore interface roughness. This model is then combined with the simplified Biot-Rosenbaum model to predict Stoneley wave attenuation and dispersion. We have applied this hybrid model to interpret laboratory measurements for which the previously suggested choice of the viscous relaxation length does not provide an accurate prediction. The results indicate that the hybrid model can provide another approach to model Stoneley wave attenuation and dispersion across the entire frequency range.

Electromagnetic Casimir effect of concentric cylinders in (D + 1)-dimensional space–time

2019 ◽  
Vol 34 (08) ◽  
pp. 1950035
Author(s):  
Chun Yong Chew ◽  
Yong Kheng Goh
Keyword(s):  
Boundary Condition ◽  
Interaction Energy ◽  
Casimir Effect ◽  
Dimensional Space ◽  
Order Term ◽  
Perturbation Technique ◽  
Space Time ◽  
Concentric Cylinders ◽  
Dimensional Minkowski Space ◽  
Dimensional Space Time

We study the electromagnetic Casimir interaction energy between two parallel concentric cylinders in [Formula: see text]-dimensional Minkowski space–time for different combinations of perfectly conducting boundary condition and infinitely permeable boundary condition. We consider two cases where one cylinder is outside each other and where one is inside the other. By solving the equation of motion and computing the TGTG formulas, explicit formulas for the Casimir interaction energy can be derived and asymptotic behavior of the Casimir interaction energy in the nanoregime is calculated by using perturbation technique. We computed the interaction energy analytically up to next-to-leading order term.

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