Theoretical study of perturbation method for acoustic multipole logging in anisotropic formation

Abstract

A new and effective analytical perturbation method is presented for the multipole acoustic logging in a transversely isotropic elastic solid whose symmetric principal axis is perpendicular to the borehole axis. The transversely isotropic elastic solid is regarded as a relevant isotropic elastic solid added to the perturbations, and three perturbation quantities about moduli deviated from the isotropic medium are introduced. By selecting a group of displacement potentials and a cylindrical coordinate system oriented along the borehole axis, the zero- and first-order perturbation solutions of the multipole acoustic field are derived for the weak transversely isotropic elastic solid which has its symmetric principal axis perpendicular to the borehole axis. The acoustic fields inside and outside the borehole excited by a multipole source are investigated. It is found that not only the nth multipole acoustic field can be excited by an nth multipole source, but the multipole fields whose orders are higher and lower than n also can be excited by an nth multipole source.

General Reflection Coefficients for Acoustic Multipole Sources on guided waves in Isotropic Tubular Structure Liquid Metal Sodium Filled

“…(1) is the field equation in isotropic medium. If the acoustic field is excited by an nth-order multipole source at r=0, the zero-order displacement potentials in the frequency wavenumber (ω,kz) domain satisfying the radiation (2) condition at infinity can be written as [10][11][12][13][14] Where Kn is the second kind of nth-order modified Bessel function, r0 is the multipole source separation and are the weighting coefficients.The acoustical potential in a tube for an nth multipole source can be written the same form as that in the isotropic medium [7][8][9][10][11][12][13][14] …”

“…However, their perturbation methods cannot be used to investigate the characteristic of the full-wave as well as P and S waves. Zhang et al [6][7][8][9] presented another perturbation method that can treat the full-wave acoustic field for TIM with the principal axis perpendicular to the borehole axis. They obtained zero-and first-order approximation solutions for the acoustic fields inside and outside the borehole.…”

A numerical investigation of the acoustic mode waves in a deviated borehole penetrating a transversely isotropic formation

“…Dispersion curves can be obtained from the results in time domain [18][19]. The perturbation method is also used when the TI formation shows weak anisotropy [3,[20][21][22]. Laboratory measurement is also realized to observe the wave propagation phenomena [23,24], especially for the flexural waves excited by a dipole source.…”

Full waveform investigation of a perturbation method for acoustical multipole logging in transversely isotropic medium

“…However, their perturbation methods cannot be used to investigate the characteristic of the full-wave as well as P and S waves. Zhang et al [7][8][9] presented another perturbation method that can treat the full-wave acoustic field a E-mail: dbcy9999@163.com, wangxt45@gmail.com 2158-3226/2014/4(9)/097130/9 C Author(s) 2014 4, 097130-1 for TIM with the principal axis perpendicular to the borehole axis. They obtained zero-and firstorder approximation solutions for the acoustic fields inside and outside the borehole.…”

“…[7][8][9] Not only the zero and the first order but also the second order approximation perturbation solutions are obtained when TIM's principal axis is parallel to the borehole axis. The asymptotic characteristic of each order of the perturbation solutions outside the borehole are discussed when the radial variable r is changed.…”