Single-valued definition of the multivalued function for borehole acoustic waves in transversely isotropic formations

Abstract

It is useful to extract all components, including compressional, shear, and guided waves, from the full waveforms when we investigate the acoustic log data. The component waves can be simulated by calculating the contributions from poles and branch points of the borehole acoustic function according to Cauchy's theorem. For such an algorithm to be implemented, the multivalued function for the borehole wave field in the frequency-axial-wavenumber domain has to be rendered single-valued first. Assuming that the borehole axis is parallel to the symmetry axis of transverse isotropy, this paper derives the branch points of the borehole acoustic function. We discover that the number and the locations of those branch points are determined by the relation among the formation parameters c 33 , c 44 , ε, and δ. Thus the single-valued definitions in the acoustic-wave computation are sorted into two different cases. After building the Riemann surface related to each radial wavenumber, we give the single-valued definition of the borehole acoustic function inside and on the integration contour based on the radiation condition.In a formation with δ > ε + c 44 /2c 33 , if we choose the integration contour and the single-valued definition of the acoustic function in the way used in isotropic cases, the simulation results of component waves will be wrong.anisotropy, acoustic logging, elastic wave, Riemann sheet PACS: 43.20.Fn, 43.40.Ph, 91.30.Fn In the theoretical research on acoustic logging, it is important to simulate and analyze the borehole full wave and the component waves within. To investigate wavefields varying with the time and space, transforming the acoustic function ϕ(ω, k z ) in the frequency-axial-wavenumber domain into ϕ(t, z) in the time-space domain is required. There are two common ways to implement such a computation process: one is the real axis integration [1] from which we can directly obtain the full waveforms; the other way is to calculate the total wavefield as a sum of contributions from both poles and branch points in the complex axial-wavenumber plane according to Cauchy's theorem [2]. By the latter method, all component waves, including the compressional, shear, and guided waves, can therefore be separated from the full waveform in the simulation. The acoustic function ϕ(ω, k z ), however, is a multivalued function which has many values with single k z and ω, so the contour integration on the complex plane cannot be evaluated unless the integrand is single-valued. Kurkjian [3] and Liu et al. [4] have discussed the characteristics of the borehole acoustic functions for wavefields in elastic and quasi-elastic media, respectively. They gave the single-valued choices of the acoustic function on the vertical integration paths related to the branch points. Zhang and Wang [5,6] analyzed the algebraic and logarithm multivalued features of the radial wavenumbers and the modified Bessel functions of the second kind, respectively. They calculated all poles on each sheet of the Riemann surface. In their numeri...

Component wave calculation and analysis of the acoustic field in a borehole within a three-phase porous medium

“…According to Cauchy theorem, the real axis integral is converted into the contribution of contour integral and pole residue. The path of integration coincides with the steepest descent path and passes through several different Riemann sheets 29 . A Riemann surface is composed of several Riemann sheets, and is single-valued on each Riemann sheet.…”

“…According to the work of Zhang et al 22,23,26,28 , He 29 and Zheng 24 , when the center frequency of the source is high, the results of branch-cut integration of S1 wave cannot be used to explain the wave. The reason is that the branch-cut integral ignores the contribution of the pole near the branch cut, and the calculation of the integral alone cannot truly reflect the waveforms of the S1 wave in the propagation.…”