Multipole borehole acoustic theory: Source imbalances and the effects of an elastic logging tool Tim W. Geerits a, ⁎, Xiaoming Tang b , Olaf Hellwig c , Thomas Bohlen c,d,1 a Baker Hughes, Celle Technology Center, 29221 Celle, Germany b Baker Hughes, Houston Technology Center, 77073 Houston (TX), USA c Institut für Geophysik, Technische Universität Bergakademie Freiberg, 09596 Freiberg, Germany d Karlsruher Institut für Technologie (KIT) Geophysikalisches Institut, Hertzstr. 16, 76187 Karlsruhe, Germany abstract article info Article history: Received 2 March 2009 Accepted 31 October 2009 Keywords: Borehole geophysics Multipole acoustic source Source imbalances Borehole guided modes Acoustic logging tool In recent years the emphasis in acoustic logging has been shifting from the wireline to the Logging While Drilling (LWD) environment, the latter being far different from the former in that both tool rigidity and tool radius are considerably greater. In this paper we present a generic mathematical formulation for the multipole borehole acoustic measurement (alternate and equal polarity case), including a detailed analysis on the effects of multipole source amplitude imbalances. It is shown that source imbalance induced mode contaminants have excitation amplitudes that are scaled by the sum of the relative source imbalances between diametrically opposed sources. Furthermore it is shown that for mode contaminants with odd modal number there can be a significant offset in the associated directivity pattern, even at low levels of source imbalance. However, it is also shown that source imbalance induced mode contaminants can be completely eliminated if the multipole source is accompanied with a ‘vertically’ (but not azimuthally) offset multipole receiver. Mathematically, it is demonstrated (for a centered tool) that a polarity weighted stack of these multipole receivers completely eliminates the source imbalance induced mode contaminants. Excitation amplitudes and phase slowness of borehole guided modes are presented for the most common excitation regimes, i.e., monopole, dipole, quadrupole and the hexapole excitation, the latter one showing to have the advantage of a higher formation shear cut-off frequency than the quadrupole mode. Special emphasis will be on the analysis of the dipole excitation and the differences that occur due to variations in tool rigidity, tool diameter and (isotropic) formation properties with the resulting conclusion that a formation flexural mode is not observable as a result of a LWD dipole excitation (this opposed to its dipole wireline counterpart). The guided mode excitation amplitudes are calculated as residues using a Laurent series expansion. This (unconventional) way of calculating the residue has the advantage that it is independent of the pole order, does not require the numerical evaluation of derivatives with respect to the vertical wavenumber and allows for an accurate and efficient FFT implementation. © 2009 Elsevier B.V. All rights reserved. 1. Introduction In recent years the emphasis in acoustic logging has been shifting from the wireline to the Logging While Drilling (LWD) environment, the latter being far different from the former in that both tool rigidity and tool radius are considerably greater (i.e., assuming the same borehole radius, formation and borehole fluid). For the LWD case, this has led to the overall conclusion that in slow formations, where the borehole fluid slowness is smaller than the formation shear slowness, conventional dipole logging (Varsamis, 1999) as practiced in the wireline environment is inadequate to obtain (true) formation shear slowness. Instead, quadrupole logging has been proposed (Tang et al., 2002) as a solution to this problem and has generally been accepted as the new standard for shear slowness determination in the LWD environment. Not only, to understand the dipole and quadrupole measurements, but also to appreciate the multipole measurements in general, this paper presents a rigorous mathematical formulation. Although the formulation is based on previous work, this formulation distinguishes itself from previous work, in the following ways: The mathematical description of the acoustic multipole source was first presented by Kurkjian and Chang (1986). In their formulation two assumptions were made that are not necessarily realistic when describing the LWD borehole acoustic multipole measurement: (1) the radial wavelength is assumed large compared to the multipole radius. (2) the source signatures of the individual volume injection sources that make up the acoustic multipole source Journal of Applied Geophysics 70 (2010) 113–143 ⁎ Corresponding author. Tel.: +49 5141 2036884. E-mail address: tim.geerits@bakerhughes.com (T.W. Geerits). 1 Tel.: +49 721 6084416. 0926-9851/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jappgeo.2009.10.004 Contents lists available at ScienceDirect Journal of Applied Geophysics journal homepage: www.elsevier.com/locate/jappgeo