Multipole borehole acoustic theory: Source imbalances and the effects of an elastic 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.