Shu-Kong Chang

Acoustic multipole sources in fluid-filled boreholes

Acoustic well logging methods historically have been based on the excitation and reception of axisymmetric wave phenomena in a fluid-filled wellbore. We consider the reception of nonsymmetric wave phenomena excited by acoustic multipole sources. The first three orders of multipoles are the monopole, dipole, and quadrupole, and we examine these particular sources in detail. Existing sonic tools make use of a monopole source, while more recently, both dipole and quadrupole sources have been explored. An exact frequency-wavenumber domain representation of the acoustic field in the borehole due to a multipole source is formulated and numerical methods are used to compute synthetic space-time domain waveforms. We consider wideband monopole, dipole, and quadrupole excitations with center frequencies of 1, 4, and 12 kHz, and treat both slow and fast formation models. Finally, we derive low-frequency, far-field asymptotic expressions for the monopole, dipole, and quadrupole waveforms. At frequencies such that the shear wavelength is on the order of the borehole diameter or less, the difference between the monopole, dipole, and quadrupole waveforms is primarily in the nature of the surface wave mode which they excite: the monopole excites a Stoneley, or tube mode; the dipole excites a flexural mode; and the quadrupole excites a screw mode. By comparison, the compressional and shear head waves and the trapped waveguide modes do not change as much as the order of the multipole is changed.At low frequencies, where the shear wavelength is much longer than the diameter of the hole, the monopole excites a dominating tube mode, while the dipole and quadrupole excites dominating shear waves. Low-frequency asymptotic expressions for the waveforms agree well with the numerically computed waveforms.

Arrays of synthetic acoustic well logging waveforms: Computation and source design

Arrays of synthetic acoustic well logging waveforms are useful in the initial development of signal processing methods, as an aid in understanding actual well logging data, and in an analysis-by-synthesis approach to data inversion. In this paper, we present a fast and accurate method for the numerical evaluation of arrays of synthetic acoustic well logging waveforms associated with a homogeneous solid formation model. The method evaluates the two-dimensional Fourier transform of the modal coefficient which is expressed in terms of the cylindrical wave reflection coefficient and cylindrical wave impedances. A table look-up and interpolation scheme for the evaluation of ratios of Hankel functions for complex-valued arguments is responsible for a factor of three reduction in the overall computation time required to generate synthetic data, relative to the evaluation of these ratios using a Hankel function evaluation algorithm. This substantial savings easily outweighs the modest increase in storage which is required for the table. A low frequency constraint on the temporal response of the source is presented to ensure that the synthetic waveforms tend to zero for large time.