Numerical evaluation of the transient acoustic waveform due to a point source in a fluid-filled borehole

A key measurement employed in oil well wireline logging is the acoustic wave traveltime over a specified formation interval, typically 1 ft. In the traditional measurement, only the compressional head wave is monitored, but for some time it has been obvious that there is significant additional information, such as the shear head wave arrival, in the received waveform. We describe two numerical methods for computing the profile and parameter dependence of the transient waveform based on a model of the acoustic logging problem consisting of a point source on the axis of a fluid-filled cylindrical borehole. The response to this excitation is determined at a distance from the source, generally on the borehole axis. In the first of the two numerical methods, called 'real axis integration', the complete acoustic waveform is obtained. The second method, called 'branch-cut integration', evaluates the first compressional and shear-pseudo-Rayleigh arrivals individually with much less computation time than the first method. The validity and accuracy of the two methods are demonstrated by their close agreement within appropriate time windows. It is also shown that the results from the ordinary asymptotic method that exist in the literature predict different behavior. The dependence of the transient arrivals on formation parameters is illustrated by various numerical results in both time and frequency domains.