Asymptotic methods for the first compressional head wave arrival in a fluid-filled borehole

Asymptotic results are obtained for the first compressional head wave arrival in a fluid-filled borehole with a modified asymptotic technique. The results are verified with direct numerical integrations and shown to be quite different from those obtained with the ordinary asymptotic techniques. The modified asymptotic technique accounts for the effect of an extraneous pole, which is neglected in the ordinary asymptotic method. The basic assumption in the modified asymptotic method is that the observation point must be far away from the source point, whereas in the ordinary asymptotic technique, the frequency is assumed to be large. The pressure responses of the first compressional arrival are illustrated and compared. It is found that the amplitudes of the compressional arrivals are strongly dependent on the Poisson’s ratio and the compressional velocity of the formation.