Higher-order modes in a fluid-filled borehole

Sonic techniques in geophysical prospecting involve elastic wave velocity measurements that are performed by placing acoustic transmitters and receivers in a fluid-filled borehole. The signals recorded at the receivers are processed to obtain compressional and shear wave velocities in the surrounding formation. These velocities are generally used in seismic surveys for the time-to-depth conversion and other formation parameters, such as porosity and lithology. Depending upon the type of transmitter used and as a result of eccentering, one can excite axisymmetric (n=0), flexural (n=1), quadrupole (n=2), and hexapole (n=3) family of modes propagating along the borehole. We present a study of various propagating and leaky modes that includes their dispersion and attenuation characteristics caused by radiation into the surrounding formation. These propagation characteristics help in a proper selection of transmitter bandwidth for suppressing unwanted modes that create problems in the inversion for the compressional and shear velocities from the dispersive arrivals. Computational results for the axisymmetric family of modes in a fast formation with its shear velocity of 2032 m/s show the existence of Stoneley, pseudo-Rayleigh, and anharmonic cutoff modes. In a slow formation with its shear velocity of 508 m/s, we find the existence of Stoneley, and the first leaky-compressional mode which cuts in at approximately the same normalized frequency wa/V_S=2.5 (a is the borehole radius), as that of the fast formation. The corresponding modes among the flexural family include the lowest-order flexural and anharmonic cutoff modes. For both the fast and slow formations, the first anharmonic mode cuts in at a normalized frequency wa/V_S =1.5 approximately