Reflection of drill-string extensional waves at the bit-rock contact

Downward propagating extensional waves are partially reflected at the bit-rock contact. The evaluation of the reflection coefficient is important to obtain while drilling information about the acoustic properties of the formations. The scope of this work is to estimate the bit-rock reflection coefficient, assuming a flat drill bit in perfect contact with the formation. Using the low-frequency approximation, which holds when the wavelength is much larger than the lateral dimensions of the borehole, the drill-string is assumed to be a laterally free rod, and the formation an homogeneous and isotropic medium. This work shows that the reflection coefficient of the extensional waves depends, along with the elastic properties of the formation, on the ratio of the cross sections of the drill-string and borehole. The impedance of the drilled rock can be calculated from the measured reflection coefficient, which is related to the amplitude of waves produced in the string and in the formation by a working drill-bit.