On the group velocity of guided waves in drill strings

A key element to verify the drill-bit signal processing is the estimation of the time delay of the guided waves transmitted through the drill string from the bit to the surface. This delay can be measured from drill-bit vibrations or calculated using the mechanical properties of the drill string. The calculation of the complete solution and the exact dispersion relations requires expensive numerical modeling. In this work, a simple approach to compute the group velocity of extensional and torsional waves by averaging the drill-string properties in the low-frequency approximation is used. The result is a generalization of the group velocity for periodic systems in which stationary distribution of properties is assumed. Results obtained with real strings show that the group velocity in the bottom-hole assembly is greater than the velocity in a uniform rod and lower than the velocity in the drill pipes.