Flexural waves in drill-string tubulars with variable loads

We revisit the equations governing the bending motions in thin rods and analyse the filtration of flexural waves in vertical drill strings pre-stressed by gravity. The aim is to study transverse drill-string vibrations at seismic frequencies for acoustic communication purposes and provide an algorithm for processing reflected and transmitted bending motions generated by downhole lateral vibrations. We obtain the dispersion equation, including attenuation due to a gravity pre-stress gradient and frequency-dependent reflection and transmission coefficients at the interface between subsequent tube intervals. We then develop a propagation-matrix algorithm to simulate flexural waves in a drill string consisting in an assembly of multiple tube sections of different dimensions. The deflection vibrations are obtained at any arbitrary recording point in the drill string. The modelling is cross-checked with a full-wave grid algorithm. The analysis shows that the waves produced by a concentrated force are partitioned in standing and propagating modes, which are calculated by using the flexural impedance of the drill string. Moreover, the reflection coefficients weakly depend on the pre-stress conditions and pre-stress has important effects for far-field signal transmission with variable weight on bit (WOB). We discuss the approximations and limits of the method with respect to realistic drilling conditions.