Synthetic waveforms of axial motion in a borehole with drill string

This work studies the wave motion in a fluid-filled borehole in the presence of drill string and geological formation. The synthetic waveforms are obtained by a three-dimensional axis-symmetric full-wave numerical simulation in a two-dimensional multi-domain where the medium is uniform with respect to the azimuth. The discretization is performed in cylindrical coordinates. In order to simulate the waves at the origin (axis of the polar radius), a very small radius is used to avoid the singularity. The free-surface and rigid boundary conditions are tested and it is shown that the rigid one constitutes the best approximation. The simulations provide the amplitude distribution and motion diagrams in the borehole vertical cross-sections and at the outer boundary, away from the borehole. Propagation in the presence of hard and soft formations is analysed. The dispersion, the amplitude, and the orbital polarization of the modes excited by a point source acting in the fluid inside a drillstring are considered and examples of comparison with literature results obtained using multi-modal analysis are shown. The proposed approach is more general than the multi-modal analysis, since it allows for arbitrary variations of the properties in the plane of symmetry.