Theoretical simulations of wave field variation excited by a monopole within collar for acoustic logging while drilling

To extract the compressional (P) wave signals from the collar modes in logging while drilling (LWD) data, it is significant to investigate the propagation of those two arrivals separately. Yet most simulation studies on LWD, however, assumed all transducers were placed in fluids, and produced much weaker collar signals than those in reality. The underestimate of collar waves was caused by the decoupling of the transducers and the tool in the computational models. In this work, we analytically investigate the wave fields generated by a monopole within the collar, an LWD model more appropriate. The collar and the P waves are simulated by evaluating the contributions from the poles and the branch point of the characteristic function, respectively. When both sources and receivers move together from the tool surface to interior, the collar waves could become much stronger especially at high frequencies, while the P wave amplitudes tend to be weaker. With a fixed source within the collar, it is shown that the collar mode energy concentrates in the collar and spreads through the whole layered structure. Meanwhile, the P waves received within the collar are even greater than those near the borehole wall, revealing that the P signals are also a kind of waveguides in the LWD model, like the collar waves. We investigate the influences of model properties on the wavefields with different physical parameters. The formation P wave reaches the resonance frequency at which its speed equals to the collar-wave phase velocity. The borehole fluid with greater impedance leads to weaker collar waves inside the tool and stronger P waves in all layers. By comparisons of collar waves in infinite fluids and in a borehole, propagation mechanisms of two kinds of collar waves in the LWD models are revealed. Besides the direct collar waves propagating all along the tool, there is an indirect collar mode contributed by reflections from the borehole wall. Both the direct and the indirect tool modes are of similar magnitudes in varying spaces. Therefore, one should take the presence of borehole into account during simulating or testing the performance of an LWD sonic tool. Otherwise, the tool waves will also be far underestimated due to the indirect collar waves being neglected.