Monopole acoustic logging while drilling (LWD) enables the direct measurement of S-wave velocity in slow formations, which has been corroborated by recent theoretical and experimental studies. However, this measurement is hampered by the weakness of the S-wave signal and the lack of techniques to amplify it. To address this challenge, we analytically compute the monopole LWD wavefields, considering centralized and off-center tools in various slow formations. Modeling analysis reveals that four parameters primarily influence the excitation of the formation S wave: the formation S-wave velocity, the source-to-receiver distance, the radial distance from the receiver to the wellbore, and the source frequency. S-wave signals can be enhanced by judiciously optimizing these parameters during tool design. Furthermore, our research suggests that the S-wave velocity can be accurately extracted through the slowness-time correlation method only when formation S-wave velocities are in a suitable range. This is because an overly high S-wave velocity causes shear arrivals to interfere with the inner Stoneley mode, whereas an ultraslow formation S-wave velocity results in S-wave signals too faint to detect. For the LWD model with an off-center tool, simulations demonstrate that tool eccentricity, especially large eccentricity, can amplify the S wave and improve its measurement accuracy, provided that waveforms received in the direction of tool movement are used. In a very slow formation, we successfully extract the S-wave velocity from synthetic full wave data at that azimuth under conditions of large eccentricity, a task not achievable with a centralized instrument.

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