A simple method for fast simulation of total reflection waves of single-well imaging with eccentric tools

In the exploration of unconventional oil and gas reservoirs, single-well acoustic imaging has become an important technique to detect geophysical structures around the borehole. In this study, we propose a simple method to calculate the total reflection wavefields of pressure and displacement for single-well imaging (SWI) with eccentric monopole or dipoles. To calculate the pressure at the eccentric site for incident waves with different angles of incidence and azimuth, reciprocity is applied to the radiation from the eccentric monopole source and the far-field forces generating the waves corresponding to the incident waves. Displacement solutions are derived by calculating the gradient of the pressure fields. When considering eccentric monopole or dipole sources for an inclined reflector outside the borehole, reciprocity is again used to derive the total reflection waves, including the same type of waves (P-P/SV-SV/SH- SH) and the converted waves (P- SV/SV-P). The analytical results agree well with the reference solutions of the 3D finite-difference time-domain method, illustrating the validation of the method. The analysis shows that the pressure amplitude in the eccentric position is greater than that in the borehole axis, proving that the eccentric monopole can increase the weak amplitude of reflection waves. We also simulate the wavefields for the acoustic imaging model for eccentric monopole and dipoles. The results show that the pressure and displacement wavefields in the eccentric position are sensitive to the azimuth angle of the reflector, proving that SWI with eccentric logging tools is capable of identifying the azimuth angle.