A fast algorithm for simulation and analysis of wavefields in acoustic single-well imaging of logging-while-drilling considering arbitrary types of sources

Acoustic single-well imaging (SWI) of logging-while-drilling (LWD) is an advanced logging method in reservoir exploration, which uses reflected waves to detect the around-borehole geological structures and quickly determines the drilling direction for enhancing the drilling-encounter ratio and reducing the drilling risk. Forward acoustic modelling is a fundamental problem for SWI in LWD. Due to the complex structures, it is a challenge to simulate the wave propagation and investigate wavefield characteristics based on the forward model. Numerical modeling is a commonly used method for calculating wavefields, however it is too computationally expensive. In this study, we propose a fast method for calculating the full reflected pressure and displacement waves (i.e., P-P, SV-SV, SH-SH, and P-SV/SV-P) in SWI of LWD considering different types of sources including arcuate, monopole and dipole transmitters. The analytical algorithm is proposed by applying the reciprocity relation between the virtual force (displacement) sources located at the receiver position and the outside-borehole virtual forces which are equivalent to the reflections from the formation interfaces. Numerical experiments show that the analytical solutions agree well with the reference solutions from 3D finite-difference time-domain method, demonstrating the accuracy and high efficiency of the analytical method. Based on the analytical solutions, we find that LWD reflected waves are much more sensitive to the azimuth than those in the wireline case, showing that the availability of LWD is important for identifying the reflector azimuth. Furthermore, to enhance the reception efficiency of reflected waves, we present the optimized LWD parameters: For slow formations, we suggest using a dipole source with dominant excitation-frequency band being from 1 kHz to 3 kHz; For fast formations, a dipole with wider excitation-frequency band from 1 kHz to 5 kHz is recommended; For all formations, recording pressure signals shows much higher reception efficiency than the displacement signals.