Analytical Simulation of the Waves Reflected from A Distant Slanted Interface Outside A Borehole

The wave field of single-well imaging is asymmetric. It takes a lot of time (hours or even longer) to calculate the wave field by numerical algorithm, which cannot meet the needs of real-time processing of logging data. In order to solve the problem, the method of steepest descent and reciprocity are used to obtain the analytic solution of the wave field in the borehole stimulated by the reflected wave field. The waves radiated by a monopole or a dipole acoustic source in a borehole is first formulated by ignoring the reflector, by applying the boundary conditions at borehole wall. These radiated waves are approximated by the steepest descent path method, which is quite exact when the waves travel far from the borehole. These waves are then reflected back by the reflector. The reflection waves, including P, SV and/or SH waves, are taken as radiated from image sources. Finally the waves arriving an array of receivers in the borehole are obtained by applying reciprocity relations. The analytical waveforms agree well with those by three dimensional finite difference time domain method.