摘要
An approach of separating individual wave arrivals for a dipole logging is presented. The branch points are treated as a type of logarithm and the Riemann surface structure of the multivalued function is studied, that is, the displacement potential within the borehole. Based on the theoretical analysis, the complex poles contributing to the wave field on various Riemann sheets are investigated in detail for the case of a fast formation. It is shown that poles on Riemann sheet (0,0) are real and form branches of modes with dispersion. Mathematically, it is demonstrated that the flexural mode has no cutoff frequency, which is different from the traditional point of view. Poles on other relevant Riemann sheets are complex and form many branches on the complex frequency-wavenumber plane. Further investigation on the pole and branch cut contributions indicates that the vertical branch cut integration method has limitations in separating wave arrivals. By properly taking into account the complex poles on various Riemann sheets together with branch cut integrations, wave arrivals are separated from the full wave-forms effectively for both the fast and slow formation models. Specially, there are complex poles on Riemann sheet (0,-1) in the vicinity of the compressional branch cut for a slow formation with a large Poisson's ratio, which have small imaginary parts and contribute a lot to the p-wave arrival.
An approach of separating individual wave arrivals for a dipole logging is presented. The branch points are treated as a type of logarithm and the Riemann surface structure of the multivalued function is studied, that is, the displacement potential within the borehole. Based on the theoretical analysis, the complex poles contributing to the wave field on various Riemann sheets are investigated in detail for the case of a fast formation. It is shown that poles on Riemann sheet (0,0) are real and form branches of modes with dispersion. Mathematically, it is demonstrated that the flexural mode has no cutoff frequency, which is different from the traditional point of view. Poles on other relevant Riemann sheets are complex and form many branches on the complex frequency-wavenumber plane. Further investigation on the pole and branch cut contributions indicates that the vertical branch cut integration method has limitations in separating wave arrivals. By properly taking into account the complex poles on various Riemann sheets together with branch cut integrations, wave arrivals are separated from the full waveforms effectively for both the fast and slow formation models. Specially, there are complex poles on Riemann sheet (0,?1) in the vicinity of the compressional branch cut for a slow formation with a large Poisson’s ratio, which have small imaginary parts and contribute a lot to the p-wave arrival.
基金
Supported by the National Natural Science Foundation of China (Grant No.10534040)
