Theoretical And Numerical Analysis For Acoustic Multipole Logging In Anisotropic Medium

A cylindrical coordinate system (r,θ, z) oriented to the borehole axis is introduced. The z axis is on the borehole axis. A transversely isotropic medium whose symmetric principal axis is parallel to the borehole axis can be gained the exact solution. In this model, SH wave whose propagating velocity is along the borehole axis equals to SV wave whose propagating velocity is along the borehole axis. So the shear wave birefringence and the propagation of the flexural wave splits in borehole can not been observed. In fact, a transversely isotropic medium whose symmetric principal axis is not parallel to the borehole axis is always in existence, SH wave whose propagating velocity is along the borehole axis not equals to SV wave whose propagating velocity is along the borehole axis. So the shear wave birefringence can be observed. But the borehole-medium model has no axial symmetry and the acoustic fields inside and outside the borehole may not be studied analytically because r andθare coupled and can not be separated.This kind of model is the best approximate to fact medium model in the borehole. Therefore, some approximate analytical methods and pure numerical simulation have to be presented. But only a few researches to study this kind of case because the borehole-medium model has no axial symmetry become the most difficult problem influence the excitation and transmission of acoustic fields.In this Thesis, the models are liquid in a borehole surrounded by a cubic crystal anisotropic medium (CCM), a transversely isotropic medium whose symmetric principal axis is parallel to the borehole axis (VTI) and a transversely isotropic elastic solid whose symmetric principal axis is perpendicular to the borehole axis (HTI). The perturbation integral method is adopted to analyze the dispersion characteristic the guided waves in a borehole surrounded by a CCM. An extended analytical perturbation method is presented for the multipole acoustic logging in VTI medium and HTI medium to the displacement fields. Full-waveforms fields are studied and analyzed by a sonic tool in a well theoretically and numerically.A cubic crystal anisotropic medium (CCM) is a kind of anisotropic medium (White, 1983). There are only three absolute elastic constants under the orthogonal coordinate system, but the borehole-medium model has no axial symmetry and the acoustic fields inside and outside the borehole may not be studied analytically. In this Thesis, the perturbation method is adopted to analyze the guided waves in a borehole surrounded by a cubic crystal medium. Based on the theory of dispersion of phase velocity of mode waves surrounded by isotropic formation in a borehole, the dispersion characteristic of Stoneley wave, Pseodu-Rayleigh wave, flexural wave, and screw wave is investigated in detail. We give the dispersion curves of the Stoneley wave for different perturbation quantities based on the fast formation especially. It can be seen that the difference of the dispersion curves between the reference and perturbation states decreases as the perturbation quantity decreases. It also shows that the dispersion curves displays as the characteristics of slow formations when the perturbation quantity is greater than 3%. It is also found that dispersion of the guided waves excited by monopole and dipole sources does not depend on the azimuth of the source but the dispersion of screw wave excited by quadrupole source has a significant relation to the azimuth of the source. It shows that screw waves propagated along different azimuth in the borehole can be split.An extended analytical perturbation method is presented for the multipole acoustic logging in a transversely isotropic medium whose symmetric principal axis is parallel to the borehole axis (VTI). Take A and L as two moduli for the reference unperturbed isotropic state and introduce three perturbation quantities, not only the zero- and first- order but also the second- order approximation perturbation solutions are obtained strictly in analytical form. And Taylor series expansions are proved practical in first- order approximation perturbation solutions. The acoustic full-waveforms inside the borehole excited by a monopole, dipole, and quadrupole sources are numerically investigated by the extended analytical perturbation method and the exact solution by programming software ourselves. It is found that the perturbation method in the second-order approximation can obtain a good result (the perturbation quantities are about 10%). It also shows that the extended analytical perturbation method is a feasible method and can apply to the complicated anisotropic acoustic logging.An effective extended analytical perturbation method is presented for the multipole acoustic logging in a transversely isotropic elastic solid whose symmetric principal axis is perpendicular to the borehole axis because of absenting the exact solution. The horizontal transversely isotropic elastic solid is regarded as a relevant isotropic elastic solid added to the perturbations, and three perturbation quantities about moduli deviated from the isotropic medium are introduced. By selecting a group of displacement potentials and a cylindrical coordinate system oriented along the borehole axis, the zero- and first-order perturbation solutions of the multipole acoustic field are derived for the weak transversely isotropic elastic solid which has its symmetric principal axis perpendicular to the borehole axis. The acoustic fields inside and outside the borehole excited by a multipole source are investigated. It shows that waves propagated excited by monopole, dipole sources and quadrupole sources along different azimuth in the borehole can be simulated numerically. It is found that not only the nth multipole acoustic field can be excited by an nth multipole source, but also multipole fields whose orders are higher and lower than n also can be excited by an nth multipole source. These conclusions benefit more the other anisotropic formations than the transmission of acoustic fields.At the same time, the dispersion of the guided waves in the second and the third models is analyzed with the method of the frequency-wavenumber (f-k) analysis. The acoustic frequency-wavenumber (f-k) analysis and dispersion characteristic inside the borehole excited by a monopole, dipole, and quadrupole sources are numerically investigated by the perturbation method in VTI and HTI medium. The acoustic frequency-wavenumber (f-k) analysis of reflective field according to different orders are calculated and analyzed. The acoustic frequency-wavenumber (f-k) analysis method benefits more the analyzing full-waveforms than the transmission of acoustic fields.