各向异性地层多极源声测井理论与数值研究

Theoretical and Numerical Analysis for Acoustic Multipole Logging in Anisotropic Medium

作者: 专业:声学 导师:张碧星 年度:2008 学位:博士 

关键词
各向异性 摄动 声学 测井

Keywords
anisotropy, perturbation, acoustics, logging
        本文用摄动积分和摄动展开方法分别研究立方晶系(CCM—Cubic Crystal Medium)、横向各向同性(TI—Transversely Isotropic )介质对称主轴与井轴平行模型( VTI - Vertical Transversely Isotropic)和横向各向同性介质对称主轴与井轴垂直模型(HTI—Horizontal Transversely Isotropic)的各向异性地层下井孔声场的理论求解和数值计算。论文首先将摄动积分方法应用到立方晶系各向异性地层-井孔模型,主要对Stoneley波的反常频散效应及螺旋波与多极源方位角的关系等问题进行了研究和分析。然后,利用摄动展开法对TI介质声测井问题进行了全波场摄动展开、摄动求解及数值模拟研究。在VTI介质模型下,得到了精确到二级近似的全波场摄动解析解,同时与将严格解作泰勒级数展开的结果进行比较,理论上验证了摄动求解一级近似的可靠性。通过数值计算发现在10%的摄动范围内,摄动解和严格解吻合较好,能给出较高的计算精度。在HTI模型下,得到了全波场的零级和一级摄动解,通过摄动解数值模拟了井中多极源的激发声场,研究了井中各分波的传播特性,分析了各分波与多极源方位的关系。论文还用频率-波数法对TI介质井孔声场进行了分析,在VTI和HTI模型下得到了井中波场的频率-波数分布图和频率-相速度分布图。
    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.
        

各向异性地层多极源声测井理论与数值研究

提要5-8
第一章 绪论8-21
    1.1 引言8-9
    1.2 多极源声波测井的发展9-13
    1.3 各向异性介质地层中的多极声测井研究13-16
    1.4 介质对称主轴与井轴不平行的地层多极源声波测井16-18
    1.5 各向异性多极源声波测井问题的进一步发展探析18-19
    1.6 本文的主要内容19-21
第二章 各向异性介质中弹性波的基本理论21-27
    2.1 均匀各向异性弹性固体介质21-22
    2.2 几种各向异性介质22-27
第三章 立方晶系各向异性地层井孔声多极模式波频散的摄动积分理论计算与数值分析27-42
    3.1 引言27
    3.2 立方晶系各向异性介质中的弹性动力学27-30
    3.3 用摄动积分计算立方晶系各向异性地层井孔导波的频散30-35
    3.4 数值计算与分析35-40
    3.5 小结40-42
第四章 摄动展开方法在 VTI 介质-井孔模型中的应用42-71
    4.1 VTI 介质-井孔模型的摄动求解45-55
    4.2 VTI 介质严格解的泰勒展开55-62
    4.3 数值模拟与分析62-69
    4.4 小结69-71
第五章 摄动展开方法在 HTI 介质-井孔模型中的应用71-92
    5.1 位移势方程71-73
    5.2 摄动展开73-75
    5.3 柱坐标中位移势的表示75-77
    5.4 应力应变关系77-78
    5.5 柱坐标系中的应力分量表示78-81
    5.6 总位移和应力81-82
    5.7 井孔中的波场82-83
    5.8 边界条件83-85
    5.9 数值分析85-90
    5.10 小结90-92
第六章 VTI 和 HTI 介质井孔中声场的频率-波数分析92-104
    6.1 VTI 模型下频率—波数分析92-96
    6.2 HTI 模型下频率—波数分析96-103
    6.3 小结103-104
第七章 全文总结104-108
附录108-115
攻博期间发表的学术论文及其他成果115-116
致谢116-118
参考文献118-124
中文摘要124-127
Abstract127-130


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