横观各向同性地层声波测井波场模拟与地层渗透率反演
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摘要
声波测井是地球物理测量和油气储层勘探的主要方法之一。为了改进测井技术、根据测井信号准确反演地层信息,必须深入研究弹性波在流-固柱面分层介质中的传播规律。鉴于地壳表层岩石通常呈现横观各向同性(TI),本文以TI地层声波测井为应用背景,在对井孔声场进行解析和数值模拟的基础上,分析柱面导波的激发、频散和衰减,研究由井内波场反演井外地层渗透率的方法。主要进行了以下几方面的工作:
首先对弹性地层中井轴平行于TI对称轴的井孔(竖直井)声场进行了分析,其中各模式导波频散曲线由井孔声场函数的极点获得。前人的算例结果显示在TI介质井孔中单极源伪瑞利波、偶极源弯曲波和四极源螺旋波在低频极限处的速度等于井孔方向上的横波速度,但缺乏理论证明。本文讨论了多值声场函数在复平面内的支点分布,并根据井内声波在流体-TI固体边界上产生透射横波的条件论述了伪瑞利波、弯曲波和螺旋波的轴向速度上限。结合TI弹性地层井孔声场的模拟结果,证明了当TI地层参数满足δ>ε+c_(44)/2c_(33)时上述导波的低频极限速度小于地层的横波速度,其中c_(33)和c_(44)分别为TI地层对称轴方向上的纵波和横波模量而ε、δ是各向异性系数。时域全波的模拟结果证实了在这类地层中由导波到达时间所测得的地层横波速度总是小于其真实值。
然后,研究了TI孔隙地层竖直井孔中由单极源激发的声场,计算了井孔声场函数全部极点的分布并由此得到了包括泄漏模式和非泄漏模式在内的导波频散、衰减以及激发强度曲线。非泄漏模式波的轴向传播速度小于等于井孔方向上的横波速度,其衰减仅由弹性孔隙骨架和粘滞流体之间的相对运动而产生;相比之下,速度快于横波的泄漏模式波衰减非常迅速,这是因为它的衰减机制同时包含介质的本征耗散和机械能的几何泄漏。考察了地层、井孔和井内流体各参数分别对非泄漏模式波和临界折射波的影响大小。基于灵敏度分析的结果,在算例中通过拟合斯通利波的时域波形用最小二乘法同时反演了地层的水平渗透率和水平剪切模量。在理论模拟波形时,孔隙度和垂直渗透率的取值对反演精度的影响很小,这体现出反演方法的稳定性。
然而在实际钻井过程中,井轴往往与TI对称轴方向不一致。为了数值模拟这种情况下的井孔偶极声场,本文采用了三维时域有限差分(FDTD)法。由于快、慢横波的速度差异,在偏离TI对称轴的井孔中偶极子能相应地激发快弯曲波和慢弯曲波,形成弯曲波分裂现象。发现弯曲波的速度和相位与声源偏振方向无关,变化声源方位角只改变两类弯曲波的相对幅度。在大多数情况下,低频偶极子激发的快、慢弯曲波在速度上分别趋近于井轴向的快、慢横波速度。算例结果表明,即使在参数满足δ>ε+c_(44)/2c_(33)的地层中,当井孔倾角足够大时根据弯曲波的波至仍可准确提取地层的横波速度。此外,分别模拟了沿深度梯度非均匀以及水平分层介质中的井孔声场。
在渗透率呈各向异性的孔隙地层中,如果渗透率主方向与坐标轴不一致,渗流速度各分量都耦合在一起,传统的应力-速度交错网格无法将描述广义达西定律的方程组离散化。本文对FDTD算法作出了改进使得达西定律方程组依然可以写成差分方程的形式。在算例中模拟了渗透率各向异性地层中的井孔声场。当井轴与渗透率的TI对称轴不一致时,可根据正交偶极接收阵列得到的波形幅度及衰减判别岩层沿井周各方向上的径向渗透率大小,并依此确定地层定向裂隙的分布平面。在灵敏度分析结果的基础上,利用弯曲波的衰减反演了TI地层的渗透率。在竖直井孔中,利用弯曲波的衰减能反演地层的水平渗透率,但在反演精度的稳定性方面弯曲波不如斯通利波。在水平井中,利用水平偏振偶极子激发的弯曲波可反演水平渗透率;而根据垂直偏振偶极子的声场则可测得地层的垂直渗透率。当待测渗透率较高时,反演结果具有较高精度;但对于渗透率较低的地层,反演精度将明显受渗透率各向异性的影响,各向异性越强则渗透率的反演误差越大。这些研究为从正交偶极声测井数据中提取储层渗透率提供了参考。
Acoustic logging is a useful method of geophysical measurements and explorations for oil or gas reservoirs. To develop the logging technique and extract reservoir informations, researches on cylindrical elastic-wave propagation along a radially layered medium is of great practical importance. Transverse isotropy (TI) of sedimentary rocks is a typical characteristic in the lithospere medium. In the present thesis, full waves and component waves excited by a multipole source in a borehole is simulated by the analytical and numerical methods. The inversion scheme for formation permeabilities is put forward after studying the influence of each factor on the wave field.
The wavefields in a vertically extending wellbore are modeled first. In this case, the bore axis is parallel to the TI symmetry axis and the wave equations inside and outside the borehole can therefore be analytically solved. The dispersion of the guided waves can be investigated by finding poles of the borehole acoustic function. Previous works have taken for granted that guided-wave velocities of the pseudo-Rayleigh, flexural, and screw modes equal the shear speed at the low-frequency limits in TI media without proof. With the results of wavefield computations, however, it is revealed that if parameters of a TI formation satisfyδ>ε+c_(44)/2c_(33), asymptotic velocities of those guided waves are lower than the shear speed, where c_(33) and c_(44) are the moduli related to the compressional and shear wave along the TI symmetry axis, respectively, andεandδare the anisotropic coefficients. The simulated full-waves further show that in such a kind of formations, the measurement result of shear speed obtained from those guided-wave arrivals are always smaller than the true value.
Borehole wavefields in a fluid-saturated porous TI formation excited by a monopole source is then simulated. The dispersion, attenuation, and excitation of all modes, including the unleaky and leaky modes, are investigated. It is revealed that the attenuation of the unleaky modes is caused by the relative motion between the porous skeleton and the viscous pore-fluid while that of the leaky modes includes both the intrinsic dissipation and the geometric leak of mechanical energy. Hence the leaky modes, whose velocities are higher than the shear speed, attenuate rapidly as they propagate. Based on the simulation results of component waves, influences of all parameters on the guided waves as well as the critical refracted waves are investigated. Sensitivity analysis theoretically confirms that Stoneley wave can be used to measure the permeability and the shear modulus on the horizontal plane. In the numerical example, those two parameters are well extracted from Stoneley waveforms in the time-domain by solving a nonlinear least-squares problem. It is shown that the initial estimations of the porosity and the vertical permeability have little effect on the inversion results.
The borehole, however, is not parallel to the TI symmetry axis in most cases. Dipole acoustic waves in such a borehole are computed by a 3-D finite-difference time-domain (FDTD) algorithm instead of the analytical method. Due to the shear anisotropy, a dipole source can excite a fast and a slow flexural mode if the bore axis deviates from the TI symmetry axis. That is the flexural-wave splitting phenomenon. The dipole orientation has no relation with the velocities or phases of flexural waves. It influences the relative amplitudes of the two modes. In most situations, the fast and slow flexural modes in the low-frequency range travel at the speeds of the fast and slow shear wave, respectively. The numerical results show that even in the formations withδ>ε+c_(44)/2c_(33), extracting the shear velocities from the flexural arrivals is still possible if the borehole deviation is large enough. Moreover, the wavefields in gradient heterogeneous or horizontally layered formations are also modeled by the FDTD code.
For a permeability anisotropic porous model, once the principal direction of the TI permeability tensor is not consistent with the coordinate axis, all velocity components of pore-fluid flow are coupled together. And the equations of generalized Darcy’s law cannot be discretized using the conventional stress-velocity staggered grids. To solve such a problem, the modified FDTD algorithm is introduced to rewrite Darcy’s equations into the finite-difference formulations. Thus the dipole wavefields in a borehole deviated from the TI symmetry axis can be simulated. According to our numerical examples, it is concluded that the distributing plane of directed cracks can be detected from the attenuation and the amplitudes of flexural-wave signals at the cross dipole array of receivers. Based on the results of sensitivity analysis, it is found that the flexural attenuation can be used to measure the formation permeability. In a vertical well, the horizontal permeability can be obtained, but the inversion accuracy depends on the precision of the porosity and vertical-permeability estimation. So the inversion stability is worse compared with the Stoneley-wave inversion process. In a horizontal well, the flexural wave excited by a horizontally polarized dipole can be used to extract the horizontal permeability; while the vertical permeability can be measured from the flexural attenuation generated by the vertically oriented source. The measurement results of permeabilities are very close to the true values in a highly permeable formation. The inversion accuracy, however, is obviously influenced by the anisotropy level if the formation permeability is lower in a horizontal borehole. The stronger the permeability anisotropy is, the worse the measurement precision is. On the basis of the inversion examples, an effective method is provided to estimate the anisotropic permeabilities from the log data of a cross-dipole sonic tool.
首先对弹性地层中井轴平行于TI对称轴的井孔(竖直井)声场进行了分析,其中各模式导波频散曲线由井孔声场函数的极点获得。前人的算例结果显示在TI介质井孔中单极源伪瑞利波、偶极源弯曲波和四极源螺旋波在低频极限处的速度等于井孔方向上的横波速度,但缺乏理论证明。本文讨论了多值声场函数在复平面内的支点分布,并根据井内声波在流体-TI固体边界上产生透射横波的条件论述了伪瑞利波、弯曲波和螺旋波的轴向速度上限。结合TI弹性地层井孔声场的模拟结果,证明了当TI地层参数满足δ>ε+c_(44)/2c_(33)时上述导波的低频极限速度小于地层的横波速度,其中c_(33)和c_(44)分别为TI地层对称轴方向上的纵波和横波模量而ε、δ是各向异性系数。时域全波的模拟结果证实了在这类地层中由导波到达时间所测得的地层横波速度总是小于其真实值。
然后,研究了TI孔隙地层竖直井孔中由单极源激发的声场,计算了井孔声场函数全部极点的分布并由此得到了包括泄漏模式和非泄漏模式在内的导波频散、衰减以及激发强度曲线。非泄漏模式波的轴向传播速度小于等于井孔方向上的横波速度,其衰减仅由弹性孔隙骨架和粘滞流体之间的相对运动而产生;相比之下,速度快于横波的泄漏模式波衰减非常迅速,这是因为它的衰减机制同时包含介质的本征耗散和机械能的几何泄漏。考察了地层、井孔和井内流体各参数分别对非泄漏模式波和临界折射波的影响大小。基于灵敏度分析的结果,在算例中通过拟合斯通利波的时域波形用最小二乘法同时反演了地层的水平渗透率和水平剪切模量。在理论模拟波形时,孔隙度和垂直渗透率的取值对反演精度的影响很小,这体现出反演方法的稳定性。
然而在实际钻井过程中,井轴往往与TI对称轴方向不一致。为了数值模拟这种情况下的井孔偶极声场,本文采用了三维时域有限差分(FDTD)法。由于快、慢横波的速度差异,在偏离TI对称轴的井孔中偶极子能相应地激发快弯曲波和慢弯曲波,形成弯曲波分裂现象。发现弯曲波的速度和相位与声源偏振方向无关,变化声源方位角只改变两类弯曲波的相对幅度。在大多数情况下,低频偶极子激发的快、慢弯曲波在速度上分别趋近于井轴向的快、慢横波速度。算例结果表明,即使在参数满足δ>ε+c_(44)/2c_(33)的地层中,当井孔倾角足够大时根据弯曲波的波至仍可准确提取地层的横波速度。此外,分别模拟了沿深度梯度非均匀以及水平分层介质中的井孔声场。
在渗透率呈各向异性的孔隙地层中,如果渗透率主方向与坐标轴不一致,渗流速度各分量都耦合在一起,传统的应力-速度交错网格无法将描述广义达西定律的方程组离散化。本文对FDTD算法作出了改进使得达西定律方程组依然可以写成差分方程的形式。在算例中模拟了渗透率各向异性地层中的井孔声场。当井轴与渗透率的TI对称轴不一致时,可根据正交偶极接收阵列得到的波形幅度及衰减判别岩层沿井周各方向上的径向渗透率大小,并依此确定地层定向裂隙的分布平面。在灵敏度分析结果的基础上,利用弯曲波的衰减反演了TI地层的渗透率。在竖直井孔中,利用弯曲波的衰减能反演地层的水平渗透率,但在反演精度的稳定性方面弯曲波不如斯通利波。在水平井中,利用水平偏振偶极子激发的弯曲波可反演水平渗透率;而根据垂直偏振偶极子的声场则可测得地层的垂直渗透率。当待测渗透率较高时,反演结果具有较高精度;但对于渗透率较低的地层,反演精度将明显受渗透率各向异性的影响,各向异性越强则渗透率的反演误差越大。这些研究为从正交偶极声测井数据中提取储层渗透率提供了参考。
Acoustic logging is a useful method of geophysical measurements and explorations for oil or gas reservoirs. To develop the logging technique and extract reservoir informations, researches on cylindrical elastic-wave propagation along a radially layered medium is of great practical importance. Transverse isotropy (TI) of sedimentary rocks is a typical characteristic in the lithospere medium. In the present thesis, full waves and component waves excited by a multipole source in a borehole is simulated by the analytical and numerical methods. The inversion scheme for formation permeabilities is put forward after studying the influence of each factor on the wave field.
The wavefields in a vertically extending wellbore are modeled first. In this case, the bore axis is parallel to the TI symmetry axis and the wave equations inside and outside the borehole can therefore be analytically solved. The dispersion of the guided waves can be investigated by finding poles of the borehole acoustic function. Previous works have taken for granted that guided-wave velocities of the pseudo-Rayleigh, flexural, and screw modes equal the shear speed at the low-frequency limits in TI media without proof. With the results of wavefield computations, however, it is revealed that if parameters of a TI formation satisfyδ>ε+c_(44)/2c_(33), asymptotic velocities of those guided waves are lower than the shear speed, where c_(33) and c_(44) are the moduli related to the compressional and shear wave along the TI symmetry axis, respectively, andεandδare the anisotropic coefficients. The simulated full-waves further show that in such a kind of formations, the measurement result of shear speed obtained from those guided-wave arrivals are always smaller than the true value.
Borehole wavefields in a fluid-saturated porous TI formation excited by a monopole source is then simulated. The dispersion, attenuation, and excitation of all modes, including the unleaky and leaky modes, are investigated. It is revealed that the attenuation of the unleaky modes is caused by the relative motion between the porous skeleton and the viscous pore-fluid while that of the leaky modes includes both the intrinsic dissipation and the geometric leak of mechanical energy. Hence the leaky modes, whose velocities are higher than the shear speed, attenuate rapidly as they propagate. Based on the simulation results of component waves, influences of all parameters on the guided waves as well as the critical refracted waves are investigated. Sensitivity analysis theoretically confirms that Stoneley wave can be used to measure the permeability and the shear modulus on the horizontal plane. In the numerical example, those two parameters are well extracted from Stoneley waveforms in the time-domain by solving a nonlinear least-squares problem. It is shown that the initial estimations of the porosity and the vertical permeability have little effect on the inversion results.
The borehole, however, is not parallel to the TI symmetry axis in most cases. Dipole acoustic waves in such a borehole are computed by a 3-D finite-difference time-domain (FDTD) algorithm instead of the analytical method. Due to the shear anisotropy, a dipole source can excite a fast and a slow flexural mode if the bore axis deviates from the TI symmetry axis. That is the flexural-wave splitting phenomenon. The dipole orientation has no relation with the velocities or phases of flexural waves. It influences the relative amplitudes of the two modes. In most situations, the fast and slow flexural modes in the low-frequency range travel at the speeds of the fast and slow shear wave, respectively. The numerical results show that even in the formations withδ>ε+c_(44)/2c_(33), extracting the shear velocities from the flexural arrivals is still possible if the borehole deviation is large enough. Moreover, the wavefields in gradient heterogeneous or horizontally layered formations are also modeled by the FDTD code.
For a permeability anisotropic porous model, once the principal direction of the TI permeability tensor is not consistent with the coordinate axis, all velocity components of pore-fluid flow are coupled together. And the equations of generalized Darcy’s law cannot be discretized using the conventional stress-velocity staggered grids. To solve such a problem, the modified FDTD algorithm is introduced to rewrite Darcy’s equations into the finite-difference formulations. Thus the dipole wavefields in a borehole deviated from the TI symmetry axis can be simulated. According to our numerical examples, it is concluded that the distributing plane of directed cracks can be detected from the attenuation and the amplitudes of flexural-wave signals at the cross dipole array of receivers. Based on the results of sensitivity analysis, it is found that the flexural attenuation can be used to measure the formation permeability. In a vertical well, the horizontal permeability can be obtained, but the inversion accuracy depends on the precision of the porosity and vertical-permeability estimation. So the inversion stability is worse compared with the Stoneley-wave inversion process. In a horizontal well, the flexural wave excited by a horizontally polarized dipole can be used to extract the horizontal permeability; while the vertical permeability can be measured from the flexural attenuation generated by the vertically oriented source. The measurement results of permeabilities are very close to the true values in a highly permeable formation. The inversion accuracy, however, is obviously influenced by the anisotropy level if the formation permeability is lower in a horizontal borehole. The stronger the permeability anisotropy is, the worse the measurement precision is. On the basis of the inversion examples, an effective method is provided to estimate the anisotropic permeabilities from the log data of a cross-dipole sonic tool.
引文
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