双侧向测井资料迭代正则化反演与各向异性地层多分量感应测井数值仿真
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摘要
本文主要分为以下两部分工作:
第一部分:根据非线性反演理论与Morozov偏差原理研究建立从双侧向测井(DLL)资料中同时重构原状地层电阻率、侵入带电阻率、侵入半径、层界面位置以及井眼泥浆电阻率的迭代正则化算法。利用Tikhonov正则化反演理论将双侧向测井资料的反演问题转化为含有稳定泛函的非线性目标函数极小化问题,并由Gauss-Newton算法确定极小化解。为得到稳定的反演结果,有效实现测井资料的最佳拟合,在迭代过程中将Morozov偏差原理和Cholesky分解技术相结合,建立了一套后验选择正则化因子的方法。此外,为了比较双侧向和双感应测井仪器的探测特性,利用几何因子理论考察了双侧向和双感应测井仪器在探测深度和纵向分辨率上的差异;由规范化的Fréchet导数矩阵对比分析了双侧向和双感应测井响应对各地层模型参数的敏感程度;针对不同理论模型对比分析了双侧向反演中不同因素对各模型参数反演效果的影响。最后对大庆油田三个不同区块五十多口井的测井资料进行了反演处理,并将解释结果与实际试油结论进行对比,验证了新算法在处理薄层或薄交互层上的双侧向测井资料时能够取得比双感应反演更为满意的效果。
第二部分:采用传播矩阵技术研究建立层状各向异性地层中多分量感应测井响应的有效算法。通过Fourier变换将频率空间域中的Maxwell方程组求解问题转化为频率波数域中关于电磁场水平分量常微分方程组的定解问题。利用该方程组系数矩阵的本征值和归一化本征向量将电磁场分解成上行波和下行波模式的组合,推导出均匀各向异性介质中由任意方向磁偶极子产生的电磁波模式解析表达式;在此基础上,利用叠加原理和边界条件研究了电磁波在层状各向异性地层中的反射和透射,给出各个界面上的广义反射系数和不同地层中电磁波振幅的递推公式,进而得到电磁波模式的解析解。为了有效确定频率空间域中的电磁场,采用二维Patterson自适应求积算法结合有限连分式展开技术计算傅氏逆变换。最后通过数值模拟结果证明了该算法的有效性,考察了层状正交各向异性地层中不同各向异性系数、不同井眼倾角以及仪器长度和工作频率变化等情况下的多分量感应测井响应特征。
The work of this paper can be divided into two parts: in the first part, according to the difficulty of interpretation and evaluation about thin reservoirs and thin interactive reservoirs in current Daqing oil field, whose thickness of individual laminate is smaller than 1 m, we present a iterative regularization algorithm to simultaneously reconstruct all the model parameters, including the virgin formation resistivity, flushed zone resistivity, invasion radius, horizontal boundary depth and mud resistivity per bed, from dual-laterolog (DLL) datum based on nonlinear inversion theory and Morozov discrepancy principle. In the second part, we present an efficient algorithm to numerically simulate the responses of multi-component induction logging (MCIL) tool in layered anisotropic formations by using propagator matrix method (PMM) for investigating the response characteristics of MCIL tool in layered general anisotropic formations, which can become a theoretical basis of new instrument design and data processing.
The electrical logging is one of the most basic and important logging method in petroleum logging. It can classify the lithology, estimate the hydrocarbon saturation and predict the oil-gas production capacity by detecting formation resistivity. The DLL tool is the most widely used logging tool in current domestic, in practical application it usually combined with microspherically focused logging (MSFL) tool to determine the position of permeability formation and identificate the oil-water bed. How to fast and efficiently simulate the logging response in complex formations has important significance to logging data interpretation and inversion algorithm establishment. In chapter 2, a fast forward algorithm for DLL in layered inhomogeneous medium has been set up. The fast algorithm and the semi-analytical expression of Green’s function in layered inhomogeneous formations have been given by numerical mode matching method (NMM) firstly. Then the integral equations on potential values and current distributions of beam electrode, shielding electrodes and monitor electrodes have been established by considering the special structure of DLL tool. Finally, we combine the integral equations with focusing conditions to calculate the logging responses of deep and shallow lateral. Besides, the differences in detecting depth and vertical resolution between dual induction logging tool (DIT) and DLL tool are investigated by geometric factor theory. The numerical results show that the DIT has deeper detecting depth than the DLL tool, and the vertical resolution of DLL tool is much higher than the DIT’s.
With the development of oil field, people find the logging response values in thin reservoirs and thin interactive reservoirs cannot completely, even incorrectly, reflect the formation parameter information because of the influence of complex environment around the borehole and the self-limiting of instrument. So the inversion processing of well-logging datum is necessary. The inversion of well-logging datum is a mathematically representative ill-posed problem for the nonuniqueness and instability. In order to improve the reliability of inversion results, we present a new iterative regularization inversion algorithm in chapter 3. First, the Born’s approximate formulae of DLL responses and the relation between perturbations of formation conductivity and small changes of model vector are given by perturbation principle. In combination with the forward modeling method, a fast algorithm of normalized Fréchet derivative matrix has been set up. Then a stabilizing function is introduced, which is a non-quadratic function including the derivative of electric conductivity and invasion radius. From the Tikhonov regularization inversion theory, the inversion problem can be transformed into the minimization problem of non-quadratic objective function with the stabilizing functional defined on model space. The minimization of objective function can be obtained by using Gauss-Newton method. In the iterative process, the combination of Morozov discrepancy principle and Cholesky decomposition is executed to propose an a-posteriori regularization factor choice strategy in order to obtain a stable and reliable inversion solution. Thus, not only the objective function and the fitting error of well-logging responses decreased continually, but also the best fit of input data with modeling logs has been realized. The numerical results of theoretical models show that the full-parameter iterative regularization inversion technique can not only reconstruct the real values of virgin formation resistivity and flushed zone resistivity effectively, but also improve the stability and reliability of inversion solutions. It is necessary to make the mud resistivity as a independent inversion parameter when inversing DLL datum, because the detecting depth of DLL tool is more shallow and its response is more sensitive to the changes of mud resistivity than DIT’s. The inversion results of field logs measured from Daqing oil field show that the inversion results of DIT and DLL datum accord with the test oil conclusion at many test oil layers, but the inversion effect of DLL datum are much better than DIT’s in thin reservoirs and thin interactive reservoirs, whose thickness of individual laminate is smaller than 1 m, and it has very important significance to inverse DLL datum in thin reservoirs identification and classification.
The electric anisotropy phenomenon hadn’t aroused attention to people in early stage of electrical prospecting. With the constrant progress of drilling and logging technology, the study on response characteristics of logging tool in anisotropic formations and influence of formation anisotropy to instrument measure have been gradually concerned by people. The multi-component induction logging (MCIL) is a newly logging technology which can simultaneously measure the formation conductivity with different directions in anisotropic formations. The study on simulation and inversion of logging response under different formation conditions is a very important research topic in current petroleum logging. The forward study in stratified medium is mostly assumed the formation is transverse isotropy (TI), there is no report about the numeirical simulation of MICL responses in layered general anisotropic formations now. In chapter 4, an efficient algorithm to numerically simulate the responses of MCIL in layered anisotropic formations by using propagator matrix method. First, the solving problem of Maxwell’s equations in frequency-spatial domain is transformed into a definite solution problem of first-order ordinary differential systems in frequency-wavenumber domain by Fourier transform. The electromagnetic (EM) field can be decomposed into upgoing and downgoing mode by using eigenvalues and normalized eigenvectors of the system matrix. And an analytical expression of electromagnetic wave emitted by an arbitrary direction of magnetic dipole in homogeneous anisotropic media is derived. Then we derive the recurrence formula of generalized reflection coefficient per interface and that of amplitude of electromagnetic wave each bed by studying the reflection and transmission of electromagnetic wave in layered anisotropic formations based on superposition principle and interface conditions. By this method, it can avoid computing lots of matrices multiple directly. In order to determine numerical solution of electromagnetic field in frequency-spatial domain, a 2D adaptive Patterson integral formula and the limitedly continued fraction expansion technique are used to compute the 2D inverse Fourier transform, which can improve the operation efficiency and the calculation accuracy. Finally, the numerical tests validate the new algorithm and investigate the response characteristics of MCIL tool in layered orthorhombic anisotropic formations: the variations of horizontal anisotropy coefficient can not change the logging response of Y axis, the variations of vertical anisotropy coefficient can not change the logging response of Z axis, the growth of tool length can lead to the vertical resolution reduction, the increase of work frequency can cause much higher skin effect, the changes in borehole dipping angles can not only affect the logging responses in three principle axes direction but also generate counter responses of X axis and Z axis.
第一部分:根据非线性反演理论与Morozov偏差原理研究建立从双侧向测井(DLL)资料中同时重构原状地层电阻率、侵入带电阻率、侵入半径、层界面位置以及井眼泥浆电阻率的迭代正则化算法。利用Tikhonov正则化反演理论将双侧向测井资料的反演问题转化为含有稳定泛函的非线性目标函数极小化问题,并由Gauss-Newton算法确定极小化解。为得到稳定的反演结果,有效实现测井资料的最佳拟合,在迭代过程中将Morozov偏差原理和Cholesky分解技术相结合,建立了一套后验选择正则化因子的方法。此外,为了比较双侧向和双感应测井仪器的探测特性,利用几何因子理论考察了双侧向和双感应测井仪器在探测深度和纵向分辨率上的差异;由规范化的Fréchet导数矩阵对比分析了双侧向和双感应测井响应对各地层模型参数的敏感程度;针对不同理论模型对比分析了双侧向反演中不同因素对各模型参数反演效果的影响。最后对大庆油田三个不同区块五十多口井的测井资料进行了反演处理,并将解释结果与实际试油结论进行对比,验证了新算法在处理薄层或薄交互层上的双侧向测井资料时能够取得比双感应反演更为满意的效果。
第二部分:采用传播矩阵技术研究建立层状各向异性地层中多分量感应测井响应的有效算法。通过Fourier变换将频率空间域中的Maxwell方程组求解问题转化为频率波数域中关于电磁场水平分量常微分方程组的定解问题。利用该方程组系数矩阵的本征值和归一化本征向量将电磁场分解成上行波和下行波模式的组合,推导出均匀各向异性介质中由任意方向磁偶极子产生的电磁波模式解析表达式;在此基础上,利用叠加原理和边界条件研究了电磁波在层状各向异性地层中的反射和透射,给出各个界面上的广义反射系数和不同地层中电磁波振幅的递推公式,进而得到电磁波模式的解析解。为了有效确定频率空间域中的电磁场,采用二维Patterson自适应求积算法结合有限连分式展开技术计算傅氏逆变换。最后通过数值模拟结果证明了该算法的有效性,考察了层状正交各向异性地层中不同各向异性系数、不同井眼倾角以及仪器长度和工作频率变化等情况下的多分量感应测井响应特征。
The work of this paper can be divided into two parts: in the first part, according to the difficulty of interpretation and evaluation about thin reservoirs and thin interactive reservoirs in current Daqing oil field, whose thickness of individual laminate is smaller than 1 m, we present a iterative regularization algorithm to simultaneously reconstruct all the model parameters, including the virgin formation resistivity, flushed zone resistivity, invasion radius, horizontal boundary depth and mud resistivity per bed, from dual-laterolog (DLL) datum based on nonlinear inversion theory and Morozov discrepancy principle. In the second part, we present an efficient algorithm to numerically simulate the responses of multi-component induction logging (MCIL) tool in layered anisotropic formations by using propagator matrix method (PMM) for investigating the response characteristics of MCIL tool in layered general anisotropic formations, which can become a theoretical basis of new instrument design and data processing.
The electrical logging is one of the most basic and important logging method in petroleum logging. It can classify the lithology, estimate the hydrocarbon saturation and predict the oil-gas production capacity by detecting formation resistivity. The DLL tool is the most widely used logging tool in current domestic, in practical application it usually combined with microspherically focused logging (MSFL) tool to determine the position of permeability formation and identificate the oil-water bed. How to fast and efficiently simulate the logging response in complex formations has important significance to logging data interpretation and inversion algorithm establishment. In chapter 2, a fast forward algorithm for DLL in layered inhomogeneous medium has been set up. The fast algorithm and the semi-analytical expression of Green’s function in layered inhomogeneous formations have been given by numerical mode matching method (NMM) firstly. Then the integral equations on potential values and current distributions of beam electrode, shielding electrodes and monitor electrodes have been established by considering the special structure of DLL tool. Finally, we combine the integral equations with focusing conditions to calculate the logging responses of deep and shallow lateral. Besides, the differences in detecting depth and vertical resolution between dual induction logging tool (DIT) and DLL tool are investigated by geometric factor theory. The numerical results show that the DIT has deeper detecting depth than the DLL tool, and the vertical resolution of DLL tool is much higher than the DIT’s.
With the development of oil field, people find the logging response values in thin reservoirs and thin interactive reservoirs cannot completely, even incorrectly, reflect the formation parameter information because of the influence of complex environment around the borehole and the self-limiting of instrument. So the inversion processing of well-logging datum is necessary. The inversion of well-logging datum is a mathematically representative ill-posed problem for the nonuniqueness and instability. In order to improve the reliability of inversion results, we present a new iterative regularization inversion algorithm in chapter 3. First, the Born’s approximate formulae of DLL responses and the relation between perturbations of formation conductivity and small changes of model vector are given by perturbation principle. In combination with the forward modeling method, a fast algorithm of normalized Fréchet derivative matrix has been set up. Then a stabilizing function is introduced, which is a non-quadratic function including the derivative of electric conductivity and invasion radius. From the Tikhonov regularization inversion theory, the inversion problem can be transformed into the minimization problem of non-quadratic objective function with the stabilizing functional defined on model space. The minimization of objective function can be obtained by using Gauss-Newton method. In the iterative process, the combination of Morozov discrepancy principle and Cholesky decomposition is executed to propose an a-posteriori regularization factor choice strategy in order to obtain a stable and reliable inversion solution. Thus, not only the objective function and the fitting error of well-logging responses decreased continually, but also the best fit of input data with modeling logs has been realized. The numerical results of theoretical models show that the full-parameter iterative regularization inversion technique can not only reconstruct the real values of virgin formation resistivity and flushed zone resistivity effectively, but also improve the stability and reliability of inversion solutions. It is necessary to make the mud resistivity as a independent inversion parameter when inversing DLL datum, because the detecting depth of DLL tool is more shallow and its response is more sensitive to the changes of mud resistivity than DIT’s. The inversion results of field logs measured from Daqing oil field show that the inversion results of DIT and DLL datum accord with the test oil conclusion at many test oil layers, but the inversion effect of DLL datum are much better than DIT’s in thin reservoirs and thin interactive reservoirs, whose thickness of individual laminate is smaller than 1 m, and it has very important significance to inverse DLL datum in thin reservoirs identification and classification.
The electric anisotropy phenomenon hadn’t aroused attention to people in early stage of electrical prospecting. With the constrant progress of drilling and logging technology, the study on response characteristics of logging tool in anisotropic formations and influence of formation anisotropy to instrument measure have been gradually concerned by people. The multi-component induction logging (MCIL) is a newly logging technology which can simultaneously measure the formation conductivity with different directions in anisotropic formations. The study on simulation and inversion of logging response under different formation conditions is a very important research topic in current petroleum logging. The forward study in stratified medium is mostly assumed the formation is transverse isotropy (TI), there is no report about the numeirical simulation of MICL responses in layered general anisotropic formations now. In chapter 4, an efficient algorithm to numerically simulate the responses of MCIL in layered anisotropic formations by using propagator matrix method. First, the solving problem of Maxwell’s equations in frequency-spatial domain is transformed into a definite solution problem of first-order ordinary differential systems in frequency-wavenumber domain by Fourier transform. The electromagnetic (EM) field can be decomposed into upgoing and downgoing mode by using eigenvalues and normalized eigenvectors of the system matrix. And an analytical expression of electromagnetic wave emitted by an arbitrary direction of magnetic dipole in homogeneous anisotropic media is derived. Then we derive the recurrence formula of generalized reflection coefficient per interface and that of amplitude of electromagnetic wave each bed by studying the reflection and transmission of electromagnetic wave in layered anisotropic formations based on superposition principle and interface conditions. By this method, it can avoid computing lots of matrices multiple directly. In order to determine numerical solution of electromagnetic field in frequency-spatial domain, a 2D adaptive Patterson integral formula and the limitedly continued fraction expansion technique are used to compute the 2D inverse Fourier transform, which can improve the operation efficiency and the calculation accuracy. Finally, the numerical tests validate the new algorithm and investigate the response characteristics of MCIL tool in layered orthorhombic anisotropic formations: the variations of horizontal anisotropy coefficient can not change the logging response of Y axis, the variations of vertical anisotropy coefficient can not change the logging response of Z axis, the growth of tool length can lead to the vertical resolution reduction, the increase of work frequency can cause much higher skin effect, the changes in borehole dipping angles can not only affect the logging responses in three principle axes direction but also generate counter responses of X axis and Z axis.
引文
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