各向异性地层中电磁场三维数值模拟的积分方程算法及其应用
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摘要
积分方程法是地球物理学中电磁场三维数值模拟的一种常见而有效的方法,但目前该方法的应用主要还是集中于各向同性地层,本文的主要目的是研究开发在各向异性地层条件下电磁场三维数值模拟的积分方程算法,并考察分析各向异性对三维电磁响应的影响特征和规律。
鉴于并矢Green函数在积分方程算法中的关键作用,本文首先讨论了水平层状各向异性介质中的并矢Green函数理论。利用传输线原理推导出水平层状各向异性介质中频率域并矢Green函数的解析式,并借助于Fourier逆变换得到了空间域并矢Green函数的Sommerfeld积分表达式。为了提高Sommerfeld积分的计算效率,本文提出了一种利用高阶窗函数结合连分式展开来计算该积分的快速算法。
文中第三章给出了各向异性地层中电磁场三维数值模拟的积分方程算法基本原理。采用等效体积单元法和表面积分技术解决了并矢Green函数的奇异积分问题;为了节省计算机内存以及计算时间,引入迭代法求解积分方程的离散化矩阵方程,并且为保证积分方程满足压缩映射条件,利用电磁能量不等式将原始积分方程转化成压缩积分方程,从而使其在任意电导率分布以及任意频率下都是迭代收敛的。
第四章利用积分方程法对各向异性地层频率测深响应进行了三维数值模拟。为了提高多频情况下的计算效率,引入一种迭代法求解积分方程的迭代初值优化选择方法。数值结果表明:地层的各向异性降低了轴向及赤道向频率测深视电阻率响应对地下三维电性异常体的探测能力。
第五章利用积分方程法研究了三维地层条件下各向异性对可控源音频大地电磁(CSAMT)场源效应与静态效应的影响。数值结果表明:地层的各向异性使CSAMT非平面波效应、影子效应和静态效应均得到不同程度的增强,并且使CSAMT视电阻率和相位响应拟断面图对地下三维目标体的反映能力降低。
第六章利用积分方程法对各向异性海底地层海洋可控源电磁(MCSEM)响应进行了三维数值模拟。为了提高计算效率,引入分区域多重网格准线性近似技术。数值结果表明:MCSEM响应对包围油气藏的海底沉积岩地层的横向和纵向电阻率都非常敏感;海底三维有限大小油气藏的横向电阻率对MCSEM响应有一定的影响,但其纵向电阻率对MCSEM响应几乎没有影响。
1Electromagnetic (EM) prospecting method is one of important sounding ways in the applied geophysics because of its many advantages such as flexible source, various mode, high efficiency and stable performance compared with other ways. It is widely used in the geophysical engineering, oil and gas exploration, evaluation of geothermal reserves, metal mineral exploration and deep geological research, etc. Recently, although many more flexible and reliable new digital instruments have continuously invented, lagging interpretation theories and methods restrict obviously further development and more application of EM method. As to my knowledge, 1-D numerical modeling has already become quite mature and 2-D numerical simulation is also gradually maturing, however the 3-D modeling is just still in the start-up and developing stage. With the continuous improvement in accuracy of EM measurement and the rapid development of computer technology, 3-D EM forward modeling has become a very important subject and attracted more and more scholars to engage in such study. 3-D EM numerical modeling has three main algorithms: the finite difference (FD) method, the finite element (FE) method and the integral equation (IE) method. Different from the full-space discrete FD method and FE method, IE method only requires to discrete the abnormal region on background formation to solve Maxwell’s equation, so smaller number of grids is required and higher accuracy of approximation can be obtained. IE method has in fact become a very important method to solve 3-D EM field and has been deeply and wildely studied. However, it must be pointed out that currently IE method is chiefly applied to solve the 3-D isotropically geoelectric problem only. In fact, many theoretical analyse and field observations show that the upper crusts and sedimentary rocks often behave obvious anisotropy. The effect of anisotropy must be taken into account to correct analyze and interprete the EM datum, otherwise neglect of anisotropy will propably lead to incorrect result. Because of its complexity, the study about 3-D EM numerical simulation in anisotropic media is still not much executed up to now. In this paper, it will try to be developed the 3-D EM IE method in layered anisotropic earth and the corresponding key techniques, and to be analyzed the influence of anisotropy on the 3-D EM responses through numerical results obtained by the IE method in various environments so as to supply some theoretical basises for correct and new methods to interpretate EM datum in complex formation.
Because dyadic Green’s function plays a key role in the the IE algorithm, an efficient algorithm on it in the horizontally layered anisotropic formation has been studied. Through Fourier transform, the EM field is decomposed into two independent transverse magnetic wave(TM) and transverse electric wave(TE) in frequency domain and their solutions are analytically gotten by using transmission line and propagation factor so that it finally produces four analytical EM dyadic Green’s functions in frequency domain. Then, the dyadic Green’s functions in space domain are expressed in form of Sommerfeld integrals by using inverse Fourier transform. Value of the Sommerfeld integral is usually computed by a digital filtering algorithm, and the high order window function and continued fraction expansion technique is advanced to further enhance efficiency of Sommerfeld integral.
Chapter 3 mainly discusses the basic principles of IE method for 3-D EM modeling in layered anisotropic earth. To discretize integral equation and obtain an approximation system, the electrical current dyadic Green’s function is decomposed into two parts: one is the singular primary wave directly from the source and another a nonsingular waves reflected and transmitted from the interfaces.Value of the singular kernel of IE each node is computed by using equivalent volume element and surface integral methods. Then Bi-Conjugate Gradient (BICGSTAB) iterative method is introduced to reduce computer memory and improve computational efficiency based on Krylov subspace because the approximation system of the IE is complex and dense. Besides, in order to efficiently solve 3-D EM responses produced by several different anomalous bodies far from each other, a block over-relaxation iterative method is applied because it require smaller computer resource only. Furthermore, the contraction IE in layered anisotropic earth is derived by EM energy inequality so that the convergence of numerical solution can be always assured for any formation. Finally, numerical examples validate the efficient algorithms.
Frequency sounding is an EM method using artificial source, it detects the change of rock resistivity with depth to understand geological structure by altering frequency of alternating EM field. This method has been applied for more than thirty years in our country and played a very important role in the geological research and national economy. In order to improve 3-D forward efficiency in the case of multi-frequency, a technique is introduced to optimize the initial solution for each frequency. Chapter 4 gives a lot of 3-D EM numerical results to investigate frequency sounding responses in layered anisotropic earth. The results show that anisotropy has obvious effects on apparent resistivity of frequency sounding and the intensitivity of both axial and equatorial system responses to 3-D underground anormal body is reduced with the increase of anisotropic coefficients.
Controlled source audio-frequency magnetotelluric(CSAMT) measurement is commonly used artificial source such as grounding electric dipole(electric source) or non-grounded loop(magnetic source) to excite EM field, and two orthogonal components of electric and magnetic fields far enough from the source is observed to determine apparent resistivity so that it can still used the interpretation method similar to that of MT and AMT. However, unlike to the response of MT or AMT approach with natural source, that of CSAMT has often such many additional effects as non-plane wave effects, shadow effects, transmitter overprint effects and static shift effects even if the distance between the transmitter and receiver might be enough far. In Chapter 5 some numerical results in 3-D buried bodies in layered anisotropic earth are given to investigate the influence of anisotropy on the field source effects and static shift effects. Numerical results show that the anisotropy of formation will reduce the far field extent and enhance shadow effects and static shift effects of CSAMT so that the ability for CSAMT to detect abnormal information in the pseudo-sections will be reduced.
In recent years, offshore oil and gas exploration has been become a new research subject with the increasing demand for oil and gas resources. Because of strong capacity to detect high resistance of the seabed strata, marine controlled-source electromagnetic (MCSEM) method has developed into an important means of exploration marine oil and gas resources. Chapter 6 establish an efficient algorithm to simulate 3-D MCSEM response in layered anisotropic seabed through the sub-domain multi-grid and quasi-linear approximation technique. Numerical results show that MCSEM responses are very sensitive to change in the horizontal and vertical resistivity of sedimentary strata undersea surrounding oil and gas reservoir. Horizontal resistivity of 3-D finite oil and gas reservoir has some effects on MCSEM responses, but vertical resistivity hardly influences MCSEM responses.
In conclusion, the 3-D responses of EM measurements in anisotropic formation have been better understood through deep study on 3-D IE numerical modeling of EM and systematically numerical results in various complex environments. So it provides essentially theoretical basis for processing and interpretation of EM datum in complex formation. Thus it is theoretically and practically very usefull.
鉴于并矢Green函数在积分方程算法中的关键作用,本文首先讨论了水平层状各向异性介质中的并矢Green函数理论。利用传输线原理推导出水平层状各向异性介质中频率域并矢Green函数的解析式,并借助于Fourier逆变换得到了空间域并矢Green函数的Sommerfeld积分表达式。为了提高Sommerfeld积分的计算效率,本文提出了一种利用高阶窗函数结合连分式展开来计算该积分的快速算法。
文中第三章给出了各向异性地层中电磁场三维数值模拟的积分方程算法基本原理。采用等效体积单元法和表面积分技术解决了并矢Green函数的奇异积分问题;为了节省计算机内存以及计算时间,引入迭代法求解积分方程的离散化矩阵方程,并且为保证积分方程满足压缩映射条件,利用电磁能量不等式将原始积分方程转化成压缩积分方程,从而使其在任意电导率分布以及任意频率下都是迭代收敛的。
第四章利用积分方程法对各向异性地层频率测深响应进行了三维数值模拟。为了提高多频情况下的计算效率,引入一种迭代法求解积分方程的迭代初值优化选择方法。数值结果表明:地层的各向异性降低了轴向及赤道向频率测深视电阻率响应对地下三维电性异常体的探测能力。
第五章利用积分方程法研究了三维地层条件下各向异性对可控源音频大地电磁(CSAMT)场源效应与静态效应的影响。数值结果表明:地层的各向异性使CSAMT非平面波效应、影子效应和静态效应均得到不同程度的增强,并且使CSAMT视电阻率和相位响应拟断面图对地下三维目标体的反映能力降低。
第六章利用积分方程法对各向异性海底地层海洋可控源电磁(MCSEM)响应进行了三维数值模拟。为了提高计算效率,引入分区域多重网格准线性近似技术。数值结果表明:MCSEM响应对包围油气藏的海底沉积岩地层的横向和纵向电阻率都非常敏感;海底三维有限大小油气藏的横向电阻率对MCSEM响应有一定的影响,但其纵向电阻率对MCSEM响应几乎没有影响。
1Electromagnetic (EM) prospecting method is one of important sounding ways in the applied geophysics because of its many advantages such as flexible source, various mode, high efficiency and stable performance compared with other ways. It is widely used in the geophysical engineering, oil and gas exploration, evaluation of geothermal reserves, metal mineral exploration and deep geological research, etc. Recently, although many more flexible and reliable new digital instruments have continuously invented, lagging interpretation theories and methods restrict obviously further development and more application of EM method. As to my knowledge, 1-D numerical modeling has already become quite mature and 2-D numerical simulation is also gradually maturing, however the 3-D modeling is just still in the start-up and developing stage. With the continuous improvement in accuracy of EM measurement and the rapid development of computer technology, 3-D EM forward modeling has become a very important subject and attracted more and more scholars to engage in such study. 3-D EM numerical modeling has three main algorithms: the finite difference (FD) method, the finite element (FE) method and the integral equation (IE) method. Different from the full-space discrete FD method and FE method, IE method only requires to discrete the abnormal region on background formation to solve Maxwell’s equation, so smaller number of grids is required and higher accuracy of approximation can be obtained. IE method has in fact become a very important method to solve 3-D EM field and has been deeply and wildely studied. However, it must be pointed out that currently IE method is chiefly applied to solve the 3-D isotropically geoelectric problem only. In fact, many theoretical analyse and field observations show that the upper crusts and sedimentary rocks often behave obvious anisotropy. The effect of anisotropy must be taken into account to correct analyze and interprete the EM datum, otherwise neglect of anisotropy will propably lead to incorrect result. Because of its complexity, the study about 3-D EM numerical simulation in anisotropic media is still not much executed up to now. In this paper, it will try to be developed the 3-D EM IE method in layered anisotropic earth and the corresponding key techniques, and to be analyzed the influence of anisotropy on the 3-D EM responses through numerical results obtained by the IE method in various environments so as to supply some theoretical basises for correct and new methods to interpretate EM datum in complex formation.
Because dyadic Green’s function plays a key role in the the IE algorithm, an efficient algorithm on it in the horizontally layered anisotropic formation has been studied. Through Fourier transform, the EM field is decomposed into two independent transverse magnetic wave(TM) and transverse electric wave(TE) in frequency domain and their solutions are analytically gotten by using transmission line and propagation factor so that it finally produces four analytical EM dyadic Green’s functions in frequency domain. Then, the dyadic Green’s functions in space domain are expressed in form of Sommerfeld integrals by using inverse Fourier transform. Value of the Sommerfeld integral is usually computed by a digital filtering algorithm, and the high order window function and continued fraction expansion technique is advanced to further enhance efficiency of Sommerfeld integral.
Chapter 3 mainly discusses the basic principles of IE method for 3-D EM modeling in layered anisotropic earth. To discretize integral equation and obtain an approximation system, the electrical current dyadic Green’s function is decomposed into two parts: one is the singular primary wave directly from the source and another a nonsingular waves reflected and transmitted from the interfaces.Value of the singular kernel of IE each node is computed by using equivalent volume element and surface integral methods. Then Bi-Conjugate Gradient (BICGSTAB) iterative method is introduced to reduce computer memory and improve computational efficiency based on Krylov subspace because the approximation system of the IE is complex and dense. Besides, in order to efficiently solve 3-D EM responses produced by several different anomalous bodies far from each other, a block over-relaxation iterative method is applied because it require smaller computer resource only. Furthermore, the contraction IE in layered anisotropic earth is derived by EM energy inequality so that the convergence of numerical solution can be always assured for any formation. Finally, numerical examples validate the efficient algorithms.
Frequency sounding is an EM method using artificial source, it detects the change of rock resistivity with depth to understand geological structure by altering frequency of alternating EM field. This method has been applied for more than thirty years in our country and played a very important role in the geological research and national economy. In order to improve 3-D forward efficiency in the case of multi-frequency, a technique is introduced to optimize the initial solution for each frequency. Chapter 4 gives a lot of 3-D EM numerical results to investigate frequency sounding responses in layered anisotropic earth. The results show that anisotropy has obvious effects on apparent resistivity of frequency sounding and the intensitivity of both axial and equatorial system responses to 3-D underground anormal body is reduced with the increase of anisotropic coefficients.
Controlled source audio-frequency magnetotelluric(CSAMT) measurement is commonly used artificial source such as grounding electric dipole(electric source) or non-grounded loop(magnetic source) to excite EM field, and two orthogonal components of electric and magnetic fields far enough from the source is observed to determine apparent resistivity so that it can still used the interpretation method similar to that of MT and AMT. However, unlike to the response of MT or AMT approach with natural source, that of CSAMT has often such many additional effects as non-plane wave effects, shadow effects, transmitter overprint effects and static shift effects even if the distance between the transmitter and receiver might be enough far. In Chapter 5 some numerical results in 3-D buried bodies in layered anisotropic earth are given to investigate the influence of anisotropy on the field source effects and static shift effects. Numerical results show that the anisotropy of formation will reduce the far field extent and enhance shadow effects and static shift effects of CSAMT so that the ability for CSAMT to detect abnormal information in the pseudo-sections will be reduced.
In recent years, offshore oil and gas exploration has been become a new research subject with the increasing demand for oil and gas resources. Because of strong capacity to detect high resistance of the seabed strata, marine controlled-source electromagnetic (MCSEM) method has developed into an important means of exploration marine oil and gas resources. Chapter 6 establish an efficient algorithm to simulate 3-D MCSEM response in layered anisotropic seabed through the sub-domain multi-grid and quasi-linear approximation technique. Numerical results show that MCSEM responses are very sensitive to change in the horizontal and vertical resistivity of sedimentary strata undersea surrounding oil and gas reservoir. Horizontal resistivity of 3-D finite oil and gas reservoir has some effects on MCSEM responses, but vertical resistivity hardly influences MCSEM responses.
In conclusion, the 3-D responses of EM measurements in anisotropic formation have been better understood through deep study on 3-D IE numerical modeling of EM and systematically numerical results in various complex environments. So it provides essentially theoretical basis for processing and interpretation of EM datum in complex formation. Thus it is theoretically and practically very usefull.
引文
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