基于有限元法的复电阻率正反演研究及应用
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摘要
复电阻率法以岩、矿石的频谱参数特性为勘探依据，多参数的组合解释能为评价异常源的性质提供更为丰富的信息。但是，反演问题一直没有得到较好的解决，已经严重影响和制约了该方法的应用和发展。因此，可靠和实用的正、反演算法已成为复电阻率法研究的关键。本文基于有限元法开展了2.5维和3维复电阻率正演模拟研究，并以实际应用为目的，对2.5维复电阻率法全区反演开展了研究。
推导了同时包含激电效应和电磁效应的复电阻率法基本边值问题。从标准变分原理出发，推导并证明了适用于地球物理场中有耗电磁介质的广义变分原理，该部分作为本文三维矢量有限元分析的基础理论。针对电磁学领域中不存在相应变分的边值问题，给出了构建有限元公式的伽辽金法，该部分作为本文2.5维节点有限元分析的理论基础。
研究并实现了基于节点有限元法的2.5维复电阻率正演模拟。从复电阻率法基本边值问题出发，推导了频率域电、磁场对耦偏微分方程，并利用伽辽金法完成离散化。考虑到地形影响，采用双二次插值的等参单元完成区域剖分。用伪Delta函数等效偶极源，以提高近区数据的模拟精度。考虑到内存占用和计算效率等问题，对总体系数矩阵中的非零元素直接进行一维压缩存储，并采用IC-CBCG法求解正演方程组。在反傅里叶变换时，给出了有效波数的取值规则。最后，模拟了二维复地电模型的视电阻率和视相位响应，并对响应特征进行了分析和总结。
研究了基于矢量有限元法的三维复电阻率正演模拟技术。针对复电阻率法的近区观测方式，采用将总场分解为背景场和二次场的求解方案。推导了长导线源激发下，层状介质的电场解析式，从而实现了背景场的高精度计算。对于二次场计算，首先利用广义变分原理推导了三维边值问题的变分方程，然后将求解区域采用规则的六面体单元剖分，并使用矢量插值函数完成单元分析，避免了常规有限元法中出现“伪解”的问题。系数矩阵存储和方程组求解采用与2.5维正演相同的方案，最终实现了大尺寸区域可细密网格剖分的三维复电阻率正演模拟。在验证算法及程序正确性的基础上，分析了三维数值模拟精度并探讨了复电阻率模型的二次场响应特征。
提出了电场振幅和电场相位联合反演的方案，基于此实现了2.5维复电阻率法全区反演。在反演算法中施加了模型光滑度约束和参数界限约束，有效降低了问题的多解性。借助电场的偏导数形式，推导出了电场振幅和电场相位灵敏度元素的解析表达式，并应用互换定理直接求解，提高了反演效率。试算了多种复电阻率理论模型，并对反演结果及分辨率问题展开分析。最后，对沙溪地区的SIP实测数据进行反演成像，并与已知钻井及其它地球物理资料进行了对比验证，取得了较好的地质效果。
Complex resistivity method is based on the characteristics of SIP parameters ofrocks and ores, and the multiple parameters provide more information for evaluatinganomalous bodies. However, the inverse problem of SIP has not been resolved well,and has seriously restricted the application and development of SIP. In this paper, the2.5D and3D SIP numerical simulations using finite element method were studied, anda large-scale2.5D SIP inversion algorithm for application was proposed and finished.
The basic boundary value problem of SIP is derived by using Maxwell’s equationand Cole-Cole model. From the standard variational principle, the generalizedvariational principle is derived which is the basic theory of3D vector finite elementanalysis in this paper. At last, the Galerkin finite element method is introduced, whichis the basic theory of2.5D node finite element analysis in this paper.
Based on the node finite element method, the2.5D complex resistivity forward isstudied. From the basic boundary value problem of SIP, the frequency domain electricand magnetic fields coupled partial differential equations were derived, and they werediscretized by Galerkin method. Considering the effect of topography, the solvedregion is divided into small units with isoparametric element，and the pseudo-deltafunction is used to improve the simulation precision of electromagnetic fields near thesource region. Considering the computational efficiency, the non-zero elements ingeneral coefficient matrix is stored directly by using1D compressing method and theIC-CBCG method is used in solving forward equations. The rule of selecting effectivewave numbers is proposed for carrying out inverse Fourier Transform. The apparentresistivity and apparent phase of2D complex resistivity model were calculated, and itscharacteristics were also analyzed.
Based on the vector finite element method, the3D complex resistivity forwardmodeling is studied. In order to improve the precision of numerical simulation in thenear source region, the total field is decomposed into the background and secondaryfields. The field analysis formula of layered media under the excitation of long wiresource is derived, thus the high-precision calculation of the background field is achieved. For the secondary field, the rectangular elements are used to split the solvedregion, and vector interpolation function is used to complete the unit analysis, whichavoids the “pseudo-solution” in node finite element method. The same scheme as2.5Dforward is used in coefficient matrix storage. The3D forward method is verified andsome typical model’s responses are analyzed by SIP3D numerical simulation.
A reliable and practical2.5D combined inversion algorithm is proposed in thispaper, which used amplitudes and phases of electric fields of multi-arrangements. Themodel’s smoothness and parameter variation range as two kinds of constraints areintroduced into2.5D SIP inversion, thus the multiplicity of inversion is efficientlyimproved. The analytical expression of the sensitivity matrix is derived with the partialderivatives of the electric field, and calculated by the reciprocity theorem. Thealgorithm can perform a large-scale inversion and take advantage of geologicalinformation. Finally, the inversion procedure was used in Anhui SIP measured data,and the result shows good consistence with known drilling data and CSAMT inversionresult.
推导了同时包含激电效应和电磁效应的复电阻率法基本边值问题。从标准变分原理出发，推导并证明了适用于地球物理场中有耗电磁介质的广义变分原理，该部分作为本文三维矢量有限元分析的基础理论。针对电磁学领域中不存在相应变分的边值问题，给出了构建有限元公式的伽辽金法，该部分作为本文2.5维节点有限元分析的理论基础。
研究并实现了基于节点有限元法的2.5维复电阻率正演模拟。从复电阻率法基本边值问题出发，推导了频率域电、磁场对耦偏微分方程，并利用伽辽金法完成离散化。考虑到地形影响，采用双二次插值的等参单元完成区域剖分。用伪Delta函数等效偶极源，以提高近区数据的模拟精度。考虑到内存占用和计算效率等问题，对总体系数矩阵中的非零元素直接进行一维压缩存储，并采用IC-CBCG法求解正演方程组。在反傅里叶变换时，给出了有效波数的取值规则。最后，模拟了二维复地电模型的视电阻率和视相位响应，并对响应特征进行了分析和总结。
研究了基于矢量有限元法的三维复电阻率正演模拟技术。针对复电阻率法的近区观测方式，采用将总场分解为背景场和二次场的求解方案。推导了长导线源激发下，层状介质的电场解析式，从而实现了背景场的高精度计算。对于二次场计算，首先利用广义变分原理推导了三维边值问题的变分方程，然后将求解区域采用规则的六面体单元剖分，并使用矢量插值函数完成单元分析，避免了常规有限元法中出现“伪解”的问题。系数矩阵存储和方程组求解采用与2.5维正演相同的方案，最终实现了大尺寸区域可细密网格剖分的三维复电阻率正演模拟。在验证算法及程序正确性的基础上，分析了三维数值模拟精度并探讨了复电阻率模型的二次场响应特征。
提出了电场振幅和电场相位联合反演的方案，基于此实现了2.5维复电阻率法全区反演。在反演算法中施加了模型光滑度约束和参数界限约束，有效降低了问题的多解性。借助电场的偏导数形式，推导出了电场振幅和电场相位灵敏度元素的解析表达式，并应用互换定理直接求解，提高了反演效率。试算了多种复电阻率理论模型，并对反演结果及分辨率问题展开分析。最后，对沙溪地区的SIP实测数据进行反演成像，并与已知钻井及其它地球物理资料进行了对比验证，取得了较好的地质效果。
Complex resistivity method is based on the characteristics of SIP parameters ofrocks and ores, and the multiple parameters provide more information for evaluatinganomalous bodies. However, the inverse problem of SIP has not been resolved well,and has seriously restricted the application and development of SIP. In this paper, the2.5D and3D SIP numerical simulations using finite element method were studied, anda large-scale2.5D SIP inversion algorithm for application was proposed and finished.
The basic boundary value problem of SIP is derived by using Maxwell’s equationand Cole-Cole model. From the standard variational principle, the generalizedvariational principle is derived which is the basic theory of3D vector finite elementanalysis in this paper. At last, the Galerkin finite element method is introduced, whichis the basic theory of2.5D node finite element analysis in this paper.
Based on the node finite element method, the2.5D complex resistivity forward isstudied. From the basic boundary value problem of SIP, the frequency domain electricand magnetic fields coupled partial differential equations were derived, and they werediscretized by Galerkin method. Considering the effect of topography, the solvedregion is divided into small units with isoparametric element，and the pseudo-deltafunction is used to improve the simulation precision of electromagnetic fields near thesource region. Considering the computational efficiency, the non-zero elements ingeneral coefficient matrix is stored directly by using1D compressing method and theIC-CBCG method is used in solving forward equations. The rule of selecting effectivewave numbers is proposed for carrying out inverse Fourier Transform. The apparentresistivity and apparent phase of2D complex resistivity model were calculated, and itscharacteristics were also analyzed.
Based on the vector finite element method, the3D complex resistivity forwardmodeling is studied. In order to improve the precision of numerical simulation in thenear source region, the total field is decomposed into the background and secondaryfields. The field analysis formula of layered media under the excitation of long wiresource is derived, thus the high-precision calculation of the background field is achieved. For the secondary field, the rectangular elements are used to split the solvedregion, and vector interpolation function is used to complete the unit analysis, whichavoids the “pseudo-solution” in node finite element method. The same scheme as2.5Dforward is used in coefficient matrix storage. The3D forward method is verified andsome typical model’s responses are analyzed by SIP3D numerical simulation.
A reliable and practical2.5D combined inversion algorithm is proposed in thispaper, which used amplitudes and phases of electric fields of multi-arrangements. Themodel’s smoothness and parameter variation range as two kinds of constraints areintroduced into2.5D SIP inversion, thus the multiplicity of inversion is efficientlyimproved. The analytical expression of the sensitivity matrix is derived with the partialderivatives of the electric field, and calculated by the reciprocity theorem. Thealgorithm can perform a large-scale inversion and take advantage of geologicalinformation. Finally, the inversion procedure was used in Anhui SIP measured data,and the result shows good consistence with known drilling data and CSAMT inversionresult.
引文
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