频率域可控源二维有限元正演数值模拟
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摘要
频率域可控源大地电磁法由于人工源的加入使其克服了大地电磁法中天然场源信号微弱的缺点,因此在矿产普查、油气勘探等方面得到了广泛的应用。本文主要对频率域可控源电磁法二维正演问题进行研究,并通过全区视电阻率对过渡区以及近区数据进行校正。
本文采用有限单元法进行数值模拟。用Matlab为程序编译工具,采用双线性插值和双二次插值的有限单元法,推导出了相对应的边值问题和变分问题满足的微分方程。为了模拟无穷远边界并满足计算机的内存需求,在保证计算精度的情况下设计了非均匀网格剖分。在程序编制过程中,只存储有限元系数矩阵的非零元素,大大节省了存储空间。通过计算几个典型模型的卡尼亚视电阻率拟断面图和相位拟断面图,得出了不同模型下电磁场响应特征的分布规律,验证了算法的正确性。
针对频率域可控源电磁法中卡尼亚电阻率在过渡区和近区畸变的问题,给出了全区视电阻率的迭代公式,并对典型的一维层状模型以及简单二维模型进行了计算。过渡区和近区数据经过校正后,可以正确反映出模型的地电特征,证明了线源下近区勘探的可能性。
Because of the artificial source, fCSEM has overcome the weakness of natural source, which is the shortcoming of MT, so it has been widely used in the mineral, oil and gas exploration and so on. This paper has done some studies on the 2D forward problem of fCSEM, and uses the whole range apparent resistivity to rectify the date which are collected near the source.
The finite element method was used for numerical simulation. Based on Matlab as programming tool, we adopted the finite element method of linear interpolation and second interpolation in a rectangular element for solving the fCSEM forward problem and derived the corresponding formulas for calculating. In order to simulate the infinity border and meet the demand of computer memory, the non-uniform gird is designed to ensuring the accuracy. In programming, we only stored the non-zero elements of the finite element matrix for saving the calculation memory. By numerical simulation of several typical models, we verified the correctness of the forward modeling algorithm.
The iterative formula of the whole range apparent resistivity was develpped to solving the problem of distortion of Cagniard apparent resistivity when the date are collected near the source. Typical 1D layered model and simple 2D model are calculated. After rectify the date, it can accurately reflect the characteristics of the model which verified the fact that the exploration near the source is possible.
本文采用有限单元法进行数值模拟。用Matlab为程序编译工具,采用双线性插值和双二次插值的有限单元法,推导出了相对应的边值问题和变分问题满足的微分方程。为了模拟无穷远边界并满足计算机的内存需求,在保证计算精度的情况下设计了非均匀网格剖分。在程序编制过程中,只存储有限元系数矩阵的非零元素,大大节省了存储空间。通过计算几个典型模型的卡尼亚视电阻率拟断面图和相位拟断面图,得出了不同模型下电磁场响应特征的分布规律,验证了算法的正确性。
针对频率域可控源电磁法中卡尼亚电阻率在过渡区和近区畸变的问题,给出了全区视电阻率的迭代公式,并对典型的一维层状模型以及简单二维模型进行了计算。过渡区和近区数据经过校正后,可以正确反映出模型的地电特征,证明了线源下近区勘探的可能性。
Because of the artificial source, fCSEM has overcome the weakness of natural source, which is the shortcoming of MT, so it has been widely used in the mineral, oil and gas exploration and so on. This paper has done some studies on the 2D forward problem of fCSEM, and uses the whole range apparent resistivity to rectify the date which are collected near the source.
The finite element method was used for numerical simulation. Based on Matlab as programming tool, we adopted the finite element method of linear interpolation and second interpolation in a rectangular element for solving the fCSEM forward problem and derived the corresponding formulas for calculating. In order to simulate the infinity border and meet the demand of computer memory, the non-uniform gird is designed to ensuring the accuracy. In programming, we only stored the non-zero elements of the finite element matrix for saving the calculation memory. By numerical simulation of several typical models, we verified the correctness of the forward modeling algorithm.
The iterative formula of the whole range apparent resistivity was develpped to solving the problem of distortion of Cagniard apparent resistivity when the date are collected near the source. Typical 1D layered model and simple 2D model are calculated. After rectify the date, it can accurately reflect the characteristics of the model which verified the fact that the exploration near the source is possible.
引文
[1]何继善编译.可控源音频大地电磁法[M].长沙：中南工业大学出版社,1990.
[2]Coggon J H. Electromagnetic and electrical modeling by the finite element method[J]Geophysics,1971,36(1):132-151.
[3]William L, Rodi. A technique for improving the accuracy of finite element solution for magnetotelluric data[J]Geophysics,1976,44(3):483-506.
[4]Rijo L. Modeling of electric and electromagnetic data [D].Ph. Ddissertation, University of Utah,1977.
[5]Wannamaker PE, Stodt JA, Rijo L. Two-dimensional topographic responses in magnetotelluric model using finite elements[J]Geophysics,1986,51(11):2131-2144.
[6]de Lugao P, Wannamaker P E. Calculating the two-dimensional magnetotelluric Jacobian in finite elements using reciprocity[J]Geophysics,1996,127(3):806-810.
[7]Franke A, Ralph B, Klaus S.2D finite element modeling of plane-wave diffusive time-harmonic electromagnetic fields using adaptive unstructured grids [A]. Proceeding of the 17th Work shop,2004,S.2:1-6.
[8]Mogi T. Three-dimensional modeling of magnetotelluric data using finite-element method [J]Journal of Applied Geophysics,1996,35(2):185-189
[9]Zyserman F I, Santos J E. Parallel finite element algorithm with domain decomposition for three-dimensional magneto telluric modeling[J] Journal of Applied Geophysics,2000,44(4):337-351.
[10]Son J S, Song y, Suh J H. High-Frequency Three-Dimensional Electromagnetic Modeling Using Vector Finite Elements [J] Three-Dimensional Electromagnetic Ⅲ, Adelaide, Australia,February,2003:20-21.
[11]Mitsuhata Y, Uchida T.3D magneto telluric modeling using the T-finite-element method[J]. Geophysics,2004,69(1):108-119.
[12]胡建德,王光愕,陈乐寿等.大地电磁二维正演计算中若干问题的讨论[J].石油地球物理勘探,1982,17(6)：47-55.
[13]赵生凯,徐世浙.有限单元法计算良导嵌入体的大地电磁测深曲线[J]物探化探计算技术，1983,5(1)：47-55.
[14]徐世浙.地球物理中的有限单元法[M].科学出版社,1994.
[15]徐世浙,于涛,李予国等.电导率连续变化的MT有限元正演(Ⅰ)[J].高校地质学 报,1995,1(2)：65-73.
[16]李予国,徐世浙，刘斌等.电导率连续变化的MT有限元正演(Ⅱ)[J].高校地质学报,1996,2(4)：448-452.
[17]陈小斌.有限元直接迭代算法及其在电磁模拟中的应用研究[D].江汉石油学院,2000.
[18]阮百尧,熊斌,徐世浙.三维地电断面电阻率测深有限元数值模拟[J].地球科学,2001,26(1)：73-77.
[19]阮百尧,熊斌.电导率连续变化的三维电阻率测深有限元数值模拟[J].地球物理学报,2002,45(1)：14-18.
[20]黄临平,戴世坤.复杂条件下3D电磁场有限元计算方法[J].地球科学-中国地质大学学报,2002,27(6)：775-779.
[2l]强建科.起伏地形三维电阻率正演模拟与反演成像研究[D].武汉：中国地质大学,2006.
[22]强建科,罗延钟.三维地形直流电阻率有限元法模拟[J].地球物理学报,2007,50(5)：1606-1613.
[23]黄俊革.三维电阻率／极化率有限元正演模拟与反演成像[D].长沙：中南大学,2003.
[24]黄俊革,阮百尧,王家林.坑道直流电阻率法超前探测的快速反演[J].地球物理学报,2007,50(2)：619-624.
[25]王若,王妙月,卢元林.三维三分量CASMT法有限元正演模拟研究初探[J].地球物理学进展,2007,22(2)：579-585.
[26]史明娟,徐世浙,刘斌.大地电磁二次函数插值的有限元法正演模拟[J].地球物理学报,1997,40(3)：421-430.
[27]王永刚.全空间电阻率法探测油藏边界有限元模拟研究[D]西安,长安大学.2006.
[28]周刚.人工源电磁法二维半问题等参有限元法数值模拟研究[D].北京,中国地质大学,2006.
[29]王胜阁.地形起伏条件下可控源音频大地电磁法2.5维正演研究[D].北京,中国地质大学,2007.
[30]蔡军涛,阮百尧,赵国泽等.复电阻率法二维有限元数值模拟[J].地球物理学报,2007,50(6)：1869-1876.
[31]陈小斌,张翔,胡文宝.有限元直接迭代算法在MT二维正演计算中的应用[J].石油地球物理勘探,2002,35(4)：487-496.
[32]刘树才,周圣武.二维电法数值模拟中的网格剖分方法[J].物化探计算技术,1995,17(1)：49-53.
[33]汤井田,何继善.可控源音频大地电磁法及其应用[M].长沙：中南大学出版社,2005.
[34]曹昌祺.水平层状大地上的交流视电阻率[J].地球物理学报,1978,21(2)：248-251.
[35]Spies, B.R. and Eggers D. E. The use and misuse of apparent resistivity in electromagnetic methods[J]. Geophysics,1986,51 (7):1462-1471.
[36]Wilt. M. and Stark. M. A simple method for calculating the apparent resistivity from electromagnetic sounding data[J]. Geophysics,1982,47(7):1100-1105.
[37]殷长春,朴化荣.时间域和频率域电磁测深法中的视电阻率全区定义[J].物探与化探,1991,15(2)：290-299.
[38]汤井田,何继善.水平电偶源频率域测深中全区视电阻率定义的新方法[J].地球物理学报,1994,37(4)：543-551.
[39]Hohmann G W. Electromagnetic scattering by conductors in the earth near a line source of current[J]. Geophysics,1971,36(1):101-131.
[40]陈小斌,胡文宝.有限元直接迭代算法及其在线源频率域电磁响应计算中的应用[J].地球物理学报,2002,45(1)：119-130.
[41]阎述,陈明生.线源频率域电磁测深二维正演(一)[J].煤田地质与勘探,1999,27(5)：60-62.
[42]阎述,陈明生.线源频率域电磁测深二维正演(二)[J].煤田地质与勘探,1999,27(6)：56-59.
[43]底青云,王若等.可控源音频大地电磁数据正反演及方法应用[M].北京：科学出版社,2008.
[44]曾国.大地电磁二维有限元正演数值模拟[D].中南大学,2008.
[45]米萨克N.纳比吉安主编.勘察地球物理电磁法(第一卷：理论).赵经祥,王艳君译.北京：地质出版社,1992.
[2]Coggon J H. Electromagnetic and electrical modeling by the finite element method[J]Geophysics,1971,36(1):132-151.
[3]William L, Rodi. A technique for improving the accuracy of finite element solution for magnetotelluric data[J]Geophysics,1976,44(3):483-506.
[4]Rijo L. Modeling of electric and electromagnetic data [D].Ph. Ddissertation, University of Utah,1977.
[5]Wannamaker PE, Stodt JA, Rijo L. Two-dimensional topographic responses in magnetotelluric model using finite elements[J]Geophysics,1986,51(11):2131-2144.
[6]de Lugao P, Wannamaker P E. Calculating the two-dimensional magnetotelluric Jacobian in finite elements using reciprocity[J]Geophysics,1996,127(3):806-810.
[7]Franke A, Ralph B, Klaus S.2D finite element modeling of plane-wave diffusive time-harmonic electromagnetic fields using adaptive unstructured grids [A]. Proceeding of the 17th Work shop,2004,S.2:1-6.
[8]Mogi T. Three-dimensional modeling of magnetotelluric data using finite-element method [J]Journal of Applied Geophysics,1996,35(2):185-189
[9]Zyserman F I, Santos J E. Parallel finite element algorithm with domain decomposition for three-dimensional magneto telluric modeling[J] Journal of Applied Geophysics,2000,44(4):337-351.
[10]Son J S, Song y, Suh J H. High-Frequency Three-Dimensional Electromagnetic Modeling Using Vector Finite Elements [J] Three-Dimensional Electromagnetic Ⅲ, Adelaide, Australia,February,2003:20-21.
[11]Mitsuhata Y, Uchida T.3D magneto telluric modeling using the T-finite-element method[J]. Geophysics,2004,69(1):108-119.
[12]胡建德,王光愕,陈乐寿等.大地电磁二维正演计算中若干问题的讨论[J].石油地球物理勘探,1982,17(6)：47-55.
[13]赵生凯,徐世浙.有限单元法计算良导嵌入体的大地电磁测深曲线[J]物探化探计算技术，1983,5(1)：47-55.
[14]徐世浙.地球物理中的有限单元法[M].科学出版社,1994.
[15]徐世浙,于涛,李予国等.电导率连续变化的MT有限元正演(Ⅰ)[J].高校地质学 报,1995,1(2)：65-73.
[16]李予国,徐世浙，刘斌等.电导率连续变化的MT有限元正演(Ⅱ)[J].高校地质学报,1996,2(4)：448-452.
[17]陈小斌.有限元直接迭代算法及其在电磁模拟中的应用研究[D].江汉石油学院,2000.
[18]阮百尧,熊斌,徐世浙.三维地电断面电阻率测深有限元数值模拟[J].地球科学,2001,26(1)：73-77.
[19]阮百尧,熊斌.电导率连续变化的三维电阻率测深有限元数值模拟[J].地球物理学报,2002,45(1)：14-18.
[20]黄临平,戴世坤.复杂条件下3D电磁场有限元计算方法[J].地球科学-中国地质大学学报,2002,27(6)：775-779.
[2l]强建科.起伏地形三维电阻率正演模拟与反演成像研究[D].武汉：中国地质大学,2006.
[22]强建科,罗延钟.三维地形直流电阻率有限元法模拟[J].地球物理学报,2007,50(5)：1606-1613.
[23]黄俊革.三维电阻率／极化率有限元正演模拟与反演成像[D].长沙：中南大学,2003.
[24]黄俊革,阮百尧,王家林.坑道直流电阻率法超前探测的快速反演[J].地球物理学报,2007,50(2)：619-624.
[25]王若,王妙月,卢元林.三维三分量CASMT法有限元正演模拟研究初探[J].地球物理学进展,2007,22(2)：579-585.
[26]史明娟,徐世浙,刘斌.大地电磁二次函数插值的有限元法正演模拟[J].地球物理学报,1997,40(3)：421-430.
[27]王永刚.全空间电阻率法探测油藏边界有限元模拟研究[D]西安,长安大学.2006.
[28]周刚.人工源电磁法二维半问题等参有限元法数值模拟研究[D].北京,中国地质大学,2006.
[29]王胜阁.地形起伏条件下可控源音频大地电磁法2.5维正演研究[D].北京,中国地质大学,2007.
[30]蔡军涛,阮百尧,赵国泽等.复电阻率法二维有限元数值模拟[J].地球物理学报,2007,50(6)：1869-1876.
[31]陈小斌,张翔,胡文宝.有限元直接迭代算法在MT二维正演计算中的应用[J].石油地球物理勘探,2002,35(4)：487-496.
[32]刘树才,周圣武.二维电法数值模拟中的网格剖分方法[J].物化探计算技术,1995,17(1)：49-53.
[33]汤井田,何继善.可控源音频大地电磁法及其应用[M].长沙：中南大学出版社,2005.
[34]曹昌祺.水平层状大地上的交流视电阻率[J].地球物理学报,1978,21(2)：248-251.
[35]Spies, B.R. and Eggers D. E. The use and misuse of apparent resistivity in electromagnetic methods[J]. Geophysics,1986,51 (7):1462-1471.
[36]Wilt. M. and Stark. M. A simple method for calculating the apparent resistivity from electromagnetic sounding data[J]. Geophysics,1982,47(7):1100-1105.
[37]殷长春,朴化荣.时间域和频率域电磁测深法中的视电阻率全区定义[J].物探与化探,1991,15(2)：290-299.
[38]汤井田,何继善.水平电偶源频率域测深中全区视电阻率定义的新方法[J].地球物理学报,1994,37(4)：543-551.
[39]Hohmann G W. Electromagnetic scattering by conductors in the earth near a line source of current[J]. Geophysics,1971,36(1):101-131.
[40]陈小斌,胡文宝.有限元直接迭代算法及其在线源频率域电磁响应计算中的应用[J].地球物理学报,2002,45(1)：119-130.
[41]阎述,陈明生.线源频率域电磁测深二维正演(一)[J].煤田地质与勘探,1999,27(5)：60-62.
[42]阎述,陈明生.线源频率域电磁测深二维正演(二)[J].煤田地质与勘探,1999,27(6)：56-59.
[43]底青云,王若等.可控源音频大地电磁数据正反演及方法应用[M].北京：科学出版社,2008.
[44]曾国.大地电磁二维有限元正演数值模拟[D].中南大学,2008.
[45]米萨克N.纳比吉安主编.勘察地球物理电磁法(第一卷：理论).赵经祥,王艳君译.北京：地质出版社,1992.