基于电场矢量波动方程的三维可控源电磁法有限单元法数值模拟
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摘要
从可控源电磁法的基本原理出发,推导了基于电场矢量波动方程的三维边值问题,利用广义变分原理,把边值问题转换为变分问题,并引入散度条件,避免了伪解的出现,使有限元计算在理论上更加完备.在准静态近似条件下,把水平电偶极子在空中和大地的远区电场闭合表达式作为有限元计算中的区域外边界条件,解决了边界条件加载的困难;把应用于地震模拟中的伪delta函数引入到可控源电磁法中的三维有限元模拟中,消除了源点的奇异性,提高了方程组的稳定性.通过对均匀大地和层状介质模型的模拟,检验了程序的正确性,并对典型的地质体模型进行了数值模拟,分析了其变化规律.
From the basic principles of controlled source electromagnetic,boundary value problem based on electric field vector wave equation is derived,and then transformed into variational problem by generalized variational principle from which weak solutions form is obtained based on vector wave equation of the electric field,and the introduction of divergence conditions avoids a spurious solution and makes electromagnetic finite element method more complete in theory.In quasi-static conditions,the closed expressions of electric fields in the far zone in the air and the earth are taken as the outer boundary conditions of the finite element method,difficulties in loading the boundary conditions are solved.The pseudo delta function in seismic method is used to simulate three-dimensional electric dipole source in finite element simulation then the singularity at source is eliminated,and also the stability of the equation is improved remarkably. Through the numerical simulation for uniform media and layered media,the validity of the procedure is tested,and for typical geological models,numerical computing is performed and its electromagnetic response is analyzed.
From the basic principles of controlled source electromagnetic,boundary value problem based on electric field vector wave equation is derived,and then transformed into variational problem by generalized variational principle from which weak solutions form is obtained based on vector wave equation of the electric field,and the introduction of divergence conditions avoids a spurious solution and makes electromagnetic finite element method more complete in theory.In quasi-static conditions,the closed expressions of electric fields in the far zone in the air and the earth are taken as the outer boundary conditions of the finite element method,difficulties in loading the boundary conditions are solved.The pseudo delta function in seismic method is used to simulate three-dimensional electric dipole source in finite element simulation then the singularity at source is eliminated,and also the stability of the equation is improved remarkably. Through the numerical simulation for uniform media and layered media,the validity of the procedure is tested,and for typical geological models,numerical computing is performed and its electromagnetic response is analyzed.
引文
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[3] 闫述.基于三维有限元数值模拟的电和电磁探测研究[博士论文].西安:西安交通大学,2003 YAN S.Studies on the electromagnetic prospecting based on three-dimensional finite element method [Ph.D.thesis],Xi' an:Xi'an Jiaotong University,2003
[4] Lee K H,Morrison H F.A numerical solution for the electromagnetic scattering by a two-dimensional inhomogeneity.Geophysics,1985,50(3) :466-472
[5] Eugene A.Badea,Mark E.Everettz,Gregory A.Newman, et al.Finite-element analysis of controlled-source electromagnetic induction using Coulomb-gauged potentials.Geophysics,2001,66(3) :786-799
[6] Mitsuhata Y,Uchida T.2． 5-D modeling and inversion of CSEM data.Expanded Abstracts,69th Ann.Int.Mtg.,Soc.Expl.Geophys.1999
[7] Mitsuhata Y.2-D electromagnetic modeling by the finite-element method with a dipole source and topography. Geophysics,2000
[8] Yuji Mitsuhata,Toshihiro Uchida.2． 5-D SEM Inversion Based on ABIC.SEG Exparded Abstract,2000
[9] 底青云,Martyn Unsworth,王妙月.复杂介质有限元法2． 5维可控源音频大地电磁法数值模拟.地球物理学报,2004,47(4) :723-730 Di Q Y,Unsworth,Wang M Y.2． 5D CSAMT modeling with the finite element method over 2D complex earth media.Chinese J.Geophys.(in Chinese),2004,47(4) :723-730
[10] 底青云,Martyn Unsworth,王妙月.有限元法2． 5维CSAMT数值模拟.地球物理学进展,19(2) :317-324 Di Q Y,Unsworth,Wand M Y.2． 5D CSAMT modeling with the finite element method.Progress in Geophysics (in Chinese),19(2) :317-324
[11] 闫述,陈明生.电偶源频率电磁测深三维地电模型有限元正演.煤田地质与勘探,2000,28(3) :50-56 Yan S,Chen M S.Finite element solution of three dimensional geoelectric models frequency electromagnetic sounding excited by a horizontal electric dipole.Coal Geology & Exploration (in Chinese),2000,28(3) :50-56
[12] Li Yuguo.2D marine controlled-source electromagnetic modeling:Partl-An adaptive finite-element algorithm.Geophysics,2007,72(2) :51-62
[13] Li Yuguo.2D marine controlled-source electromagnetic modeling:Part 2-The effect of bathymetry.Geophysics,2007,72(2) :63-71
[14] 徐世浙.地球物理中的有限单元法.北京:科学出版社,1994 Xu S Z.The Finite Element Method in Geophysics ( in Chinese).Beijing:Science Press,1994
[15] 金建铭.电磁场有限单元法.西安:西安电子科技大学出版社,1998 [15] Jin J M.Finite Element Method of Electromagnetic Field (in Chinese).Xi'an:Xi'an Electronic University Press,1998
[16] 考夫曼AA,Keller G V.频率域和时间域电磁测深.北京:地质出版社,1987 Kaufman A A,Keller G V.Frequency and Transient Sounding.Beijing:Geological Publishing House,1987
[17] 张秋光.场论.北京:地质出版社,1983 Zhang Q G.Field Theory.Beijing:Geological Publishing House,1983
[2] 何继善.可控源音频大地电磁法.长沙:中南大学出版社,1990． He J S.Controlled Source Audio-frequency Magnetotellurics Method (in Chinese).Changsha;Central South University Press,1990
[3] 闫述.基于三维有限元数值模拟的电和电磁探测研究[博士论文].西安:西安交通大学,2003 YAN S.Studies on the electromagnetic prospecting based on three-dimensional finite element method [Ph.D.thesis],Xi' an:Xi'an Jiaotong University,2003
[4] Lee K H,Morrison H F.A numerical solution for the electromagnetic scattering by a two-dimensional inhomogeneity.Geophysics,1985,50(3) :466-472
[5] Eugene A.Badea,Mark E.Everettz,Gregory A.Newman, et al.Finite-element analysis of controlled-source electromagnetic induction using Coulomb-gauged potentials.Geophysics,2001,66(3) :786-799
[6] Mitsuhata Y,Uchida T.2． 5-D modeling and inversion of CSEM data.Expanded Abstracts,69th Ann.Int.Mtg.,Soc.Expl.Geophys.1999
[7] Mitsuhata Y.2-D electromagnetic modeling by the finite-element method with a dipole source and topography. Geophysics,2000
[8] Yuji Mitsuhata,Toshihiro Uchida.2． 5-D SEM Inversion Based on ABIC.SEG Exparded Abstract,2000
[9] 底青云,Martyn Unsworth,王妙月.复杂介质有限元法2． 5维可控源音频大地电磁法数值模拟.地球物理学报,2004,47(4) :723-730 Di Q Y,Unsworth,Wang M Y.2． 5D CSAMT modeling with the finite element method over 2D complex earth media.Chinese J.Geophys.(in Chinese),2004,47(4) :723-730
[10] 底青云,Martyn Unsworth,王妙月.有限元法2． 5维CSAMT数值模拟.地球物理学进展,19(2) :317-324 Di Q Y,Unsworth,Wand M Y.2． 5D CSAMT modeling with the finite element method.Progress in Geophysics (in Chinese),19(2) :317-324
[11] 闫述,陈明生.电偶源频率电磁测深三维地电模型有限元正演.煤田地质与勘探,2000,28(3) :50-56 Yan S,Chen M S.Finite element solution of three dimensional geoelectric models frequency electromagnetic sounding excited by a horizontal electric dipole.Coal Geology & Exploration (in Chinese),2000,28(3) :50-56
[12] Li Yuguo.2D marine controlled-source electromagnetic modeling:Partl-An adaptive finite-element algorithm.Geophysics,2007,72(2) :51-62
[13] Li Yuguo.2D marine controlled-source electromagnetic modeling:Part 2-The effect of bathymetry.Geophysics,2007,72(2) :63-71
[14] 徐世浙.地球物理中的有限单元法.北京:科学出版社,1994 Xu S Z.The Finite Element Method in Geophysics ( in Chinese).Beijing:Science Press,1994
[15] 金建铭.电磁场有限单元法.西安:西安电子科技大学出版社,1998 [15] Jin J M.Finite Element Method of Electromagnetic Field (in Chinese).Xi'an:Xi'an Electronic University Press,1998
[16] 考夫曼AA,Keller G V.频率域和时间域电磁测深.北京:地质出版社,1987 Kaufman A A,Keller G V.Frequency and Transient Sounding.Beijing:Geological Publishing House,1987
[17] 张秋光.场论.北京:地质出版社,1983 Zhang Q G.Field Theory.Beijing:Geological Publishing House,1983
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