﻿ 电性任意各向异性且分块连续变化CSAMT三维有限元数值模拟
 地球物理学报  2019, Vol. 62 Issue (10): 3923-3933 PDF

1. 中国地质科学院地球物理地球化学勘查研究所, 河北廊坊 065000;
2. 国土资源部地球物理电磁法探测技术重点实验室, 河北廊坊 065000;
3. 中国自然资源航空物探遥感中心, 北京 100083

Three-dimensional CSAMT FEM modeling on electrical medium with arbitrary anisotropy and continuous variation
LI Yong1,2, LIN PinRong1,2, LIU ZuJian3
1. Institute of Geophysical and Geochemical Exploration, Chinese Academy of Geological Science, Langfang Hebei 065000, China;
2. Laboratory of Geophysical Electromagnetic Probing Technologies, Ministry of Land and Resources, Langfang Hebei 065000, China;
3. China Aero Geophysical Survey and Remote Sensing Center for Natural and Resources, Beijing 100083, China
Abstract: Considering that the conductivity of the earth medium is arbitrarily anisotropic and continuously changes with spatial position,this work makes three-dimensional CSAMT finite element numerical simulation to resolve the electromagnetic field directly. Firstly,we give the governing equation and its corresponding variational problem of CSAMT secondary electric field in an arbitrary anisotropic conductive medium. Then,the study area is divided into many elements of arbitrary hexahedral shape. In each element,the tri-line interpolation is performed for the arbitrary anisotropic conductivity parameter to simulate the arbitrary anisotropy and continuous change of rock and ore conductivity. The variational problem is transformed into solution of a linear algebraic equation set. Three-dimensional finite element numerical simulation results of an one-dimension conductivity model with anisotropy and continuous change are compared with analytical solution results of an anisotropic and layered progressive conductivity-model to verify the effectiveness of the method. Forward calculation is carried out on other three-dimensional geoelectric models,in which the electrical conductivity varies linearly with position and the medium is isotropic,principal axis anisotropic,azimuthal anisotropic and oblique anisotropic,respectively. Results show that the electrical conductivity anisotropy and continuous change have prominent influence on CSAMT apparent resistivity and phase data.
Keywords: CSAMT    Medium with arbitrary anisotropic conductivity    Continuous variation of conductivity within each block    Finite element method    Three-dimensional modeling
0 引言

1 三维CSAMT变分问题

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 图 1 电导率各向异性且连续变化三维CSAMT模型示意图 Fig. 1 Schematic diagram of 3-D CSAMT model for electrical medium with arbitrary anisotropy and continuous variation

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2 有限元方程

2.1 网格剖分及插值

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 图 2 任意形状的六面体单元示意图 (a)子单元；(b)母单元. Fig. 2 Schematic diagram for an arbitrary hexahedron element (a) Sub-element; (b) Parent element.

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2.2 单元分析

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2.3 线性方程组

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3 模型算例

3.1 电导率各向异性且连续变化一维模型

 图 3 电导率各向异性且连续变化一维模型示意图 Fig. 3 Schematic diagram of 1-D CSAMT model for arbitrary anisotropic and layered homogeneous electrical medium

 图 4 电导率各向异性且分层均匀一维模型示意图 (a) σx-z变化模型；(b) σy-z变化模型；(c) σz-z变化模型. Fig. 4 Schematic diagram of 1-D CSAMT model for electrical medium with arbitrary anisotropy and layered homogeneous (a) σx changes in the z direction; (b) σy changes in the z direction; (c) σz changes in the z direction.

 图 5 电导率各向异性且连续变化一维模型三维有限元计算结果与渐进模型解析解结果对比 (a)视电阻率ρxy；(b)相位φxy. Fig. 5 Comparison between the analytical solutions of the progressive model and the 3-D FEM solutions for electrical medium with arbitrary anisotropy and continuous variation (a) Apparent resistivity ρxy; (b) Phase φxy.
3.2 电导率各向异性且连续变化三维模型

 图 6 电导率各向异性且连续变化三维模型示意图 (a)三维模型示意图；(b)平面图；(c)断面图. Fig. 6 Schematic diagram for the 3D model for electrical medium with arbitrary anisotropy and continuous variation (a) Schematic diagram for the 3D model; (b) Planar view; (c) Section view.

 图 7 不同电导率各向异性类型不同剖面的三维有限元数值模拟结果 (a1，a2) x=-200 m剖面的视电阻率和相位；(b1，b2) x=0 m剖面的视电阻率和相位；(c1，c2) x=400 m剖面的视电阻率和相位. Fig. 7 Numerical simulation results of 3D finite element method of different profiles for different anisotropic conductivity models (a1, a2) Show the apparent resistivity and phase when the section x-axis is a -200 meter; (b1, b2) Show the apparent resistivity and phase when the section x-axis is a 0 meter; (c1, c2) Show the apparent resistivity and phase when the section x-axis is a 400 meter.
 图 8 频率为10 Hz时地表视电阻率(Ωm)和相位(°)等值线图 (a1, a2, a3, a4)分别为异常体⑤为电导率各向同性、主轴各向异性、方位各向异性和倾斜各向异性的视电阻率; (b1, b2, b3, b4)分别为异常体⑤为电导率各向同性、主轴各向异性、方位各向异性和倾斜各向异性的相位. Fig. 8 Contour plots of the apparent resistivity and phase on the ground at 10 Hz (a1, a2, a3, a4) are the apparent resistivity when the conductivity of abnormal body ⑤ is isotropic, principal axial anisotropic, azimuthal anisotropic, and oblique anisotropic respectively; (b1, b2, b3, b4) are the apparent phase when the conductivity of abnormal body ⑤ is isotropic, principal axial anisotropic, azimuthal anisotropic, and oblique anisotropic respectively.
4 结论

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