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Mathematical Problems in Engineering Volume 2017 ,2017-05-15
Finite Element Method for Modeling 3D Resistivity Sounding on Anisotropic Geoelectric Media
Research Article
Tao Song 1 , 2 Yun Liu 1 Yun Wang 1
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DOI:10.1155/2017/8027616
Received 2017-01-11, accepted for publication 2017-04-09, Published 2017-04-09
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摘要

A 3D DC finite element method forward program is developed in this paper for anisotropic geoelectric media. Both total and secondary field approaches have been implemented. In this paper, we focused on the structured grid scheme. The modeling shows that the symmetry of the structure grid determines the symmetry of the response potential around the point source for an anisotropic half-space. Through numerical modeling with three kinds of coarse meshes by the total field approach and secondary field approach separately, a higher accuracy can be achieved via the secondary field approach. And the relative error between the numerical solution and the analytical solution is less than 2%. Modeling results contrasting with previous scholars also verify the correctness of the algorithm. Then, the numerical results of 3D models with anisotropic properties were presented and compared to models with isotropic properties. These results clearly illustrated the strong effect of anisotropy and the problems in interpretation if anisotropy was not properly addressed. This work will establish the foundation for our future effort of building a 3D inversion program with arbitrary anisotropy.

授权许可

Copyright © 2017 Tao Song et al. 2017
This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

图表

The global region subdivision with coarse area. (a) 2D global mesh design; (b) 3D global mesh design.

The global region subdivision with coarse area. (a) 2D global mesh design; (b) 3D global mesh design.

Subdivision of the region. (a) Five-tetrahedron scheme A; (b) five-tetrahedron scheme B; (c) 3D global mesh with scheme A; (d) 3D global mesh with scheme B; (e) 3D global mesh with schemes A and B.

Subdivision of the region. (a) Five-tetrahedron scheme A; (b) five-tetrahedron scheme B; (c) 3D global mesh with scheme A; (d) 3D global mesh with scheme B; (e) 3D global mesh with schemes A and B.

Subdivision of the region. (a) Five-tetrahedron scheme A; (b) five-tetrahedron scheme B; (c) 3D global mesh with scheme A; (d) 3D global mesh with scheme B; (e) 3D global mesh with schemes A and B.

Subdivision of the region. (a) Five-tetrahedron scheme A; (b) five-tetrahedron scheme B; (c) 3D global mesh with scheme A; (d) 3D global mesh with scheme B; (e) 3D global mesh with schemes A and B.

Subdivision of the region. (a) Five-tetrahedron scheme A; (b) five-tetrahedron scheme B; (c) 3D global mesh with scheme A; (d) 3D global mesh with scheme B; (e) 3D global mesh with schemes A and B.

Forward modeling accuracy in percentage for three kinds of subdivision relative to analytical solutions by total potential approach: (a) by grid shown in Figure 2(c), (b) by grid shown in Figure 2(d), and (c) by grid shown in Figure 2(e).

Forward modeling accuracy in percentage for three kinds of subdivision relative to analytical solutions by total potential approach: (a) by grid shown in Figure 2(c), (b) by grid shown in Figure 2(d), and (c) by grid shown in Figure 2(e).

Forward modeling accuracy in percentage for three kinds of subdivision relative to analytical solutions by total potential approach: (a) by grid shown in Figure 2(c), (b) by grid shown in Figure 2(d), and (c) by grid shown in Figure 2(e).

Forward modeling accuracy in percentage for three kinds of coarse meshes relative to analytical solutions by total potential approach: (a) by Mesh 1, (b) by Mesh 2, and (c) by Mesh 3.

Forward modeling accuracy in percentage for three kinds of coarse meshes relative to analytical solutions by total potential approach: (a) by Mesh 1, (b) by Mesh 2, and (c) by Mesh 3.

Forward modeling accuracy in percentage for three kinds of coarse meshes relative to analytical solutions by total potential approach: (a) by Mesh 1, (b) by Mesh 2, and (c) by Mesh 3.

Forward modeling accuracy in percentage relative to analytical solutions with three kinds of sparse meshes by secondary potential approach: (a) by Mesh 1, (b) by Mesh 2, and (c) by Mesh 3.

Forward modeling accuracy in percentage relative to analytical solutions with three kinds of sparse meshes by secondary potential approach: (a) by Mesh 1, (b) by Mesh 2, and (c) by Mesh 3.

Forward modeling accuracy in percentage relative to analytical solutions with three kinds of sparse meshes by secondary potential approach: (a) by Mesh 1, (b) by Mesh 2, and (c) by Mesh 3.

A two-layer model.

A two-layer model with azimuthal anisotropy.

Apparent resistivity of pole-pole array along x-direction for model shown in Figure 7.

A two-layer model consisting of a TTI overburden layer with variable dip over an isotropic half-space.

Apparent resistivities along x- and y-direction for pole-pole array with changing dipping angles.

A 2D anisotropic model.

Dipole-dipole array apparent resistivity of model shown in Figure 11. The resistivity of the isotropic model is described as follows: the resistivity of the half-space is 20 Ω⋅m and the resistivity of the anomaly is 1 Ω⋅m. (a) Dipole spacing r=4 m. (b) Dipole spacing r=20 m.

Dipole-dipole array apparent resistivity of model shown in Figure 11. The resistivity of the isotropic model is described as follows: the resistivity of the half-space is 20 Ω⋅m and the resistivity of the anomaly is 1 Ω⋅m. (a) Dipole spacing r=4 m. (b) Dipole spacing r=20 m.

Comparison of 3D and 2D FEM numerical solutions of the model shown in Figure 11. (a) Apparent resistivity pseudosections along the central line by 3D FEM. (b) Apparent resistivity pseudosections by 2D FEM.

Comparison of 3D and 2D FEM numerical solutions of the model shown in Figure 11. (a) Apparent resistivity pseudosections along the central line by 3D FEM. (b) Apparent resistivity pseudosections by 2D FEM.

(a) A 3D anisotropic cube in a homogeneous isotropic half-space. The principal resistivity of the cube is given by ρx/ρy/ρz = 10/10/100 Ω⋅m and the background resistivity is 100 (Ω⋅m). (b) 21 survey lines were deployed along both x- and y-direction and the solid lines outline the boundary of the buried cube.

(a) A 3D anisotropic cube in a homogeneous isotropic half-space. The principal resistivity of the cube is given by ρx/ρy/ρz = 10/10/100 Ω⋅m and the background resistivity is 100 (Ω⋅m). (b) 21 survey lines were deployed along both x- and y-direction and the solid lines outline the boundary of the buried cube.

Apparent resistivity pseudosection of the model shown in Figure 14. (a) Apparent resistivity pseudosection of line X11 for the isotropic model. (b) Apparent resistivity pseudosection of line X07 for the isotropic model. (c) Apparent resistivity pseudosection of line X11 for the anisotropic model. (d) Apparent resistivity pseudosection of line X07 for the anisotropic model.

Apparent resistivity pseudosection of the model shown in Figure 14. (a) Apparent resistivity pseudosection of line X11 for the isotropic model. (b) Apparent resistivity pseudosection of line X07 for the isotropic model. (c) Apparent resistivity pseudosection of line X11 for the anisotropic model. (d) Apparent resistivity pseudosection of line X07 for the anisotropic model.

Apparent resistivity pseudosection of the model shown in Figure 14. (a) Apparent resistivity pseudosection of line X11 for the isotropic model. (b) Apparent resistivity pseudosection of line X07 for the isotropic model. (c) Apparent resistivity pseudosection of line X11 for the anisotropic model. (d) Apparent resistivity pseudosection of line X07 for the anisotropic model.

Apparent resistivity pseudosection of the model shown in Figure 14. (a) Apparent resistivity pseudosection of line X11 for the isotropic model. (b) Apparent resistivity pseudosection of line X07 for the isotropic model. (c) Apparent resistivity pseudosection of line X11 for the anisotropic model. (d) Apparent resistivity pseudosection of line X07 for the anisotropic model.

Pseudosection of the apparent resistivity for dipole-dipole array when the dipping angle α=45°. (a) Pseudosection of apparent resistivity at X11. (b) Pseudosection of apparent resistivity at X07. (c) Pseudosection of apparent resistivity at Y11. (d) Pseudosection of apparent resistivity Y07.

Pseudosection of the apparent resistivity for dipole-dipole array when the dipping angle α=45°. (a) Pseudosection of apparent resistivity at X11. (b) Pseudosection of apparent resistivity at X07. (c) Pseudosection of apparent resistivity at Y11. (d) Pseudosection of apparent resistivity Y07.

Pseudosection of the apparent resistivity for dipole-dipole array when the dipping angle α=45°. (a) Pseudosection of apparent resistivity at X11. (b) Pseudosection of apparent resistivity at X07. (c) Pseudosection of apparent resistivity at Y11. (d) Pseudosection of apparent resistivity Y07.

Pseudosection of the apparent resistivity for dipole-dipole array when the dipping angle α=45°. (a) Pseudosection of apparent resistivity at X11. (b) Pseudosection of apparent resistivity at X07. (c) Pseudosection of apparent resistivity at Y11. (d) Pseudosection of apparent resistivity Y07.

Pseudosection of the apparent resistivity for dipole-dipole array when the dipping angle α=90°. (a) Pseudosection of apparent resistivity at X11. (b) Pseudosection of apparent resistivity at X07. (c) Pseudosection of apparent resistivity at Y11. (d) Pseudosection of apparent resistivity at Y07.

Pseudosection of the apparent resistivity for dipole-dipole array when the dipping angle α=90°. (a) Pseudosection of apparent resistivity at X11. (b) Pseudosection of apparent resistivity at X07. (c) Pseudosection of apparent resistivity at Y11. (d) Pseudosection of apparent resistivity at Y07.

Pseudosection of the apparent resistivity for dipole-dipole array when the dipping angle α=90°. (a) Pseudosection of apparent resistivity at X11. (b) Pseudosection of apparent resistivity at X07. (c) Pseudosection of apparent resistivity at Y11. (d) Pseudosection of apparent resistivity at Y07.

Pseudosection of the apparent resistivity for dipole-dipole array when the dipping angle α=90°. (a) Pseudosection of apparent resistivity at X11. (b) Pseudosection of apparent resistivity at X07. (c) Pseudosection of apparent resistivity at Y11. (d) Pseudosection of apparent resistivity at Y07.

通讯作者

Tao Song.State Key Laboratory of Ore Deposit Geochemistry, Institute of Geochemistry, Chinese Academy of Sciences, Guiyang 550081, China, cas.cn;University of Chinese Academy of Sciences, Beijing 100049, China, ucas.ac.cn.songtao@mail.gyig.ac.cn

推荐引用方式

Tao Song,Yun Liu,Yun Wang. Finite Element Method for Modeling 3D Resistivity Sounding on Anisotropic Geoelectric Media. Mathematical Problems in Engineering ,Vol.2017(2017)

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