﻿ 双轴各向异性介质中回线源瞬变电磁三维拟态有限体积正演算法
 地球物理学报  2018, Vol. 61 Issue (1): 368-378 PDF

Research on the 3D mimetic finite volume method for loop-source TEM response in biaxial anisotropic formation
ZHOU JianMei, LIU WenTao, LI Xiu, QI ZhiPeng, LIU Hang
College of Geology Engineering and Geomatics, Chang'an University, Xi'an 710054, China
Abstract: In this paper, a 3D forward modeling of loop-source transient electromagnetic (TEM) response in biaxial anisotropic formation is proposed by using the mimetic finite volume method (MFVM). Firstly, the definition of inner product is introduced, and the governing equation of TEM method is transformed into a weak form under a natural boundary condition. The computational domain is divided into a series of control volume units. The staggered grid is used to simulate the finite volume space discretization of the governing equations, including the curl operator discretization and the spatial inner product discretization. Discretization of the curl operator is based on Stokes' Theorem. Discretization of inner product in the biaxial anisotropic formation is based on midpoint average. The backward Euler time stepping method, which is unconditionally stable, is chosen to discretize the governing equation in the time domain. Electromagnetic field distribution at the initial time is obtained by solving the magnetic field of the stable current loop source in the uniform space. When solving the governing equations in the time domain, the MFVM adopts double time step size for each m constant time steps (here m is an input parameter) and uses the direct method solver PARDISO to ensure the accuracy and efficiency where the coefficient matrix just needs to decompose only once for each m time steps. Finally, the computational accuracy and computational efficiency of the proposed algorithm are verified by comparing the results of the forward modeling with a layered model and the anisotropic half-space model. The forward response of the 3D biaxial anisotropy model shows that the horizontal conductivity has a significant impact on the electromagnetic response, while the vertical direction of the conductivity change has little effect. The main reason is that the induced current generated by the loop source is mainly horizontal, so the response is primarily affected by the conductivity in the horizontal direction, while the conductivity in the vertical direction has little effect.
Key words: Transient electromagnetic    3D forward modeling    Finite volume method    Anisotropy
0 引言

1 正演算法

1.1 空间离散的模拟有限体积算法 1.1.1 控制方程

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1.1.2 弱形式表示

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1.1.3 空间离散

 图 1 网格单元(i, j, k) Fig. 1 Grid cell (i, j, k)

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nx, ny, nz分别表示x, y, z方向网格数.

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 图 2 网格单元中E与σ位置分布图 Fig. 2 Location of E and σ in grid cell

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vσxvσyvσz分别为包含所有网格单元体积与该网格单元不同方向电导率的乘积的对角矩阵，Aer(r=x, y, z)代表定义在网格棱边中心的Er平均到网格中心的转换矩阵.

1.1.4 矩阵表示

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1.2 隐式时间步迭代算法

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1.2.1 初始场

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1.2.2 线性方程组的求解

2 数值实例

2.1 层状地层模型

 图 3 层状地层模型 Fig. 3 Layered earth model
 图 4 层状模型3D解与1D解的对比 (a) dBz/dt；(b) dBz/dt相对误差；(c) Bz；(d) Bz相对误差. Fig. 4 Comparison of 3D results with 1D solutions for layered earth model (a) dBz/dt; (b) Relative errors for dBz/dt; (c) Bz; (d) Relative errors for Bz.
2.2 各向异性半空间模型

 图 5 各向异性地层模型 Fig. 5 Anisotropic earth model
 图 6 MFVM解与Yin(2016)解的对比 (a) dBz/dt；(b) Bz. Fig. 6 Comparison of results by MFVM with Yin (2016)
2.3 三维垂直接触带模型

 图 7 三维模型(Li, 2017) (a)模型图；(b)网格剖分图. Fig. 7 3D earth model (Li, 2017) (a) Model; (b) Grid for model.
 图 8 MFVM解与Li(2017)解的对比 Fig. 8 Comparison of results by MFVM with Li(2017)
2.4 各向异性分析

 图 9 三维地层模型 (a)模型示意图；(b)网格剖分图. Fig. 9 Layered earth model (a) Model; (b) Grid for model.

 图 10 S1处不同各向异性σ2模型的瞬变电磁响应 (a) dBz/dt；(b) Bz. Fig. 10 TEM response in S1 for different anisotropic σ2.

 图 11 S1处不同各向异性σ1模型的瞬变电磁响应 (a) dBz/dt；(b) Bz. Fig. 11 TEM response in S1 for different anisotropic σ1

 图 12 S2处不同各向异性σ2模型的瞬变电磁响应 (a) dBz/dt；(b) Bz. Fig. 12 TEM response in S2 for different anisotropic σ2
 图 13 S2处不同各向异性σ1模型的瞬变电磁响应 (a) dBz/dt；(b)Bz. Fig. 13 TEM response in S2 for different anisotropic σ1
3 结论

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