| 电导率各向异性的海洋电磁三维有限单元法正演 |
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| 引用本文： | 蔡红柱,熊彬,Michael Zhdanov.电导率各向异性的海洋电磁三维有限单元法正演[J].地球物理学报,2015,58(8):2839-2850. |
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| 作者姓名： | 蔡红柱 熊彬 Michael Zhdanov |
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| 作者单位： | 1. College of Mines&Earth Sciences, University of Utah, Salt lake city, UT 84112, USA;
2. 桂林理工大学地球科学学院, 桂林 541004 |
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| 基金项目： | 国家自然科学基金项目(40974077、41164004),广西自然科学基金项目(2011GXNSFA018003、2013GXNSFAA019277)联合资助. |
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| 摘 要： | 本文提出了一种基于非结构化网格的海洋电磁有限单元正演算法.为了回避场源奇异性,文中选用二次场算法,将背景电阻率设置为水平层状且各向异性,场源在水平层状各向异性介质中所激发的一次场通过汉克尔积分得到.基于Coulomb规范得到二次矢量位和标量位所满足的Maxwell方程组,通过Galerkin加权余量法形成大型稀疏有限元方程,采用不完全LU分解(ILU)预条件因子的quasi-minimum residual(QMR)迭代解法对有限元方程进行求解得到二次矢量位和标量位;进而,利用滑动平均方法得到二次矢量位和标量位在空间的导数,由此得到二次电磁场;通过一维模型对算法的可靠性进行验证,与此同时,针对实际复杂海洋电磁模型,比较有限元模拟结果与积分方程模拟结果,进一步验证算法精度.若干计算结果均表明,文中算法具有良好的通用性,适用于井中电磁、航空电磁,环境地球物理等非均匀且各向异性介质中的电磁感应基础研究.
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| 关 键 词： | 海洋电磁 非结构化网格 Coulomb规范 三维正演 有限单元法 滑动平均 |
| 收稿时间： | 2014-07-02 |
Three-dimensional marine controlled-source electromagnetic modelling in anisotropic medium using finite element method |
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| CAI Hong-Zhu,XIONG Bin,Michael Zhdanov.Three-dimensional marine controlled-source electromagnetic modelling in anisotropic medium using finite element method[J].Chinese Journal of Geophysics,2015,58(8):2839-2850. |
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| Authors: | CAI Hong-Zhu XIONG Bin Michael Zhdanov |
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| Institution: | 1. College of Mines & Earth Sciences, University of Utah, Salt lake city, UT 84112, USA;
2. College of Earth Sciences, Guilin University of Technology, Guilin 541004, China |
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| Abstract: | In recent years, we have observed a strong interest in applying controlled-source electromagnetic(CSEM)method in hydrocarbon exploration to reduce the uncertainty resulting from seismic method, especially in the marine environment. Comparing to seismic method, CSEM method can be potentially used to distinguish the oil-bearing and water bearing reservoir. In practical application, the subsurface conductivity is characterized by three-dimensional inhomogeneity and anisotropy. In the meantime, the CSEM data can also be distorted significantly by the seafloor bathymetry in marine environment. To correctly interpret the subsurface structure with the observed CSEM data, it is desirable to develop full three-dimensional forward modeling algorithm which can address the conductivity anisotropy and the variation of seafloor bathymetry.
Generally, there are three main numerical methods for the modeling of CSEM data:integral equation, finite difference, and finite element methods. Among these three methods, the finite element method with unstructured tetrahedral mesh is more suitable for simulating complex geological structures such as seafloor bathymetry. Comparing to the integral equation method, the finite element stiffness matrix is much sparser and as such the method can be used for large scale modeling of CSEM data. We adopt the node-based finite element method in this paper. For CSEM modeling, the accuracy is affected significantly by the representation of source. In order to avoid the source singularity problem caused by the total field formulation, we use the secondary field formulation and consider that the background conductivity is formed by a series of infinite horizontal layers and each with a constant conductivity value. We consider that the background conductivity is characterized by transverse anisotropy which is typically the case for marine CSEM. For the secondary field formulation, one needs to compute the background field on the mesh. We use the fast Hankel transform method to compute the background field. We adopt the classic node-based finite element method for the forward modeling. In order to avoid the spurious solution which is usually encountered in solving the Maxwell's equation directly using node-based finite element, we solve the secondary scalar and vector electromagnetic potential based on the Coulomb gauge. We re-formulated the original Maxwell's equation for the secondary scalar and vector potential. The computed background field from fast Hankel transform is converted to the primary scalar and vector potential. After applying the Galerkin finite element analysis, one can obtain a large sparse system of finite element equations. We applied the effective Dirichlet boundary condition and consider that both the scalar and vector secondary electromagnetic potential vanish on the boundary of the study domain. We solve the sparse finite element system of equations using quasi-minimum residual(QMR)method with an incomplete LU decomposition(ILU)preconditioner. Once the secondary scalar and vector potential is solved, we can compute the secondary electric and magnetic field by taking the differential of the secondary potential. The general numerical differential method can introduce significant noise. In order to solve this problem, we use a weighted moving least square method to calculate the secondary electric and magnetic field from the computed secondary potential.
To validate the effectiveness and correctness of the developed algorithm, we have implemented several model studies to compare the electromagnetic field computed from our method with the 1D analytical solution and integral equation solution. We first consider a 1D model with air and half-space as the background. We consider that there exists an infinite thin layer embedded in the half-space background. The anomalous field caused by this infinite thin layer can be computed either with finite element or fast Hankel transform. Our numerical result shows that the solution from finite element method is practically the same as the 1D analytical solution obtained from fast Hankel transform either for secondary electric or magnetic field. We have also done a comparison study for the QMR solver with Jacobian and ILU preconditioner.The result demonstrates that the convergence can be speeded up significantly by adopting the ILU preconditioner. We also consider a 3D marine reservoir model with flat bathymetry. For this model the secondary field can be effectively computed by integral equation method. As such, we have compared our results with integral equation solution for this model. In the model, we also considered the conductivity anisotropy for both reservoir and the marine sediment.Numerical result shows that the finite element solution for the problem with conductivity anisotropy is practically the same as integral equation solution. Following this, we consider a marine reservoir model with simple seafloor bathymetry represented by a trapezoidal hill. For integral equation method, we simulate this trapezoidal hill model by a staircase approximation. The numerical results show that the bathymetry effect simulated from finite element method is very close to the integral equation method especially when the receivers are far away from the bathymetry. The difference between the finite element and integral equation method can be attributed to the staircase approximation of the bathymetry for integral equation modeling. Finally, we consider a complex seafloor bathymetry model with reservoir and conductivity anisotropy. Our numerical modeling result using the finite element method have demonstrated that the observed CSEM data can be significantly distorted by the seafloor bathymetry in practical application and this effect should be addressed by robust forward modeling algorithm for the correct interpretation of subsurface geo-electric structures.
We have implemented a three-dimensional numerical modeling method for marine CSEM data using node-based finite element method based on a fully unstructured tetrahedral mesh. Comparing to the conventional structured mesh, the computation load can be significantly reduced by adopting the unstructured mesh. In the meantime, the unstructured mesh is capable of simulating complex geo-electric structures such as seafloor bathymetry. We have formulated our problem for the scalar and vector potential instead of the electric and magnetic field to address the spurious solution. In order to avoid the source singularity problem for CSEM modeling, we adopt a secondary field formulation. The sparse finite element system of equations is solved using a quasi-minimum residual method with incomplete LU decomposition result as the preconditioner to speed up the convergence. The developed algorithm was tested for several synthetic and realistic models. The numerical results have demonstrated the effectiveness of the algorithm for the modeling of complex geo-electric structures with conductivity anisotropy. |
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| Keywords: | Marine CSEM Unstructured mesh Coulomb gauge 3D modeling Finite element Moving least-squares interpolation |
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