用边有限元法计算三维各向异性介质的电磁响应

Modeling of the 3-D Electromagnetic Responses to the Anisotropic Medium by the Edge Finite Element Method

用边有限元基函数导出了各向异性介质中麦克斯韦 (Maxwell)方程的边有限元关系式 ,计算了三维各向异性介质中多分量电磁测井的响应。将总场分离成背景场和二次场的处理技术 ,使该方法适应于有限尺寸和任意方向的源以及倾斜井眼。离散化得到的方程用加不完全乔累斯基分解预处理的Krylov子空间迭代算法实现。利用多层模型检验了该方法。模拟结果表明 ,不同的发射和接收方式在各向异性介质中电磁测井响应有较大差别 ,共面垂直线圈发射和接收 ,视电导率介于各向异性的垂直电导率和水平电导率之间 ;同轴水平线圈发射和接收 ,视电导率等于各向异性的水平电导率 ,用多分量电磁测井可以给出更多和更准确的地层电阻率信息。

The finite element formulation of the Maxwell equation in 3-D anisotropic formation was derived by using the edge finite element interpolation and it was used to simulate the 3-D electromagnetic well logging responses to the subsurface anisotropic formation. The separation of the total electric field into the background and the secondary fields makes it possible that the technique proposed is suitable for the magnetic dipole in any direction and finite size and the deviated borehole. The discrete large sparse linear equation was solved by using the Krylov subspace iterative solver, and its convergence rate was improved by the in-complete Cholesky decomposition preconditioner. Calculation of the simple model shows that, different types of transmitters and receivers have obvious differences of the electromagnetic responses to the anisotropic formation. In thick bed, configuration of co-planar coils in vertical plane shows apparent conductivity between the vertical conductivity and horizontal conductivity in the anisotropic formation; configuration of co-axial coils responses to horizontal conductivity in the anisotropic formation. By using all the information of the multi-component electromagnetic well logging tool, it is possible to derive the true conductivity in the anisotropic formation.