点源场井-地电位测量三维有限元模拟

Three-dimensional FEM modeling of point source borehole-ground electrical potential measurements

井地电法是深部地质矿产勘查的重要方法之一,研究高效率、高精度、适用于起伏地形等复杂条件下的正演模拟算法具有理论和实际意义.本文研究了点源场井地电位测量的三维有限元正演模拟,采用与井轴一致倾斜的计算区域,使剖分单元随地形起伏,用预处理共轭梯度法求解线性方程组,利用行压缩存储和改进的行压缩存储方式来存储刚度矩阵和预处理矩阵,实现了起伏地形和倾斜井情形下的正演计算.此外,本文利用仿射坐标变换技术,给出了平行五面体单元精确的单元积分公式,这种处理技术同样可推广应用于其他平行多面体单元的单元分析,相比于采用等参变换和高斯数值积分,能极大地提高计算效率.文中构建了一些模型算例,其计算结果验证了程序的可靠性和高效性.

Borehole-ground electrical method is one of the most important methods to exploit the deep mineral resources. There is theoretical and practical significance in the study of efficient and high accuracy simulations regarding a deviated well and topography. In this paper, the threedimensional FEM modeling of point source borehole-ground electrical potential measurements is studied. The computational region remains titled in the same direction of a borehole axial, and meshes are designed to undulate with the topography. Iterative solver based on pre-conditioned conjugate gradient and compressed storage scheme-Row Compressed Storage（CSR） and Modified Row Compressed Storage （MSR）-was employed to solve the linear system. By adopting these techniques, we implemented the borehole-ground electrical potential logging simulation for a deviated well and undulating topography. Furthermore, a 3-D parallel tri-prism elements discretization based on an affine transformation to map the global Cartesian coordinates into the local coordinates is used. In comparison to the isoparametric form and Gauss numerical integration, the computational time was greatly shortened. Performance and accuracy of the simulations are demonstrated by some examples given in this paper.