基于电流密度连续性条件的直流电阻率各向异性问题自适应有限元模拟

摘要

The direct current (DC) resistivity method has been widely applied in engineering,environmental,and hydrological geophysics as well as field mining,geothermal exploration and so forth.Because of existence of bedding surfaces or fissures with directional dependence,underground rocks often exhibit resistivity anisotropy.Therefore it provides an urgent impetus to develop a high-accuracy algorithm to deal with such a complex problem.This work has conducted a high-precision forward modeling for the complex DC anisotropic resistivity problem with arbitrary topography by combining the goal-oriented adaptive finite-element method with an unstructured grid.Different from previous work,we applied an extraordinary secondary virtual potential strategy to simulate the DC problem with arbitrary topography and sources.Furthermore,we built an a-posteriori error estimating algorithm based on the continuity of the normal component of current density adjusting to the DC anisotropic resistivity problem to effectively drive the goaloriented adaptive grid refinement.Finally,three synthetic anisotropic resistivity models were designed to verify the accuracy and effectiveness of the developed algorithm.The results show that this new algorithm has high accuracy and robust effectiveness.In addition,we found the continuity condition of normal component of electric current density can be adopted to design effective a-posteriori error estimating algorithms for the DC resistivity problem.%直流电阻率法被广泛应用在工程和环境及水文地球物理、野外采矿、地热探测等领域.地下岩石常具有层理面和裂缝等具有方向依赖性的结构,岩石电导率常常呈现各向异性特征,因此研究复杂直流电阻率各向异性问题的高精度正演算法具有迫切的理论和学术需求.本文利用面向目标的自适应有限元算法和非结构化网格相结合的方式,解决了带地形任意复杂直流电阻率各向异性问题的高精度正演这一难题.有别于前人的研究成果,本文提出了一种特别的二次虚拟场算法实现带源的任意起伏地形问题模拟;另外,本文第一次基于电流密度连续性条件构建适合直流电阻率各向异性问题的后验误差估计算法,有效地驱动面向目标有限元网格自适应加密过程.最后,通过三组电阻率各向异性模型验证本文提出算法的正确性和适应性,测试结果表明:对于任意复杂直流电阻率各向异性问题,本文提出的算法具有精度高、适应性强等特点;另外,我们还发现电流密度连续性条件可用于设计直流电阻率问题的有效后验误差估计算法.