3D modeling of direct-current anisotropic resistivity using the adaptive finite-element method based on continuity of current density
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摘要:
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.
Abstract: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 goal-oriented 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.
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图 1
各向异性电导率主轴坐标系和测量坐标系框架
Figure 1.
The measuring coordinate frame and principal axis frame of anisotropic conductivity
图 5
两层各向异性模型上面向目标自适应加密算法结果与解析解对比
Figure 5.
Comparison of the apparent resistivity computed by goal-oriented adaptive algorithm for two-layer anisotropic model against analytical solution
图 7
山脉峡谷模型上不同迭代网格沿测线上的视电阻率收敛情况及相对误差曲线
Figure 7.
Convergence of apparent resistivity and relative error curves along survey profile computed on different mesh levels for mountain-valley model
图 8
山脉峡谷模型自适应加密网格(第一次网格(a), 第五次网格(b))上单元相对误差分布图，第五次网格影响函数W的分布图(c)
Figure 8.
Adaptive refining mesh of relative elemental error for mountain-valley model(the first mesh (a), the fifth mesh (b)), the distribution of influence function W based on the fifth mesh
图 11
A和B为各向同性，C和D为各向异性情况下的张量视电阻率的P2旋转不变量等值线图
Figure 11.
P2 rotation variable contour lines of tensor apparent resistivity with A and B of isotropy, C and D of anisotropy
图 12
A, B, C, D均为各向异性情况下的张量视电阻率的P2旋转不变量等值线图
Figure 12.
P2 rotation variable contour lines of tensor apparent resistivity with A, B, C and D of anisotropy
表 1
山脉峡谷模型三次网格参数对比
Table 1.
Mesh parameters of three mesh levels for mountain-valley model
网格水平 节点数 单元数 平均误差 网格(1st) 23882 126106 8.72 网格(3rd) 94865 536472 3.59 网格(5th) 120032 682638 0.82 
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