3D forward modeling of DC resistivity method based on finite element with mixed grid

Authors

WANG Xinyu, Key Laboratory of Exploration Technologies for Oil and Gas Resources, Ministry of Education, Yangtze University, Wuhan 430100, China; Cooperative Innovation Center of Unconventional Oil and Gas, Wuhan 430100, ChinaFollow
WANG Cheng, Xi’an Research Institute Co. Ltd., China Coal Technology and Engineering Group Corp., Xi’an 710077, China
MAO Yurong, Key Laboratory of Exploration Technologies for Oil and Gas Resources, Ministry of Education, Yangtze University, Wuhan 430100, China; Cooperative Innovation Center of Unconventional Oil and Gas, Wuhan 430100, ChinaFollow
YAN Liangjun, Key Laboratory of Exploration Technologies for Oil and Gas Resources, Ministry of Education, Yangtze University, Wuhan 430100, China; Cooperative Innovation Center of Unconventional Oil and Gas, Wuhan 430100, China
ZHOU Lei, Key Laboratory of Exploration Technologies for Oil and Gas Resources, Ministry of Education, Yangtze University, Wuhan 430100, China; Cooperative Innovation Center of Unconventional Oil and Gas, Wuhan 430100, China
GAO Wenlong, Key Laboratory of Exploration Technologies for Oil and Gas Resources, Ministry of Education, Yangtze University, Wuhan 430100, China; Cooperative Innovation Center of Unconventional Oil and Gas, Wuhan 430100, China

Abstract

The DC resistivity method is widely used in the mineral resources exploration and geological survey industries because of its high efficiency and low cost, such as non-ferrous metals and coal fields. However, in the process of borehole–to–surface and surface–to–borehole resistivity method exploration, the influence of borehole factors (borehole fluid resistivity and borehole diameter) on the apparent resistivity response in different exploration modes is unclear, and whether it will affect the interpretation of apparent resistivity data is an issue worth exploring. Therefore, we proposed a mixed grid finite element method to realize the 3D forward modeling of DC resistivity. Then we gave the boundary value problem and the finite element variational problem satisfied by the abnormal electric potential method, applied the trigonal and tetrahedral grid to achieve a fast discretization of the computational region, and established the linear interpolation basis function and the element matrix of the two kinds of the grids. Finally, we used the SSOR–PCG iterative algorithm to solve the large-scale linear equations satisfied by the secondary potential and obtained the response of every observation point. On the premise of ensuring the calculation accuracy, we applied the mixed grid to discrete the borehole geoelectric model and explored the characteristics of the influence of borehole factors on the apparent resistivity data of the borehole-to-surface and surface-to-borehole observation methods. For the borehole-to-surface observation method: the apparent resistivity near the borehole is most influenced by the borehole factor, which seriously affects the reasonable interpretation of the apparent resistivity data. However, with the increase of the transmit-receive distance of the observation point, the apparent resistivity response gradually inclines to the resistivity of the surrounding rock. And with the depth of the emission source increases, the influence of borehole on the apparent resistivity response of the surface is also gradually reduced. For surface-to-borehole observation method: the borehole factor has a greater influence on shallow data and less influence on deep data, and the apparent resistivity response is more susceptible to the influence of borehole diameter than borehole fluid resistivity. The 3D forward modeling of mixed grid finite element method of DC resistivity method will provide theoretical guidance for practical borehole-to-surface and surface-to-borehole resistivity exploration, geophysical workers can combine the borehole information with the 3D forward modeling and select the appropriate transmit-receive distance to effectively suppress the influence of borehole on the observed data.

Keywords

DC resistivity method, borehole fluid resistivity, borehole diameter, mixed grid finite element method, prismatic grid, tetrahedron grid, 3D forward modeling

DOI

10.12363/issn.1001-1986.21.06.0338

Recommended Citation

WANG Xinyu, WANG Cheng, MAO Yurong, et al. (2022) "3D forward modeling of DC resistivity method based on finite element with mixed grid," Coal Geology & Exploration: Vol. 50: Iss. 5, Article 18.
DOI: 10.12363/issn.1001-1986.21.06.0338
Available at: https://cge.researchcommons.org/journal/vol50/iss5/18

Reference

[1] 解海军,李志强,栗升. 线源直流电法有限元二维正演模拟[J]. 煤田地质与勘探,2019,47(1):194−199. XIE Haijun,LI Zhiqiang,LI Sheng. Finite element 2D forward modeling of DC method with line source[J]. Coal Geology & Exploration,2019,47(1):194−199.

[2] 占文锋,武玉梁,李文. 矿井直流电法全空间电场分布数值模拟及影响因素[J]. 煤田地质与勘探,2018,46(1):139−147. ZHAN Wenfeng,WU Yuliang,LI Wen. Simulation and analysis of electric field distribution and its influence factors in coal mine direct current method[J]. Coal Geology & Exploration,2018,46(1):139−147.

[3] 刘斌,李术才,李树忱,等. 基于预条件共轭梯度法的直流电阻率三维有限元正演研究[J]. 岩土工程学报,2010,32(12):1846−1853. LIU Bin,LI Shucai,LI Shuchen,et al. 3D FEM numerical forward modeling of direct current electrical resistivity based on PCG algorithm[J]. Chinese Journal of Geotechnical Engineering,2010,32(12):1846−1853.

[4] 张钱江,戴世坤,陈龙伟,等. 多源条件下直流电阻率法有限元三维数值模拟中一种近似边界条件[J]. 地球物理学报,2016,59(9):3448−3458. ZHANG Qianjiang,DAI Shikun,CHEN Longwei,et al. An approximate boundary condition for FEM−based 3D numerical simulation with multi–source direct current resistivity method[J]. Chinese Journal of Geophysics,2016,59(9):3448−3458.

[5] 王智,吴爱平,李刚. 起伏地表条件下的井中激电井地观测正演模拟研究[J]. 石油物探,2018,57(6):927−935. WANG Zhi,WU Aiping,LI Gang. Forward modeling of borehole–ground induced polarization method under undulating topography[J]. Geophysical Prospecting for Petroleum,2018,57(6):927−935.

[6] 李勇,林品荣,徐宝利,等. 复杂地形三维直流电阻率有限元数值模拟[J]. 地球物理学进展,2009,24(3):1039−1046. LI Yong,LIN Pinrong,XU Baoli,et al. FEM numerical modeling of 3D DC resistivity under complicated terrain[J]. Progress in Geophysics,2009,24(3):1039−1046.

[7] 任政勇,汤井田. 基于局部加密非结构化网格的三维电阻率法有限元数值模拟[J]. 地球物理学报,2009,52(10):2627−2634. REN Zhengyong,TANG Jingtian. Finite element modeling of 3D DC resistivity using locally refined unstructured meshes[J]. Chinese Journal of Geophysics,2009,52(10):2627−2634.

[8] 任政勇,邱乐稳,汤井田,等. 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[J]. 地球物理学报,2018,61(1):331−343. REN Zhengyong,QIU Lewen,TANG Jingtian,et al. 3D modeling of direct–current anisotropic resistivity using the adaptive finite–element method based on continuity of current density[J]. Chinese Journal of Geophysics,2018,61(1):331−343.

[9] DEY A,MORRISON H F. Resistivity modeling for arbitrarily shaped three–dimensional structures[J]. Geophysics,1979,44(4):753−780.

[10] FOX R C,HOHMANN G W,KILLPACK T J,et al. Topographic effects in resistivity and induced–polarization surveys[J]. Geophysics,1980,45(1):75−93.

[11] LOWRY T,ALLEN M B,SHIVE P N. Singularity removal:A refinement of resistivity modeling techniques[J]. Geophysics,1989,54(6):766−774.

[12] ZHAO Shengkai,YEDLIN M J. Some refinements on the finite–difference method for 3D DC resistivity modeling[J]. Geophysics,1996,61(5):1301−1307.

[13] RÜCKER C,GÜNTHER T,SPITZER K. Three–dimensional modeling and inversion of DC resistivity data incorporating topography−I. modelling[J]. Geophysics,2006,166:495−505.

[14] 强建科,罗延钟. 三维地形直流电阻率有限元法模拟[J]. 地球物理学报,2007,50(5):1606−1613. QIANG Jianke,LUO Yanzhong. The resistivity FEM numerical modeling on 3D undulating topography[J]. Chinese Journal of Geophysics,2007,50(5):1606−1613.

[15] 汤井田,公劲喆. 三维直流电阻率有限元–无限元耦合数值模拟[J]. 地球物理学报,2010,53(3):717−728. TANG Jingtian,GONG Jinzhe. 3D DC resistivity forward modeling by finite−infinite element coupling method[J]. Chinese Journal of Geophysics,2010,53(3):717−728.

[16] 王志刚,何展翔,魏文博. 积分方程法三维模拟井地电法并行算法研究[J]. 物探化探计算技术,2007,29(5):425−430. WANG Zhigang,HE Zhanxiang,WEI Wenbo. Parallel algorithm research of integral equation method to 3D borehole–surface electromagnetic modeling[J]. Computing Techniques for Geophysical and Geochemical Exploration,2007,29(5):425−430.

[17] 徐凯军,李桐林. 垂直有限线源三维地电场有限差分正演研究[J]. 吉林大学学报(地球科学版),2006,36(1):137−141. XU Kaijun,LI Tonglin. The forward modeling of three–dimensional geoelectric field of vertical finite line source by finite difference method[J]. Journal of Jilin University(Earth Science Edition),2006,36(1):137−141.

[18] 柯敢攀,黄清华. 井地电法的三维正反演研究[J]. 北京大学学报(自然科学版),2009,45(2):264−272. KE Ganpan,HUANG Qinghua. 3D forward and inversion problems of borehole–to–surface electrical method[J]. Acta Scientiarum Naturalium Universitatis Pekinensis,2009,45(2):264−272.

[19] 李长伟,熊彬,吕玉增. 电法测井的三维有限元模拟[J]. 物探与化探,2012,36(4):585−590. LI Changwei,XIONG Bin,LYU Yuzeng. Three–dimensional finite element modeling of electrical well logging[J]. Geophysical & Geochemical Exploration,2012,36(4):585−590.

[20] 王智,潘和平. 三维井地电阻率法异常响应特征增强算法模拟研究[J]. 石油物探,2014,53(4):491−500. WANG Zhi,PAN Heping. Research on the enhanced algorithms of the abnormal response characteristics for 3D borehole–to–surface resistivity method[J]. Geophysical Prospecting for Petroleum,2014,53(4):491−500.

[21] 徐世浙. 地球物理中的有限单元法[M]. 北京：科学出版社，1994.

[22] 林绍忠. 用预处理共轭梯度法求解有限元方程组及程序设计[J]. 河海大学学报,1998,26(3):112−115. LIN Shaozhong. Application of preconditioned conjugated gradient method to finite element equations and programme design[J]. Journal of Hohai University,1998,26(3):112−115.