随钻正交方位电磁波三维有限体积法快速模拟*

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摘要本文应用柱坐标系下有限体积法(FVM)结合开窗技术建立了一套针对各向异性介质随钻正交方位电磁波(OAEMT)三维数值模拟方法。首先应用柱坐标系下电场耦合势Helmholtz 方程克服数值模拟中的低感应数问题，用电场切向与法向分量在金属钻铤面上的奇偶延拓逼近完全电导率边界条件(PEC)，设法提高钻铤表面耦合势方程的离散精度。在此基础上，借助柱坐标Lebedev 交错网格、非均质单元中等效电导率标准化以及三维有限体积法建立与钻铤表面完全协调的离散方程，由直接法求解计算器Pardiso 获得稳定可靠的大型离散方程数值解。同时，为提高层状模型中正演模拟效率，采用开窗技术实现多发射源耦合势方程的并行运算。最后，通过数值计算分析了该仪器的共轴和共面响应特性。

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关键词：柱坐标有限体积法正交方位电磁测井麦克斯韦方程各向异性地层随钻测井(LWD)

Abstract： In this study, the cylindrical finite-volume method (FVM) is advanced for the efficient and high-precision simulation of the logging while drilling (LWD) orthogonal azimuth electromagnetic tool (OAEMT) response in a three-dimensional (3D) anisotropic formation. To overcome the ill-condition and convergence problems arising from the low induction number, Maxwell’s equations are reformulated into a mixed Helmholtz equation for the coupled potentials in a cylindrical coordinate system. The electrical fi eld continuation method is applied to approximate the perfectly electrical conducting (PEC) boundary condition, to improve the discretization accuracy of the Helmholtz equation on the surface of metal mandrels. On the base, the 3D FVM on Lebedev’s staggered grids in the cylindrical coordinates is employed to discretize the mixed equations to ensure good conformity with typical well-logging tool geometries. The equivalent conductivity in a non-uniform element is determined by a standardization technique. The direct solver, PARDISO, is applied to efficiently solve the sparse linear equation systems for the multi-transmitter problem. To reduce the number of calls to PARDISO, the whole computational domain is divided into small windows that contain multiple measuring points. The electromagnetic (EM) solutions produced by all the transmitters per window are simultaneously solved because the discrete matrix, relevant to all the transmitters in the same window, is changed. Finally, the 3D FVM is validated against the numerical mode matching method (NMM), and the characteristics of both the coaxial and coplanar responses of the EM field tool are investigated using the numerical results.

Key words： finite-volume method orthogonal azimuth electromagnetic measurement Maxwell’s equation anisotropic formation logging while drilling (LWD)

收稿日期: 2020-02-25;

基金资助:

本研究是由中国科学院A 类战略性先导科技专项(XDA14020102)，国家重点研发计划(2017YFC0601805)，国家自然科学基金项目(41574110)，河北省教育厅青年基金项目(QN2018217)，河北省高等教育教学改革研究与实践项目(2018GJJG328)、张家口市科技局项目(1821011B)，河北建筑工程学院博士启动金项目(B-201606)，河北建筑工程学院学术团队创新能力提升项目（TD202011）联合资助。

通讯作者: 汪宏年(E-mail: wanghn@jlu.edu.cn) E-mail: wanghn@jlu.edu.cn

作者简介: 王浩森，讲师，河北建筑工程学院。2016 年于吉林大学获博士学位。研究方向:电法测井，电磁场数值模拟。目前主要致力于电磁场数值模拟与反演成像方面的研究与教学工作。汪宏年，教授/博导,1994 年于长春地质学院获博士学位，1995-05 至1997-06 在吉林大学物理系从事博士后研究，1997-07 至2001-09吉林大学物理学院副教授，2001-10 至今吉林大学物理学院教授，2003 至2004 University of Victoria 电子工程系高级访问学者，目前主要致力于电磁场数值模拟与反演成像方面的研究与教学工作。

引用本文:

. 随钻正交方位电磁波三维有限体积法快速模拟*[J]. 应用地球物理, 2020, 17(2): 192-207.

. Efficient finite-volume simulation of the LWD orthogonal azimuth electromagnetic response in a three-dimensional anisotropic formation using potentials on cylindrical meshes*[J]. APPLIED GEOPHYSICS, 2020, 17(2): 192-207.