1.College of Earth Sciences, Guilin University of Technology, Guilin 541004, China 2.Department of Earth and Space Sciences, Southern University of Science and Technology, Shenzhen 518055, China
Fund Project:Project supported by National Natural Science Foundation of China (Grant No. 41604097) and the Research Foundation from Guilin University of Technology, China (Grant No. 002401003503).
Received Date:08 March 2019
Accepted Date:12 April 2019
Available Online:01 July 2019
Published Online:20 July 2019
Abstract:As an efficient geophysical exploration technology, well electromagnetic method is particularly applicable to oil and gas exploration in China's complex terrain areas (deserts, mountains, etc.). A serious influence of topographic relief area on the electromagnetic response of well is inevitable but challenging. To the best of our knowledge, there is no literature on modeling the electromagnetic response of three-dimensional (3D) topography with well electromagnetic method. Based on the domain decomposition, an integral equation method is presented to simulate the electromagnetic response of 3D topography in frequency domain via the well electromagnetic method. Compared with the finite difference and finite element method based on partial differential equation, this method is very efficient in simulating topographic response without huge computation or truncation boundary error accumulation or special boundary condition requirements. Firstly, an induction coefficient is defined according to the topographic relief situation. Then the computational domain consisting of the target body, background medium and 3D topography is divided into reference model, background medium and the distribution of target body medium area. According to the characteristics of each sub-region, Anderson algorithm is an analytic solution based on Gaussian filtering, which is used to provide the primary field from the excited sources in surface. And then, the stable double conjugate gradient-fast Fourier transform is incorporated into integral equation algorithm to obtain the fast 3D terrain shaft frequency domain electromagnetic responses. By comparing the calculation results using the new algorithm presented in this paper with the analytical solutions of Anderson algorithm for half-space model with surface electromagnetic method, the precision and the efficiency of this new algorithm are demonstrated. And the ability to model the electromagnetic responses of 3D topography is shown by comparing with the published results of 3D boundary integral equation. Thus, the high accuracy and high efficiency of the new algorithm presented in this paper are validated. Finally, the influence of 3D valley topography on electromagnetic field response of surface to borehole electromagnetic (SBEM) observation system is presented and analyzed. It is observed that the response of SBEM is seriously disturbed by the field of 3D valley topography which is necessarily removed. The research results presented in this paper are of significance for guiding the identification and correction of electromagnetic topographic effect from 3D SBEM. Keywords:domain decomposition based integral equation/ response of 3-dimensional topography/ well electromagnetic/ surface to borehole electromagnetic
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2.1.积分方程
理论上积分方程模拟电磁场求解的过程为: 在电性参数有别于背景介质的异常区域的网格剖分基础上, 推导满足勘探问题的可控源电磁三维体积分方程, 采用稳定型双共轭梯度法求解目标区域离散化后的大型线性方程组. 三维积分方程正演模拟问题的实质为均匀半空间介质或水平层状背景介质中异常区域的三维地面可控源电磁场模拟问题的转化. 即拟采用地面发射源激发一次场, 在电磁场远区定义范围之外面积性观测异常电磁场. 考虑多方位Walkaround地井电磁观测系统(图1), 第n层的节点处总电场可表示为该点入射场与散射场之和[11]: 图 1 积分方程模拟三维地井电磁场观测系统示意图(未显示地形) Figure1. Sketch of 3D (three-dimensional) SBEM (surface to borehole electromagnetic) measurement system using IE (integral equation) without topography.
表1均匀半空间介质地电结构模型的不同算法的电磁场模拟效率对比 Table1.Comparison of computational effectiveness of modeling electromagnetic field via different algorithms for a half homogeneous medium.
23.2.三维地形响应模拟验证 -->
3.2.三维地形响应模拟验证
由于三维起伏地形频率域的地井电磁场响应的模拟结果发表的较少, 本文将三维积分方程法的模拟算法用于三维起伏地形频率域地面电磁场响应的模拟计算, 并与三维边界积分的模拟结果对比. 如图5所示, 收发装置采用地面电磁观测系统, 垂直磁偶源位于山谷地形底部, 激发频率为1000 Hz; 接收点位于y = 0剖面, 点距为非均匀间距; 地下半空间介质导电率为0.01 S/m. 三维山谷地形为倒梯形体, 上顶面为200 m × 200 m, 下底面为40 m × 40 m, 纵向高差为50 m. 采用本文提出的区域划分方法, 将上述含山谷地形的模拟算例划分为层状介质参考模型(图3(b)), 即包含空气层和地下电导率为0.01 S/m的介质层; 将山谷地形作为区域划分异常目标体(图3(c)). 于是, 在参考模型中对比度函数为零, 而包含空气层的山谷地形目标区域的对比度为1, 需要求解的区域介质与参考模型介质在对比度函数上具有显著的差异, 保障了积分方程模拟的精度. 图 5 三维山谷地形及地面电磁观测系统示意图, Tx为场源位置, Rx为接收点位置 Figure5. Sketch of 3D valley terrain with surface electromagnetic. Tx denotes transmitter and Rx is receiver.
首先采用Anderson算法求解参考模型分布在y = 0剖面上各接收点的一次场响应, 将一次场响应作为(9)式右端项入射场, 采用稳定型双共轭梯度-快速傅里叶变换算法求解积分方程组, 即可获取三维地形电磁场响应. 三维边界积分方程模拟将地形问题转化为空气空间及介质空间的矢量面积分问题, 简化了三维边界积分求解过程; 针对地形区域采用加密三角单元积分(相应计算量增大), 并将地形影响视为常数项因(地形响应模拟精度有限). 图6所示为三维山谷地形的三维积分方程模拟、三维边界积分模拟地面水平磁场分量的归一化响应及二者模拟地形响应差值的对比情况. 如图6(a)和图6(b)所示, 两组磁场分量模拟结果的衰减变化趋势一致性较好, 地形起伏区域(x轴–100— –40 m, 40—100 m)在相应磁场响应曲线上均有所反映, 表明了本文区域积分方程模拟起伏地形的可行性. 基于上述两种算法模拟地形响应精度不同的问题, 绘制了三维区域积分方程算法相对三维边界积分方程模拟地形水平磁场响应的差值曲线(图6(c)和图6(d)). 差值曲线表明, 相对三维边界积分方程算法, 本文提出的区域积分方程算法模拟地形响应的幅值最小值达7% (Hy分量), 最大值达20% (Hx分量), 验证了本文正演模拟方法的有效性. 图 6 三维山谷地形三维积分方程模拟、三维边界积分模拟地面磁场分量归一化响应及其差值对比图 Figure6. Magnetic field of 3D valley terrain calculated by 3D IE and 3D BIE: (a), (b) Total magnetic field; (c), (d) difference of magnetic field between IE and BIE.
23.3.多方位地井电磁算例 -->
3.3.多方位地井电磁算例
为分析本文提出的区域积分方程方法在地井电磁观测系统上模拟的可行性, 设计均匀半空间层状介质模型, 导电率为0.01 S/m; 垂直电偶源位于地面, 其水平位置与接收井口距离为100 m, 激发频率为1000 Hz; 接收井深度方向0—100 m内布置若干纵向电场分量接收点, 间距为5 m. 图7所示三维区域积分方程方法(3D IE)与Anderson算法的模拟结果完全吻合, 数据间拟合误差小于1%, 验证了本文算法用于地井电磁场响应模拟计算的可行性, 为后续将Anderson算法嵌入三维地形地井电磁多方位观测的电磁场模拟降低计算代价奠定了基础. 图 7 均匀半空间地井电磁观测三维积分方程法、Anderson算法模拟电场响应对比 Figure7. Electric field of reference model calculated by 3D IE and Anderson code for 3D SBEM.
进一步, 设计考虑三维地形条件下地井电磁多方位观测方式的算例分析, 探索地形对地井电磁场的影响规律. 如图8(a)所示, 假设三维山谷地形三方位地井电磁观测系统采用三方位水平径向电偶源, 分别为主剖面观测(Tx1位于地形上升中段)、旁侧观测(Tx2位于地形旁侧)及对侧观测(Tx3与地形在井孔的两侧); 为便于对比分析, 各场源与接收井水平距离均为300 m, 激发频率为25 Hz. 主剖面观测场源位于山谷地形内, 其余方位观测场源位于地面, 接收井0—100 m内布置若干纵向电场分量接收点, 间距为5 m, 如图8(b)—(d)所示. 图 8 三维山谷地形及多方位地井电磁观测示意图 Figure8. Sketch of 3D valley terrain with multi-azimuth SBEM.
本文采用Anderson算法计算区域划分参考模型的一次场响应, 利用区域积分方程方法模拟计算上述地井电磁多方位电磁场响应. 当场源与接收井孔之间存在地形空间时, 在地形空间深度范围内, 场源与接收点的传播空间受阻, 在观测总电场响应(黑色曲线)上体现为幅值低于散射场响应(红色曲线)的畸变现象(图9(a)). 在场源、地形底部与接收井孔测点为连线区域, 地形影响产生的上述畸变现象减弱, 但引起显著的高幅值异常, 揭示了地形存在. 当接收点深度大于地形深度范围, 地形影响基本无效, 总电场、散射场响应趋于正常分布, 并与Anderson算法提供的均匀空间电场响应曲线(蓝色曲线)分布吻合. 旁侧观测模拟结果如图9(b)所示, 由于地形与场源位置关于井孔位置为正交关系, 地形深度范围场源激发一次场占主导, 但与地形底部深部的相同位置接收点仍受到上述畸变现象影响; 大于地形深度接收点电场的响应则受频率域电磁法体积效应、旁侧影响干扰, 导致总场响应较Anderson算法提供的一次场响应的幅值增加. 由于井孔位于地形与场源中间, 对侧观测方式下地形引起的散射电场较微弱, 如图9(c)所示, 总场响应与Anderson算法提供的一次场的响应基本一致, 大于地形深度接收点电场的响应仍受体积效应、旁侧影响干扰. 图 9 三维山谷地形三方位地井电磁场响应 (a) Tx1场源; (b) Tx2场源; (c) Tx3场源 Figure9. Electric field of 3D valley terrain with multi-azimuth SBEM: (a) Tx1; (b) Tx1; (c) Tx1.