基于矢量有限元法的三维带地形大地电磁正演模拟研究

发布时间：2018-05-18 11:21

本文选题：矢量有限元 + 三维　；参考：《吉林大学》2016年硕士论文

【摘要】：大地电磁测深法是一种频率域电磁测深法,它以天然存在的交变电磁场为场源进行深部地质研究,因此野外施工比较方便,可以减小成本,另外这种方法还有不会被高阻层屏蔽以及对低阻地层的分辨率比较高等优点,所以它在地球物理勘探中得到了广泛的应用。进行大地电磁的正演数值模拟,能让我们对各种不同电性介质下场的规律有个很好的认识,为最终要实现的反演提供基础和保证。本文应用矢量有限元方法,实现了带地形三维大地电磁的正演数值模拟。首先,从MAXWELL方程组出发,根据三维大地电磁所满足的边界条件,利用加权余量法推导了与三维大地电磁边值问题等价的变分方程。采用规则的六面体单元对计算区域进行剖分,使用矢量插值函数进行单元分析,插值函数简单,精度高,满足法向方向场的不连续性,能够避免节点有限元法中“伪解”的问题,可以不用做散度校正。最后形成的单元刚度矩阵巨大而且非常稀疏,考虑到计算速度和内存占用的问题,将总体系数矩阵中采用一维非零元素存储,将不完全乔列斯基分解与双复共轭梯度算法相结合对方程进行求解,在保证计算速度的同时,高效、稳定地收敛。其次,建立均匀半空间模型和层状模型,并与解析解进行对比,电阻率和相位的计算精度都很高,从而验证了算法及程序的正确。在验证的过程中发现,利用传统的三维大地电磁边界条件时,精度不高,计算速度慢,针对此情况还需给出计算区域前后左右四个侧面的边界条件。另外也认识到网格剖分对计算结果的影响很大,得到了一些关于网格剖分的结论。在此基础上,本文又给出了三维异常体(包含高阻体和低阻体)模型,分别计算了XY模式和YX模式下异常体模型的视电阻率和阻抗相位的异常响应,并且分析了它们的响应特征。最后,利用矢量有限元方法,实现了带地形的三维大地电磁正演模拟,为了验证结果,分别建立了一个大家常用的三维梯形山峰模型和山谷模型。计算了频率f=2Hz时XY模式与YX模式下的视电阻率和阻抗相位平面图,以及山峰地形上不同测线的响应曲线,并对结果进行了分析;为了研究不同频率下地形对大地电磁响应的影响规律,对2Hz、50Hz和2000Hz三个频点在两种不同模式下进行数值模拟,并对其响应规律进行了讨论;为了确定是否可以用二维地形的模拟结果去近似三维地形,将沿着X剖面(Y=0)三维模拟结果与二维模拟结果进行了比较,在这里它们的横截面是一样的。将模拟结果进行对比,得到如下结论:XY模式下三维地形的响应结果与TM模式下二维地形的响应结果比较接近,YX模式下三维地形的响应结果与TE模式下二维地形的响应结果的差别较大。
[Abstract]:Magnetotelluric sounding is a frequency-domain electromagnetic sounding method. It takes the natural alternating electromagnetic field as the source for deep geological research, so the field construction is convenient and the cost can be reduced. In addition, this method has the advantages of not being shielded by high resistivity layer and high resolution of low resistivity formation, so it has been widely used in geophysical exploration. The forward numerical simulation of magnetotelluric can give us a good understanding of the law of the end of various electric media and provide the foundation and guarantee for the final inversion. In this paper, the forward numerical simulation of 3D magnetotelluric with terrain is realized by using vector finite element method. Firstly, according to the boundary conditions of 3D magnetotelluric, the variational equation equivalent to 3D magnetotelluric boundary value problem is derived by using weighted residual method from MAXWELL equations. The regular hexahedron element is used to divide the calculation area, and the vector interpolation function is used for element analysis. The interpolation function is simple and accurate, which satisfies the discontinuity of the normal direction field. The problem of "pseudo solution" in the finite element method can be avoided, and the divergence correction can be avoided. The resulting element stiffness matrix is large and sparse. Considering the problem of computing speed and memory occupation, the one-dimensional nonzero element is used in the total coefficient matrix. The incomplete Cholesky decomposition and the double complex conjugate gradient algorithm are combined to solve the equation, which ensures the calculation speed and converges efficiently and stably. Secondly, the uniform half-space model and the layered model are established, and compared with the analytical solution, the calculation accuracy of resistivity and phase is very high, which verifies the correctness of the algorithm and the program. In the process of verification, it is found that the accuracy is not high and the calculation speed is slow when the traditional 3D magnetotelluric boundary condition is used. In addition, it is recognized that mesh generation has a great influence on the calculation results, and some conclusions about mesh generation are obtained. On the basis of this, the 3D anomalous body model (including high resistivity and low resistivity) is given, and the abnormal responses of apparent resistivity and impedance phase of anomalous volume model under XY mode and YX mode are calculated, respectively. Their response characteristics are analyzed. Finally, the 3D magnetotelluric forward modeling with terrain is realized by using vector finite element method. In order to verify the results, a commonly used three-dimensional trapezoidal peak model and a valley model are established respectively. The apparent resistivity and impedance phase plane maps of XY mode and YX mode at frequency f=2Hz are calculated, and the response curves of different surveying lines on the mountain topography are also calculated, and the results are analyzed. In order to study the influence of terrain on magnetotelluric response at different frequencies, the two different modes of 2Hz / 50Hz and 2000Hz are numerically simulated, and the response law is discussed. In order to determine whether it is possible to approximate the three-dimensional terrain with the simulation results of two-dimensional terrain, the results of three-dimensional simulation along the X section are compared with the results of two-dimensional simulation, where their cross sections are the same. Compare the simulation results, The results obtained are as follows: the response results of 3D terrain in XY model and 2D terrain in TM mode are close to those of 3D terrain in TM-mode and the response result of 3D terrain in te mode is different from that in TM-mode.
【学位授予单位】：吉林大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：P631.325

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