【作者】 杨军；

【导师】 吴小平； 蔡伟；

【作者基本信息】 中国科学技术大学 ， 固体地球物理， 2016， 博士

【摘要】 电磁法是地球物理勘探中的重要分支。该方法以地球导电媒质的物性差异为基础,通过观测和分析在其中传播的电磁场,从而预测测区的地下电性结构。经过一百多年的发展,电磁法勘探技术己广泛应用于各个勘探领域。随着油气和固体矿产资源的日益减少,勘探深度越来越大,地形地质条件越来越复杂,勘探难度也越来越大。在工程应用中,施工工艺和质量要求的提高也对电磁法勘察提出了更高的要求。除了需要有更高精度的电磁法勘探仪器外,也要求正演算法能应对更复杂的地质模型,能提供更高的计算精度和效率。本文研究的正是地球电磁场的高效、高精度算法,具体研究内容包括直流电场、频率域可控源电磁场和时间域电磁场的三维正演。采用的数值计算方法是基于非结构四面体网格的有限单元法,包括连续和间断有限元方法,二者的本质区别在于前者采用连续的有限维函数空间逼近真实解,而后者的解空间不连续。点电源直流电场的正反演技术已相当成熟,而长电极源电场的正演计算却依然存在许多难点。其中最大的难点是长电极源穿过非均匀媒质时其表面的电流密度难以准确计算。此外,在大测区中构建穿过非均匀媒质的细长电极模型是比较困难的,而任意形状和倾角的长电极则更难构造。同时,长电极的电阻率远远低于围岩电阻率,若考虑长电极的电阻率进行正演计算,将会带来一个非常病态的系统,影响求解的收敛速度和数值解的精度。我们在长电极源直流电场正演中将长电极源近似为电阻率可忽略不计的有限长线电流源,既避免了悬殊的电阻率差异带来的困难,也免去了构建长电极网格模型的麻烦。利用电磁场基本原理推导的长电极源表面电流密度近似公式允许正演模型中长电极穿过非均匀媒质。此部分研究采用基于节点基函数的连续有限元方法实现了任意形状多段长电极源的直流电场,基于总电位和二次电位的正演算法均由解析解验证了数值精度。我们首次提出了任意长电极源观测装置的视电阻率公式,分析了长电极源和点电源电场的异同。通过地面和井中的数值模拟算例发现长电极源装置对地下异常体的响应比点电源装置更敏感,具有更强的勘探能力。文中还提出了混合双极供电源装置(即长电极和点电极配合供电)以弥补测区中长电极不足的情况,数值模拟结果显示混合供电极同样能获得比点电极装置更强的异常响应。最后还进行了油田水力压裂模型的长电极直流电法正演,预示了油田套管可用作长电极源进行水力压裂过程的监测。基于有限元方法的频率域电磁场正演难点在于电场法向分量不连续且散度为零,尽管节点有限元方法的应用相当成熟,却不能直接用于求解非均匀媒质中的电磁场问题,而只能先求解连续的矢量势和标量势,再转换成电磁场,而且在求解矢量势和标量势时依然要做散度校正才能得到可靠的数值解。考虑到以上问题,我们在频率域可控源电磁场的三维正演中采用的是基于非结构网格的矢量有限元方法。矢量有限元方法将求解的自由度设置在剖分单元的棱边上,只要求待求物理量切向连续,同时矢量基函数自动满足散度为零的条件,因此是直接求解非均匀媒质中电磁场的极佳选择。此部分正演中以电场双旋度方程作为控制方程,由广义变分原理得到对应的泛函形式再利用矢量有限元离散和求解。截断边界采用了索末菲辐射边界条件,可以得到比齐次边界条件更好的截断效果。通过与一维层状模型的解析解和二维模型的有限元数值解的对比,验证了矢量有限元算法的稳定性和计算精度。我们还研究了海底地形对海洋可控源电磁勘探的影响,数值模拟结果显示,海底的起伏地形会造成较大的异常,可能掩盖由海底油气藏产生的异常,在数据处理时,应考虑海底地形的影响。由于采用的是非结构的四面体网格,我们的频率域可控源电磁正演算法可进行复杂海底模型中低频电磁场的正演计算。间断有限元方法是最近二十年来非常热门的数值计算方法,该方法易于实现高阶插值,具有天然的并行性,具有处理间断问题的能力,在计算电磁学中广受欢迎,但该方法目前还没有引入到低频地球电磁场的计算中。我们完成的第三部分研究正是利用间断有限元方法实现三维时间域高频和低频电磁场的正演,因此既可用于探地雷达正演,也能用于瞬变电磁场的计算,可实现任意电流输入波形的电偶极子、磁偶极子和任意形状大回线源的瞬变电磁正演模拟。在边界的处理上,采用的是各向异性的完全匹配层。引入完全匹配层后出现了频率依赖关系,利用12个辅助变量可以将依赖关系解耦,从而推导了完全匹配层中的时域麦克斯韦方程组。完全匹配层的存在可以高效的吸收外传电磁波而不产生明显反射,因此不需要像常规的齐次边界那样需要将计算域设得非常大。空间离散采用的是层级型的高阶正交基函数,而时间离散则采用高阶龙格-库塔格式。该基函数的正交性使得单元矩阵变成单元对角阵,而高阶的刚度矩阵依然是良态矩阵,计算复杂度大大降低。数值模拟结果表明数值解的误差随着所采用的基函数的阶数增加而指数降低。当采用k阶基函数时,数值解具有接近k+1阶的数值精度。为了将间断有限元的优势最大化,还利用OpenMP库实现了并行计算,测试结果显示其并行效率非常高。更多还原

【Abstract】 As an important branch of geophysical prospecting techniques, electromagnetic (EM) sounding detects the underground electrical structure by measuring and analyz-ing the EM field propagating in the conductive earth media, in which reflection, re-fraction and attenuation of the EM field occur where heterogeneity exists. The EM sounding techniques, which were first applied for near surface exploration more than a century ago, have been widely used in various prospecting areas. With the dwindling of oil, gas and solid mineral reserves, geophysical prospecting techniques are facing new challenges,such as greater prospecting depth and more complicated geological cir-cumstance. New construction techniques and quality control in engineering also require more precise detection methods. These new trends require not only high-precision in-struments for EM sounding, but also accurate and efficient simulation algorithms which are able to handle complicated geological models. This thesis is focusing on the latter, the advanced algorithms for direct-current (DC) resistivity modeling and frequency-and time-domain EM simulations. Finite element (FE) methods on unstructured grids, including continuous and discontinuous FE methods, have been applied in the simu-lations. The main difference between the continuous and discontinuous FE is that the former requires a continuous finite-dimensional solution space while the latter allows discontinuity in the solution.There have been vast researches in the simulation and application for point elec-trode (PE) sources. The numerical simulation for long electrode (LE) source, however, is rare, as some theoretical obstacles are not yet overcome. The biggest problem is that the current density distribution on the surface of the electrode is extremely hard to determine when it is buried in heterogeneous media. In addition, the construction of a vimineous vertical electrode in a big mesh model is tedious, let along arbitrary LEs penetrating through heterogeneous blocks. When FE method is applied, the sharp re-sistivity contrast between the metallic electrode and the surrounding media will also lead to an ill-posed linear system and result in slow convergence and degraded accu-racy. In this thesis, the LE source is regarded as a perfectly conducting line source with finite length, which not only gets rid of the convergence problem resulting from the resistivity contrast but also leaves the construction of the electrode out of the mesh model. A continuous FE method with nodal basis is applied to the simulation for ar-bitrary LE sources on unstructured grids, combined with an approximated equation for determining the current density along the electrode which allows the electrode model to penetrate through heterogeneity in the simulations. Both total- and secondary field methods are applied in the simulation and the numerical solution is verified using the analytic solutions. The apparent resistivity formulas of LE source arrays are proposed for the first time. The electrical fields excited by PE and LE sources are compared and the simulation of ground and borehole models show that the arrays with LE sources are more sensitive to the resistivity anomalies than those with PE sources. The mixed bipole source, which is a compromise in the case that the LEs are insufficient in work sites, is also tested and the simulation result indicates an enhanced detection ability of the mixed bipole array. A time-lapsed LE source simulation in a hydraulic fracturing model is also performed to investigate the capability of the LE array for oil field application and the result shows that the LE array can be used for monitoring the hydrofracturing process.The main difficulties in the frequency-domain EM simulation with FE method are the normal discontinuity and divergence-free condition of the electrical field. The the-ory of the nodal FE method is solid and it is widely used in numerical simulations, but it is not suitable for simulating EM field directly in heterogeneous media. Instead, the equations of the continuous vector and scalar potentials need to be solved first and then the solutions can be converted to EM field components. While solving the potentials, the divergence condition still needs to be satisfied in order to eliminate the spurious solution. In a word, it is cumbersome to solve the EM field problem in heterogeneous media using nodal FE method. To overcome these difficulties, the vector FE method is chosen for the 3D simulation of frequency-domain controlled-source electromagnetics (CSEM). The degree of freedoms is defined on the edges of the elements in vector FE method, which only requires the continuity of the tangential components. Additionally, the edge vector basis satisfies the divergence-free condition naturally. Consequently, it is a better choice to use vector FE method for EM simulation. The curl-curl equation of the electrical field is used as the governing equation and the general variation principle is applied to obtain the equivalent variational problem. The vector FE method is then applied to the discretization of the variational problem and the linear system is solved to obtain the frequency-domain EM solution. Sommerfeld radiation boundary condition, which offers better absorbing efficiency, is imposed for the domain truncation. The stability and accuracy of the vector FE algorithm are verified by comparing with a ID analytic solution and a 2D FE solution. The influence of the submarine topography to the CSEM exploration is investigated with numerical simulations and the results show that submarine topography will give rise to significant anomalous signal and may cover the signal from the oil and gas reservoir. With the use of unstructured tetrahedral grids, the vector FE algorithm is able to handle extremely complicated submarine models.As it is easy to implement high order interpolation and has the capability to handle discontinuous solutions, together with high parallel performance, discontinuous finite method (also known as discontinuous Galerkin method) has become one of the most at-tractive numerical methods during the last two decades. However, this advanced numer-ical method is not yet employed to the simulation of low frequency geo-electromagnetic field simulation. We have accomplished the discontinuous FE algorithm for time do-main EM field simulation in high and low frequency regimes. For the geophysical purpose, the algorithm can be applied to the simulations of ground penetrating radar (GPR) and transient electromagnetics (TEM). The TEM fields excited by electric and magnetic dipoles and arbitrary large loops loaded with various current pulses can be simulated. The uniaxial perfectly matched layer (UPML) is employed to truncate the calculation domain, which leads to a frequency dependency in the governing equation. In order to decouple the dependency,12 auxiliary variables are introduced and the cor-responding Maxwell’s equation in UPML is derived. The UPML is able to absorb the out-going wave and reduce the reflection from the artificial boundary efficiently. As a result, the calculation domain can be shrank greatly. A class of hierarchical orthogonal bases is chosen for spatial discretization while the Runge-Kutta method is applied for time integration. The computational complexity is greatly diminished with the use of the orthogonal bases, as the elemental mass matrix become a diagonal and the stiffness matrix maintains well-conditioned even for high order basis functions. The simulation result shows that the accuracy of the discontinuous FE solution is close to order k+1 when k-th order basis functions are applied. The discontinuous FE algorithm is imple-mented in parallel with OpenMP library and the parallel test shows superior parallel performance of the algorithm.更多还原