摘要
在电磁场的分析与探测应用中,非均匀介质中低频近场问题的分析一直是研究和应用中的重点和难点。一方面,该类问题的研究具有广阔的前景和应用价值,如电磁波测井与地球物理探测分析等；然而另一方面,该类问题的研究却一直是数值分析和计算方法中的难题,特别是大规模多尺度问题,更对现有的分析方法提出了严峻的挑战。针对上述挑战,本文首先从近场问题的物理本质出发,将其归类为极低频准静态位场问题和时变电磁场问题,针对不同的问题采用不同的建模策略和分析方法。然后,本文重点针对时变的低频近场问题,研发了结合树-叉树策略和高阶传输条件的有限元区域分解方法,并将其应用于电磁波测井前沿问题的分析。在该算法的基础上,本文还实现了低频近场探测前端的分析与设计,以获得多维度的地层信息。最后,为了实现低频近场问题中三维目标的探测与参数的快速反演,本文在对已有的反演算法进行研究总结的基础上,提出一种结合有限元区域分解框架的三维全波反演算法,以便在对复杂未知目标的低频近场探测问题中实现快速有效的反演。
首先,本文首先针对低频准静态位场测井模型,以拉普拉斯方程作为控制方程,用标量有限元方法对低频准静态位场问题进行研究,提出了低频准静态位场理论建模的几个重要问题。根据准静态场的位场分布规律,本文首次提出,时变场中无限远边界条件并不适用于位场分析,并给出了更符合实际位场分布的截断边界条件。其次,为了减少测井仪器复杂激励模型导致的计算需求,本文提出了周期边界条件以及金属和绝缘材料边界条件。最后,以位场测量回路中引入取样电阻为例,本文提出了场与路一体化分析的建模思路,以精确模拟实际探测仪器对于匹配单元的高斯积分,唯一的区别就在于步骤(3),(4),(5)步中的坐标变换,而这些基于面的朗格朗日插值所花的时间很少,与整体建模求解时间比较起来,
其次,本文针对时变电磁场问题,主要采用了矢量有限元及其区域分解算法进行分析。通过对矢量有限元方法的基础研究,本文比较、分析了有限元方法中基于不同单元的基函数,引入了一种基于任意六面体单元的高阶叠层插值基函数,该基函数能适用于复杂结构多尺度目标的建模,因此能很好地胜任复杂低频近场问题的分析。然而,在复杂大规模的问题分析中,特别是因扫描探测需要激励端不断移动时,传统有限元方法的一体化建模剖分仍将导致极大的工作量、计算量和计算资源的占用。另外在低频近场问题中,有限元等数值计算方法往往会出现低频崩溃问题,即数值误差过大甚至无法求解。为了解决上述问题,本文首先引入了区域分解算法,将待求问题区域分为若干子区域,并行计算,大大减少内存和计算时间,提高了计算效率。接着,本文在区域分解算法已有的传输条件的基础上,提出了一种改进的二阶传输条件,该传输条件相对以往传输条件,具有在本征值单位圆内更聚合于圆心的本征值分布,因此也具备在任意频率下更快的迭代收敛速度,从而进一步减少了问题的求解分析时间。为了解决低频问题,本文在非匹配区域分解算法中还引入了树-叉树分割(tree-cotree splitting)技术。树-叉树分割是一种高效快速的消除低频奇异性的方法,通过去除基函数中的冗余,并在基函数的展开中加入位势的梯度,从而使得低频近场的求解更加精确。由于引用了该技术,本文对在低至零频的任意频段内的问题分析都能保持很高的效率和精度。基于上述研究,本文通过一些典型的地球物理探测应用,如介电扫描成像测井,随钻方位电阻率测井,油基泥浆微电阻率扫描成像测井以及三分量感应测井等问题,对本文提出方法的应用效率和精度进行了验证。结果证明了上述方法和技术在分析低频近场问题时的高效性和正确性,同时也体现出了该算法在分析这类问题时的优势。
在低频近场探测中,为了多维度地获取地层信息,需要对复杂的低频近场探测系统前端的激励-响应进行设计和分析。以上述算法作为分析工具,本文首先在三分量感应测井应用中设计了几种三分量线圈系结构。该设计方案因其在三个相互正交的极化方向的探测使得仪器可以探测倾斜地层和各向异性层等复杂地层。其中,三个正交方向的探测使得仪器可以探测薄互层和各向异性层等复杂地层,线圈系的共位缠绕大大缩短了仪器长度,而直耦对消则大大提高了仪器的信噪比。不仅如此,本文还进行了设计线圈系的加工,绕制,系统调试与实验井实验等,并最终推出了能用于工程生产的三分量感应测井仪器。其次,为了获取地层探测的方位向分辨力,本文首先推导出了倾斜磁偶极子在球坐标系下的解析表达,从中提取出其方位向因子。然后通过单个倾斜探头不同倾角和不同极化方向探头组合在水平面和垂直面的电场分布,本文提出了提高随钻电测井方位向分辨力的思路和策略,并在此基础上提出了新型随钻发射-接收组合设计。最终,采用有限元区域分解算法的仿真结果证明了新型设计在方位向分辨力上的优势,同时也表明随钻电测井的方位向分辨力仍有很大的提升空间。
最后,在总结已有的近似反演方法的基础上,为了使得低频近场目标探测中三维复杂目标的反演更加快速和精确,本文在区域分解的框架下提出了一种新型的三维全波反演算法。该反演算法对求解区域的分解将使得区域分解框架下的反演算法所需内存更少,并行效率更高,计算时间更快。不仅如此,与基于FEM的其它反演算法相比,该算法在每次反演迭代中只用重新计算需要反演的目标区域,而不用重复计算整个区域。因此,当本文将该反演算法应用到磁感应成像,地下目标探测等问题的反演分析中时,收到了很好的效果。证明了该算法确为一种快速和稳健的三维全波反演算法。
In the analysis of electromagnetic field and its application in detection, the research of low-frequency near field is always challenging and very important in real-life applications. On the one hand, it has broad future and pratical value, such as the application of electromagnetic well-logging problems and investigation of the geophysical structures which have been put into much research effort. On the other hand, the analysis of low-frequency near field has always been a hard zone in numerical computation and analysis. The existing methods have been seriously chanllenged especially when one has to face the large-and multi-scale problems. To deal with these challenges, in this dessertation, these problems are first classified into quasi-static potential field problems and time harmonic field problems based on the physic principle of the near field. And then each kind of problem is handled with different modeling strategy and analysis method. To deal with the time harmonic low-frequency near field problems, this dessertation developed the finite element domain decomposition method combined with tree-cotree splitting strategy and higher order transmission conditions, and then implemented it into the analysis of the forward electromagnetic well-logging problems. Based on the method, the modeling, design and realization of the low-frequency near-field detecting sensors are accomplished easily to gain multi-dimensional information of the earth. Finally, to have a fast inversion of the parameters of3D objects in complex problems, this dessertation proposed a fast3D full-wave inverse method combined with domain decomposition framework based on the research of the existing inverse methods. The inverse method proposed in this dessertation can perform fast inversion of the unknown objects in the low-frequency near-field detection problems.
First of all, to model the quasi-static potential field well-logging problems, this dessertation uses scalar finite element method to analysis the quasi-static potential field problems using Laplace equation as the constraint equation, and put forward several important issues in the modeling of quasi-static potential field problems. This dessertation finds out that the traditional far-field truncating boundary condition is no longer suitable for the potential field analysis for the first time and thus put forward a new far-field truncating boundary condition for the real application. Additionally, this dessertation implements another two boundary conditions to optimize the electromagnetic model which are the periodic boundary condition and equal potential field condition. Finally this dessertation build up a unified model both containing the field parameters and circuit parameters and get the numerical result of the slight and important changes of the response for the tools. The numerical results have proved the accuracy and advantage of our proposed quasi-static potential field modeling techniques.
At the second part, the vector finite element method and its non-conformal domain decomposition method (NC-FEM-DDM) are developed and optimized for the analysis of the time harmonic field problems. Based on the research of the vector FEM and comparison of basis functions based on different element types, this dessertation introduces a hexahedral hierarchical basis function, which can be very adaptive to the modeling of a complex multi-scale problem and consequently capable of modeling complex low-freqeuncy near field problems. However, in the application of a large-scale problem, the traditional FEM will suffer from large amounts of work load, memory cost and computational time because of its unified modeling strategy, especially when the excitation sources has to move in the need of scanning detection. Additionaly, numerical methods such as FEM will suffer from low-frequency break down and accurarcy loss. To solve the above problems, domain decomposition method is first developed to divide the original problems into several subdomains so that one can compute each subdomain independently and improve parallel efficiency. Then, based on the existing transmission conditions, an improved transmission condition is used to couple fields from different subdomains efficiently. This transmission condition has a more clustered eigenvalue distribution compared with the existing ones and thus it has faster convergence in both high-frequency band and low-frequency band. To solve the low-frequency break down problem, a tree-cotree splitting (TCS) strategy is then applied to eliminate the low-frequency breakdown at very low frequencies. Numerical results of the advanced well-logging problems such as the dielectric scanning tool, directional resistivity logging-while-drilling (LWD) tool, the micro-resistivity scanning tool and the three induction array tool all show an excellent convergence and accuracy obtained with the DDM with TCS at any frequencies.
In the low-frequency near-field problems, the analysis and design of the detecting sensors are very important because one need to get multi-dimensional information of the earth. Based on the development and improvement of the NC-FEM-DDM with TCS, series of3D induction sensor arrays have been designed at first. These designs have three orthometric polarized components so that the dip formation and the anisotropic bed can be interpreted. In these designs, the compact wind of the coils greatly shortened the length of the tool and the easy-implementing cancellation of the direct coupling largely improved the signal noise ratio (SNR). Based on the designings, this dessertation accomplished the fabrication, coil winding, system debugging and experiments in the experimental well and finally put out the3D induction array tool. Besides, to gain azimuthal resolution in the investigation, the analytical expression for the electric field of the tilted sensor as magnetic dipole is first derived. The directional component of the dipole was extracted. Then, through the field distribution of single transmitter with different tilt angles and the field distribution of combination of transmitters with different polarizations, clear strategies are illustrated on improving the azimuthal resolution of directional sensors. Based on it, novel transmitter-receiver designs are put forward and modeled using an efficient and accurate finite-element-based domain decomposition method. The results not only testify the designing strategies with better azimuthal resolution, but also indicate that there is still great potential in optimizing the azimuthal resolution of the directional propagation tools.
Finally, this dessertation has taken an over view of the existing inverse method to deal with the low-frequency and near-field inverse problems. A fast full-wave inverse scheme is developed combined with a domain decomposition method (DDM) so that the3D objects in complex real-life applications can be easily inversed as well. The division of the solution domain will bring huge advantage in the memory cost, parallel efficiency and computation time. Besides that, the object domain, whose parameters are unknown, is separated from the peripheral devices and outer boundaries so that only the object domain has to be recalculated, which will cut down the redundant computational burden in the inversion compared with other methods based on FEM. This nonlinear inverse scheme is implemented successfully for3-D full-wave problems such as the magnetic induction tomography (MIT) and ground penetrating radar, showing that this method is a robust and efficient3-D full-wave inverse method.
引文
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