非均质各向异性地层中Maxwell方程的3D耦合势有限体积算法研究及其在多分量感应测井中的应用
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摘要
随着人类对油气资源需要的不断增加以及勘探技术的不断发展，砂-泥岩薄交互储层、页岩气等非常规油气资源的勘探和开发已成为当前非常重要的研究课题。这种非常规储层的典型特点是电参数的各向异性，为了同时测量地层各向异性电导率以便对非常规油气进行有效评价，需要一种新型的多分量感应测井技术。与常规的感应测井不同，多分量感应测井仪器采用三分量正交发射线圈和接收线圈系，同时测量出完整的磁场强度张量，因此，可以提取出地层纵横向电导率、地层倾角和方位角等参数。然而，由于共面线圈系电磁场的非轴对称性以及对井眼泥浆和侵入带电导率、水平界面深度等变化的敏感性，复杂地层条件下多分量感应测井响应的有效数值模拟与软件研制仍然面临诸多挑战。本文针对各向异性储层中多分量感应测井响应数值模拟过程中，三维数值效率低且在大反差电导率地层中难以取得满意数值结果等问题，建立了一套基于电场矢势-标势的3D有限体积算法，从Maxwell方程的矢势-标势分解、有限体积法离散、非对称矩阵的压缩存储、不完全LU分解预处理以及稳定双共轭梯度法(BICGSTAB)快速求解大型复代数方程等方面展开了全面深入研究，并研制开发出相应的计算软件。通过对垂直井模型和倾斜井模型中多分量感应测井响应的大量数值计算，系统考察了井眼泥浆电导率、层厚、各向异性系数、侵入深度、地层倾角、仪器偏心效应、裂缝、源距以及工作频率等变化对多分量感应测井响应的影响，此外，利用交叉分量合成技术深入探讨了快速提取层界面深度、地层倾角以及各向异性系数等处理方法，取得非常有意义的结果，为多分量感应测井仪器参数的优化设计、资料处理等提供理论基础和有效的方法手段。
为克服低频发射或高阻地层导致的低感应数问题，有效提高三维电磁数值模拟算法效率与应用范围,首先利用电场标势与矢势将Maxwell方程转化为满足库仑规范条件的耦合势Helmholtz方程,以Yee氏交错网格中不同位置上的节点为中心建立四种控制体积单元,通过对控制体积单元中矢势、标势以及电导率的体积积分平均实现耦合势方程和磁偶极子旋度的离散,得到一个关于各个离散节点上电场标势与矢势的大型稀疏代数方程。此外,针对多分量感应仪器结构以及地层模型特点,规定了非均匀网格的剖分规则以及数值计算区域的范围,给出了离散电磁场和多分量感应测井响应的计算表达式。
针对离散化过程得到的大型稀疏矩阵中非零元素的分布特点,利用行格式压缩方法(CSR)对离散方程系数矩阵进行压缩存储,节省计算机的存储空间、减少无用的计算量。为有效降低系数矩阵的条件数提高Krylov子空间法的迭代速度,采用不完全LU分解预处理技术对系数矩阵进行优化,并结合稳定双共轭梯度法(BICGSTAB)有效地提高了大型离散方程的求解效率与精度。并利用数值结果对频率变化对迭代收敛速度的影响进行了考察,结果证明基于电场标势与矢势3D有限体积算法,由于释放了▽×▽算子的零空间,使得在较低频率下仍然能够保持较快的收敛速度和满意的数值结果。
为了对基于电场标势与矢势3D有限体积算法以及开放出的相应软件的有效性进行检验,分别用一维、二维垂直井模型中解析方法(ANA)和数值模式匹配方法(NMM)的数值结果与3D有限体积算法的结果进行了充分对比分析。在此基础上,进一步利用3D数值模拟软件系统研究考察了地层厚度、各向异性系数、线圈源距、井眼泥浆电阻率、泥浆侵入半径、以及仪器工作频率等因素对多分量感应测井响应的影响。此外,为有效模拟倾斜井眼中多分量感应的响应,借助于旋转矩阵实现了地层电阻率由地层坐标系向仪器坐标系的变换,系统考察倾斜井眼中多分量测井响应的三个主分量以及两个非零的交叉分量的响应特征,并对仪器偏心效应的成因以及井眼垂直裂缝对测井响应的影响等进行了深入研究。最后,利用数值结果对井眼倾斜多分量感应测井中两个非零的交叉分量的组合(σ(？)-σ(?))和(σ(?)+σ(?))的探测能力进行了检验,结果证明组合曲线(σ(?)-σ(?))能够提高对地层边界探测能力,并且可根据(σ(?)-σ(?))的正负判断层边界两侧电导率的高低,而组合曲线(σ(?)+σ(?))能够提高对各向异性地层的判断能力，且通过数值结果进一步系统考察了仪器倾角、线圈系源距、井眼泥浆电阻率对组合曲线的影响，并试探性的研究了含方位角变化的多分量感应测井响应特征。
With the increasing needs of oil and gas resource and the development ofexploration technology, exploration and development of unconventional oil and gasresource such as the laminated sand and shale reservoir, shale gas has become animportant research topic. The typical feature of the unconventional reservior iselectrical anisotropy. In order to simultaneously measure the anisotropicconductivity of the formations for effective evaluation of unconventional oil and gas,we need a new type of multicomponent induction logging (MCIL) technique.Different from the conventional induction logging, MCIL’s tool has a triaxial coilstransmitter and a triaxil coils receivers in order to measure the full tensor ofmagnetic field. Therefore, the parameters of the horizontal and vertical conductivityof formation, dipping angle of formation, azimuth angle of MCIL’s tool could beextracted by means of magnetic field tensor. However, because of the asymmetry ofelectromagnetic fields inducted by coplanar coils and sensitivity of change ofborehole and invasion conductivity horizontal interface depth, many challengessuch as the numerical simulation and software development of MCIL’ response incomplex anisotropic fornation modle should be faced. In the paper, we establish afinite volume method based on coupled vector-scalar potentials in order to solve thelow efficiency problem and difficulty of large conductivity contrast in3D numerical simulation of MCIL’s response. We comprehensively study the contents ofMaxwell’s vector-scalar potentials decomposition, finite volume discrete, compressstorage of ansymetric matrix, ILU(0), BICGSTAB. We also develop thecorresponding software. By a large number computation of MCIL’S response invertical an dipping borehole model, we systematically investigate the effect onMCIL’s response such as mud resistivity, thickness, anisotropy coefficient, invasionradius, borehole dipping angle, eccentricity, vertical fracture and so on. In addition,we apply the composite technique of cross components to study the method toextract the boundary locaton, borehole dipping anlge and anisotropic coefficient andobtain many useful numerical results. These results provide the theoretical basis andeffective methods to optimize MCIL’s tool and data processing.
In recent years, the research on induction logging response in anisotropicformations has become more and more active. However，as a result of thecomplexity of electromagnetic responses in anisotropy formation, there is still a lackof fast simulation algorithm so far. And then，we establish a3dimensional finitevolume algorithm (FVM) based on vector and scalar potentials to simulate theresponse of multi-component induction logging(MCIL) tool. And the correspondingcomputation codes are developed on the basis of FVM. Applying these codes, we doa large number of numerical calculations on MCIL’s response in vertical anddipping borehole models. We can obtain many useful numerical results that providethe theoretical basis on MCIL’s data analysis and tool’s improvements.
In order to overcome the low induction numbers problems (LINs), extend theapplication range of the3D EM modeling and enhance algorithm efficiency in thelow frequency domain and high resistivity formation, we first reformulateMaxwell’s equation into Helmholtz equations in terms of coupled scalar-vectorpotentials with Coulomb gauge. According to the different location of node in theYee’s non-uniform staggered grids, we set up four kinds of control volume cells.With volume integral averaging of vector potential, scalar potential and conductivitytensor on the control volume, we discrete the Helmholtz equations and the rotationof magnetic dipole sources successfully. The discrete form of the rotations of magnetic dipole sources can be approximately considered as discrete electricaldipole sources. The expression of discrete electromagnetic fields and MCIL’sresponse are also given in this paper. According to characteristic of MCIL’sinstrument structure and formation model, we define the rules of dividingnon-uniform grid and the size of numerical calculation area.
The choice of appropriate computation method is the crucial for solvingdiscrete equation efficiently. Through the analysis of distribution of non-zero matrixelements in discrete equation’s coefficient matrix, we can conclude that thecoefficient matrix is a large, sparse and complex matrix. In order to save thecomputer storage space and reduce the useless computation, we apply thecompressed sparse row (CSR) method to save the coefficient matrix of discretesystem. The condition number of coefficient matrix must be reduced in order toaccelerate the iterative convergence speed of Krylov subspace method. And then,the incomplete LU-decomposition preconditioner is applied to optimize thecoefficient matrix, and combine with the Bi-conjugate gradient stabilization(BICGSTAB) method to solve the discrete system efficiently. In the paper, we alsoinvestigate the effect of frequency change on iterative speed; the numerical resultsshow that FVM based on coupled potentials deflate the null space of curl-curloperator so that it can maintain fast iterative speed in low frequency.
In the Chapter4, we investigate characteristics of MCIL response in1D and2Dvertical borehole model. FVM is validated by means of Comparing numerical resultwith the analytical method and numerical mode matching method. Wesystematically investigate the effect of the anisotropic coefficient, spacing of coils,borehole mud resistivity, invasion radius, tool’s frequency on MCIL’s response. Inaddition, we also analyze the physical cause of eccentricity and investigate theeffect of the vertical fracture.
In the Chapter5, we achieve to transform the conductivity tensor from themedia coordinate system to the instrument coordinate system by means of rotationmatrices. The FVM based on coupled potentials are applied to calculate andinvestigate the response characteristics of MCIL in dipping well model. Numerical results of finite element method software are used to verify the validity of FVM in the complex3D Formation model. Because of the appearance of dipping borehole, there exist two non-zero cross components (σAXZ,σ) in MCIL's responses. Composite curves (σAXZ-σAZX) of cross component improve the detection capability of boundary in the formation. According the positive and negative of (σAXZ-σAZX), we can determine the difference of conductivity in the both sides of layer boundaries. Composite curves (σAXZ+σAZX)can be used to determine anisotropy in the formation. We also systematically studied the effect of the dipping angle, spacing of coils, borehole mud resistivity, on (σAXZ-σAZX) and (σAXZ-σAZX).In the end, the characteristics of MCIL's response with azimuth change are tentatively studied.
为克服低频发射或高阻地层导致的低感应数问题，有效提高三维电磁数值模拟算法效率与应用范围,首先利用电场标势与矢势将Maxwell方程转化为满足库仑规范条件的耦合势Helmholtz方程,以Yee氏交错网格中不同位置上的节点为中心建立四种控制体积单元,通过对控制体积单元中矢势、标势以及电导率的体积积分平均实现耦合势方程和磁偶极子旋度的离散,得到一个关于各个离散节点上电场标势与矢势的大型稀疏代数方程。此外,针对多分量感应仪器结构以及地层模型特点,规定了非均匀网格的剖分规则以及数值计算区域的范围,给出了离散电磁场和多分量感应测井响应的计算表达式。
针对离散化过程得到的大型稀疏矩阵中非零元素的分布特点,利用行格式压缩方法(CSR)对离散方程系数矩阵进行压缩存储,节省计算机的存储空间、减少无用的计算量。为有效降低系数矩阵的条件数提高Krylov子空间法的迭代速度,采用不完全LU分解预处理技术对系数矩阵进行优化,并结合稳定双共轭梯度法(BICGSTAB)有效地提高了大型离散方程的求解效率与精度。并利用数值结果对频率变化对迭代收敛速度的影响进行了考察,结果证明基于电场标势与矢势3D有限体积算法,由于释放了▽×▽算子的零空间,使得在较低频率下仍然能够保持较快的收敛速度和满意的数值结果。
为了对基于电场标势与矢势3D有限体积算法以及开放出的相应软件的有效性进行检验,分别用一维、二维垂直井模型中解析方法(ANA)和数值模式匹配方法(NMM)的数值结果与3D有限体积算法的结果进行了充分对比分析。在此基础上,进一步利用3D数值模拟软件系统研究考察了地层厚度、各向异性系数、线圈源距、井眼泥浆电阻率、泥浆侵入半径、以及仪器工作频率等因素对多分量感应测井响应的影响。此外,为有效模拟倾斜井眼中多分量感应的响应,借助于旋转矩阵实现了地层电阻率由地层坐标系向仪器坐标系的变换,系统考察倾斜井眼中多分量测井响应的三个主分量以及两个非零的交叉分量的响应特征,并对仪器偏心效应的成因以及井眼垂直裂缝对测井响应的影响等进行了深入研究。最后,利用数值结果对井眼倾斜多分量感应测井中两个非零的交叉分量的组合(σ(？)-σ(?))和(σ(?)+σ(?))的探测能力进行了检验,结果证明组合曲线(σ(?)-σ(?))能够提高对地层边界探测能力,并且可根据(σ(?)-σ(?))的正负判断层边界两侧电导率的高低,而组合曲线(σ(?)+σ(?))能够提高对各向异性地层的判断能力，且通过数值结果进一步系统考察了仪器倾角、线圈系源距、井眼泥浆电阻率对组合曲线的影响，并试探性的研究了含方位角变化的多分量感应测井响应特征。
With the increasing needs of oil and gas resource and the development ofexploration technology, exploration and development of unconventional oil and gasresource such as the laminated sand and shale reservoir, shale gas has become animportant research topic. The typical feature of the unconventional reservior iselectrical anisotropy. In order to simultaneously measure the anisotropicconductivity of the formations for effective evaluation of unconventional oil and gas,we need a new type of multicomponent induction logging (MCIL) technique.Different from the conventional induction logging, MCIL’s tool has a triaxial coilstransmitter and a triaxil coils receivers in order to measure the full tensor ofmagnetic field. Therefore, the parameters of the horizontal and vertical conductivityof formation, dipping angle of formation, azimuth angle of MCIL’s tool could beextracted by means of magnetic field tensor. However, because of the asymmetry ofelectromagnetic fields inducted by coplanar coils and sensitivity of change ofborehole and invasion conductivity horizontal interface depth, many challengessuch as the numerical simulation and software development of MCIL’ response incomplex anisotropic fornation modle should be faced. In the paper, we establish afinite volume method based on coupled vector-scalar potentials in order to solve thelow efficiency problem and difficulty of large conductivity contrast in3D numerical simulation of MCIL’s response. We comprehensively study the contents ofMaxwell’s vector-scalar potentials decomposition, finite volume discrete, compressstorage of ansymetric matrix, ILU(0), BICGSTAB. We also develop thecorresponding software. By a large number computation of MCIL’S response invertical an dipping borehole model, we systematically investigate the effect onMCIL’s response such as mud resistivity, thickness, anisotropy coefficient, invasionradius, borehole dipping angle, eccentricity, vertical fracture and so on. In addition,we apply the composite technique of cross components to study the method toextract the boundary locaton, borehole dipping anlge and anisotropic coefficient andobtain many useful numerical results. These results provide the theoretical basis andeffective methods to optimize MCIL’s tool and data processing.
In recent years, the research on induction logging response in anisotropicformations has become more and more active. However，as a result of thecomplexity of electromagnetic responses in anisotropy formation, there is still a lackof fast simulation algorithm so far. And then，we establish a3dimensional finitevolume algorithm (FVM) based on vector and scalar potentials to simulate theresponse of multi-component induction logging(MCIL) tool. And the correspondingcomputation codes are developed on the basis of FVM. Applying these codes, we doa large number of numerical calculations on MCIL’s response in vertical anddipping borehole models. We can obtain many useful numerical results that providethe theoretical basis on MCIL’s data analysis and tool’s improvements.
In order to overcome the low induction numbers problems (LINs), extend theapplication range of the3D EM modeling and enhance algorithm efficiency in thelow frequency domain and high resistivity formation, we first reformulateMaxwell’s equation into Helmholtz equations in terms of coupled scalar-vectorpotentials with Coulomb gauge. According to the different location of node in theYee’s non-uniform staggered grids, we set up four kinds of control volume cells.With volume integral averaging of vector potential, scalar potential and conductivitytensor on the control volume, we discrete the Helmholtz equations and the rotationof magnetic dipole sources successfully. The discrete form of the rotations of magnetic dipole sources can be approximately considered as discrete electricaldipole sources. The expression of discrete electromagnetic fields and MCIL’sresponse are also given in this paper. According to characteristic of MCIL’sinstrument structure and formation model, we define the rules of dividingnon-uniform grid and the size of numerical calculation area.
The choice of appropriate computation method is the crucial for solvingdiscrete equation efficiently. Through the analysis of distribution of non-zero matrixelements in discrete equation’s coefficient matrix, we can conclude that thecoefficient matrix is a large, sparse and complex matrix. In order to save thecomputer storage space and reduce the useless computation, we apply thecompressed sparse row (CSR) method to save the coefficient matrix of discretesystem. The condition number of coefficient matrix must be reduced in order toaccelerate the iterative convergence speed of Krylov subspace method. And then,the incomplete LU-decomposition preconditioner is applied to optimize thecoefficient matrix, and combine with the Bi-conjugate gradient stabilization(BICGSTAB) method to solve the discrete system efficiently. In the paper, we alsoinvestigate the effect of frequency change on iterative speed; the numerical resultsshow that FVM based on coupled potentials deflate the null space of curl-curloperator so that it can maintain fast iterative speed in low frequency.
In the Chapter4, we investigate characteristics of MCIL response in1D and2Dvertical borehole model. FVM is validated by means of Comparing numerical resultwith the analytical method and numerical mode matching method. Wesystematically investigate the effect of the anisotropic coefficient, spacing of coils,borehole mud resistivity, invasion radius, tool’s frequency on MCIL’s response. Inaddition, we also analyze the physical cause of eccentricity and investigate theeffect of the vertical fracture.
In the Chapter5, we achieve to transform the conductivity tensor from themedia coordinate system to the instrument coordinate system by means of rotationmatrices. The FVM based on coupled potentials are applied to calculate andinvestigate the response characteristics of MCIL in dipping well model. Numerical results of finite element method software are used to verify the validity of FVM in the complex3D Formation model. Because of the appearance of dipping borehole, there exist two non-zero cross components (σAXZ,σ) in MCIL's responses. Composite curves (σAXZ-σAZX) of cross component improve the detection capability of boundary in the formation. According the positive and negative of (σAXZ-σAZX), we can determine the difference of conductivity in the both sides of layer boundaries. Composite curves (σAXZ+σAZX)can be used to determine anisotropy in the formation. We also systematically studied the effect of the dipping angle, spacing of coils, borehole mud resistivity, on (σAXZ-σAZX) and (σAXZ-σAZX).In the end, the characteristics of MCIL's response with azimuth change are tentatively studied.
引文
[1]张庚骥.电法测井[M].山东东营：石油大学出版社，1996.
[2]张建华，刘振华，仵杰.电法测井原理与应用[M].西安：西北大学出版社，2002.
[3]汪宏年,杨善德,常明澈.各向异性地层中电阻率测井的响应特征[J]石油地球物理勘探,1999,34(6):649-656.
[4]汪宏年，杨善德，常明澈.层状各向异性介质中侧向电阻率测井的快速数值模拟与应用[J].测井技术，1998,，22(1):28－31.
[5]王昌学,杨韦华，楚昭坦,等.多分量感应测井响应得交错网格有限差分法模拟[J].石油大学学报,2005.29(3):35-40.
[6] MORAN J H. GIANZEROS. Effects of formation anisotropy on resistivitylogging measurements. Geophysics.44(7):1266~1286,1979.
[7] RABINOVICH M, GOFALINI M, ROCQUE T, et al. a Multicomponentinduction logging:10years after, SPWLA48thAnnual Logging Symp.,2007Paper CC.
[8] KRIEGSHIUSER B, FANINI O, FORGANG S, et al. A new multi-componentinduction logging tool to resolve anisotropic formations [C]. SPWLA40th.Annual Logging Symposium,2000, paper D.
[9] KRIEGSHIUSER B, FANINI O, FORGANG S, et al. Improved shaly sandinterpretation in highly deviated and horizontal wells using multicomponentinduction log data [C]. SPWLA42th. Annual Logging Symposium,2001, paperS.
[10]YU L, FANINI O N, KRIEGSHAUSER B F. Enhanced evaluation oflow-resistivity reservoirs using multi-component induction log data [J].Petrophysics,2001,42(6):611-622.
[11]HUSSAIN F, MUBARK S, ALDRED R, et al. Contributions ofmulti-component induction logs in difficult well bore environments in northKuwait low resistivity pay evaluation [C]. SPWLA46ndAnnual LoggingSymposium,2005, Paper VV.
[12]RABINOVICH M, TOBAROVSKY L, CORLEY B, et al. Processingmulticomponent induction data for formation dips and anisotropy [J].Petrophysics,2006,47(6):506-526.
[13]ZHDZNOV M S, KENNEDY D M, PEKSEN E, et al. Foundations of tensorinduction well-logging [J]. Petrophysics,2001,42(6):588-61
[14]HANMING WANG, TOM BARBER, KOUCHIANG CHEN, et al. Triaxialinduction logging: theory,modeling, inversion, and interpretation, SPEInternational Oil&Gas Conference.2006.
[15]RABINOVICH M, TABAROVSKY L, CORLEY B, et al. Processingmulti-component induction data for formation dip and azimuth in AnisotropicFormation [C]，SPWLA46thAnnual Logging Symposium,2005, Paper XX.
[16]ZHONG L L, LI J, SHEN L C. Computation of triaxial induction logging toolsin layered anisotropic dipping formations [J]. IEEE Transactions on Geoscienceand Remote Sensing,2008,46(4):1148-1163.
[17]姚东华，汪宏年，杨守文等.用传播矩阵法研究层状正交各向异性地层中多分量感应测井响应.地球物理学报,2010,53(12):3026~3037
[18]WANG H N, SO P, YANG S W, et al. Numerical modeling of multicomponentinduction well logging tools in the cylindrically stratified anisotropic media [J].IEEE Transactions on Geoscience and Remote Sensing,2008,46(4):1134-1147.
[19]汪宏年，陶宏根，杨守文.用模式匹配算法研究层状各向异性倾斜地层中多分量感应测井响应，地球物理学报.51（5）：1591-1599,2008.
[20]WANG H N, HU P, TAO H G. Fast Numerical Simulation of Responses ofArray Multicomponent Induction Logging Tool in Horizontally StratifiedInhomogeneous Ti Formation by NMM, The30thPIERS Progress InElectromagnetics Research Symposium. Suzhou, China, September2011
[21]WANG H N, YANG S W. Fast modeling of micro-spherically focused log inhorizontally layered formation. IEEE Trans. On Geosci. And Remote Sensing,39(10),2275-2282,2001
[22]汪宏年，胡平，陶宏根.水平层状非均质横向同性地层中阵列多分量感应测井的快速计算，地球物理学报，55(2):717-726
[23]WANG H N, TAO H G. An Efficient and reliable simulation ofmulticomponent inducion logging response in horizontally stratifiedinhomogeneous TI formations by numerical mode-matching method, IEEETrans. On Geosci. And Remote Sensing,编号：IEEE-TGRS-2010-00755R（已接收）
[24]YEE K S. Numerical solution of initial boundary value problem involvingMaxwell's equations in isotropic media [J]. IEEE Transactions on Antennas andPropagation,1966，14(3)：302-307.
[25]沈金松.用有限差分法计算各向异性介质中多分量感应测井的响应[J].地球物理学进展，2004，19(1):101-107.
[26]沈金松.用交错网格有限差分法计算三维频率域电磁响应.地球物理学报,2003,46(2):281~289.
[27]汪功礼,张庚骥，崔锋修等.三维感应测井响应计算的交错网格有限差分法，[J].地球物理学报，2003，46(4):561-567.
[28]杨守文，汪宏年，陈桂波等.倾斜各向异性地层中多分量电磁波测井响应三维时域有限差分（FDTD）算法[J].地球物理学报，2009，52(3):833-841.
[29]NEWMAN G A, ALUMBAUGH D L. Three-dimensional induction loggingproblems, Part2: A finite difference solution [J]. Geophysics,2002,67(2):484-491.
[30]WEISS C J, NEWMAN G A. Electromagnetic induction in a fully3-Danisotropic earth [J]. Geophysics,2002,67[4]:1104-1114.
[31]WEISS C J, NEWMAN G A. Electromagnetic induction in a generalized3Danisotropic earth, Part2: The LIN precondition [J]. Geophysics,2003,68(3):922–930.
[32]NEWMAN G A, ALUMBAUGH D L. Frequency-domain modelling ofairborne electromagnetic responses using staggered finite differences.Geophysics,1995,43:1021–1042.
[33]WEISS C J, NEWMAN G A. A strategy for rapid solutions in modeling EMinduction in anisotropic media. SEG Int’l Exposition and72ndAnnual Meeting.Salt Lake City, Utah. October6-11,2002
[34]ALUMBAUGH D L, NEWMAN G A, PREVOST, L. Three-dimensionalwideband electromagnetic modeling on massively parallel computers.RADIO SCIENCE,31(1),1-23.
[35]ALUMBAUGH D L, NEWMAN G A. Three-dimensional massively parallelelectromagnetic inversion—I. Theory. Geophys. J. Int.128,345-354,1997
[36]ALUMBAUGH D L, NEWMAN G A. Three-dimensional massively parallelelectromagnetic inversion II. Analysis of a crosswell electromagneticexperiment. Geophys. J. Int.(1997)128,355-363.
[37]WANG T, HOHMANN G W. A finite-difference, time-domain solution forthree dimensional electromagnetic modeling, Geophysics,199358,797-809.
[38]WANG T, FANG S.3-D electromagnetic anisotropic modeling using finitedifferences, Geophysics,2001,66(5):1386-1398.
[39]WANG T, Yu L M, Fanini O. Multicomponent induction response in boreholeenvironment Geophysics,2003,68(5):1510-1518
[40]MOSKOW S, DRUSKIN V, HABASHY T, LEE P, DAVYDYCHEVA S A.Finite difference scheme for elliptic equations with rough coefficients using aCartesian grid noncoforming to interfaces. SIAM J. NUMER. ANAL,36442-464.
[41]DAVYDYCHEVA S, DRUSKIN V, HABASHY T. An efficientfinite-difference scheme for electromagnetic logging in3D anisotropicinhomogeneous media [J]. Geophysics,2003,68(5):1525-1536.
[42]DAVYDYCHEVA S, HOMANN D, MINERBO G. Triaxial induction toolwith electrode sleeve: FD modeling in3D geometries. Journal of AppliedGeophysics,2009,67,98-108.
[43]DAVYDYCHEVA S. Two triaxial induction tools: sensitivity to radial invasionprofile, Geophysical Prospecting,1-18.
[44]DAVYDYCHEVA S.3D modeling of new-generation (1999-2010) resistivitylogging tools. The Leading Edge.29(7),780-789.
[45]DAVYDYCHEVA S.Separation of azimuthal effects for new-generationresistivity logging tools—Part I. Geophysics,2010,75(1): E31-E40.
[46]DAVYDYCHEVA S.Separation of azimuthal effects for new-generationresistivity logging tools—Part2. Geophysics,2010,76(3): F185-F202.
[47]DAVYDYCHEVA S, FRENKE. Review of3D EM Modeling andInterpretation Methods for Triaxial Induction and Propagation Resistivity WellLogging Tools. PIERS Proceedings, Cambridge, USA, July5-8,2010,390-396.
[48]DAVYDYCHEVA S. DRUSKIN V. Staggered grid for Maxwell's equations inarbitrary3-Dinhomogeneous anisotropic media. Three-DimensionalElectromagnetics, M. Oristaglio and B. Spies, Eds.,119-137, SEG,1999.
[49]DAVYDYCHEVA S, RYKHLINSKI N. Focused-source electromagneticsurvey versus standard CSEM:3D modeling in complex geometries.Geophysics,2011,76(1): F27-F41.
[50]ZASLAVSKY M, DRUSKIN V, DAVYDYCHEVA S. Hybrid finite-differenceintegral equation solver for3D frequency domain anisotropic electromagneticproblems. Geophysics,2011,76(2): F123-F137.
[51]DAVYDYCHEVA S, RYKHLINSKI N, LEGEIDO P. Electrical-prospectingmethod for hydrocarbon search using the induced polarization effect.Geophysics,2011,71(4): G179-G189.
[52]DAVYDYCHEVA S, RYKHLINSKI N. Focused source EM survey versustime-domain and frequency-domain CSEM. The Leading Edge,28,944–949.
[53]SMITH J. Conservative modeling of3-D electromagnetic field, Part one:properties and error analysis. Geophysics,1996,61:1308-1318.
[54]Conservative modeling of3-D electromagnetic fields, Part two: Biconjugategradient solution and an accelerator. Geophysics,1996,61:1319-1324.
[55]HOU J S, MALLANL R K, TORRES V C. Finite-difference simulation ofborehole EM measurements in3D anisotropic media using coupledscalar-vector potentials [J]. Geophysics,71(5): G225-233.
[56]AVDEEV D B, KUVSHINOV A V, PANKRATOV O V et al.Three-dimensional induction logging problems, Part I: An integral equationsolution and model comparisons. Geophysics,2002,67(2):413-426.
[57]AVDEEV D B, KNIZHNIK S.3D integral equation modeling with a lineardependence on dimensions, Geophysics,2009,74(5): F89-F94.
[58]NIE X C, YUAN N, LIU C R. Simulation of LWD tool response using a fastintegral equation method. IEEE Trans. On Geosci. And Remote Sensing,48(1),2010.
[59]陈桂波,汪宏年,姚敬金等.用积分方程法模拟各向异性地层中三维电性异常体的电磁响应,地球物理学报，2009，52(8):2174-2181.
[60]陈桂波,汪宏年,姚敬金等.各向异性海底地层海洋可控源电磁响应三维积分方程法数值模拟,物理学报,2009,58(6):3848-3857.
[61]ZUNOUBI, M.R., JIN J. M., DONEPUDI K. C., et al., A spectral Lanczosdecomposition method for solving3-D low-frequency electromagnetic diffusionby the finite-element method. Antennas and Propagation, IEEE Transactions on,1999.47(2): p.242-248.
[62]NAM, M.J., KIM H. J., LEE T. J., et al.,3D magnetotelluric modellingincluding surface topography. Geophysical prospecting,2007.55(2): p.277-287.
[63]SUN, X Y, NIE Z P. Analysis and correction of borehole effect on the responsesof multicomponent induction logging tools. Progress In ElectromagneticsResearch,2008.85: p.211-226.
[64]孙向阳,聂在平,赵延文等.用矢量有限元方法模拟随钻测井仪在倾斜各向异性地层中的电磁响应.地球物理学报,2008,51(5):1600～1607
[65]SUN X Y, NIE Z P. Vector finite analysis of multicomponent induction responsein anisotropic formations. Progress In Electromagnetics Research,2008.81: p.21-39.
[66]李荣华.偏微分方程数值解法,吉林长春,高等教育出版社,2005.
[67]张文生,科学计算中的偏微分方程有限差分解法,高等教育出版社,2006.
[68]HABER E, ASCHER U M, OLDENBURG D, et al. Fast simulation of3Delectromagnetic problems using potentials [J]. Journal of ComputationalPhysics,2000,163:150–171.
[69]HABER E, ASCHER U M. Fast finite volume simulation of3Delectromagnetic problems with highly discontinuous coefficients. SIAM J. SCI.COMPUT.2001,22,1943-1961.
[70]ARULIAH D, ASCHER U M, HABER E A, et al. A method for the forwardmodeling of3D electromagnetic quasi-static problems. Mathematical Modelsand Methods in Applied Sciences,11(1),1-21,2001.
[71]NOVO M S, SILVA L C, TERIXEIRA L. Application of CFS-PML to finitevolume analysis of EM well-logging tools in multilayered geophysicalformations. Electromagnetic Field Computation,200612th Biennial IEEEConference, APRIL,2006.
[72]NOVO M S, SILVA L C, TERIXEIRA L. Finite volume modeling of boreholeelectromagnetic logging in3-D anisotropic formations using coupledscalar-Vector potentials. IEEE ANTENNAS AND WIRELESSPROPAGATION LETTERS, VOL.6,2007.
[73]NOVO M S, SILVA L C, TERIXEIRA L. Comparison of coupled-potentialsand field-based finite-volume techniques for modeling of borehole EM Tools.IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL.5, NO.2,2008.
[74]NOVO M S, SILVA L C, TERIXEIRA L. Three-dimensional finite-volumeanalysis of directional resistivity logging sensors. IEEE TRANSACTIONS ONGEOSCIENCEAND REMOTE SENSING, VOL.48, NO.3,2010
[75]NOVO M S, SILVA L C, TERIXEIRA L. A comparative analysis of krylovsolvers for three-dimensional simulations of borehole sensors. IEEEGEOSCIENCEAND REMOTE SENSING LETTERS, VOL.8, NO.1,2011.
[76]SAAD Y. Iterative Method for Sparse Liner Systems.(PWS, Boston,1996).
[77]VORST H V D. Iterative krylov methods for large linear systems. Cambridgeuniversity press,2003.
[78]SAAD Y. LANCZOS C, Solutions of systems of linear equations by minimizediterations, J. Res. Natl. Bur.Stand.49,33(1952)
[79]HESTENES M. R. and STIEFEL E. L., Methods of conjugate gradients forsolving linear systems, J. Res. Natl.Bur. Stand.1952,49:409
[80]SAAD Y. Krylov subspace methods for solving large unsymmetric linearsystems, Math. Comp.37(1981)105–126.
[81]SAAD Y and SCHULTZ M H. GMRES: A generalized minimal residualalgorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput,1986,7:856-869
[82]FLETCHER R. Gradient methods for indefinite systems, V506OF LECTURENOTES MATH.73-89.NEWYORK,1976
[83]FREUND, R. and NACHTIGAL N.. QMR: A quasi-minimal residual methodfor non-Hermitian linear systems. Numer. Math.,1991,60:315-339.
[84]SONNEVELD P. CGS: a fast Lanczos-type solver for nonsymmetric linearsystems. SIAM J. Sci. Statist Comput.10:36-52,1989.
[85]VORST H V D. Bi-CGSTAB: Afast and smoothy converging variant of Bi-CGfor the soution of nonsymmetric linear systems. SIAM J. Sci. Statist. Comput.,1992,13:631-644.
[86]SLEIJPEN G L G, VORST H V D. Reliable Updated Residuals in HybridBi-CG Methods. Computing,1996,56,141-163.
[87]SLEIJPEN G L G, FOKKEMA D. R. BICGSTAB(L) for linear equationsinvolving unsymmetric matrices with complex spectrum. ElectronicTransactions on Numerical Analysis. Volume1, pp.11-32.1993.
[88]SLEIJPEN G L G, VORST H V D. BiCGstab(l) and other hybrid Bi-CGmethods. Numerical Algorithms.7,75-109,1994
[89]SLEIJPEN G L G, VORST H V D. Maintaining convergence properties ofBiCGstab methods in finite precision arithmetic. Numerical Algorithms.10,203-223,1995
[90]SAAD Y. ILUT: A dual threshold incomplete LU factorization. Numer. LinearAlgebra Appl,1994,1:387-402.
[91]SAAD Y. and ZHANG J. BILUM: Block versions of multielimination andmultilevel preconditioner for general sparse linear systems. SIAM J. Sci.Comput.,1999,20:2103-2121.
[92]CAI X C, GROPP W D, KEYES D E. A comparison of some domaindecomposition and ILU preconditioned iterative methods for nonsymmetricelliptic problems.
[93]TREFETHEN L N. and BAU N.,1996, Numerical linear algebra: Soc. Ind.Appl. Math.
[94]LABRECQUE D.1999, Finite difference modeling of3-D EM felds withscalar and vector potentials, in Oristaglio, M., and Spies, B., Eds.,Three-dimensional electromagnetics: Soc. Expl. Geophys.148–160.
[95]LINDELL I V, SIHVOLA A. H. Perfect electromagnetic conductor, J.Electromagn. WavesAppl.,2005,19(7):861–869
[96]陈桂波,汪宏年,姚敬金,韩子夜.利用积分方程法的各向异性地层频率测深三维模拟.计算物理,27(2):274-280,2010.
[97]GOLUB G H, LOAN F CHARTES. Matrix computations (Third edition). TheJohns Hopkins University Press.1996
[98]黄廷祝,钟守铭,李正良.矩阵理论,高等教育出版社,2003.
[99]李庆杨，王能超.数值分析,华中理工大学出版社,1986.
[100]洪德成.三轴感应测井资料处理方法研究[博士论文].长春:吉林大学物理学院,2009
[101]居于马.线性代数（第二版）,清华大学出版社.2002.
[102] WANG G L, TORRES-VERDIN C, and GIANZERO S. Fast simulation oftriaxial borehole induction measurements acquired in axially symmetrical andtransversely isotropic media. Geophysics,2009,74(6): E233~E249.
[103] Wang H N, Tao H G, and Yao J J, et al. Fast multiparameter reconstructionof multicomponent induction well-logging datum in a deviated well in ahorizontally stratified anisotropic formation. IEEE Trans. On Geosci. andRemote Sensing,2008,46(5):1525~1534.
[104] Wang H N. Adaptive regularization iterative inversion of arraymulticomponent induction well logging datum in a horizontally stratifiedinhomogeneous TI formation, IEEE Trans. on Geosci. and Remote Sensing,2011,49(11):4483~4491.
[105]汪宏年,陶宏根，王桂萍，等.双感应测井资料的快速近似迭代反演[J]..地球物理学报，2007，50(5):1614-1622.
[106]汪宏年,陶宏根,其木苏荣等.层状介质中双侧向测井资料正则化全参数迭代反演与应用[J]，地球物理学报，2002,45（增刊）:387-399.
[107] WANG T L, TANG X M, Yu L M, et al. Characterizing fracture withmulticomponent induction measurements. Petrophysics,2005,46(1):42~51.
[108]姚东华,汪宏年,陶宏根,杨守文.水平层状介质中双侧向测井资料的迭代Tikhonov正则化反演，2010,53（9）:2227-2236.
[109]周建美,汪宏年,姚敬金等.水平层状横向同性地层中频率测深资料全参数快速迭代反演.物理学报(已接收).
[110] HOU J S, BITTAR M. Correction for the borehole effect ofmulti-component array induction log data. Progress In ElectromagneticsResearch Symposium Proceedings, Cambridge, USA, July,2010.
[111] MALLAN R K, TORRES-VERDIN C. Effects of petrophysicalenvironmental and geometrical parameters on multicomponent inductionmeasurements. SPWLA47thAnnual Logging Symposium, June.2006.
[112] XUE D, RABINOVICH M, WANG T. Detection of tool eccentricitydirection from multi-component induction responses.2006SEG AnnualMeeting,2006, New Orleans. Louisiana.
[113] WU P, WANG H, MINERBO G, ET AL. Borehole effects and correction inOBM with dip and anisotropy for triaxial induction tools, SPE AnnualTechnical Conference and Exhibition, November2007, Anaheim, Colifornia.U.S.A.
[114] ONEGOVA E. V., Effect of multicoil electromagnetic tool eccentricity onmeasured signals. Russian Geology and Geophysics.51(4),2010,423-427.
[115]张烨,汪宏年,陶宏根,杨守文.基于耦合标势与矢势的有限体积法模拟非均匀各向异性地层中多分量感应测井三维响应,地球物理学报（已接收）
[116] ZHANG Y, WANG H N, YANG S W. Finite volume algorithm to simulateresponse of multicomponent induction tools in3D inhomogeneous anisotropicformation using coupled scalar-vector potentials, The30thPIERS Progress InElectromagnetic Research Symposium, Suzhou, China, September.
[117] LIU G S. Analysis of directional logging tools in anisotropic andmultieccentric cylindrically-layered earth formations, IEEE Transactions onAntennas and propagation.2012,60(1):318-327.
[118] TOMKINS M J, ALUMBAUGH D L, STANLEY D T. Numerical analysisof near-borehole and anisotropic layer effects on the response ofmulticomponent induction logging tools [J]. Geophysics,2004,69(1):140-151.
[119] HUE Y K, TEIXEIRA F L. Three-dimensional simulation of eccentricLWD tool response in boreholes through dipping formations [J]. IEEETransactions on Geoscience and Remote Sensing,2005,43(2):257-268.
[120] WANG T L, TANG X M, YU L M, et al. Characterizing fracture withmulticomponent induction measurements. Petrophysics,2005,46(1):42~51
[121]沈金松,苏本玉,郭乃川.裂缝性储层的电各向异性响应特性研究.地球物理学报,2009,52(11):2904~2912.
[122] YU L M, RABINOVVICH M, TABAROVSKY L, et al. The reduction ofnear zone effects on the multi-component induction logging tool. SPE AnnualTechnical Conference and Exhibition, October2003, Denver, Colorado.
[123] CHEW W C著,聂在平,柳清伙译.非均匀介质中的场与波[M].北京：电子工业出版社,1995.
[2]张建华，刘振华，仵杰.电法测井原理与应用[M].西安：西北大学出版社，2002.
[3]汪宏年,杨善德,常明澈.各向异性地层中电阻率测井的响应特征[J]石油地球物理勘探,1999,34(6):649-656.
[4]汪宏年，杨善德，常明澈.层状各向异性介质中侧向电阻率测井的快速数值模拟与应用[J].测井技术，1998,，22(1):28－31.
[5]王昌学,杨韦华，楚昭坦,等.多分量感应测井响应得交错网格有限差分法模拟[J].石油大学学报,2005.29(3):35-40.
[6] MORAN J H. GIANZEROS. Effects of formation anisotropy on resistivitylogging measurements. Geophysics.44(7):1266~1286,1979.
[7] RABINOVICH M, GOFALINI M, ROCQUE T, et al. a Multicomponentinduction logging:10years after, SPWLA48thAnnual Logging Symp.,2007Paper CC.
[8] KRIEGSHIUSER B, FANINI O, FORGANG S, et al. A new multi-componentinduction logging tool to resolve anisotropic formations [C]. SPWLA40th.Annual Logging Symposium,2000, paper D.
[9] KRIEGSHIUSER B, FANINI O, FORGANG S, et al. Improved shaly sandinterpretation in highly deviated and horizontal wells using multicomponentinduction log data [C]. SPWLA42th. Annual Logging Symposium,2001, paperS.
[10]YU L, FANINI O N, KRIEGSHAUSER B F. Enhanced evaluation oflow-resistivity reservoirs using multi-component induction log data [J].Petrophysics,2001,42(6):611-622.
[11]HUSSAIN F, MUBARK S, ALDRED R, et al. Contributions ofmulti-component induction logs in difficult well bore environments in northKuwait low resistivity pay evaluation [C]. SPWLA46ndAnnual LoggingSymposium,2005, Paper VV.
[12]RABINOVICH M, TOBAROVSKY L, CORLEY B, et al. Processingmulticomponent induction data for formation dips and anisotropy [J].Petrophysics,2006,47(6):506-526.
[13]ZHDZNOV M S, KENNEDY D M, PEKSEN E, et al. Foundations of tensorinduction well-logging [J]. Petrophysics,2001,42(6):588-61
[14]HANMING WANG, TOM BARBER, KOUCHIANG CHEN, et al. Triaxialinduction logging: theory,modeling, inversion, and interpretation, SPEInternational Oil&Gas Conference.2006.
[15]RABINOVICH M, TABAROVSKY L, CORLEY B, et al. Processingmulti-component induction data for formation dip and azimuth in AnisotropicFormation [C]，SPWLA46thAnnual Logging Symposium,2005, Paper XX.
[16]ZHONG L L, LI J, SHEN L C. Computation of triaxial induction logging toolsin layered anisotropic dipping formations [J]. IEEE Transactions on Geoscienceand Remote Sensing,2008,46(4):1148-1163.
[17]姚东华，汪宏年，杨守文等.用传播矩阵法研究层状正交各向异性地层中多分量感应测井响应.地球物理学报,2010,53(12):3026~3037
[18]WANG H N, SO P, YANG S W, et al. Numerical modeling of multicomponentinduction well logging tools in the cylindrically stratified anisotropic media [J].IEEE Transactions on Geoscience and Remote Sensing,2008,46(4):1134-1147.
[19]汪宏年，陶宏根，杨守文.用模式匹配算法研究层状各向异性倾斜地层中多分量感应测井响应，地球物理学报.51（5）：1591-1599,2008.
[20]WANG H N, HU P, TAO H G. Fast Numerical Simulation of Responses ofArray Multicomponent Induction Logging Tool in Horizontally StratifiedInhomogeneous Ti Formation by NMM, The30thPIERS Progress InElectromagnetics Research Symposium. Suzhou, China, September2011
[21]WANG H N, YANG S W. Fast modeling of micro-spherically focused log inhorizontally layered formation. IEEE Trans. On Geosci. And Remote Sensing,39(10),2275-2282,2001
[22]汪宏年，胡平，陶宏根.水平层状非均质横向同性地层中阵列多分量感应测井的快速计算，地球物理学报，55(2):717-726
[23]WANG H N, TAO H G. An Efficient and reliable simulation ofmulticomponent inducion logging response in horizontally stratifiedinhomogeneous TI formations by numerical mode-matching method, IEEETrans. On Geosci. And Remote Sensing,编号：IEEE-TGRS-2010-00755R（已接收）
[24]YEE K S. Numerical solution of initial boundary value problem involvingMaxwell's equations in isotropic media [J]. IEEE Transactions on Antennas andPropagation,1966，14(3)：302-307.
[25]沈金松.用有限差分法计算各向异性介质中多分量感应测井的响应[J].地球物理学进展，2004，19(1):101-107.
[26]沈金松.用交错网格有限差分法计算三维频率域电磁响应.地球物理学报,2003,46(2):281~289.
[27]汪功礼,张庚骥，崔锋修等.三维感应测井响应计算的交错网格有限差分法，[J].地球物理学报，2003，46(4):561-567.
[28]杨守文，汪宏年，陈桂波等.倾斜各向异性地层中多分量电磁波测井响应三维时域有限差分（FDTD）算法[J].地球物理学报，2009，52(3):833-841.
[29]NEWMAN G A, ALUMBAUGH D L. Three-dimensional induction loggingproblems, Part2: A finite difference solution [J]. Geophysics,2002,67(2):484-491.
[30]WEISS C J, NEWMAN G A. Electromagnetic induction in a fully3-Danisotropic earth [J]. Geophysics,2002,67[4]:1104-1114.
[31]WEISS C J, NEWMAN G A. Electromagnetic induction in a generalized3Danisotropic earth, Part2: The LIN precondition [J]. Geophysics,2003,68(3):922–930.
[32]NEWMAN G A, ALUMBAUGH D L. Frequency-domain modelling ofairborne electromagnetic responses using staggered finite differences.Geophysics,1995,43:1021–1042.
[33]WEISS C J, NEWMAN G A. A strategy for rapid solutions in modeling EMinduction in anisotropic media. SEG Int’l Exposition and72ndAnnual Meeting.Salt Lake City, Utah. October6-11,2002
[34]ALUMBAUGH D L, NEWMAN G A, PREVOST, L. Three-dimensionalwideband electromagnetic modeling on massively parallel computers.RADIO SCIENCE,31(1),1-23.
[35]ALUMBAUGH D L, NEWMAN G A. Three-dimensional massively parallelelectromagnetic inversion—I. Theory. Geophys. J. Int.128,345-354,1997
[36]ALUMBAUGH D L, NEWMAN G A. Three-dimensional massively parallelelectromagnetic inversion II. Analysis of a crosswell electromagneticexperiment. Geophys. J. Int.(1997)128,355-363.
[37]WANG T, HOHMANN G W. A finite-difference, time-domain solution forthree dimensional electromagnetic modeling, Geophysics,199358,797-809.
[38]WANG T, FANG S.3-D electromagnetic anisotropic modeling using finitedifferences, Geophysics,2001,66(5):1386-1398.
[39]WANG T, Yu L M, Fanini O. Multicomponent induction response in boreholeenvironment Geophysics,2003,68(5):1510-1518
[40]MOSKOW S, DRUSKIN V, HABASHY T, LEE P, DAVYDYCHEVA S A.Finite difference scheme for elliptic equations with rough coefficients using aCartesian grid noncoforming to interfaces. SIAM J. NUMER. ANAL,36442-464.
[41]DAVYDYCHEVA S, DRUSKIN V, HABASHY T. An efficientfinite-difference scheme for electromagnetic logging in3D anisotropicinhomogeneous media [J]. Geophysics,2003,68(5):1525-1536.
[42]DAVYDYCHEVA S, HOMANN D, MINERBO G. Triaxial induction toolwith electrode sleeve: FD modeling in3D geometries. Journal of AppliedGeophysics,2009,67,98-108.
[43]DAVYDYCHEVA S. Two triaxial induction tools: sensitivity to radial invasionprofile, Geophysical Prospecting,1-18.
[44]DAVYDYCHEVA S.3D modeling of new-generation (1999-2010) resistivitylogging tools. The Leading Edge.29(7),780-789.
[45]DAVYDYCHEVA S.Separation of azimuthal effects for new-generationresistivity logging tools—Part I. Geophysics,2010,75(1): E31-E40.
[46]DAVYDYCHEVA S.Separation of azimuthal effects for new-generationresistivity logging tools—Part2. Geophysics,2010,76(3): F185-F202.
[47]DAVYDYCHEVA S, FRENKE. Review of3D EM Modeling andInterpretation Methods for Triaxial Induction and Propagation Resistivity WellLogging Tools. PIERS Proceedings, Cambridge, USA, July5-8,2010,390-396.
[48]DAVYDYCHEVA S. DRUSKIN V. Staggered grid for Maxwell's equations inarbitrary3-Dinhomogeneous anisotropic media. Three-DimensionalElectromagnetics, M. Oristaglio and B. Spies, Eds.,119-137, SEG,1999.
[49]DAVYDYCHEVA S, RYKHLINSKI N. Focused-source electromagneticsurvey versus standard CSEM:3D modeling in complex geometries.Geophysics,2011,76(1): F27-F41.
[50]ZASLAVSKY M, DRUSKIN V, DAVYDYCHEVA S. Hybrid finite-differenceintegral equation solver for3D frequency domain anisotropic electromagneticproblems. Geophysics,2011,76(2): F123-F137.
[51]DAVYDYCHEVA S, RYKHLINSKI N, LEGEIDO P. Electrical-prospectingmethod for hydrocarbon search using the induced polarization effect.Geophysics,2011,71(4): G179-G189.
[52]DAVYDYCHEVA S, RYKHLINSKI N. Focused source EM survey versustime-domain and frequency-domain CSEM. The Leading Edge,28,944–949.
[53]SMITH J. Conservative modeling of3-D electromagnetic field, Part one:properties and error analysis. Geophysics,1996,61:1308-1318.
[54]Conservative modeling of3-D electromagnetic fields, Part two: Biconjugategradient solution and an accelerator. Geophysics,1996,61:1319-1324.
[55]HOU J S, MALLANL R K, TORRES V C. Finite-difference simulation ofborehole EM measurements in3D anisotropic media using coupledscalar-vector potentials [J]. Geophysics,71(5): G225-233.
[56]AVDEEV D B, KUVSHINOV A V, PANKRATOV O V et al.Three-dimensional induction logging problems, Part I: An integral equationsolution and model comparisons. Geophysics,2002,67(2):413-426.
[57]AVDEEV D B, KNIZHNIK S.3D integral equation modeling with a lineardependence on dimensions, Geophysics,2009,74(5): F89-F94.
[58]NIE X C, YUAN N, LIU C R. Simulation of LWD tool response using a fastintegral equation method. IEEE Trans. On Geosci. And Remote Sensing,48(1),2010.
[59]陈桂波,汪宏年,姚敬金等.用积分方程法模拟各向异性地层中三维电性异常体的电磁响应,地球物理学报，2009，52(8):2174-2181.
[60]陈桂波,汪宏年,姚敬金等.各向异性海底地层海洋可控源电磁响应三维积分方程法数值模拟,物理学报,2009,58(6):3848-3857.
[61]ZUNOUBI, M.R., JIN J. M., DONEPUDI K. C., et al., A spectral Lanczosdecomposition method for solving3-D low-frequency electromagnetic diffusionby the finite-element method. Antennas and Propagation, IEEE Transactions on,1999.47(2): p.242-248.
[62]NAM, M.J., KIM H. J., LEE T. J., et al.,3D magnetotelluric modellingincluding surface topography. Geophysical prospecting,2007.55(2): p.277-287.
[63]SUN, X Y, NIE Z P. Analysis and correction of borehole effect on the responsesof multicomponent induction logging tools. Progress In ElectromagneticsResearch,2008.85: p.211-226.
[64]孙向阳,聂在平,赵延文等.用矢量有限元方法模拟随钻测井仪在倾斜各向异性地层中的电磁响应.地球物理学报,2008,51(5):1600～1607
[65]SUN X Y, NIE Z P. Vector finite analysis of multicomponent induction responsein anisotropic formations. Progress In Electromagnetics Research,2008.81: p.21-39.
[66]李荣华.偏微分方程数值解法,吉林长春,高等教育出版社,2005.
[67]张文生,科学计算中的偏微分方程有限差分解法,高等教育出版社,2006.
[68]HABER E, ASCHER U M, OLDENBURG D, et al. Fast simulation of3Delectromagnetic problems using potentials [J]. Journal of ComputationalPhysics,2000,163:150–171.
[69]HABER E, ASCHER U M. Fast finite volume simulation of3Delectromagnetic problems with highly discontinuous coefficients. SIAM J. SCI.COMPUT.2001,22,1943-1961.
[70]ARULIAH D, ASCHER U M, HABER E A, et al. A method for the forwardmodeling of3D electromagnetic quasi-static problems. Mathematical Modelsand Methods in Applied Sciences,11(1),1-21,2001.
[71]NOVO M S, SILVA L C, TERIXEIRA L. Application of CFS-PML to finitevolume analysis of EM well-logging tools in multilayered geophysicalformations. Electromagnetic Field Computation,200612th Biennial IEEEConference, APRIL,2006.
[72]NOVO M S, SILVA L C, TERIXEIRA L. Finite volume modeling of boreholeelectromagnetic logging in3-D anisotropic formations using coupledscalar-Vector potentials. IEEE ANTENNAS AND WIRELESSPROPAGATION LETTERS, VOL.6,2007.
[73]NOVO M S, SILVA L C, TERIXEIRA L. Comparison of coupled-potentialsand field-based finite-volume techniques for modeling of borehole EM Tools.IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL.5, NO.2,2008.
[74]NOVO M S, SILVA L C, TERIXEIRA L. Three-dimensional finite-volumeanalysis of directional resistivity logging sensors. IEEE TRANSACTIONS ONGEOSCIENCEAND REMOTE SENSING, VOL.48, NO.3,2010
[75]NOVO M S, SILVA L C, TERIXEIRA L. A comparative analysis of krylovsolvers for three-dimensional simulations of borehole sensors. IEEEGEOSCIENCEAND REMOTE SENSING LETTERS, VOL.8, NO.1,2011.
[76]SAAD Y. Iterative Method for Sparse Liner Systems.(PWS, Boston,1996).
[77]VORST H V D. Iterative krylov methods for large linear systems. Cambridgeuniversity press,2003.
[78]SAAD Y. LANCZOS C, Solutions of systems of linear equations by minimizediterations, J. Res. Natl. Bur.Stand.49,33(1952)
[79]HESTENES M. R. and STIEFEL E. L., Methods of conjugate gradients forsolving linear systems, J. Res. Natl.Bur. Stand.1952,49:409
[80]SAAD Y. Krylov subspace methods for solving large unsymmetric linearsystems, Math. Comp.37(1981)105–126.
[81]SAAD Y and SCHULTZ M H. GMRES: A generalized minimal residualalgorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput,1986,7:856-869
[82]FLETCHER R. Gradient methods for indefinite systems, V506OF LECTURENOTES MATH.73-89.NEWYORK,1976
[83]FREUND, R. and NACHTIGAL N.. QMR: A quasi-minimal residual methodfor non-Hermitian linear systems. Numer. Math.,1991,60:315-339.
[84]SONNEVELD P. CGS: a fast Lanczos-type solver for nonsymmetric linearsystems. SIAM J. Sci. Statist Comput.10:36-52,1989.
[85]VORST H V D. Bi-CGSTAB: Afast and smoothy converging variant of Bi-CGfor the soution of nonsymmetric linear systems. SIAM J. Sci. Statist. Comput.,1992,13:631-644.
[86]SLEIJPEN G L G, VORST H V D. Reliable Updated Residuals in HybridBi-CG Methods. Computing,1996,56,141-163.
[87]SLEIJPEN G L G, FOKKEMA D. R. BICGSTAB(L) for linear equationsinvolving unsymmetric matrices with complex spectrum. ElectronicTransactions on Numerical Analysis. Volume1, pp.11-32.1993.
[88]SLEIJPEN G L G, VORST H V D. BiCGstab(l) and other hybrid Bi-CGmethods. Numerical Algorithms.7,75-109,1994
[89]SLEIJPEN G L G, VORST H V D. Maintaining convergence properties ofBiCGstab methods in finite precision arithmetic. Numerical Algorithms.10,203-223,1995
[90]SAAD Y. ILUT: A dual threshold incomplete LU factorization. Numer. LinearAlgebra Appl,1994,1:387-402.
[91]SAAD Y. and ZHANG J. BILUM: Block versions of multielimination andmultilevel preconditioner for general sparse linear systems. SIAM J. Sci.Comput.,1999,20:2103-2121.
[92]CAI X C, GROPP W D, KEYES D E. A comparison of some domaindecomposition and ILU preconditioned iterative methods for nonsymmetricelliptic problems.
[93]TREFETHEN L N. and BAU N.,1996, Numerical linear algebra: Soc. Ind.Appl. Math.
[94]LABRECQUE D.1999, Finite difference modeling of3-D EM felds withscalar and vector potentials, in Oristaglio, M., and Spies, B., Eds.,Three-dimensional electromagnetics: Soc. Expl. Geophys.148–160.
[95]LINDELL I V, SIHVOLA A. H. Perfect electromagnetic conductor, J.Electromagn. WavesAppl.,2005,19(7):861–869
[96]陈桂波,汪宏年,姚敬金,韩子夜.利用积分方程法的各向异性地层频率测深三维模拟.计算物理,27(2):274-280,2010.
[97]GOLUB G H, LOAN F CHARTES. Matrix computations (Third edition). TheJohns Hopkins University Press.1996
[98]黄廷祝,钟守铭,李正良.矩阵理论,高等教育出版社,2003.
[99]李庆杨，王能超.数值分析,华中理工大学出版社,1986.
[100]洪德成.三轴感应测井资料处理方法研究[博士论文].长春:吉林大学物理学院,2009
[101]居于马.线性代数（第二版）,清华大学出版社.2002.
[102] WANG G L, TORRES-VERDIN C, and GIANZERO S. Fast simulation oftriaxial borehole induction measurements acquired in axially symmetrical andtransversely isotropic media. Geophysics,2009,74(6): E233~E249.
[103] Wang H N, Tao H G, and Yao J J, et al. Fast multiparameter reconstructionof multicomponent induction well-logging datum in a deviated well in ahorizontally stratified anisotropic formation. IEEE Trans. On Geosci. andRemote Sensing,2008,46(5):1525~1534.
[104] Wang H N. Adaptive regularization iterative inversion of arraymulticomponent induction well logging datum in a horizontally stratifiedinhomogeneous TI formation, IEEE Trans. on Geosci. and Remote Sensing,2011,49(11):4483~4491.
[105]汪宏年,陶宏根，王桂萍，等.双感应测井资料的快速近似迭代反演[J]..地球物理学报，2007，50(5):1614-1622.
[106]汪宏年,陶宏根,其木苏荣等.层状介质中双侧向测井资料正则化全参数迭代反演与应用[J]，地球物理学报，2002,45（增刊）:387-399.
[107] WANG T L, TANG X M, Yu L M, et al. Characterizing fracture withmulticomponent induction measurements. Petrophysics,2005,46(1):42~51.
[108]姚东华,汪宏年,陶宏根,杨守文.水平层状介质中双侧向测井资料的迭代Tikhonov正则化反演，2010,53（9）:2227-2236.
[109]周建美,汪宏年,姚敬金等.水平层状横向同性地层中频率测深资料全参数快速迭代反演.物理学报(已接收).
[110] HOU J S, BITTAR M. Correction for the borehole effect ofmulti-component array induction log data. Progress In ElectromagneticsResearch Symposium Proceedings, Cambridge, USA, July,2010.
[111] MALLAN R K, TORRES-VERDIN C. Effects of petrophysicalenvironmental and geometrical parameters on multicomponent inductionmeasurements. SPWLA47thAnnual Logging Symposium, June.2006.
[112] XUE D, RABINOVICH M, WANG T. Detection of tool eccentricitydirection from multi-component induction responses.2006SEG AnnualMeeting,2006, New Orleans. Louisiana.
[113] WU P, WANG H, MINERBO G, ET AL. Borehole effects and correction inOBM with dip and anisotropy for triaxial induction tools, SPE AnnualTechnical Conference and Exhibition, November2007, Anaheim, Colifornia.U.S.A.
[114] ONEGOVA E. V., Effect of multicoil electromagnetic tool eccentricity onmeasured signals. Russian Geology and Geophysics.51(4),2010,423-427.
[115]张烨,汪宏年,陶宏根,杨守文.基于耦合标势与矢势的有限体积法模拟非均匀各向异性地层中多分量感应测井三维响应,地球物理学报（已接收）
[116] ZHANG Y, WANG H N, YANG S W. Finite volume algorithm to simulateresponse of multicomponent induction tools in3D inhomogeneous anisotropicformation using coupled scalar-vector potentials, The30thPIERS Progress InElectromagnetic Research Symposium, Suzhou, China, September.
[117] LIU G S. Analysis of directional logging tools in anisotropic andmultieccentric cylindrically-layered earth formations, IEEE Transactions onAntennas and propagation.2012,60(1):318-327.
[118] TOMKINS M J, ALUMBAUGH D L, STANLEY D T. Numerical analysisof near-borehole and anisotropic layer effects on the response ofmulticomponent induction logging tools [J]. Geophysics,2004,69(1):140-151.
[119] HUE Y K, TEIXEIRA F L. Three-dimensional simulation of eccentricLWD tool response in boreholes through dipping formations [J]. IEEETransactions on Geoscience and Remote Sensing,2005,43(2):257-268.
[120] WANG T L, TANG X M, YU L M, et al. Characterizing fracture withmulticomponent induction measurements. Petrophysics,2005,46(1):42~51
[121]沈金松,苏本玉,郭乃川.裂缝性储层的电各向异性响应特性研究.地球物理学报,2009,52(11):2904~2912.
[122] YU L M, RABINOVVICH M, TABAROVSKY L, et al. The reduction ofnear zone effects on the multi-component induction logging tool. SPE AnnualTechnical Conference and Exhibition, October2003, Denver, Colorado.
[123] CHEW W C著,聂在平,柳清伙译.非均匀介质中的场与波[M].北京：电子工业出版社,1995.