Studying On Maxwell Equations3D Finite Volume Algorithm In Inhomogeneous Anisotropic Formation And Its Application In Multicomponent Induction Logging

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.