井中激发极化法正反演及快速迭代求解技术研究
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摘要
井中激发极化法是勘查多金属和贵金属硫化物矿床,尤其是寻找深部盲矿体优先选用的井中物探方法。研究适用于起伏地形和复杂井眼环境条件下高效率、高精度的正演模拟算法以及稳健可靠的快速反演算法具有理论意义和应用价值。有限元法具有网格剖分灵活,求解过程规范的优点,适合复杂地球物理模型的模拟。在正演的基础上根据正则化原理建立反演目标函数,对目标函数求极小实现反演问题的求解。三维正反演问题一般采用迭代法来求解其中的线性方程组和优化问题,迭代求解技术是影响正反演计算效率的关键因素。本文研究了井中激发极化法的三维正反演,以及求解线性方程组和优化问题的快速迭代求解技术。
考虑到井中激发极化法的不同测量方式(井-地,地-井,井-井),以及深度方向上大尺度的网格剖分及井眼的影响等,在正演模拟中采用放射状三棱柱单元结构化网格,结合非结构化网格对模拟区域离散,提高了网格质量,减少了网格单元数,解决了考虑井眼影响时的模拟问题。进一步利用仿射坐标变换对三棱柱单元做单元分析,实现了显式的单元积分。其与采用等参变换和高斯数值积分相比,可以大大缩短获得刚度矩阵所需的时间。对有限元方程作右端项校正,在保证计算精度的前提下有效地减小了计算区域和剖分网格单元数。
在边值问题的处理中,采用人工截断边界方式,在边界上施加混合边界条件能获得较好的模拟效果。在实际勘探工作中通常涉及到多个源电极布置,不同的源电极对应着不同的有限元方程,从而生成了一序列线性方程组。根据Krylov子空间迭代法的收敛性分析,将体积分项和边界积分项分开进行计算和存储,并选择一个适当的线性方程组作为种子系统预先求解,利用在此过程中生成的子空间信息以加速其余线性方程组的求解过程,得到了求解序列线性方程组的循环Krylov子空间预条件共轭梯度法(PCG)算法。该算法的收敛性与序列线性方程组各系数矩阵之间的差异程度和右端项的靠近程度相关,通过构造新的右端项更靠近的序列线性方程组可进一步提高方法的计算效率。
根据正则化原理实现了反演问题的求解。讨论了正则化参数的选择,光滑约束矩阵的构造方法。为降低反演问题的不适定性,正反演采用不同的网格系统,反演用尽量少的网格单元,正演网格在反演网格的基础上进行加密。利用Jacobian-free Krylov子空间技术,不直接求取和存储Jacobian矩阵,只计算Jacobian矩阵与向量的乘积。用不精确的Krylov子空间法求解Gauss-Newton模型修正量方程,可减少迭代次数,降低计算量。在视电阻率反演的基础上,进一步实现了视极化率的正反演。
Induced polarization well logging is a preferred method for surveying deep metal mineral resources. There is important theoretical and practical significance in the study of efficient and high accuracy simulations and inversions regarding complicate well logging environment and topography. The forward modeling with FEM (Finite Element Method) leads to a large, sparse system by discretizing partial differential equations. To solve the inverse problem, it is first to build an objective function with the regularization technique, then to solve the functional minimization through an optimization iterative algorithm. Iterative solvers play a critical role in the process of solving the forward and inverse problems. The main task in this paper is to solve the forward and inverse problem of induced polarization well logging, and to study varieties of iterative algorithms involving in the modeling and inverse problems.
Forward modeling provides the basis of the solution of inverse problem. The techniques of mesh generations were discussed because it's important for the large computational domain and complexity in3D FEM modeling of the well logging. It improved the mesh quality and reduced the number of elements by employing a radial grid with parallel tri-prism. The accurate element integration of parallel tri-prism was implemented through an affine transformation, which shortens the element integral computational time greatly compared with isoparametric form and Gauss numerical integration. Incorporated with the unstructured mesh techniques, the forward modeling under complicated conditions was implemented. A correction technique to the right hand side could not only reduce the effects of the source singularity, but also reduce the errors associated with the influence of boundaries.
A typical method for the boundary problem is to impose mixed boundary condition on an artifical boundary, which has been proved to be a reasonable option. Typical3D resistivity surveys may involve several hundred electrode positions. With domain discretization, the numerial simulation with FEM for the problem results in a sequence of large sparse linear systems. Based the analysis of the convergence of Krylov subspace method, the global stiff matrices were considered to be assembled and stored into two parts separately. The equivalent multiple linear systems with more correlative right-hand sides were constructed. And the recycling Krylov subspace technique for the multiple linear systems arising from FEM modeling was implemented by solving a seed system firstly, and the descent direction and the solution of the seed system was reused to accelerate the process of finding the solution of the subsequent systems.
The regularization technique for ill-posed inverse problem has been discussed and the regularization parameters, model weighting matrix and line search techniques were analyzed with numerical tests. A multi-grid inverse technique was applied to reduce the ill-posedness of the inverse problem. Relatively coarse inverse grids define the elements whose electrial characters are to be determined, while in each process of inversion, refined grids in the forward calculations were carried out. An inexact Gauss-Newton solver has been implemented for the inverse algorithm. The search directions are determined with an inexact preconditioned conjugate gradient(PCG) algorithm. The Jacobian-free Krlov method was discussed. The method computes the Jacobian-vector product without forming and storing the elements of the true Jacobian matrix. Based on the apparent resistivity inversion, apparent polarizability inverse techniques were discussed and tested with numerical simulations.
考虑到井中激发极化法的不同测量方式(井-地,地-井,井-井),以及深度方向上大尺度的网格剖分及井眼的影响等,在正演模拟中采用放射状三棱柱单元结构化网格,结合非结构化网格对模拟区域离散,提高了网格质量,减少了网格单元数,解决了考虑井眼影响时的模拟问题。进一步利用仿射坐标变换对三棱柱单元做单元分析,实现了显式的单元积分。其与采用等参变换和高斯数值积分相比,可以大大缩短获得刚度矩阵所需的时间。对有限元方程作右端项校正,在保证计算精度的前提下有效地减小了计算区域和剖分网格单元数。
在边值问题的处理中,采用人工截断边界方式,在边界上施加混合边界条件能获得较好的模拟效果。在实际勘探工作中通常涉及到多个源电极布置,不同的源电极对应着不同的有限元方程,从而生成了一序列线性方程组。根据Krylov子空间迭代法的收敛性分析,将体积分项和边界积分项分开进行计算和存储,并选择一个适当的线性方程组作为种子系统预先求解,利用在此过程中生成的子空间信息以加速其余线性方程组的求解过程,得到了求解序列线性方程组的循环Krylov子空间预条件共轭梯度法(PCG)算法。该算法的收敛性与序列线性方程组各系数矩阵之间的差异程度和右端项的靠近程度相关,通过构造新的右端项更靠近的序列线性方程组可进一步提高方法的计算效率。
根据正则化原理实现了反演问题的求解。讨论了正则化参数的选择,光滑约束矩阵的构造方法。为降低反演问题的不适定性,正反演采用不同的网格系统,反演用尽量少的网格单元,正演网格在反演网格的基础上进行加密。利用Jacobian-free Krylov子空间技术,不直接求取和存储Jacobian矩阵,只计算Jacobian矩阵与向量的乘积。用不精确的Krylov子空间法求解Gauss-Newton模型修正量方程,可减少迭代次数,降低计算量。在视电阻率反演的基础上,进一步实现了视极化率的正反演。
Induced polarization well logging is a preferred method for surveying deep metal mineral resources. There is important theoretical and practical significance in the study of efficient and high accuracy simulations and inversions regarding complicate well logging environment and topography. The forward modeling with FEM (Finite Element Method) leads to a large, sparse system by discretizing partial differential equations. To solve the inverse problem, it is first to build an objective function with the regularization technique, then to solve the functional minimization through an optimization iterative algorithm. Iterative solvers play a critical role in the process of solving the forward and inverse problems. The main task in this paper is to solve the forward and inverse problem of induced polarization well logging, and to study varieties of iterative algorithms involving in the modeling and inverse problems.
Forward modeling provides the basis of the solution of inverse problem. The techniques of mesh generations were discussed because it's important for the large computational domain and complexity in3D FEM modeling of the well logging. It improved the mesh quality and reduced the number of elements by employing a radial grid with parallel tri-prism. The accurate element integration of parallel tri-prism was implemented through an affine transformation, which shortens the element integral computational time greatly compared with isoparametric form and Gauss numerical integration. Incorporated with the unstructured mesh techniques, the forward modeling under complicated conditions was implemented. A correction technique to the right hand side could not only reduce the effects of the source singularity, but also reduce the errors associated with the influence of boundaries.
A typical method for the boundary problem is to impose mixed boundary condition on an artifical boundary, which has been proved to be a reasonable option. Typical3D resistivity surveys may involve several hundred electrode positions. With domain discretization, the numerial simulation with FEM for the problem results in a sequence of large sparse linear systems. Based the analysis of the convergence of Krylov subspace method, the global stiff matrices were considered to be assembled and stored into two parts separately. The equivalent multiple linear systems with more correlative right-hand sides were constructed. And the recycling Krylov subspace technique for the multiple linear systems arising from FEM modeling was implemented by solving a seed system firstly, and the descent direction and the solution of the seed system was reused to accelerate the process of finding the solution of the subsequent systems.
The regularization technique for ill-posed inverse problem has been discussed and the regularization parameters, model weighting matrix and line search techniques were analyzed with numerical tests. A multi-grid inverse technique was applied to reduce the ill-posedness of the inverse problem. Relatively coarse inverse grids define the elements whose electrial characters are to be determined, while in each process of inversion, refined grids in the forward calculations were carried out. An inexact Gauss-Newton solver has been implemented for the inverse algorithm. The search directions are determined with an inexact preconditioned conjugate gradient(PCG) algorithm. The Jacobian-free Krlov method was discussed. The method computes the Jacobian-vector product without forming and storing the elements of the true Jacobian matrix. Based on the apparent resistivity inversion, apparent polarizability inverse techniques were discussed and tested with numerical simulations.
引文
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