Study On Forward And Inversion Of Induced-polarization Well Logging And Fast Iterative Solvers

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.