Application Of The Vector Finite Element Method To 3-D Induction Well Logging Problems

Induction logging is one of the most important resistivity logging method which uses the electromagnetic induction principle to measure the resistivity of formation. Since the invention of induction well logging, the vertical or similar to vertical drilling technology was wildly used, and the well logging response calculation procedure is usually based on the vertical well models. However, with the development of the oil exploration technology, the vertical wells have been gradually replaced by deviated or horizontal wells. Therefore, the calculation of well logging response in full three-dimension deviated or horizontal wells becomes a hot issue for logging workers.In this paper, the response of induction well logging in deviated and horizontal wells have been studied. The tetrahedral elements based vector finite element method (VFEM) is used to compute electromagnetic fields in 3-D inhomogeneous media with arbitrary distribution of conductivity. Because of the cylindrical symmetry of the well logging system, a cylindrical grid is used for the simulation. The total field is separated into the background and the secondary fields, which reduce the mesh size and improve the simulation accuracy effectively.The difficult in solving the large-scale equations discrete by finite element method is a bottleneck of the large scale application of finite element in engineer field, and the development of an effective solver have always been a hot topic in numerical calculation field. Furthermore, the equations discrete by the vector finite element method are complex ones in ill condition, finding their solution is a huge challenge. Direct method and iterative method are two ways to solving linear equations, the traditional direct method can find the accurate solution but it cost too much compute resource, the iterative method is simple and cost little computer resource, but it usually has a bad iterative convergence rate due to the bad property of the coefficient matrix. In this paper, we developed a front solver and a multi front solver to overcome the shortcomings of traditional direct method, on the other hand, We combine the iterative stabilized biconjugate-gradient (BICGSTAB) technique with a Algebraic Multigrid (AMG) Preconditioning algorithm to solve the large scale complex linear system, which improve the convergence rate efficiently.The introduction section of this article describes the background, significance and innovation points of this study. In chapter 2 and chapter 3, we describe the basic principles of induction logging and finite element method.In chapter 4, a detail exposition of the application of tetrahedral edge element was given, including the deviation of the tetrahedral edge element basis functions, the computer of the stiff matrix, and the algorithm to implement the tetrahedral edge element procedure. After then, we studied the methods to solve the linear systems discrete by the finite element method. Both the direct method such as frontal solver, multi frontal solver and the iterative methods: algebraic multi-grid pre conditioned BICGSTAB solver are discussed.In chapter 5, cases for the use of tetrahedral edge element are given, including the application of the developed algorithm to the simulation of normal induction well logging tools, as well as the immediate widely used array induction and LWD(logging while drilling) well logging tools. Finally, summarization and expectation of this paper is given. The VFEM is successfully applied to simulate the resistivity responses of deviated and horizontal wells in inhomogeneous formation. Numerical results showthe efficiency of the developed method in 3-D induction well loggingproblems. The results of this research have good prospects in well longingapplications.