3-D Numerical Mode Matching Method for Off- Centered Electromagnetic Well Logging Tools in Noncircular Vertical Borehole and Invasion Zones in Multilayered Media
Jie Liu, Jianliang Zhuo, Wei Jiang, Qing Liu
Abstract
In electromagnetic (EM) well logging for petroleum exploration, off-centered tools within a noncircular borehole and invasion zones in multilayered media represent a challenging 3-D problem for traditional numerical methods. The 3-D finite-element numerical mode-matching (FNMM) method and the spectral numerical mode-matching (SNMM) method are developed to address this problem in this work. The numerical mode-matching (NMM) methods reduce the original 3-D well logging problem into a series of 2-D open waveguide eigenvalue problems plus a 1-D layered medium problem, which can be analytically solved by a recursion procedure. These waveguide eigenvalue problems with anisotropic inhomogeneous media in the horizontal directions are solved numerically by the mixed finite-element method (MFEM) with the flexibility for the complex geometry and the mixed spectral element method (MSEM) with the exponential convergence. A general operator form of Maxwell’s equations is also derived to obtain the source excitation vector independent of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula> -axis. Therefore, the NMM solvers can effectively and accurately simulate the response of off-centered EM well logging tools in an irregular borehole in complex formations. The NMM methods are applied to simulate three formations including a five-layer isotropic homogeneous formation, an off-centered logging tool in a noncircular borehole with anisotropic invasion zones in a five-layer anisotropic formation, and the Oklahoma formation. Numerical experiments indicate that the NMM methods are highly efficient and accurate for the EM well logging models.