3‐D Numerical Mode‐Matching (NMM) Method for Resistivity Logging Responses in Nonsymmetric Conditions

Abstract

AbstractThree‐dimension (3‐D) Numerical Mode‐Matching (NMM) is selected to deal with resistivity logging responses in nonsymmetric conditions. First, an appropriate Descartes's reference frame is set up: the plane of deviated borehole and normal of formation is regarded as plane XOY , and formation strike is regarded as Z axis direction. In the plane XOY the finite‐element method (FEM) is used and a series of two‐dimension (2‐D) generalized eigenvalue problems are solved; whereas vertical analytic is adopted along Z axis. In analytical section, the medium is sliced into some layers artificially, and the relationship between adjacent layers is derived according to the electromagnetic field continuity, namely boundary transition matrices rather than reflection and transmission matrices theory in the 2‐D NMM method. Finally, a system of linear equations is constructed according to the natural boundary conditions that the topmost layer bed has no up‐going wave or the bottommost layer has no down‐going wave, and the potential of every point of formation is solved and the apparent resistivity is computed. In the given formation model, the results of 3‐D NMM agree well with previous ones of 2‐D NMM and FEM in symmetry conditions. Furthermore, the algorithm also applies to with formations titled angle and elliptical borehole; and some examples illustrate how all kinds of formation models with tilt angles affect logging response results.