Fast one-dimensional finite element approximation of geophysical measurements

Abstract

When inverting Logging-While-Drilling (LWD) resistivity measurements, it is a common practice to consider a one-dimensional (1D) layered media to reduce the problem dimensionality using a Hankel transform. Using orthogonality of Bessel functions, we arrive at a system of Ordinary Differential Equations (ODEs); one systema of ODEs per Hankel mode. The dimensionality of the resulting problem is referred to as 1.5D since the computational cost to resolve it is in between that needed to solve a 1D problema and a 2D problem. When material properties are piecewise-constant, we can solve the resulting ODEs either (a) analytically, which leads to a so-called semi-analytic method, or (b) numerically. Semi-analytic methods are faster, but they also have important limitations, for example, (a) the analytical solution can only account for piecewise constant material properties, and other resistivity distributions cannot be solved analytically, which prevents to accurately model, for example, and OWT zone when fluids are considered to be inmiscible; (b) a specific set of cumbersome formulas has to be derived for each physical process (e.g. electromagnetism, elasticity, etc.), anisotropy type, etc.; (c) analytical derivatives of specific models (e.g. cross-bedded formations, or derivatives with respect to the bed boundary positios) are often diffcult to obtain and have not been published to the best of our knowledge.