Finite-element modeling of the normal resistivity tool in azimuthally inhomogenous formations

A finite-element, three-dimensional, theoretical model of resistivity logging is presented. The method is based on minimizing the total power, or energy, of the system. The derivation of the power equation is outlined. The Rayleigh-Ritz procedure is used for numerical minimization of the total power. For each individual element, the element matrix is obtained first. After that, the overall system of equations is assembled from the individual element matrices. Then, the boundary conditions are enforced and finally, the electric potential is calculated at each node by solving the overall system of equations. The finite-element method is tested for a limiting case with the integral equation method and the agreement is excellent. The finite-element method is used to analyze the response of the 16-inch normal tool in azimuthally inhomogenous formations. The finite-element theoretical model presented can be used to study the responses of all electrode-type resistivity tools, such as the laterolog and the spherically focused log in complex logging situations.