Nonlinear inversion in electrode logging in a highly deviated formation with invasion using an oblique coordinate system

Electrode logging as known in the oil industry is a method for determining the electrical conductivity distribution around a borehole or between two boreholes from the static-field (dc) measurements in the borehole. The authors discuss the reconstruction of the three-dimensional (3D) conductivity around a borehole in a highly deviated formation with invasion. At this moment, they have not included the borehole effect. To solve this problem, the full vector analysis is required. In most available algorithms, for the forward and inverse modeling of the resistivity data, the dipping bed environment is approximated using the staircase-discretization grid. In contrast, they have modeled the dipping-bed environment by introducing an oblique (nonorthogonal) coordinate system. By using the oblique coordinate system, they have gained some advantages over the usual approach. First, the use of the staircasing approximation for the dipping-bed environment can be avoided. This means that they reduce the discretization error, and they can suffice with less discretization points to obtain the results with the same degree of accuracy as the problem formulated in the Cartesian coordinate system. Secondly, the horizontally-symmetry constraints of the conductivity distribution can be included easily in the inversion procedure. Several numerical results are presented to demonstrate the performance of the inversion method using the synthetic "measured" data, which are generated by solving a forward-scattering problem numerically with the help of the conjugate gradient fast Fourier transform (CGFFT) method.