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 b oreholes from the static-held (dc) measurements in the borehole, In this pa per, we discuss the reconstruction of the three-dimensional (3-D) conductiv ity around a borehole in a highly deviated formation with invasion. At this moment, we-have not included the borehole effect. To solve this problem, t he full vector analysis is required. In most available algorithms, for the forward and inverse modeling of the resistivity data, the dipping bed envir onment is approximated using the staircase-discretization grid. In contrast , we have modeled the dipping-bed environment by introducing an oblique (no northogonal) coordinate system. By using the oblique coordinate system, we have gained some advantages over the usual approach. First, the use of the staircasing approximation for the dipping-bed environment can be avoided. T his means that we reduce the discretization error and we can suffice with l ess discretization points to obtain the results with the same degree of acc uracy as the problem formulated in the Cartesian coordinate system. Secondl y, the horizontally-symmetry constraints of the conductivity distribution c an 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-sc attering problem numerically with the help of the conjugate gradient fast f ourier transform (CGFFT) method.