Three-dimensional nonlinear inversion in cross-well electrode logging

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. In this paper we discuss the reconstruction of the conductivity in a three-dimensional domain between two boreholes. The electric current density integral equation will be taken as point of departure to develop a nonlinear inversion scheme based on a new inversion method, the so-called contrast source inversion method (CSI). The CSI method considers the inverse scattering problem as an inverse Source problem in which the unknown contrast sources in the object domain are reconstructed by minimizing the object and data error using a;conjugate gradient Step, after which the contrast is updated by minimizing only the error in the object. The robustness and the abilities of this full-vector three-dimensional inversion method have been demonstrated with a number of numerical examples where the synthetic "measured" data used were generated by solving a forward scattering problem using the conjugate gradient fast Fourier transform method.