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

Abstract

Abstract-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 three-dimensional (3-D) conductivity around a borehole in a highly deviated formation with invasion. At this moment, we 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, we have modeled the dipping-bed environment by introducing an oblique (nonorthogonal) 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. This means that we reduce the discretization error, and we 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.Index Terms-Electrode logging, nonlinear inversion, oblique coordinate system, three-dimensional method.

A hybrid method for inversion of 3D AC resistivity logging measurements

“…Let E i ( ), E j ( ), i > j be two consecutive solutions of the Galerkin primal forward problem (33) and G i ( ), G j ( ) the corresponding solutions to the Galerkin dual forward problems (34). Then,…”

“…Within this group, we encounter algorithms for inversion of general geometries where the forward problem is simulated using finite differences [35], finite elements [44][45][46], and integral equations [47]. In order to increase the speed of computations and also to obtain a more stable inverse problem, some authors employ physics based model reduction techniques [33,34,48], which also restrict the allowable geometries that are considered during the inversion process. All algorithms considered within this group are only capable of finding a local minima, and for the case of multi-modal inverse problems, it can be rarely determined when the found local minima is indeed the global one.…”

Iterative forward and inverse algorithms based on domain integral equations for three-dimensional electric and magnetic objects

“…Among the huge amount of literature we only list a few without intention of review: Abubakar and co-authors [2][3][4][5][6][7]9,30], Barkeshli and Journal of Computational Physics 195 (2004) www.elsevier.com/locate/jcp Lautzenheiser [10], Chew and Wang [13], Franchois and Pichot [14], Habashy and co-authors [15,16,27], Isernia et al [18], Kleinman and van den Berg [19], Liu and co-authors [20,36], Zhdanov and Hursan [33]. Furthermore, a more recent development on the three-dimensional electromagnetic forward and inverse modeling for geophysical applications can be found in [22,34].…”

Contrast Source Inversion Method: State of Art

“…Similar to the modified gradient method, in each iteration there is no full inversion of the object equations involved. The CSI method outperforms the modified gradient method, because it is computationally faster, has less memory as well as data requirements and accommodates easily a priori information, see e.g., [1].…”