HR: 14:20h
AN: NS33A-04 INVITED     [Abstracts]
TI: Near Well-bore Imaging Algorithm for the Multi-Array Triaxial Induction Logging Measurements
AU: * Abubakar, A
EM: aabubakar@slb.com
AF: Schlumberger-Doll Research, 36 Old Quarry Road, Ridgefield, CT 06877 United States
AU: Habashy, T M
EM: habashy1@slb.com
AF: Schlumberger-Doll Research, 36 Old Quarry Road, Ridgefield, CT 06877 United States
AB: Induction tools have been the standard resistivity devices for borehole geophysics for more than fifty years. Conventional induction tools are built with coils that have their magnetic moments directed along the tool axis. The resulting sensitivity to the formation is in a direction perpendicular to the well-bore axis. When the beds are dipping, the response is more complicated. For over thirty years a different design for an induction array has appeared in the literature. The most recent design of the induction tool has three orthogonal transmitter coils, all located at the same position on the tool axis, and three orthogonal receiver coils similarly arranged. This device is referred to as a triaxial induction tool. Each transmitter couples to each of the orthogonal receivers, hence in an arbitrary formation this tool produces nine independent measurement data. Further in order to obtain a different depth of investigation from the well-bore a multiple triaxial receiver arrays are used (the so-called multi-array induction tool). One of the important applications of the data from the multi-array triaxial induction tool is to form an image around the well-bore. To carry out this quantitative inversion (imaging) of this data we propose to use the so-called Multiplicative Regularized Contrast Source Inversion algorithm. In this algorithm the resistivity contrast and the contrast sources (the product of the resistivity contrast and the electric field) are updated by an iterative minimization of an appropriate cost functional. The non-regularized cost functional consists of two terms. The first term represents the misfit in the data (data equation) and the second term represents the error of the contrast and the total field in satisfying the integral equation over the computational domain (object equation). In each iteration step, we first update the contrast sources using the Conjugate Gradient (CG) direction so as to minimize the cost functional and then we update the contrast by minimizing again the cost functional using the updated contrast sources. Since all the updating parameters in the inversion procedure are available in closed form, we do not carry out any inversion of an ill-posed operator. The latter significantly increases the robustness of the algorithm. To enhance the ability of the algorithm to handle noisy data an extra regularization term is included in the algorithm. As known in the literature the drawback of adding an extra regularization term to the cost functional is the presence of an artificial weighting parameter in the cost functional, which can only be determined through considerable numerical experimentation and a priori information about the desired unknown. Therefore, we take the regularization as a multiplicative constraint and as a consequence the weighting parameter is now completely prescribed by the error norm in the data and the object equation. Since the weighting parameter is related to the error norm of the data equation, this multiplicative procedure suppress automatically the effect of noise in the reconstruction results. Finally we remark that since in each iteration we do not solve any full forward problem, the computational complexity of the algorithm is equivalent to the complexity of solving two forward problems using the CG method. In the presentation some simulated inversion results will be shown to demonstrate the power of the multi-array triaxial induction data and its imaging algorithm.
DE: 0600 ELECTROMAGNETICS
DE: 0629 Inverse scattering
DE: 3260 Inverse theory
SC: Near-Surface Geophysics [NS]
MN: 2005 Joint Assembly