Petrophysical inversion of borehole array-induction logs: Part I — Numerical examples Faruk O. Alpak 1 , Carlos Torres-Verdín 2 , and Tarek M. Habashy 3 ABSTRACT We have developed a new methodology for the quantitative petrophysical evaluation of borehole array-induction measure- ments. The methodology is based on the time evolution of the spatial distributions of fluid saturation and salt concentration at- tributed to mud-filtrate invasion. We use a rigorous formulation to account for the physics of fluid displacement in porous media resulting from water-base mud filtrate invading hydrocarbon- bearing rock formations. Borehole array-induction measure- ments are simulated in a coupled mode with the physics of fluid flow.We use inversion to estimate parametric 1D distributions of permeability and porosity that honor the measured array-induc- tion logs. As a byproduct, the inversion yields 2D axial-sym- metric spatial distributions of aqueous phase saturation, salt concentration, and electrical resistivity. We conduct numerical inversion experiments using noisy synthetic wireline logs. The inversion requires a priori knowledge of several mud, petrophys- ical, and fluid parameters. We perform a systematic study of the accuracy and reliability of the estimated values of porosity and permeability when knowledge of such parameters is uncertain. For the numerical cases considered in this paper, inversion re- sults indicate that borehole electromagnetic-induction logs with multiple radial lengths of investigation array-induction logs en- able the accurate and reliable estimation of layer-by-layer abso- lute permeability and porosity. The accuracy of the estimated values of porosity and permeability is higher than 95% in the presence of 5% measurement noise and 10% uncertainty in rock- fluid and mud parameters. However, for cases of deep invasion beyond the radial length of investigation of array-induction log- ging tools, the estimation of permeability becomes unreliable. We emphasize the importance of a sensitivity study prior to in- version to rule out potential biases in estimating permeability re- sulting from uncertain knowledge about rock-fluid and mud properties. INTRODUCTION Robust and accurate determination of fluid-flow related petro- physical parameters from borehole measurements is a fundamental objective of quantitative geophysical exploration. Geoelectrical measurements are sensitive to the spatial distributions of porosity, fluid saturation, and salt concentration. Therefore, it is reasonable to hypothesize that incorporating the physics of fluid flow in porous media into the analysis of geoelectrical borehole measurements will significantly improve current interpretation algorithms based solely on the estimation of electrical resistivity. The phenomena of multiphase fluid flow and electromagnetic in- duction in porous media can be linked readily by means of an appro- priate saturation equation when a priori information is available about the properties of the flowing fluids i.e., viscosity, density, compressibility. Two-phase or, occasionally, three-phase multi- component fluid displacement, which takes place during mud-fil- trate invasion, provides a basis for the quantitative petrophysical in- terpretation of electrical conductivity around the wellbore. Tobola and Holditch 1991, and Yao and Holditch 1996 successfully used a history matching method based on time-lapse array-induction logs to estimate absolute permeability for the case of water-base mud fil- trate invading low-permeability gas formations. Semmelbeck et al. 1995 attempted to estimate absolute permeability for low-perme- ability gas sands from array-induction measurements. Dussan et al. 1994 advanced a similar procedure to estimate vertical formation permeability using forward modeling and experimental data. Ra- makrishnan and Wilkinson 1997, 1999 developed a method to esti- Manuscript received by the Editor September 17, 2004; revised manuscript received December 6, 2005; published online August 15, 2006. 1 Formerly Department of Petroleum and Geosystems Engineering, University of Texas at Austin, University Station, Mail Stop C0300, Austin, Texas 78712; presently Shell International E & P, 3737 Bellaire Boulevard, P. O. Box 481, Houston,Texas 77001. E-mail: omer.alpak@shell.com. 2 Department of Petroleum and Geosystems Engineering, University of Texas at Austin, University Station, Mail Stop C0300, Austin, Texas 78712. E-mail: cverdin@uts.cc.utexas.edu. 3 Schlumberger-Doll Research, Mathematics and Modeling Department, 36 Old Quarry Road, Ridgefield, Connecticut 06877. E-mail: thebashy@ridge field.oilfield.slb.com. © 2006 Society of Exploration Geophysicists. All rights reserved. GEOPHYSICS, VOL. 71, NO. 4 JULY-AUGUST 2006; P. F101–F119, 20 FIGS., 6 TABLES. 10.1190/1.2213358 F101