Sigma measurements are useful in formation evaluation because they can be used to calculate water saturation independently of resistivity. One of the most advantageous features of the logging-while-drilling (LWD) Sigma tool under study is the use of three detectors to measure time decays from which three Sigmas are calculated: one near thermal-neutron detector, one short-spaced gamma-ray detector, and one long-spaced gamma-ray detector. Each detector has a different volume of investigation. Analogous to array resistivity logs, the multidetector Sigma logs, also called multi-depth of investigation Sigma logs, can indicate the presence of invasion and can be used to estimate true formation Sigma. However, the efficacy of the multidetector Sigma interpretation is conditioned by a high-Sigma contrast between invasion and virgin zones. We used a nonlinear gradient-based fast inversion method that uses the measured three-detector time decays and a reference value for invasion-zone Sigma to estimate the radial length of invasion and virgin formation Sigma. Synthetic logs are generated with realistic noise for numerous realizations that are then used to study the stability of the inversion routine and the estimation of error bars. We used a confidence index derived from the multiple-realization study to define the conditions under which the multiple time-decay problem is not stable. The reliability of the Sigma inversion method was verified in a set of test-pit measurements, two synthetic examples, and a field case. Synthetic results indicate that the inversion of multidetector time decays enables the correction of LWD Sigma measurements for invasion effects and improves the calculation of water saturation whenever there is contrast between invasion- and virgin-zone Sigma of more than 5 capture units. Field example results suggest that the inversion can accurately reproduce three-detector time decays honoring invasion effects evidenced in neutron-density logs while managing the presence of noise.

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