Water Cut and Fractional Flow Logs from Array Induction Measurements
Authors: Ramakrishnan T.S. , Wilkinson D.J. 
Published: December 2007

Abstract
Despite the importance of relative permeabilities in reservoir simulation, no information regarding them is available from current logs. In this paper, for the first time, we demonstrate a continuous log of multi-phase flow properties.
Mud filtrate invasion is usually regarded as a process that corrupts the true logs. In reality, the multiphase flow characteristics that influence filtrate flow also determine the subsequent reservoir performance. We propose the notion that invasion is an experiment, albeit uncontrolled, that may be used to invert for multiphase flow properties. The inversion is based on a theorem according to which, a single snapshot of radial conductivity variation is sufficient to reconstruct exactly the two-phase flow behavior of the formation. Thus, in principle, inversion of array induction measurements in terms of the fractional flow curve is possible.
The forward model for filtrate invasion is based on two-phase (aqueous and oleic), three-component (oil, water and salt) radial transport. Hysteretic behavior of oil and water relative permeability functions is included. Porosity, fluid properties, and cementation/saturation exponents are assumed known from other logs or independent data. The radial conductivity profiles calculated from the flow model are converted to induction logs using radial response functions. An algorithm that permits rapid calculations of the forward logs by combining the electromagnetic and flow models is developed. A weighted nonlinear least squares method with simple bounds, along with an initialization routine to avoid false minima, is used for parameter inversion from the measured logs. Additional data of near-wellbore resistivity, or logs obtained during drilling, may be included.
Based on the inverted parameters, presentations for several output logs have been developed: a reserves estimate that partitions porosity into residual and movable saturations, initial water cut in the production stream, the fractional flow curve as a function of saturation, filtrate loss per unit depth, and a quality indicator. A field example of the processing, and its comparison with production data is also discussed.

Introduction
Drilling mud is usually weighted to maintain the wellbore hydrostatic pressure above that of the formation. This prevents the well from blowing out, but leads to invasion of borehole fluids into the formation, during which a mudcake is deposited on the borehole surface. The invasion process may consist of beneath-the-bit loss, dynamic filtration during mud circulation and finally static mud loss. While filtration beneath the bit may be important while drilling, at the time of wireline logging most of the invasion is due to radial loss from the borehole wall. Except in tight formations, this loss is largely controlled by the mudcake, Owing to its low permeability of about 1 nm2 [1 D].
One of the main objectives of logging is to determine the native formation resistivity in order to estimate oil reserves accurately. But the presence of an invaded region around the borehole distorts the electromagnetic logs and makes interpretation difficult. For understanding logs in the presence of invasion, a model based on a step resistivity change has been widely used, beginning with the work of Dumanoir et al.
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