Summary
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 multiphase 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. 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) transport. Hysteretic behavior of relative permeability functions is included. The radial conductivity profiles calculated from the flow model are converted to induction logs using radial response functions. An algorithm for rapid calculations of the forward logs by combining the electromagnetic and flow models is developed. A nonlinear least squares method is used for parameter inversion from measurements. Additional data of near-wellbore resistivity, or logs obtained during drilling, may be included. 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.1 While filtration beneath the bit may be important at the time of 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 µD].2
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 can make 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.3 The step model consists of two zones of resistivity Rxo and Rt with the zone boundary at some distance ri Charts have been developed based on this model for various shoulder and mud resistivities to help the analyst deduce Rt
For economic viability, in addition to knowing the reserves, it is important to know the recoverable amount. Here invasion has been regarded as representative of a waterflood. Thus, Rxo is a direct measure of the residual oil saturation Sor and tools to measure shallow resistivity have been built. Another unanticipated benefit of invasion has been discussed by Campbell and Martin 4 where a resistivity annulus is used as a pay zone indicator. The depth of invasion has also been believed to be related to permeability, although given the ultralow mudcake permeability, the correlation is probably weak.
The motivation for the present work is provided by Ramakrishnan and Wilkinson,5 who developed the notion of interpreting conductivity profiles around the borehole by using fluid-flow physics. Based on these profiles, a rigorous and useful inversion result was proved. It was shown that with an ideal logging tool that could measure radial conductivity variation, the fractional flow curve could be exactly inverted provided the assumptions of the invasion model are met. This was true with just a single snapshot of the profile. The filtrate loss volume at every depth is also determined. A resistivity contrast between the mud filtrate and the connate water is required. Thus, for the first time in the history of logging, the possibility of obtaining multiphase flow properties was demonstrated.
Although there is no ideal logging tool that measures conductivity profiles, tools that have multiple depths of investigation are becoming available. With the array induction imaging (AIT**) 28 raw measurements (not all independent), or more appropriately, five resolution matched channels are available. These may be combined with a shallow log and one which measures resistivity such as a log while drilling, e.g., MicroSFL** and compensated dual resistivity (CDR**). Then seven channels are obtained. The main purpose of this paper is to utilize such measurements that have different depths of investigation and demonstrate the practical utility of the inversion theorem 5,6 for obtaining fractional flow. From this, one is also able to obtain the initial water cut upon production, at any depth of interest. Rather than simply obtaining a resistivity profile based on one or two steps,7 the present work computes profiles that are constrained by the laws of fluid transport. Since the inverted flow parameters have restricted physical ranges, quality checks may be imposed. All of the familiar logs, such as Rt and Rxo can also be computed with little extra effort. Here we note that the work of Semmelbeck et al.8 done in parallel with ours, is an attempt to estimate single phase permeability (for low permeability gas sands) from array logs, quite different from the aim of this paper. Finally, it is important to point out that the principles behind the work presented here are applicable to any set of array logs that have multiple depths of investigation and are not restricted to the logging tools discussed in this paper.
Analytical Model of Waterflood Sweep Efficiency in Vertical Heterogeneous Reservoirs under Constant Pressure