How Much Core Sample Variance Should a Well-Log Model Reproduce?
Authors: Jennings James W. 
Published: December 2007

Abstract
A new log calibration technique presented in this paper determines how much core sample variance should be reproduced by a calibrated log. The linear calibration parameters are then calculated from this variance and the core-sample mean. The method relies on the idea that the correlation structure of core data and a calibrated log should be the same for distances longer than the log resolution, after accounting for the variance reduction due to log averaging. The procedure is illustrated using synthetic data, followed by example applications using real data.

Introduction
The quantification of subsurface rock properties is essential for modeling petroleum reservoir flow processes. Because most reservoirs have a limited amount of core, well logs are a basic source of petrophysical data. The accuracy of well logs is usually improved by calibration to cores in the same reservoir, but traditional calibration methods ignore inherent differences in measurement volume between the two data types.
The calibrations are typically linear correlations with two free parameters, giving independent control of the predicted mean and variance of log measurements. Clearly, the calibrated logs should reproduce the core-sample mean, if the core sampling is unbiased. However, calibrated logs should usually produce smaller variances than core data, because the logging tools measure averages over larger volumes, and most reservoirs have significant variance below the log resolution. The traditional calibration methods all reproduce the core-sample mean, but they represent different choices for predicted variance. Choosing an appropriate calibration amounts to determining an appropriate variance for the prediction.
A natural approach to this problem is to calibrate the log after averaging the core data up to the log measurement scale. This has been done successfully using closely spaced mini-permeametry and "strip samples" analysis. However, core samples are usually not spaced closely enough for the averaging to eliminate much of the smoothing-effect scatter; there are not enough samples within the log measurement volume to compute an accurate average. Williams and Sharma outline a procedure that uses high-frequency information from an FMS log to estimate the required sample density for a given accuracy in averaging.
Greder, Marion, and Pellerin describe a method involving the construction of a "core log," an estimate of a core-sample scale property at close spacing, constructed with kriging conditioned to the primary core data and some additional secondary data. The closely spaced estimate can then be smoothed, giving an estimated core property at the log scale for use in calibration. To be helpful the secondary data must be closely spaced, for example, mini-permeametry or acoustic measurements on the whole core.
In this paper a new technique is presented to calibrate logs to core data when only the usual core data are available. No additional finely spaced measurements are required.
A Mathematical Model for Log Calibration Well-log models are often calibrated by fitting a line to a cross plot of the log measurement and the corresponding core data. An example using synthetic data is shown in Figure 1. An appropriate mathematical model for this problem is
(1)
(2)
(3)
where a and b are constants determined by the calibration, × and y are the (possibly transformed) log and core data, and dx and dy are deviations of those data from the "true," but unknown rock properties, xt and yt. The calibration is used to estimate yt in wells with no core as follows:
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