Are the Oil Recoveries Distributed Lognormally? If Not, There May Be Information in the Differences

Wattley, George Stephen (Professional Petroleum Svcs)

OnePetro 

Abstract
The recoveries from the wells in an oilfield do not usually form a perfectly lognormal distribution. So the log probability plots of the recoveries usually do not exactly conform to the straight line that signifies lognormally distributed data. Certain types of deviation from the straight line can provide useful information about the oilwell recoveries.

A set of recoveries may differ from the lognormal

  • because the frequency distribution is asymmetric,
  • because the frequency distribution has a peak higher or lower than the ideal lognormal distribution,
  • because some of the recoveries are missing, or
  • because significantly different collections of data have been mixed.

and all these cases have characteristic effects on the shape of the log probability plot. Some illustrative field cases are presented.

A quick review of the quality of a set of oil recovery figures can be made by preparing a log probability plot. Certain characteristic shapes of the plot may indicate that the data is incomplete or that it is not homogenous and should be disaggregated for further analysis.

Introduction
A variable is said to be lognormally distributed if the logarithms of the variable are normally distributed1,2,3. Many reservoir properties have been found to be approximately lognormally distributed, for example formation thickness, core permeabilities, oil recoveries and field sizes1. This paper explores the significance of certain types of deviations from the lognormal.

A log probability plot is similar to a normal probability plot except that a logarithmic scale is used to plot values of the variable. The cumulative frequency distribution of a lognormal distribution forms a straight line when displayed on a log probability plot. But for real oilfields, the oil recoveries usually are not exactly lognormal, and deviate from the straight line.

A plot may deviate from the straight line simply because the data is not exactly lognormal: the distribution may be skewed, with one tail longer than the other, or the peak of the distribution may be higher or lower than the lognormal. But the data may deviate from the straight line because it is incomplete. For example, the field may have had several owners, not all of whom were equally careful about recording oil recoveries. Also, the data may deviate from the straight line because it is not homogenous. For example, if the oilfield includes different reservoirs with very different producing characteristics, then it may be appropriate to separate the oil recovery data into sub groups before making cumulative distribution plots.

This paper presents sets of data which show how some of these types of deviation can be identified from the log probability plot.

Ideal Normal Distributions
Figure 1 shows the frequency distribution of three normal distributions with different means but the same standard deviation. These are examples of the famous bell curve or Gaussian distribution. Since the distributions are symmetric, for each distribution the peak represents the mean. It can be seen that distributions with different means are displaced laterally on the plot. Figure 2 shows the frequency distribution of three normal distributions with different standard deviations but the same mean. It can be seen that distributions with different standard deviations have different widths.