It is shown herein that current practice of petroleum reserve estimates can understate expected values. The uncertainty distribution for single-well production is lognormal. Current practice estimates the geometric mean value for reserves - which can be much less than the average values. Current practice also adds single well reserve estimates, at some confidence level to estimate cumulative (corporate) reserves. This again, can result in consistent understated values for aggregated corporate reserves.

If the central limit theorem of statistics is applied (supported by examples of dice products and sums) the implications to oil and gas reserves are evident. A new, logical method to estimate reserves is suggested.


Like many companies Wascana Energy estimates reserves largely based on single-well decline curves and a subjective judgement of cumulative production probability. We profess "proven reserves" are those with an 80 percent or greater probability of production (an 80% confidence factor), interpreted on a pool basis. We then add up these reserves to arrive at a corporate reserve estimate.

There are two issues related to this method that are addressed in this paper - namely:

  1. Our current decline-curve estimation technique for single-well reserves is a method that will give log-average (geometric mean) reserves. For an underlying log-normal uncertainty distribution the geometric mean can be significantly lower than the arithmetic average value. We may be consistently underestimating our single-well reserves.

  2. The second issue relates to how we add reserves.

The only additive property of uncertainty distributions is the average value. By adding up 80 percent confidence levels (or other target confidence levels), we are either understating our reserves or we are underestimating the cumulative confidence factor for our aggregated reserves.

These issues are not specific to Wascana, but are endemic to our industry.

The purpose of this paper is to demonstrate and discuss the above effects that can cause us to understate our reserves and to suggest a blue print for a remedy.


The central limit theorem of statistics is a powerful concept. Basically it says that uncertain things that are added will approach a normal uncertainty distribution, independent of the uncertainty shape of each element of addition. Since multiplication can always be changed to addition by taking logs, this also implies that uncertain things that are multiplied will approach a log-normal distribution, independent of the uncertainty shape of each element of multiplication. More specifically, the theorem and its spin-offs are as follows:

  1. Sums (includes subtractions) of stochastic variables with the same kind of distribution, (no matter what it is) always approach a sum with a normal (Gaussian) uncertainty distribution.

  2. Products (includes divisions) of stochastic variables with the same kind of uncertainty distribution always approach a product with a log-normal uncertainty distribution.

[In practice if. we add or multiply a lot of numbers it doesn't matter what the shape of the individual-variable uncertainties are since we can always lump groups of similar shapes together - each group of which produces a normal or log-normal distribution].

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