Put a volumetrics deck and an economics deck side by side and there is a fair chance that "P90" means the pessimistic case in one and the optimistic case in the other. Both are used correctly by their own convention. The trouble starts when a number crosses from one to the other without its convention, which is exactly what happens when a volume range is handed to an economist.
Two conventions, one label
The exceedance convention is the one in reserves and resources reporting. Under the Petroleum Resources Management System, when probabilistic methods are used there should be at least a 90 percent probability that the quantity recovered will equal or exceed the proved estimate, at least 50 percent for proved plus probable, and at least 10 percent for proved plus probable plus possible. So P90 is the low case, P10 the high case, and the numbers run P90 ≤ P50 ≤ P10.
The cumulative convention is the one in statistics and in most software. Pn is the n-th percentile: the value with an n percent probability of being less than or equal to it. So P10 is the low case, P90 the high case, and the numbers run P10 ≤ P50 ≤ P90. A spreadsheet percentile function, a Monte Carlo add-in and most NPV distributions follow this convention.
The two conventions describe the same distribution. The exceedance P90 and the cumulative P10 are the same number. Nothing is wrong with either; what goes wrong is an unlabelled "P90".
A worked example
Take an oil-in-place estimate with a low case of 50 MMbbl and a high case of 200 MMbbl, the low case being the one with a 90 percent chance of being exceeded. Assume, as is common for volumes, that the distribution is lognormal.
A lognormal is symmetric in the logarithm, so the median is the geometric mean of the two ends:
P50 = √(50 × 200) = 100 MMbbl
The spread of the logarithm follows from the ends, which sit 1.2816 standard deviations either side of the median (the normal 90th percentile):
σ = ln(200 / 50) / (2 × 1.2816) = 1.3863 / 2.5631 = 0.5409
The P50 is not the expected value
For a lognormal, the mean is larger than the median:
mean = P50 × exp(σ² / 2) = 100 × exp(0.1463) = 115.8 MMbbl
The mean is the number that adds up across a portfolio and the one an expected-value calculation needs. Using the P50 in its place understates the expected volume by about 14 percent in this case (100 against 115.8), and the more skewed the distribution, the larger the gap. Swanson's rule, 0.3 × low + 0.4 × P50 + 0.3 × high, is a quick approximation for the mean of a moderately skewed distribution: 0.3 × 50 + 0.4 × 100 + 0.3 × 200 = 115, close to the exact 115.8.
Percentiles of a product are not products of percentiles
Now take a recovery factor that is also lognormal, independent of the volume, with a low case of 0.20 and a high case of 0.40. Its median is √(0.20 × 0.40) = 0.283 and its log spread is ln(2) / 2.5631 = 0.2704.
The tempting shortcut is to multiply the ends:
low EUR = 50 × 0.20 = 10 MMbbl
high EUR = 200 × 0.40 = 80 MMbbl
Both are wrong as percentiles. For independent lognormal inputs, the logarithm of the product is the sum of the logarithms, and the variances add:
σ_EUR = √(0.5409² + 0.2704²) = √(0.2925 + 0.0731) = √0.3657 = 0.6047
P50 = 100 × 0.2828 = 28.3 MMbbl
low = 28.3 × exp(−1.2816 × 0.6047) = 28.3 × 0.4607 = 13.0 MMbbl
high = 28.3 × exp(+1.2816 × 0.6047) = 28.3 × 2.1705 = 61.4 MMbbl
The correct low-to-high ratio is about 4.7, not 8. The shortcut's 10 MMbbl is not a 90 percent exceedance case at all; on this distribution only about 4 percent of outcomes fall below it. A Monte Carlo of 400,000 draws gives 13.0, 28.3 and 61.4 MMbbl, which agrees with the closed form to within sampling noise.
The reason is simple once seen: for the product to be at its low case, both inputs would have to be at their own low cases together, and with independent inputs that is much rarer than one chance in ten.
Sums behave the same way
Add two independent fields, each with the 50 / 100 / 200 MMbbl distribution above. Summing the cases gives 100, 200 and 400 MMbbl. A Monte Carlo of the sum gives about 130, 214 and 355. The sum of the low cases understates the low case of the total, and the sum of the high cases overstates its high case, because a portfolio of independent fields is less likely to be all-bad or all-good than any one field. Only the means add exactly: 115.8 + 115.8 = 231.5, and the simulated mean of the sum is 231.5.
This is why reserves reporting distinguishes arithmetic from probabilistic aggregation, and why a portfolio range built by adding field ranges is wider than it should be.
Economics adds a second twist
An NPV distribution is usually reported in the cumulative convention, so "P10 NPV" is the downside. But NPV is not a volume: it can be negative, it is often left-skewed when cost overruns and delays dominate the downside, and it is not a simple product of its inputs. Two habits keep it straight.
- Name the percentile in words. "10th percentile NPV" or "downside NPV" cannot be inverted by a reader who grew up with reserves; "P90 NPV" can.
- Run the economics on the distribution, not on three cases. Pushing the volumetric low, mid and high cases through a cash-flow model gives three NPVs, but the NPV percentiles are not those three numbers, for the same product-of-percentiles reason. Sampling the volume, recovery, price and cost distributions together and reading the percentiles off the NPV result is the consistent way.
A short checklist
- State the convention once, at the top of every table and chart, and label the cases low, best and high as well as with Pn.
- Never multiply or add percentiles. Simulate the product or the sum.
- Use the mean for expected value and for anything that is added up across projects.
- Check that the range is not too narrow. People asked for a range they are 90 percent sure of routinely give one that contains the true value far less often than that (Capen, 1976); wide ends are usually the honest ones.
- When a range is handed from one discipline to another, hand over the convention with it.
The Monte Carlo economics module of the Decision Economics track works through NPV distributions, the mean and the percentiles in detail, the STOIIP-to-cash-flow module carries the product rule through a full cash-flow model, and the probabilistic inputs module of Intermediate Petrophysics covers building the ranges in the first place.
In the platform
Volumetrics reports in-place and recoverable volumes in the reserves convention, P90 being the low case, from a Monte Carlo of the full input distributions. The Investment module labels NPV percentiles in words, 10th percentile being the downside, so the two cannot be confused on the same screen. The course states its own convention at the top of every economics module.
References
- Society of Petroleum Engineers et al. (2018). Petroleum Resources Management System, revised June 2018.
- Capen, E. C. (1976). The difficulty of assessing uncertainty. Journal of Petroleum Technology, 28(8), 843–850.
- Hurst, A., Brown, G. C., and Swanson, R. I. (2000). Swanson's 30-40-30 rule. AAPG Bulletin, 84(12).