149december2008

case where, say, two forces of 3 and 4 units are applied at right angles to each other, and represents a practical illustration of Pythagoras’s theorem.

But the examples do not end there. In alternating current applications, the individual voltage drops meas-ured across the diff erent components along an electrical circuit invariably add up arithmetically to more than the overall drop. Th is is because the voltages are vectors, which are typically out of phase with the current within each component.

Whereas these relationships are well known to many working in mathematics, there are some other terms, used extensively in the public domain, where few query the context and it is merely assumed that aggrega-tion is by simple addition.

How much oil is really there?

Making correct statistics bring reality to global

planning

Write 1 + 1 on a blackboard in the classroom, and most schoolchildren will shout out 2 automatically as the an-swer. But there may be one bright spark who says, “Not always! It could be 10.” Similarly, 3 + 4 will attract the overwhelming answer 7, but someone—probably the same individual—may say it could be 5 or, indeed, just about anything.

What lies behind this ambiguity is that the aggre-gation of any group of entities depends on the relevant mathematical rules that are to be applied. We are all accustomed to simple addition and often adopt this without more scrutiny of the context.

Th e answer 10 above follows immediately it be-comes clear that the addition is using binary notation. Conversely, 5 refl ects vector addition for the special

The oil and gas industry annually reports global proven reserves, which investment, energy and environmental analysts incorporate into future strategies. But a major statistical error has been made, reports Richard Pike. True “proven reserves” may be twice the commonly accepted figure. It is essential that this is addressed to avert climate catastrophe by the middle of the century and to compel us to a sustainable low carbon economy.

The way that oil reserves are totalled is extraordinary—and leads to huge underestimates.

150 december2008

One extraordinary example is the way that “proven reserves” of oil and gas deposits are totalled throughout the world. Even within the energy business, few are aware—and this includes chief executives—that the term has a probabilistic basis, and that individual num-bers cannot be added up simply as if in a list. Unlike many other cases, the accurate aggrega-tion leads to a global fi gure that far exceeds the notionally accepted value. Th is has profound implications for energy planning and long-term environmental management, particularly in the drive to reduce emissions of carbon di-oxide.

Problems of aggregation

According to defi nitions used in the industry, the reserves of an individual reservoir have to satisfy four criteria; that is, they have to be quantities that are discovered, remaining, recoverable and commercially available. Th e volume within this quantity that has a 90%

probability of being exceeded is defi ned as proven reserves or P90. When aggregating the data from individual reservoirs to the fi eld level, many companies merely add their P90s arithmetically, although some apply a probabi-listic approach.

Consolidating to the company level is al-most always arithmetical, and this is considered acceptable by the fi nancial markets because its principal purpose is to provide a conservative estimate for reserves, that gives investors a high degree of confi dence of being recovered. Th is form of aggregation continues to the country and world level. Th is is equivalent to assum-ing that the outcome, simultaneously, of every fi eld in the world will be its “downside”, with no scope for “upside” beyond the P90 threshold. Th e box shows the elementary mathematics of this error.

With the compartmentalisation of the oil and gas industry, most investment, energy and environmental analysts dealing at the strategic, global level seem unaware that the public report-

ing of proven reserves alone—and the way they are added up—is purely a historic convention. It bears little relevance to what will be actually produced. Such analysts are constrained, how-ever, by proven reserves being the only indicator widely available in the public domain.

In practice, companies developing their fi elds place far more emphasis on the proven plus probable reserves—P50—as this largely determines what hydrocarbons are to be pro-duced, the number of wells to be drilled and the extent of the surface facilities that will be constructed.

P50 is that quantity within the reserves category that has a 50% probability of being exceeded and is regarded as an indicator of the most likely outcome, given the shape of typical probability distributions in the industry. Th e expected future production profi les of those fi elds already on stream are usually fi led with the government of the country in question but are seldom readily available to the public.

Definitions and conventions

It is the joint defi nitions of the World Pe-troleum Council and Society of Petroleum Engineers that categorise certain oil deposits as reserves according to the criteria referred to above.

In practice, the extent and properties of each reservoir (the “building blocks” of an oil fi eld) are determined initially through the dis-covery well and subsequent appraisal wells, in addition to seismic results, which yield a range of values for porosity, permeability, volume, fl uid saturations and other parameters. Th ese enable the stock tank oil originally in place (STOOIP) to be estimated, as well as the ultimate recovery (less than the STOOIP), based on the proposed development plan and representative historic oil price.

Terms like STOOIP refl ect the practical approach taken by the industry in wanting to understand how hydrocarbons underground would translate, if wholly recovered, into bar-rels of stabilised (largely degasifi ed) oil residing as stock in storage tanks, ready for export by pipeline, or by road or sea tanker. Th e reserves are always less than the STOOIP because of physical and chemical limitations in recovering oil from the pores within the geological struc-tures within the reservoir.

Once the fi eld is in production, these data are reassessed as more detailed operational information becomes available on reservoir performance. Importantly, there is not a sin-gle fi gure for the STOOIP or reserves for the fi eld; instead, there is a probability dis-

Lessons from two dice

Intuitively, we would expect that arithmetical adding is incorrect. Our everyday experience of lower bounds is that they do not all occur simultaneously. Some events yield an upside while others give a downside. Figure 1 illustrates the lessons to be learned from playing with two dice. Throw a single die, and the probability of the outcome exceeding 1 is fi ve out of six, or 83%. Using the above terminology, the P83 fi gure is 1. Throw two dice together, and the P83 fi gure is 4, or twice the simple arithmetic aggregation of the two separate distributions. This is because 30 out of 36 permutations give a sum greater than 4. This is really a case of 1 + 1 = 4, instead of 2.

In each independent case,5 out of 6 outcomes (83%) have

mean40y %

P83mean40y

%

P83

5 out of 6 outcomes (83%) have a value greater than 1

1 2 3 4 5 60

20

Pro

babi

lity P83

1 2 3 4 5 60

20

Pro

babi

lity P83

+1 2 3 4 5 6

Value1 2 3 4 5 6

Value

20When throwing two dice there

are 36 different permutations ofmean20

% P83

are 36 different permutations, ofwhich 35 out of 36 (97%) have

a value greater than 2, and 30 out of 36 (83%) greater than4. The 6 out of 36 permutations

ith l 4 l 1 1

10

roba

bilit

y

P97

with value 4 or less are 1+1,1+2, 1+3, 2+1, 2+2 and 3+1.

In this example adding the

2 3 4 5 60

P

7 8 9 10 11 12

In this example, adding theP83 values of the separatedistributions (1+1=2) under-

estimates the P83 of the 2 3 4 5 6Value

7 8 9 10 11 12combined distribution (4)

Figure 1. Lessons from two dice

151december2008

tribution, which refl ects the variability in the actual measurement of some parameters and the theoretical derivation of others, from well to well and throughout the reservoirs, which govern the calculations for these two key num-bers. Th is distribution looks very much like the normal distribution or “bell-jar” curve familiar to students of statistics, although it is typically somewhat skewed.

Conventionally, as indicated above, the re-serves fi gure that has a 90% probability of being exceeded (P90) is designated proven reserves. Th is lies to the far left of the bell-jar shown in Figure 2. Th e most likely reserves fi gure has a 50% probability of being exceeded (P50) and typically lies near the mid-point of the distri-bution. Th e reserves fi gure that has only a 10% probability of being exceeded (P10) lies to the far right of the bell-jar. By way of example, the P50 for a reservoir prior to production may be twice the P90 fi gure, and the P10 around three times the P90, although there are marked vari-ations across the industry.

Although the terms proven, probable and possible were once applied somewhat loosely to reserves, there is now general agreement that they should have a probabilistic basis. In addition to “proven” defi ned as above, “prob-able” represents the diff erence between P50 and P90, while “possible” is P10 minus P50, as shown in Figure 2.

Mathematical interpretation

What does this all mean? Take two identical normal distributions each with mean μ and

standard deviation σ, and add them together. Provided that the two contributing distribu-tions are independent of each other, the mean of the combined distribution is 2μ and the standard deviation becomes not 2σ but σ√2, or, roughly, 1.41σ, as shown in Figure 3. In general, for diff erent normal distributions the means are added arithmetically, whereas the square of the new standard deviation is the sum of the squares of the contributing stand-ard deviations.

As more distributions are added in this way, the mean increases cumulatively but the bell-jar curve becomes squeezed relatively be-cause the standard deviation grows more slow-

ly as a square root rather than linear function. P90 always lies at 1.282 standard deviations to the left of the mean in such a distribution.

Usually, an oil company aggregates the distributions for the reservoirs to determine the distribution for the fi eld, and then aggre-gates these to calculate the overall distribution for its total hydrocarbon assets. Other than for simple curves such as a normal distribu-tion, probabilistic addition requires Monte Carlo computational techniques, and many companies or state corporations for simplic-ity, therefore, merely add arithmetically the columns for P90, P50 and P10 in combining distributions.

As we have seen above, this signifi cantly understates the true P90 of the assets; it also overstates the true P10. Th e reported P50 will usually have a relatively small error, compared with the true P50 because of the fair degree of symmetry of typical distributions found in practice. Whatever technique is used to deter-mine the distribution for the fi eld, government statisticians conventionally report the coun-try’s proven reserves by arithmetically adding the P90 fi gure for each fi eld, and the reported world fi gure follows, again, by simply adding. What is actually being derived is the P99.99… at the global scale, a lower bound that has an infi nitesimally small probability of being breached. It has a correspondingly immense probability of being exceeded—and by a very large amount.

Numerical implications

Th e quantitative implications of the confusion over defi nitions and aggregation of probabili-

Figure 3. Probability applied to the addition of STOOIP or reserves

P90 P90 P90P90 P90

+

P90

1.282 1.282

1.41

1.2821 813

μ μ2μ1.813 2μ

If two identical independent normal distributions (mean μ standard deviation ) areIf two identical, independent normal distributions (mean μ, standard deviation ) are added together, the resulting mean M is 2μ and standard deviation is 2, or 1.41 .

Adding the two P90 figures (2μ - 2.564 ) underestimates the probabilisticsum (2μ - 1.813 ). P90 always lies 1.282 standard deviations from the mean, so that

the new P90 must lie at 1.282 x 1.41 , or 1.813 away. Each probabilistic addition makes P90 a larger proportion of the new mean.

Figure 2. Comparability of reserves terms

P90 P10

ctio

nen

sity

func P50

obab

ility

de

Value

Pro

Proven P90

P90< Probable P50

P50< Possible P10

152 december2008

ties are signifi cant. Peter Odell1 summarises the views of others that the initial oil resources of the planet, before any of them were exploit-ed, were approximately 3000 billion barrels (bbl), an estimate that has remained largely unchanged for the last 10–15 years.

In 1995, in details on oil given to support this fi gure, this 3000 billion bbl was subdivid-ed. Around 788 billion bbl was estimated to have been produced up to that time, a further 1095 billion bbl were deemed proven reserves, 750 billion bbl were yet to be discovered, and the rest, an implied 367 billion bbl, was attrib-uted to current and new technologies applied to known fi elds. Th is excludes unconventional oil resources such as tar sands.

Th e true P90 of the reserves category will have been signifi cantly more than the 1095 billion bbl indicated (and the 1189 bil-lion bbl reported elsewhere for end-2004) for the reasons we have seen above; but this, and the crucial, higher, P50, cannot be quantifi ed because of the limited data in the public do-main. Similar observations can be made for the reporting of gas resources. Th ere is the prospect, therefore, that, far from running out of oil reserves, hydrocarbons will dominate future energy even beyond Odell’s projections. Th is also raises the question of the relevance, to energy and environmental management, of the ratio of proven reserves to current world production.

Proven reserves as presently calculated are enough to supply about 40 years use of oil. True proven reserves, though, and total likely recoverable resources, are very much greater. An entire generation is being misled into thinking that oil is rapidly running out.

Reporting requirements and other indicators

Th e overall proven reserves fi gure reported by a company is a necessary requirement by the Securities and Exchange Commission for quo-tation on the New York Stock Exchange. Sim-plifi cations that underestimate this (such as by arithmetic addition of individual reservoir and fi eld data) are considered acceptable because in being conservative they are generally in the interests of investors. Companies not quoted on the New York Stock Exchange, and some national oil corporations, also tend to report proven reserves in their annual reports and promotional material, but nowhere is there an obligation to divulge publicly P50 or P10 fi gures. Th ese are kept confi dential within the company or corporation, though such data are usually audited for each fi eld by third-party

consultants to ensure a consistent approach. Aggregation beyond the fi eld level, however, is invariably arithmetic rather than probabilistic.

In addition to the strict defi nition of re-serves, two other categories cover remaining hydrocarbon resources. Contingent resources are discovered and remaining hydrocarbons that have yet to meet one or both of two remain-ing criteria for characterising reserves: they are either not recoverable or not commercial at the representative oil price. Th eir develop-ment is contingent on these constraints being addressed. Lastly, undiscovered resources are those estimated to exist solely on the basis of geological or seismic inference.

Each of these categories is divided into low, mean and high subcategories, so that, with reserves included (proven, probable and pos-sible), there are eff ectively nine discrete levels of certainty into which hydrocarbon resources can be placed. Only one of these, the proven reserves fi gure, is reported publicly.

Practicalities

Th e contrast between procedural require-ments and operational practicality can be best illustrated in the following way. On the day the P90 for a fi eld is determined for statutory re-porting, the operator may well have started up production facilities that are invariably based on the higher P50 case, as this is the most likely scenario. It governs the number of wells that are drilled, the process plant capacity, the life of the fi eld and its economics. It also, crucially, determines the actual carbon dioxide emissions that will result.

Remedies

Rather than there being general agreement on the quantitative limits of oil resources, which would encourage all parties to address rem-edies and to seek out alternative energy routes, the relative abundance of oil recognised by individual producers themselves may inhibit this search.

Despite transparency and innovative ap-proaches by some of the leading oil companies, the overall global response has been inconsist-ent and unco-ordinated. In practice, those countries dependent on petroleum for national revenues will be under domestic pressure to continue to develop their signifi cant assets.

Th ere is also anecdotal evidence that some countries are under-reporting proven reserves to maintain a high oil price. Alto-gether, as a result, the world is understating the environmental challenge facing generations to

come and appears unprepared for the diffi cult compromises that will have to be made.

For more coherent global energy and environmental planning, it will be essential to use estimates that refl ect proven plus probable reserves, and to address the issues of openness and confi dentiality that this raises.

We are faced with numerous decisions in the near future, which lie on a spectrum of increasing diffi culty. Th ey are likely to in-clude selective curtailment of coal, oil and gas production, sequestration of carbon dioxide, expansion of alternative energy sources (in-cluding biofuels, wind, geothermal, tidal and solar, as well as hydrogen derived through vari-ous routes), revitalising the nuclear industry, chemical removal of carbon dioxide from the atmosphere (including artifi cial photosynthe-sis) and reforestation.

It will also require economic, technical and infrastructural support to emerging coun-tries that would otherwise rely on their own hydrocarbon industries. In what seems like an echo from the nuclear industry for any carbon dioxide “stored”, physically or chemically, it will be crucial to minimise the risk of release into the atmosphere in the centuries to come.

Solutions will need to cross technical, social and geopolitical boundaries, and rarely can there have been a greater need for a broad understanding of such complex issues, their possible solutions and consequences. Informed and fact-based views will be essential for get-ting the right balance.

Education and training

Th e public and industry confusion over just how much oil and gas there is suggests the need for more focus on the quantifi cation of environmental issues. Th is must start with set-ting much higher expectations on what edu-cation and training deliver in terms of science and mathematics skills, and the ability to chal-lenge eff ectively the status quo—and this status quo includes the traditional conventions found in the oil and gas industry. We could then all be assured that the solutions being proposed were the right ones and that they could be im-plemented successfully.

Reference1. Odell, P. (2004) Why Carbon Fuels will

Dominate the 21st Century’s Global Energy Economy. Brentwood: Multi-Science.

Richard Pike is Chief Executive of the Royal Society of Chemistry.