A couple of weeks ago I read somewhere that we have got enough coal left for over 200 years. So despite the greenhouse gas emissions that it entails, we needn’t worry about our immediate energy supply. Coal can be converted to oil through the Fischer-Tropsch process [1], securing even oil for the next decades. Good news I thought, there is more time to develop renewables without jeopardizing an already frail economy.

Yet then the inquisitive merry-go-round in my brain got going: how did they get that number? For over 200 years of coal? Is this a lower bound? An upper bound? What if the coal price rises? What if there are new discoveries? How can they predict how much coal we will use for the next 200 years? In short, they can’t. In long, they still can’t. But hey, let’s give it a try.

Energy companies, governments and since short also bloggers around the world estimate how much fossil fuels are left. The remaining technically extractible volume of a certain fossil fuel, be it oil, coal or natural gas, is called the *remaining resource*. Some resources are *proven* (90% certainty), others just *probable* (50%) or *possible* (10%). The *reserves* constitute that part of the resources that is not only technically but also economically extractible. Hence the proven reserves are those resources available profitably with a 90% probability. Changing economical and technical conditions impact the amount of fuel in each category.

Once you know the size of the reserves (R) and production (P), you are just a division away from the number of years those reserves would last *if production wouldn’t change*. This is called the R/P-ratio. The R/P-ratios at the end of 2009 were:

- Oil: 46 years (depleted in 2055)
- Natural gas: 63 years (depleted in 2072)
- Coal: 119 years (depleted in 2128)

**The effect of growth**

The numbers above are for the proven reserves. Accordingly, today’s proven reserves of oil would be depleted in 2055 if today’s production of oil wouldn’t change. In fact production *does *change. In the last 45 years the annual growth rate of oil production has swayed between 10% and -5.8%. Clearly the growth rate affects how long the fossil fuel reserves last. How large is this effect? Knowledge of the growth rate straightforwardly leads to the depletion date, but I’ll spare you the math.

The average growth rates since 1981 amount to 1%, 2.6% and 2.2% for oil, natural gas and coal respectively. These points are marked on the graph. If these rates would be constant, the proven reserves would be depleted in 2046 (oil and natural gas) and 2068 (coal), much earlier than if there would be no growth.

Luckily (or not), considerable uncertainty surrounds the depletion dates, arising from (1) the uncertainty in growth rate and (2) the uncertainty in reserves. The reserves in the figure displayed above were only the proven reserves. Potential reserves might be up to a factor 4 larger. What happens to the depletion dates if we take this into account? Let’s consider oil.

**Oil depletion: earliest and latest date**

For an estimate of when oil reserves might be depleted, take a look at the figure above. To calculate the earliest possible date, I use the proven reserves (the least amount of oil we expect to have, blue curve), with a high growth rate of 4.7% annually. There is nothing to indicate that actual production might grow this fast, but it is warranted as an upper bound as it is the largest growth on record since 1986. Thus the earliest date we arrive at is 2033.

For the latest date, we need to know how large potential reserves might be. The U.S. Geological Survey estimates the total remaining oil reserves to be at least twice as big as currently proven reserves with a 95% probability, and even up to 3-4 times as big with just a 5% chance (green curve). Consequently, I assume we can maximally extract about 3 times the currently proven reserves, with on average no growth in production. That’s how I determine the upper bound of 2146. The shaded area indicates all possible depletion dates in the range 2033-2146. The green curve is dashed because it is based on less sound numbers.

Except for Fischer-Tropsch, we can be confident the world will stop using oil in 2033-2146. Alas, in return for this confidence we have given up precision. The range 2033-2146 is quite certain and strikingly unsatisfactory. Surely we can do better! If so, it is only by reducing either (1) the uncertainty in growth rate or (2) the uncertainty in reserves.

**Fighting uncertainty: the world energy model**

The growth rate depends on complex socioeconomic factors such as policy-making, fuel prices, demographics, technical advancements and so on. Obviously, no one has figured out how to predict these, right? Forecasting one person’s actions is out of reach, let alone the blend of thousands of people’s fears, expectations and decisions.

Yet this is precisely what the *World Energy Model*, developed by the International Energy Agency, tries to do. It is a mathematical construct, comprising nearly 16000 equations, whose main goal is to replicate how energy markets work. Let’s not take the credibility of this model for granted, it produces projections – not predictions. As for any model, its output is no more trustworthy than its input. In this case one inputs some assumptions about economic growth, fuel prices and technological development. In return one gets quantities like energy demand, production growth and carbon dioxide emissions.

Putting these remarks aside, let’s just go ahead and use the model’s numbers. For oil it projects an average annual growth rate of 1% in the period 2007-2030. Remarkably this is the same rate as in the last 30 years. Let’s also allow for the possibility that this growth rate is off by a certain margin. Not having a clue how to estimate this margin, I just take the interval 0.7%-1.3% and regard it as reasonable. Over the period 2007-2030 this interval corresponds to a total growth of 17%-35%, compared to the 26% the IEA projects. The figure below illustrates this method.

Abusing the world energy model’s numbers, I find some evidence for the conclusion that oil will be depleted in 2070-2105. I get the lower bound by taking a growth rate of 1.3% (assuming the model is 0.3% wrong on the high side), based on a total potential reserve twice as big (purple curve) as the currently proven reserves. According to the USGS, there is a 95% chance that there is at least that much oil, but I still drew the curve dashed since there might be more oil by unconventional means: oil sands, oil shales, deepwater oil, etc. are not included in these graphs.

The upper bound is calculated by assuming a 0.7% growth (the model is 0.3% wrong on the low side) with potential reserves three times bigger than currently proven reserves (green curve again).

**Summary of the numbers**

Here’s a summary of the numbers. After each date I provide the assumed growth rate and reserves in the form *(average growth rate, size of reserves relative to currently proven reserves)*. For instance “2146 (0%, 3)” means that I calculated the date 2146 assuming a 0% average annual production growth, with remaining reserves 3 times bigger than currently proven reserves.

Depletion dates:

- Earliest: Oil in 2033 (4.7%, 1) // Gas in 2038 (4.9%, 1) // Coal in 2049 (5%, 1)
- Latest: Oil in 2146 (0%, 3) // Gas in 2165 (0%, 2.5) // Coal in 2448 (0%, 4)
- Reasonable range: Oil in [2070 (1.3%, 2) – 2105 (0.7%, 3)] // Gas in [2061 (2.1%, 1.5) – 2095 (1.3%, 2.5)] // Coal in [2079 (2.5%, 1.5) – 2155 (1.4%, 4)]

**Concluding remark**

So far I have concentrated on obtaining reliable ranges for the depletion dates. In reality, fossil fuels will probably never be truly depleted. A couple of years, a decade, several decades – who knows – before a depletion date is reached, prices most likely go through the roof. We will run out of cheap oil before we run out of oil. In this sense, the depletion date is irrelevant. Still, it might help us get an idea of the timescale we are faced with.

**Footnote**

[1] At the moment, little coal is converted to oil because of the large investments needed for the plants. The OPEC ensures that the oil price stays low enough to price Fischer-Tropsch plants out of the market (and high enough to make money).

**References**

International Energy Agency Statistics

Europe’s energy portal

U.S. Energy Information Administration

U.S. Geological Survey: World Petroleum Assessment

U.K. Energy Research Centre

BP Statistical Review

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