Peak oil – Predictive Model

Petroleum and other natural resources have finite size.

We approach the Peak Oil problem of resource management from a slightly different viewpoint.  Our model looks at the consequences of usage that increases exponentially; the math tools are derived on a separate Page.  This is in contrast with other models that attempt to apply a special curve of some kind (for example, the derivative of the Logistics curve) to the known issue.  Our principal results are in the table at the end of this post.

“Peak oil.” Is it inevitable? Is it already past? Will it arrive in the near future? Does it even make any sense?  The end of abundantly available petroleum is a topic of real concern, one that has driven my own career.

“Peak Oil,” to my mind, is a phrase indicating the consequence of drawing on a finite-sized reservoir.  Such use goes through the classic three phases of existence:

  • Start-up This is the period of exuberant growth, unconstrained by external effects. The resource supplies all is every required and appears unbounded in its largess.  Usage follows an exponential growth curve. (Examples of exponential model growth: Chess Board Fable, Australian Rabbits.)

Copper Production, 1900-2000Models for the drain-down of finite reservoir follow pretty clear historic patterns (example,  growth in copper production, shown here).   Something becomes a resource when early adopters start to use it.  New applications inevitably arise, and even more users are enticed to enter in. The current increase in usage is fully proportional to the current amount being utilized.
Data are World usage of copper 1900-2000. Blue trendline is ert with r = 3.3% annual growth rate (best fit value). Copper production is still in its unconstrained growth period.

  • Contraction Unconstrained growth continues until the easy portion is used up and the reservoir/user system enters into its final period. More expensive techniques must be used to extract the next bit, and even more aggressive, expensive methods for the following. Deeper wells, angled cuts, forced fracture techniques, tar sands for rendering, bio-oil, etc. Usage can not grow freely due to extraction economics; it must begin to decline.
  • Peak This is the period between initial exuberant expansion and ultimate collapse due to exhaustion of the reservoir. Usage loses its exponential growth pattern; ultimately it must transition from increasing to decreasing and this leads to the mountain-top shape of this intermediate period.

For human-based utilizations, only the start-up phase can be modeled with any real accuracy. The peak and contraction phases will be heavily influenced by the politics of the situation.

Start-up growth differs from the linear, “straight line” thinking people are used to (if we use 10 gal today, 100 gal should last us 10 days).  Exponential drain on a reserve volume cannot be sustained and ultimately must stop. This ending will have a similar “feeling” as walking about a public park when the solid ground collapses underfoot.

We developed the algebra behind exponential usage of a finite reservoir and will now apply it to the M. King Hubbert problem.

US Production 45-08Background M. King Hubbert (1903-1989) was a geophysicist working for Shell Oil, Company, then the U.S. Geological Survey.  Hubbert published an early warning ( Science v104 Feb 4 1949 MKHubbert ) that the oil reserves were finite and will be drained for times short compared with human history.  In 1956, he published his prediction (AmPetrInds Mar 7-9 1956 MKHubbert ) that US oil would peak, probably in the late 1960’s.  This proved to be the case.  US production peak occurred in November of 1970.

Full disclosure: I attended Hubbert’s presentation at a Physics Colloquium at the University of Wisconsin – Madison in late 1972.  His case was well made and he showed us the industry graph with the oil peak in it.  He said that at that time, the oil companies were in denial of this fact.

Although Hubbert is the best known of the those how have predicted the end of easily available petroleum, he was not the first.  Alexander Graham Bell published Prizes For The Inventor in Feb 1917 The National Geographic Magazine (pp 131-146) stating that coal and petroleum are finite resources and we (the US) must move away from dangerous dependence and toward a source of renewable energy. He proposed ethanol as a safe, cheap substitute.

Example Situation Suppose we estimate that we have 200 years left in our petroleum reservoir, given a constant draw at the current rate (linear usage).  Suppose the growth rate is not actually zero, but is about 4% per year.  Each year we would use about 4% more than the previous.

The first question is, how long would such a reservoir last (exponential draining from beginning to end)? We apply Eqn 3 in Using a Finite Resource. The fraction f removed is  f=0  at start up (reservoir full) and f=1 when reservoir is completely empty. The time needed to achieve the target f value is:  t = ln(1+rYf)/r. Forf = 1.0, t = ln(1+0.04*200*1]/0.04 … about 55 years.

Actually, few reservoirs could supply to a growing demand to the point that it runs dry. In reality, the more that is removed, the harder it becomes to take more out. Hubbert’s proposal (the Hubbert Rule) is that at the 1/2 usage point, production should peak and output fall thereafter. It will probably be difficult to see the roll-off in production when it happens. This is because actual commodity data has a lot of random fluctuations and deviations from exponential growth can be hard to spot, especially true when the fate of the supplying industry is at stake. It is very possible that continuously increasing growth could continue right up the the turn-over point. We will probably recognize production peaking only in the rear view mirror of 5 to 8 years, post event.

So the better question is: When will the half-drained point (f=0.5) occur? Same situation – start with a 200 year supply, let usage grow continuously at 4% per year. Calculate: t = ln(1+0.04*200*0.5]/0.04  …  about 40 years (not 27.5).

Petroleum supply and peak production estimates My best data for this is from my notes of the 1972 M. King Hubbert physics colloquium, mentioned above.  I am aware that there are various numbers being tossed about as to what Hubbert did not did not project for the future.  Here is what my notes indicate.  Hubbert spent time showing estimates as to proven petroleum reserves.  He gave a value (not in my records) and said that with at then-current use, the probable life time of the reserve was about 200 years.  He returned to the estimates and said that even with future discoveries, even monster sized ones like North Sea bed, the effective results would not change.

I have no record that he used the algebra/calculus use-function derived here.  However, he did an application using the derivative of the logistics function as the model peak function.  This has the advantage that it shows solid exponential growth in the early phases.  His comment to the audience was that even with a factor of 2 uncertainty in the world reservoir (this is my noun, he  used “reserves”), the results would have minimal effect on the life style of some of those in the audience.

Hubbert  said that the peak might come before 2010 but it actually might be delayed for another decade.  Either way, our oil driven growth was nearly over for all of history.

Let us look at these numbers, in light of Eqn 3.  Assumption 1: In 1972, (date of the talk) there were between 150 and 300 years of estimated and proven reserves in the Earth.  Assumption 2: the treandline growth rate over the past decades has been r=4% per year.  Question: How long after 1972 were we  to wait until the inevitable  peak in world production occurred?

Initial lifetime Y0 Years ln(1+rY0 f)/r Date
Minimum 150 35 2007
Mean 225 43 2015
Maximum 300 49 2021

Summarize this: The peak in world production was estimated to occur in the first half of the 21st century, most likely during 2010 — 2020.

How robust is this analysis? To test, suppose the (steady draw) lifetime of the world petroleum reserve suddenly jumped from 225 to 1,000 years, due to some extraordinary discovery. The model is that usage continues to show exponential growth up to its peak value, which would happen 76 years after our start time (1972), or 2048, before mid century and certainly during the lifespan of most readers, here.

A second test: Suppose our 225 yr supply had its use rate reduced to an annual growth of 1%, down from 4%. This yields a peak time of 75 years, or 2048.

Observation: By the model, it is much better to reduce the growth in annual oil use to 1% than to rely on the gods and their magic lightning bolts to uncover a pool of oil 4× greater than the entire known current reserve.

Validity of math model The model assumes a clean growth curve, without fluctuations.  Real life is chaotic and realized usage will be highly fluctuating.  It is harder to modify an exponential growth curve during robust growth, but near the peak, the potential exists for large variance of data from analysis.

My Analysis: As this is written (2011) the world is experiencing a major economic turn-down, with many jobs lost in the first world countries, taken by 3rd world manufacturing.  Manufacturing is energy intense, as the US ships is capability to China, we should expect a reduction in our own energy growth rate.  (Partly due to the fact that we have shifted to a low pay/low expectation assembly operation, away from high value-added experience-added manufacturing operation.

click for all our discussions about oil resources

So shift focus.  China shut its plants down about 6 months in 2008 for the Olympic glory.  This had effects on all other world economies.  By the time China brought manufacturing back to life, the world was in the depths of a huge near-depression.  Not caused by Chinese influences, necessarily, but hugely due to our Wild Cat upper class who were successful at their business scams until found out.   The turbulent business climate has dropped world growth in energy demand to about 2%, and even that is mostly due to the Chinese ramp-up in demands.   Economic reality is currently masking any supply roll-off that might be visible with a more robust world-wide demand.

Back to top


Charles J Armentrout,  Ann Arbor
2011 Jan 27,  Minor modification  2011 Jun 03
Listed under   Natural Resources     …thread  Natural Resources > Oil
Find related posts on page   INDEX

About LastTechAge

I am a physicist with years of work in fusion labs, industry labs, and teaching (physics and math). I have watched the tech scene, watched societal trends and am alarmed. My interest is to help us all improve or maintain that which we worked so hard to achieve.
This entry was posted in Natural Resources and tagged , , , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.