J.H. LaherrereThe universe is made up of cycles. Everything that is born will die: stars, days, species, humans, and civilizations. A stone thrown into the air follows a parabolic trajectory. If its velocity is more than 11 km/sec it will leave the gravitational field of the earth but will become part of the solar system, returning on an elliptic orbit. What goes up must come down. The question is: when will it peak?
King Hubbert, in his famous paper of 1956,1 predicted that U.S. oil production (from the Lower 48) would peak in the 1970s at the top of a bell-curve. The area of the curve represented the total endowment of oil, which he estimated at 200 billion bbl. He was vilified at the time as being a pessimist but was amply vindicated when the country's production indeed did peak in 1970.
Some theoryThe Hubbert curve is a derivative of the logistic curve which the Belgian mathematician Verhulst introduced in 1845 as a law of population growth. It is based on the relationship:
CP = U/(1+EXP( 2 b(t2 tm))
where: CP is cumulative production, U is an asymptote representing ultimate recovery, and tm is the inflexion point, namely peak time of annual production.
Fig. 1 [40,072 bytes] shows the relationship between the logistic curve and its derivative (multiplied by ten to be more visible).
The equation of the Hubbert curve for annual production P (being CP/t) is simple when related to peak annual production Pm occurring in year tm:
P = 2Pm/(1+COSH(2 b(t2 tm)))
The constant b is equal to 4Pm/U and also 5/c where c is the half width of the curve on the time axis when production started and has fallen to a very low level (Pm/100 as LN(100) # 5).
So a quick way to compute the ultimate of a Hubbert curve is:
U = 0.8cPm
It is interesting to compare with the Gauss bell-shape curve and a parabola (stone thrown in the air):
The normal curve is generally called the Gaussian curve. The normal law represents the probability of randomness, as the sum of a very large number of small independent causes, giving the probability. In this distribution, the mode (where the most likely equals the peak probability), the median (where there is an equal number on either side or a 50% probability) and the mean (weighted average) are the same.
As applied to oil production, a bell-shape curve is:
P = Pm EXP(- (t - tm)2/2s2)
A comparison (Fig. 2 [48,711 bytes] ) between a Hubbert curve and a Gauss curve with the same peak and the best fit shows that the difference is quite small, when the upper part of the Hubbert curve is close to a parabola. A harmonic curve (sine wave) is also displayed with the best fit and is also very close for most of the curve. In modeling, harmonic or normal (Gauss) could be used, but Hubbert is the easiest to handle and corresponds to one single cycle.
One of the arguments advanced against Hubbert's proposal was that the depletion profile of individual fields is normally asymmetrical, being skewed to the left. An explanation is offered by the Central Limit Theory, a well known tenet of statistics, which states that the sum of a large number of independent asymmetrical curves tends towards a symmetrical normal distribution.
Production mimics discoveryDiscovery is also usually symmetrical because of the cyclical nature of exploration and the law of diminishing returns.
The symmetrical production curve in fact reflects the corresponding discovery curve after a time-lag. Fig. 3 [46,237 bytes] displays the annual oil production for the U.S. Lower 48 and the annual discovery2 3 after being smoothed (3 year average) and shifted by 35 years in order to have the best fit with the production curve. The lag of 35 years corresponds to the gap between the peak of discovery during the 1930s and the peak of production in 1970. The Hubbert curve fits quite well with the production curve and also with the shifted discovery, in particular for now to 2010.
The Hubbert curve is thus a robust model of U.S. Lower 48 production because it is based on a large population of fields, exceeding 25,000. The fit is not perfect because the natural order was affected by external events such as the Depression of the 1930s, the imposition and subsequent lifting of proration in the 1950s, and the sharp increase in oil price in the early 1980s.
Fig. 4 [59,317 bytes] shows a remarkable similarity between the profile of the Lower 48 and the Former Soviet Union despite the very different environments. The FSU also has a large population of fields and was subject to a continuous and uniform exploration effort until the collapse of Communism. The profile has a steeper rise and decline. The production profile correlates well with the discovery profile with a 15 year time shift. In effect, the FSU was depleting its resources faster than the U.S., with its 35 years time shift, and the decline in production will be steeper at 10% vs. 6% in the U.S.
The FSU production curve correlates well with the 15 years shifted discovery curve. FSU oil reserves are produced faster (15 years lag against 35 for the U.S. Lower 48) and harder (decline should be around 10%/year against 6%/year for the U.S. Lower 48).
The Hubbert curve also well models world production, outside the five so-called swing producers of the Middle East (Abu Dhabi, Iran, Iraq, Kuwait, and Saudi Arabia) which are not producing at capacity (Fig. 5 [51,316 bytes]). There is a very close fit with the discovery curve with a 15 year time shift. From this correlation it is obvious that nonswing production is about to peak. Even if major new investment should delay peak by a few years, it will simply mean that the subsequent decline is that much steeper.
North Sea production, which has been largely responsible for the world surplus in recent years, will peak within a year or two for the same reasons, carrying wide implications that may well influence actions of the swing producers and world oil prices. But if the price stays at the low level of $20/bbl, nonconventional oil will stay marginal and the decline will start.
Multiple peaksMany countries with a smaller number of basins and fields have more than one peak in their production profiles, but almost all of the peaks are individually symmetrical in their upper parts. Hubbert did not envisage (neither have other previous authors) that the modeling could be with several cycles. We have modeled every production country of the world4 with several (multi-Hubbert) cycles, as Fourier analysis models sound with a few harmonics.
In fact, almost every country can be well modeled by at most four cycles in which discovery peaks are correlated with corresponding production peaks after a time-lag giving the best fit. The case of France can be modeled by two similar symmetrical cycles (Fig. 6 [48,639 bytes]).
For the Netherlands an early smooth onshore cycle is followed by a more irregular offshore cycle, with a third cycle now beginning (Fig. 7 [86,007 bytes]). In both cases, a 7 year time shift gave a good correlation.
U.S. drilling cyclesU.S. drilling has shown several cycles, and these cycles are symmetrical, so it is easy to model the all-wells curve with four cycles.
One basic cycle peaks in 1970, and three temporary cycles peak in the 'crazy years' of 1920, 1956, and 1982 (Fig. 8 [50,555 bytes]).
The question is whether a new cycle will occur?
Official oil supply forecastsMost of the world's recent oil scenarios forecast large increase of supply for the next 25 years without any drastic price increase. It means staying within what is called optimistic conventional reserves or $20-25/bbl reserves (Fig. 9 [39,664 bytes]). But almost all fail to see when the peak of production will be reached and compute how much oil is needed until exhaustion to supply their scenarios.
The 1998 scenario by the U.S. Department of Energy2 shows no sign of peak, but the 1998 International Energy Agency scenario5presented at the G8 meeting in Moscow displays a peak of liquids conventional and identified unconventional between 2010 and 2020.
The U.S. DOE forecast displays a very narrow range. We have drawn the Hubbert curves with the closest peak (year 2025-2030) to these scenarios and compute the future production of these scenarios until exhaustion (year 2200). The DOE future production is around 3,900 billion bbl, which added to the present cumulative production of 800 billion bbl gives an ultimate of 4,700 billion bbl of $20-25/bbl reserves. The IEA future production is around 2,000 billion bbl, giving an ultimate of 2,800 billion bbl, very close to our6 liquids ultimate of 2,700 billion bbl: 1,800 billion bbl for conventional oil, 200 billion bbl for NGL, and 700 billion bbl for unconventional oil.
How much oil would have to be discovered to delay the peak by 1year?
In a Hubbert curve as the ultimate is 0.8 times peak production multiplied by half width, the peak is shifted by one year when the ultimate increases by 0.8 times the annual peak production. If the peak production is about 45 billion bbl/year (or 120 million b/d as in the DOE/EIA forecast2), 36 billion bbl of new discoveries will delay the peak by only 1 year. A 1 billion bbl discovery delays the peak by 10 days.7
World populationIt has become clear that many things besides oil can be modeled in the same way, for example, population and climate.
It has been shown that the parabolic fractal, developed to model the distribution of oil fields, also models the distribution of other objects in a natural domain, such as physical (as opposed to administrative) towns (agglomerations), galaxies, spoken languages, size of species, etc.
National populations change over time. Population growth has been the norm, but history shows that civilizations wax and wane both in power and numbers: the Incas, Mayas, Greeks, and Romans, to name a few examples. It is the same with stars, the dinosaurs, and eventually humans, too. Population declines when the fertility rate of a country falls below 2.1 children/woman. There is a time lag because of the "pyramid of age," but populations do peak and decline. The curves can be modeled in the same way as discussed above.
According to Bourgeois-Pichat,7 industrial populations will soon peak, to be followed by the developing countries some 40 years later as they try to emulate the industrial countries. Europe's population is expected by the Observatoire D?mographique Europ?en8 to peak around 2025. Fig. 10 [38,971 bytes] shows Hubbert curves for three categories of people: a basic (uneducated) population, an industrial population, and a developing population.
The UN 1998 forecast with a large range of scenarios is added. The scenario of low/medium fertility is in a good agreement with our multi-Hubbert model.
The world's population will peak around 2050 at 8.5 billion people, which is the last figure forecast by the United Nations. If no other cycle occurs, the world will sharply decline staying only with people refusing education and progress.
Another example is provided by data on North Atlantic cod landings,9 which can be readily modeled by three cycles (Fig. 11 [45,748 bytes]) with the possibility of a fourth if conservation should be practiced.
Climate changeA 3,350 m deep borehole has been sunk in the Antarctica ice cap at Vostok10 to recover cores of ice that contain evidence of climate for the past 400,000 years. The content of deuterium in the ice is a measure of world climate (proxy temperature) at the time of formation.
A similar ice core project in Greenland has shown that these variations happen all around the earth and correlate. The geological glaciations (Wurm 18,000 years ago and Riess 150,000 years ago) and for example the maximum of the sea in Barbados are in agreement with the Vostok and Greenland results.
Four climatic long cycles occurred during this period (A to D, Fig. 12 [51,535 bytes]), and a new one (E) has just started. Each long cycle consists in a series of five successive decreasing peaks. It is possible to use this information to predict future climatic changes. A shift of 120,000 years places the first peak of Cycle E in juxtaposition with the first peak of Cycle D, and another shift of 235,000 years will line up Cycle E with Cycle C. The correlation provides a convincing forecast of future climate for Cycle E.
Evidently, we have passed the interglacial peak (first peak of Cycle E) and are now moving towards a new glaciation not more than 5,000 years away, to be followed in turn by a mild interglacial and yet another glaciation, another interglacial, and so on. The last fifth glaciation will occur about 100,000 years from now, being the most intense. It will be followed by a sharp and strong recovery, comparable with today's conditions, about 120,000 years into the future. The glaciations of Cycle E, which is now opening, will have a longer duration.
Consider the pattern in greater detail (Fig. 13 [49,007 bytes]). It is evident that the climate curve may be modeled by 21 Hubbert curves of the same width (c=16) but with decreasing peaks in a cycle of five peaks and with different intervals. Cycles A and B seem different from C and D, but it is due to the fact that the intervals are smaller and that the asymmetrical peak at 335,000 years ago is the sum of two Hubbert cycles B1 (red) and cycle B2 (green). It is the same for the large first peak of Cycle A made of two close Hubbert cycles.
As there are 21 Hubbert cycles for 400,000 years, there is a cycle every 20,000 years. Where is this 20,000 year cycle coming from?
The climate variation depends first on the three astronomical parameters of the orbit of the earth around the sun:
- Excentricity of the ellipse going from 0 to 7% with a period of around 100,000 years;
- Precession of the axis of the poles (axis of rotation), which describes a cone with a period of 21,000 years; and
- Obliquity of the polar axis with the perpendicular to the ecliptic plane going from 22% to 25% with a period of 40,000 years.11 Its precession variation of 21,000 years was discovered in 1842 by Adhemar, and the other parameters were largely known in 1930 as the Milankovitch periodicities. The combination of these three orbital variations, combined with the internal phenomena as the ice sheet dynamics and the deep oceanic currents (El Niño and the North Atlantic Deep Waters), controls the size and intervals of the cycles. There are many glaciations occurring during the Quaternary but few during the Tertiary and the Secondary!
ConclusionOur civilization is accustomed to growth, and it is difficult to imagine that growth is a transitory phenomenon. But the one sure thing I know about the future is that one day I shall die.
We do not like to think about our own demise any more than we like to accept that oil production will peak and decline to eventual exhaustion. The U.S. has already witnessed its peak and is well into decline, but thanks in part to its military power it has been able to ignore the consequences by being able to import large quantities of cheap oil, especially from Saudi Arabia.12 This arrangement cannot last as production in the world as a whole and later the Middle East itself is about to peak. However, this unassailable observation is no more popular than that which greeted Hubbert's correct prediction of the situation in the U.S. itself. Gas will become increasingly important after oil peaks, but it too is due to peak within 20 years.
Within 5,000 years, the world will experience another glaciation. New York will be covered with ice, and the channel between U.K. and France will be dry. But 5,000 years is a long time. There are more immediate problems with oil production to peak within the decade: the transition to a low energy economy will be difficult after our experience of abundant cheap supply. The official forecasts that ignore the elementary resource constraints do us no service.
Never listen to those who tell you only about rise without talking about peak.
Better ask "when will it peak?" without forgetting that usually there are several peaks.
AcknowledgmentThanks to Petroconsultants for allowing use of their data.
- Hubbert, M.K, Nuclear energy and fossil fuels, API Drilling & Production Practice, proc. spring meeting, San Antonio, 1956, pp. 7-25.
- U.S. DOE/EIA, International Energy Outlook, April 1998.
- Root, D.H., and Attanasi, E.D., The enigma of oil and gas field growth, AAPG Bull., March 1994, pp. 321-332.
- Campbell, C.J., and Laherrère, J.H., The world's oil supply-1930-2050, Petroconsultants report, October 1995, 650 p.
- International Energy Agency, World energy prospects to 2020, prepared for the G8 Energy Ministers meeting, Moscow, Mar. 31, 1998 (website: iea.gov).
- Perrodon, A., Laherrere, J.H., and Campbell, C.J., The world's nonconventional oil and gas, Petroleum Economist, 1998.
- Bourgeois-Pichat, J., Du XXe siecle au XXIe siecle, l'Europe et sa population apres l'an 2000, Popul 1, 1988.
- Observatoire Démographique Européen, 1997: website: http://schwyz.ined.fr/ revues/pop_et_soc/pes321/pes3212.html
- North Atlantic Fisheries Organization, 1997, website: http://www.usglobec. berkeley.edu/usglobec/Reports/rep2/rep2.fig8-1.html
- Petit, J.R. et al., Four climates cycles in Vostok ice core, Nature, Vol. 387, May 22, 1997, p. 359.
- Labeyrie, J., L'homme et le climat, Denoël, 1993, 342 p.
- Schweizer, P., Victory, Atlantic Monthly Press, 1994.
Jean H. Laherrere is a petroleum consultant residing in Paris, France. He retired in 1991 after 37 years with Total CFP and subsidiaries. He participated with Total in the discovery of supergiant Hassi Messaoud and Hassi R'Mel fields in the Algerian Sahara and also explored Australia, the Labrador Sea, and Michigan. He was a member of the Ocean Drilling Program safety panel. He was graduated from Ecole Polytechnique and Ecole Nationale du Pétrole. E-mail: [email protected]
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