Draft of an article for Oil and Gas Journal Nov. 1998
What goes up must come down: when will it peak?
The 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/s it will leave the gravitational field of the Earth, but will become part of the solar system returning back 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, predicted that US oil production (from the Lower 48) would peak in the 1970s at the top of a bell-curve, the area of which represented the total endowment of oil, which he estimated at 200 billion barrels (Gb as GB=gigabytes). He was vilified at the time as being a pessimist, but was amply vindicated when the country’s production indeed did peak in 1970.
The Hubbert curve is a derivative of the logistic curve which had introduced in 1845 by the Belgian mathematician Verhulst as a law of population growth. It is based on the following relationship
CP = U/(1+EXP(-b(t-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)
Figure 1 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(-b(t-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 independant 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 (Figure 2) 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 the 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 Discovery
Discovery 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. Figure 3 displays the annual oil production for the US Lower 48 and the annual discovery (USDOE 1990 and Root et al 1993) after being smoothed (3 years 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 30s 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 US Lower 48 production, because it is based on a large population of fields, exceeding 25 000. There is not a perfect fit 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.
Figure 4 shows a remarkable similarity between the profile of the Lower 48 and the FSU 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 USA, with its 35 years time shift, and the decline in production will be steeper at 10% versus 6% in the USA.
The FSU production curve correlates very well with the 15 years shifted discovery curve. FSU oil reserves are produced faster (15 years lag against 35 for US 48 states) and harder (decline should be around 10%/a against 6%/a for US 48)
As shown in Figure 5, the Hubbert curve also well models world production, outside the five so-called Swing producers of the Middle East (Abu Dhabi, Iran, Iraq, Kuwait, Saudi Arabia) which are not producing at full capacity. There is a very close fit with the discovery curve with a 15 year time shift. From this correlation it is obvious that Non-Swing 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$/b, non-conventional oil will stays marginal and the decline will start.
But many countries with a smaller number of basins and fields have more than one peak in their production profiles, but almost all of the peaks are individually symmetrical in their upper parts. Hubbert did not envisage (neither have other previous authors) that the modeling could be with several cycles. We have modeled every production country of the world (Campbell, Laherrere 1995) with several (multi-Hubbert ) cycles, as Fourier analysis models sound with a few harmonics.
In fact, almost every country can be well modelled by at most four cycles in which discovery peaks are correlated with corresponding production peaks after a time-lag giving the best fit. Figure 6 illustrates the case of France which can be modelled by two similar symmetrical cycles.
Figure 7 shows the Netherlands where an early smooth onshore cycle is followed by a more irregular offshore cycle, with a third cycle now beginning. In both cases, a 7 year time shift gave a good correlation.
US drilling has shown several cycles and these cycles are symmetrical, so it is easy to model the allwells curve with four cycles, one basic peaking in 1970 and three temporary with peaks in 1920, 1956 and 1982 (crazy years): figure 8 . The question is to know if a new cycle will occur?
Official Oil Supply Forecasts
Most of recent world’s oil scenarios forecast large increase of supply for the next 25 years without any drastic price increase, it means staying within what is called optimistic conventional reserves or 20-25$/b reserves. But almost all fail to see when the peak of production will be reached and to compute how much oil is needed until exhaustion to supply their scenarios. The 1998 scenario by USDOE shows no sign of peak but the 1998 IEA (International Energy Agency) scenario presented at the G8 meeting in Moscow displays a peak of liquids conventional and identified unconventional between 2010 and 2020.
The USDOE 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 3900 Gb which the present cumultive production of 800 Gb gives an ultimate of 4700 Gb of 20-25$/b reserves ultimate. The IEA future production is around 2000 Gb giving an ultimate of 2800 Gb, very close to our (Perrodon, Laherrere & Campbell 1998) liquids ultimate of 2700 Gb: 1800 Gb for conventional oil, 200 Gb for NGL and 700 Gb for unconventional oil.
How much to discover to delay the peak by one year?
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 Gb/a (or 120 Mb/d as in the forecast DOE/EIA IOE 98), 36 Gb new discoveries will delay the peak by only one year. A 1 Gb discovery delays the peak by ten days!
Examples of how other things peak in nature
It has become clear that many things besides oil can be modelled in the same way, for example population and climate.
It has been shown that the parabolic fractal, developed to model the distribution of oilfields, 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 quote 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 per woman. There is a time lag because of the “pyramid of age”, but populations do peak and decline. The curves can be modelled in the same way as discussed above.
According to Bourgeois-Pichat 1988, 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 Démographique Européen” 1997 to peak around 2025. Figure 9 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 with about 8.5 billions people. 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 (North Atlantic Fisheries Organization 1997) which can be readily modelled by three cycles as shown in Figure 10, with the possibility of a fourth if conservation should be practised.
400 000 years of climate change
A 3350 m deep borehole has been sunk in the Antarctica icecap at Vostok (Petit 1997) to recover cores of ice which 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. Similar icecore project in Greeenland has shown that these variations happen all around the earth and correlate. The geological glaciations (Wurm 18 000 years BP and Riess 150 000 years BP) and for example the maximum of the sea in the Barbades are in agreement with the Vostok and Greenland results.
Figure 12 shows that there were four climatic long cycles over this period (A to D) 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 5000 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, to be followed by a sharp and strong recovery, comparable with to-day’s conditions, about 120 000 into the future. The glaciations of Cycle E, which is now opening, will have a longer duration
Figure 13 considers the pattern in greater detail. It is evident that the climate curve may be modelled by twenty-one 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 the 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.
With twenty-one Hubbert cycles over 400 000 years, each has a duration of 20 000 years.
As there are 21 Hubbert cycles for 400 000 years, there is a cycle every 20 000 years.
Where is this 20 000 years cycle coming from?
The climate variation depends first on the three astronomical parameters of the orbit of the earth around the sun: 1) excentricity of the ellipse going from 0 to 7%: with a period of around 100 000 years; 2) precession of the axis of the poles (axis of rotation), which describes a cone with a period of 21 000 years: 3) obliquity of the polar axis with the perpendicular to the ecliptic plane going from 22 to 25 with a period of 40 000 years (Labeyrie 1993). Its precession variation of 21 000 years was discovered in 1842 by Adhemar and the other paremeters were largely known in 1930 as the Milankovitch periodicities. The combinaison of these three orbital variations, combined with the internal phenomena as the ice sheet dynamics and the deep oceanic currents (El Nino 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!
Our civilization is accustomed to growth, and it is difficult to imagine that growth is a transitory phenomenon. Bur 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 United States 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 (P.Schweizter 1994). This arrangement cannot last as production in the world as a whole and later the Middle East itself are about to peak. However, this unassailable observation is no more popular than that which greeted Hubbert’s correct prediction of the situation in the United States itself. Gas will become increasingly important after oil peaks, but it too is due to peak within twenty years.
Within 5000 years, the world will experience another glaciation. New York will be covered with ice and the Channel between UK and France will be dry. But 5000 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.
Thanks to Petroconsultants to allow to use their data
-Bourgeois-Pichat J. 1988 “Du XXe siecle au XXIe siecle, l’Europe et sa population apres l’an 2000” Popul 1
-Campbell C.J., J.H.Laherrère 1995 “The world’s oil supply -1930-2050” Petroconsultants report -Oct. -650p
-Hubbert M.K 1956 “Nuclear energy and fossil fuels ” Am. Petrol. Inst. Drilling & Production Practice, Proc. Spring Meeting San Antonio Texas p7-25.
-IEA 1998, “World energy prospects to 2020”; prepared for the G8 Energy Ministers’ meeting Moscow 31 March, Internet (iea.gov)
-Ivanhoe L.J. 1995 “Future world oil supplies: there is a finite limit ” World Oil Oct 1995
-Labeyrie J. 1993 “L’homme et le climat” Denoël 342p
-Laherrère J.H.1996 “Discovery and production trends” OPEC Bull.Feb.p7-11
-Perrodon A. J.H.Laherrere & C.J.Campbell 1998 “The world’s non-conventional oil and gas ” Petroleum Economist
-Petit JR. et al 1997 “Four climates cycles in Vostok ice core” Nature, v387, 22 May p359.
-North Atlantic Fisheries Organization 1997:
-Observatoire Démographique Européen 1997:
-Root D.H., E.D.Attanasi 1994 “The enigma of oil and gas fied growth ” AAPG 78/3 March, p321-332
-Schweizer P.1994 “Victory” The Atlantic Monthly Press
-USDOE/EIA, 1998, “International Energy Outlook”; April Internet