**by J.H. Laherrère**

M.King Hubbert was a distinguished scientist who published on many aspects of petroleum geology, but he is most remembered for his 1956 work on forecasting production, in which he introduced a special bell-curve that bears his name. He was also remarkable for his interest in the social implications of resource depletion.

In 1974, he presented several production curves for both the World and the United States, but was somewhat reticent in explaining the mathematical basis of his work. He referred to a bell-shaped curve, of which the most commonly used are the Normal or Gauss curve, and also to the derivative of the logistic curve (Bartlett 1999), but he gave no equations. He based his study on Ultimate Recovery, taking 170 Gb (billion barrels) for the USA; and low cases and high cases for the World of respectively 1350 Gb and 2100 Gb. His initial study concerned the US Lower 48 States, which had a single cycle of continuous exploration in a large number of basins. He referred also to the relationship between discovery and production. The discovery cycle peaked in the late 1930s and was followed by a corresponding production cycle peaking around 1970. But as explained below not all countries are characterised by a single discovery cycle; and there are other constraints to the Hubbert model that need to be better understood. It is to be noted in particular that it is a symmetrical curve whereas the production curve of an individual field is generally asymmetrical. As discussed below, the Hubbert curve is in fact the derivative of a logistic curve.

**Constraints**

A simple Hubbert curve may be ideally applied only in the following cases:

-Where there is a large population of fields, such that the sum of a large number of asymmetrical distributions becomes symmetrical (normal) under the Central Limit Theorem of statistics. There are several examples of inadequate populations, as Illinois and Ohio with few fields (highlighted by McCabe in his criticism of Hubbert) or Alaska and the North Sea, where a few giant fields came on production simultaneously when the pipeline connections were made.

-Where exploration follows a natural pattern unimpeded by political events or significant economic factors, as for example when OPEC artificially cut production: Hubbert modelling should not be used for the swing producers (Persian Gulf).

-Where a single geological domain having a natural distribution of fields is considered, political boundaries should be avoided.

Hubbert himself did not appreciate these constraints since he worked on the US Lower 48 and the world as a whole, prior to significant OPEC intervention.

**Methodology**

It is well to remember at the outset that what is born must die: even the Sun and our Planet have a life ending in death. What goes up must come down. Biological growth starts with an exponential multiplication (1- 2- 4- 8-16…….) as each cell divides, but because of constraints (mainly space and family planning for human beings) growth declines and population reaches a maximum and declines to death. Ultimately nothing is linear, least of all producing a finite natural resource formed but rarely in the geological past.

The classic logistic curve was discovered by Verhulst in 1845 in connection with population studies. It was used to propose that population growth increases to a midpoint (**tm**) and then decreases to zero, giving what is known as an S-curve. In this application, where there is no negative growth, total population stays constant at the asymptote (**U**). In the 1920s Pearl and Reed used the logistic curve to model the US population.

It can also be used in modelling oil cumulative production under the formula

Q = U/1+EXP(b(t-tm))

where

tis the reference date (year)

Qis Cumulative Production at the reference date (t)

Uis Ultimate Recovery

tmis the date at midpoint

bis a factor describing the slope

But in practice it is more convenient to use the derivative of the logistic curve to model how annual production starts and ends at zero with a peak in between. It is in effect the Hubbert curve, although there are variants such as the Gauss curve, the Cauchy curve, the sine wave and even the parabola. These alternatives give similar results for the upper part of the curve which is the more important part, but the Hubbert curve is the easiest to construct using one of the following formulae:

P = 2Pm/(1+COSH(b*(t-tm))or

P = 2Pm/(1+COSH(5(t-tm)/c))or

P = 2 Pm/(1+COSH(6(t-tm)/d))

(COSH is the hyperbolic cosine equalling (

EXP(x) + (EXP(-x))/2)where (see Figure 1)

Pis annual production (=dQ/dt)

Pmis production at peak (midpoint of Ultimate)

tmis the year of peak (at half life)

cis the duration of the half life from a cut-off at 0.027 Pm

dis the duration of the half life from a cut-off at 0.01 PmThe factors

b, c or dare computed variously from the following relationships:

U=4Pm/b=0.8c*Pm = 2d*Pm/3b= 5/c and Pc = 2Pm/(1+COSH(5))= 0.027 Pm

b= 6/d and Pd = 2Pm(1+COSH(6))= 0.01 Pm

The **Gauss** formula is

P= U/s*(2pi)using the three parameters:^{1/2}*EXP(-(tm-t)^{2}/2s^{2})U, tm and s(standard deviation)

it can be written:

P=Pm Exp(-(tm-t)^{2}/2s^{2})with

Pm= U/s*(2pi), (strictly^{1/2}= 0.4 U/s1/(2pi))^{1/2}=0.3989

In figure 1, the Hubbert curve is plotted together with a Gauss curve of the same ultimate (3200), peak and peak-time. The curves are very close to each other, Gauss being a little fatter at the top and leaner at the base. The maximum of the derivative is the inflection point.

Figure 1:

Some authorities prefer to use the following formula:

P= bU*EXP(b(t-tm)/(1+EXP(b(t-tm))^{2}

which is equal to

P= bU/2/(1+COSH(b(t-tm))

as the cumulative production is

Q=U/(1+EXP(-b(t-tm))orExp(-b(t-tm))=(U-Q)/Q

P = bQ-bQ^{2}/U

The Hubbert equation is a parabola in the cumulative production (**Q**) domain.

It is interesting to note that the Hubbert graph in his famous 1956 paper (Figure 1b) has a fatter top than computed with the above formula. Hubbert did not publish a formula for the graph , and it is likely that in those before computers he simply drew the graph by hand (with templates) and calculated the area below the curve to obtain the Ultimate.

Figure 1b:

To obtain the same “fat” top as in Figure 1b, it is necessary to add another parameter **k**. in constructing the graph in Figure 1c. A value of **k** around 0.1 (**k** is equal to 1 in our model) gives a similar fat top.

Figure 1c:

Procedure

The procedure for constructing the Hubbert curve depends on the status of depletion as follows:

1 Post Production Peak

The simplest case is when the peak date (**tm**) and the annual production at peak (**Pm**) are known. All that needs to be done to construct the curve is to calculate the slope factor **b,** by applying the formula **tm-tc=c=5/b with tc when Pc=0.027 Pm. **It may be further checked by making sure that **b** is the average of 5/c and 6/d, where **tm-td=d=6/b **with** td **when** Pd=0.01Pm.** The Ultimate should also be equal to **0.8c*Pm**. If it does not meet this check with the Ultimate estimate, it probably means that the production profile has to be modelled with more than one Hubbert Curve as explained below.

2. Pre Production Peak – Post Inflection

It is also possible to construct a Hubbert curve in cases where peak has not been reached by relying on the inflection point (see Figure 1), namely when the production increase reaches a maximum, but it is much less reliable. The inflection point in time (**ti**) and amount (**Pi**) corresponds with the maximum of the derivative and allows to compute (Al-Fattah, Startzman 1999):

Pm = 1.5 Pi

tm = ti + 1.317/b = ti + 0.263c

The production data give the amount of production (**Pi**) and the date (**ti**) at the inflection point being the maximum of **Pn-Pn-1**, with **Pm** being derived from the above equation. Thus,** tc** corresponds with the date when production is equal to

Pc = 0.027 Pm = 0.04 Pi; giving

c = tm – tc = 1.36 (ti-tc)tm = 1.36 ti – 0.36 tc

Accordingly, **Pm, tm **and** c** are derived from **Pi, ti **and** tc**.

The Ultimate **U = 0.8cPm = 1.63 Pi (ti-tc)**

The cumulative production up to the inflection point is

CPi= U/(1+EXP(1.317)=0,21U or U=4,7*CPi

Applying the formula to the point d is another way to compute **tm**, as **td (Pd=Pm/100)** gives **tm=1.28 ti – 0.28 td**,** d=1.28 (ti-td)**

The same procedure can be applied in constructing a Gauss curve. The inflection point occurs for **tm-ti = s **and **Pi = Pm*EXP(-1/2) **or **Pm = 1.65 Pi**. The point **td** (where **Pd = 0.01 Pm**) occurs at **d ~ 3s**. So, **ti-td=2s** or **s = 0.5(ti-td)** and **U=2.5 s*Pm**. The Gauss inflection point occurs sooner than with the Hubbert curve, as the curvature at the top is larger.

Although the Hubbert and Gauss curves are close, the inflection points occur at different times, so estimating the peak value from the inflection point of actual data give different values, being 1.5 or 1.65 times the inflection value. This is a substantial difference. Using the same inflection point, as in figure 2, Gauss peak will be in value 10% higher than Hubbert peak, and in time 5 years later and the Ultimate 33% higher. As there is no reason to prefer one to the other (except that the Central Limit Theorem corresponds to a Gauss distribution), this difference demonstrates the uncertainty of forecasting peak on the basis of inflection point.

3 Pre Production Peak – pre Inflection

Forecast can not be made in this case because the growth is still exponential, except by the use of reserve data as follows.

3a Using Annual discovery data

Since oil has to be found before it can be produced, it is obvious that the production curve must in some degree mirror the discovery curve. Hubbert assumed a good correlation, but in fact discovery curve needs to be smoothed. In the event that discovery has peaked, it may be readily modelled with a Hubbert curve. A time shift may then be introduced to match the onset of discovery with the onset of production, so that the discovery curve serves as a proxy for the production curve. If the discovery peak has not yet been reached, the forecast becomes very unreliable.

3b Using Ultimate Recovery estimates

The Ultimate Recovery from a natural domain, such as a basin with a single petroleum system, or from many domains which combine to become a single domain, may be determined by creaming curves (cumulative discovery versus cumulative wildcats) or field size distributions (parabolic fractal). It is even possible to use the risked reserves of untested prospects as assessed on strictly geological and geophysical criteria.

In the case of a single cycle, peak is reached at the midpoint of depletion (**U/2**). The date of peak may then be assessed by extrapolation of the past production trend. Such an approach allows the half-life **c** to be computed from **c = U/0.8Pm**

Applications

The foregoing explains the theory and practice of constructing Hubbert curves, drawing attention to the importance of distinguishing places with a single cycle of discovery from those with more than one cycle, which are the norm.

A set of examples illustrating how the method may be applied in practice is considered below.

1. US Lower 48

1a . US-Lower 48 as estimated in 1960 prior to Peak

Hubbert made his celebrated estimate in 1956 forecasting that the US production would peak around 1970, as indeed it did. By coincidence, 1956 was approximately the Inflection Point (ti). Figure 2 plots production up to 1960 to consider how the methodology could have been applied then.

Annual production (Pi) at the 1956 Inflection Point (ti) was 2.67 Gb/a, meaning that peak (Pm) would be 1.5 times Pi, namely 4 Gb/a. The year corresponding to the effective beginning tc is1903 (Pc=0.04 Pi = 0.11 Gb/a). The peak time is tm=1.36ti-0.36tc=1975 and c=72.

This method, which is based on the inflection point, gives a midpoint production (Pm) of 4 Gb/a in 1975 and a half-life (c) of 72 years. The resulting curve is plotted in Figure 3 and yields an Ultimate of 230 Gb.

Using the cumulative production to inflection point in 1956 CPi= 55 Gb, giving U=4.7*CPi=258 Gb, it is higher than the previous estimate because the aborted cycle by the depression..

Using a Gauss curve, with an inflection point at 1956, Pi= 2.67, Pm=1.65 Pi= 4.4 Gb/a and Pd=0.04 and td= 1890, ti-td=2s, s=33, tm= 1956+33= 1990 and U=2.5 s Pm = 360 Gb which is unrealistic.

Taking for the sake of argument an alternative inflection point at ti=1950, Pi=2 Gb/a

-for Hubbert Pm=3, Pc=0.04Pm=0.12= 1904, tm=1967, c=63, U=150 Gb = too low

-for Gauss Pm=3.3, Pd=0.033=1889, s=0.5(ti-td)=30,5, U=250 Gb = too high

The wide range of estimates based on unreliable inflection points is evident.

**Figure 3:**

**1b. US-Lower 48 as estimated in 1997** **after Peak**

Figure 4 plots annual production up to 1997 showing a very close fit with the 1960 Hubbert model based on an inflection point occurring in 1956, save for the last three years when non-conventional oil and increasing amounts of NGL cause some departure. It is at the same time curious to find matching shoulders in the late 1950s and early 1980s on either side of peak due to unrelated economic influences which were respectively proration and price rises.

**Figure 4:**

We have seen above that using the Gauss curve with an inflection point in 1956 gives an unrealistic Ultimate of 360 Gb. As annual production varies greatly because of economic and political factors, such the Great Depression of the 1930s and prorationing of the 1950s the inflection point cannot be easily identified. However, 1952 is the global maximum and a fair estimate of inflection. The inflection point ti=1952 and Pi= 2.15 Gb/a gives

-for Hubbert Pm=3.2 Gb/a, tc=1902, tm= 1970, c=68, U= 175 Gb

-for Gauss Pm=3.5 Gb/a, td=1889, s=31.5, tm=1983, U= 275 Gb

These two different models are plotted on figure 5 both as could have been drawn in 1960 and with the present data. Both fit the data fairly well to1960 but depart afterwards; Gauss being too high and too late when Hubbert is much better. Such modelling is clearly very sensitive to identifying of the inflection point, which is not easily done in practice.

The famous forecast of Hubbert of 1956 giving a peak in 1970 was based more on his knowledge on discovery and its ultimate than on the shape of the curve (or the inflection point).

Figure 5:

**1c. Discovery – Production link**

It is instructive to compare discovery and production curves. The United States suffers from an archaic reporting practice that recognizes only *Proved *reserves, when for this purpose we need *Proved & Probable *reserves (strictly having a Mean probability), with revisions properly backdated to the discovery of the fields containing them. Accordingly, it has been necessary to adjust the USGS (Attanasi & Root 1994) reported annual discoveries, by assuming a growth factor whereby one reported barrel becomes five barrels 150 years later, as based on MMS findings in the Gulf of Mexico. Thus, for example, 1930 discovery is estimated in 1991 at 8.36 Gb, and the growth factor is 4.3; giving an Ultimate of 9.72 Gb (8.36/4.3*5). Total Mean discovery through 1997 works out to be 208 Gb, based on USGS data and the MMS model. Figure 6 plots the resulting backdated discovery pattern. A trend-line (Excel) and a Hubbert curve are plotted on the basis of a peak of 3.5 Gb/a in 1938 and a half life of 70 years. Shifting the production curve by 33 years gives a remarkably close fit with the discovery curve. But a new cycle, possible from deepwater exploration, may yet arise.

**Figure 6**:

2. Alaska

Production in Alaska comes from a small number of fields (1 supergiant; 3 giants and 4 major fields) in two separate provinces : the Cook Inlet and the North Slope. These circumstances mean that Hubbert modeling should not be expected to work well. Figure 7 plots production which, unlike in the Lower 48, is markedly asymmetrical; and there is not a good correlation between the discovery and production curves. The explanation is probably related to the construction of the trans-Alaskan pipeline, which required a certain throughput before going into operation, precluding the normal build-up of production. A Hubbert curve could however be used to model the post-peak decline, assuming a 1987 peak. The respective estimates for 2010 and 2020 are Hubbert 0.1 and0.03 Gb/a; DOE 0.27 and 0.17 Gb/a; and State of Alaska 0.27 and 0.1 Gb/a. A second cycle, or the addition of natural gas liquids and synthetic oil from gas, could of course change the picture, but production from the North Slope will end when the pipeline is shut, possibly when production has fallen below 0.1 Gb/a..The cumulative production up to 1997 is 13 Gb, and the cumulative Hubbert to the close of production is 4.3 Gb, giving an Ultimate of 17.3 Gb. The Ultimate for US Lower 48, as estimated above, is 190 Gb, giving an Ultimate for the United States as a whole of around 210 Gb. Bartlett (1999) using a single Gauss curve gives a best fit of 222 Gb, which is not too far from our estimate.

**Figure 7:**

3 Former Soviet Union

3a FSU as estimated in 1980 prior to peak

Figure 8 shows production from the FSU up to 1980, which totalled 68 Gb. The derivative curve shows an inflection point (ti) in 1975 at 3.5 Gb/a (Pi) and CPi=47 Gb. Accordingly, peak production should be 5.2 Gb/a (1.5*3.5 =5.2). The c point is at 0.14 Gb/a in 1940 and the peak at 1988. The cumulative total (Ultimate) from the Hubbert curve is 200 Gb. From CPi, U= 4.7*47= 220 Gb, which is higher than the previous estrimate because the aborted cycle by the war.

It is interesting to note that the derivative of annual production, namely the growth, is much smoother than in the US(48) because there was no economic interference as the Soviet oil industry was not driven by economic criteria prior to 1990. The inflection point is accordingly more reliable than in the Unites States, but in fact it is not enough.

**Figure 8:**

3b FSU as estimated in 1997 after peak

Figure 9 shows production up to 1997, amounting to 136 Gb, on which is superimposed the 1980 assessed Hubbert curve (U=200 Gb), showing a poor fit. A best fit Hubbert curve is drawn with a peak in 1985 and an ultimate of 170 Gb. Production was set fall under a normal decline, but was exaggerated by the consequences of the collapse of the Soviet Union from 1990 to 1995. It is now back to the level as predicted by the Hubbert curve, and set to continue to decline.

**Figure 9:**

3c Discovery – Production Link

Cumulative discovery for oil and condensate through 1996 was reported as 290 Gb, but in the same way as reserves are under-reported in the USA, they were over-reported in the FSU (Khalimov 1993). Decline curve analysis of the major fields shows that their true remaining reserves are in fact approximately half as much as reported. This is confirmed by the timing of peak. Actual peak was in 1985 when 91 Gb had been produced, suggesting an Ultimate of about 180 Gb, assuming that peak coincided with depletion midpoint. Accordingly it is necessary to reduce FSU discovery to 190 Gb to obtain its mean discovery, about two-thirds of its reported value. Figure 10 shows a close match between the “mean” discovery (oil & condensate) and production curves after a shift of 17 years. Note that the Gauss curve is very close to the Hubbert curve.

**Figure 10:**

The US-48 and FSU are good examples of countries with a single cycle discovery, which is well modeled by Hubbert curves. But most of countries display multi-Hubbert curves.

4 UK

4a UK- First Cycle: discovery and production as estimated in 1990

The United Kingdom is sometimes quoted by detractors of Hubbert modelling as an example of a country where it does not work because an early peak in 1990 was followed by a saddle and an unexpected second peak. In fact, there were two discovery cycles, which may be satisfactorily modelled with the multi-Hubbert approach. The first cycle of discovery peaked in at 0.9 Gb/a in 1974; to be followed by a second discovery cycle peaking at 1.3 Gb/a in 1984.

Figure 11 plots the two cycles with a good correlation between the discovery and production curves with a shift of 11 years. It clearly indicates that a second production peak was already developing by 1990. The rapid rise at the end of the 1970s obscures the natural inflection point, which cannot accordingly be used for modelling. The explanation is that whereas onshore production can start with the first well, offshore production rises rapidly once the facilities have been installed. In the case of the United Kingdom, several giant oilfields had been discovered after the opening of North Sea grabens in 1968 and were placed on production more or less simultaneously ten years later.

**Figure 11:**

4b UK Second Cycle : discovery and production as estimated in 1998

The eleven year shift between the discovery and production curves, shown on Figure 12, fits well with the present status of production: 17.4 Gb of oil and condensate have been produced so far, and 31.2 Gb have been discovered, of which 5.1 Gb is yet to be developed. The model shows clearly that production is now close to peak and will be already in decline by 2001. Figure 10 shows modelling with two Hubbert cycles respectively peaking in 1985 and 1998. The inflection point for the second cycle was in 1994, but as in the case of the first peak, the rise offshore is too sharp (772 Mb/a in 1993), so it is better to take it at the average 1993-1994, namely 873 Mb/a . The value of the first cycle decline should be removed (100), so the second peak should be around 1015 Mb/a (775*1.31). In fact a good fit for the second Hubbert curve is 1025 Mb/a for 1998. Production from the first cycle will be almost exhausted by 2000, and the second cycle will be approaching exhaustion by 2010. A new cycle could arise in frontier offshore areas to the West..

**Figure 12:**

5 World Production

In 1974, Hubbert modelled world oil production with a low curve based on an Ultimate of 1350 Gb ( peak at 24 Gb in 1991) and an high curve of 2100 Gb (peak 37 Gb at 2000). The present liquids production is plotted on Figure 13 with a Hubbert curve with an ultimate of 2000 Gb (1800 for oil and 200 for gas liquids). It is drawn to match production to 1973, when prorationing was imposed by OPEC. This Hubbert curve shows what production would have been but for OPEC prorationing, having a peak in 1996 at 38 Gb/a. The inflection point would have been at 1979. Obviously, the oil shock of 1973 upset the production trend, and accordingly it is not possible to draw one Hubbert curve for present world oil production.

To make a correlation with discovery, we have corrected the world database (Petroconsultants, USGS and Canadian data) in order to obtain the mean values (about P50) for oil and condensate. Cumulative discovery is about 1800 Gb including the non-developed fields. The annual discovery curve was shifted by 30 years to fit the annual production curve prior to the oil shocks of 1970s. The discovery fits well with the unconstrained Hubbert.

In addition, we have tried to model present production with several cycles. H1 for the peak of 1979, H2 for a second peak of conventional liquids, based on an combined Ultimate 2000 Gb, made up of 1800 Gb for oil and 200 Gb for NGL. We have added a third cycle for unconventional oil with an Ultimate of 750 Gb (Perrodon, Laherrere & Campbell 1998). The sum of these three cycles is fitted to the present data. Our forecast is that the liquids production will peak before 2010. (Our previous forecast of 1998 referred to conventional oil only) Accordingly, pproduction in 2020 would be a little less than now.

**Figure 13**:

6. US Population

In the mid-1920’s, Pearl and Reed modelled US population with the logistic curve (termed Verhulst-Pearl) and predicted the maximum at less than about 200 million. The Hubbert modelling technique is the derivative of the logistic curve and should give better results because population can decrease. Using Hubbert modelling and the inflection point in 1992 when maximum growth occurred when the population was 255 millions, the peak could be forecasted at 380 millions (1.5*255). Taking **Pc** at 10,2, **tc** is 1810 and c is 247, giving the peak date **tm** in 2057. A good fit is provided by taking a **tm** of 2040, a **Pm** of 330 and a **c** of 240, as shown on Figure 14. It is obvious that Pearl in the 1920s could not achieve a good forecast because the maximum growth had not yet been reached. But in the 1930s the Great Depression interrupted the growth of population, and the present curve is lower than before.

We have plotted the US Bureau of Census forecast which is higher, as they predict about 400 millions in 2050 compared with our forecast of only 330. But the US Bureau of Census forecasts an unrealistic fertility rate in 2050 of 2.25 child per woman against 2.0 for Mexico, 2.02 for India, 1.3 for France and 1.6 for Canada.

Figure 14:

7. World Population

As demonstrated in an Oil & Gas Journal article of February 1^{st} , world population can also be well modelled by summing three Hubbert curves representing respectively the industrialized world, the developing world, and a “basic” world lacking birth control. Figure 15 illustrates the application of the inflection point method both to the first two groups and the world as a whole, based on UN statistics (Data on the third group are not available). The inflection points are respectively 1990, 1990 and 1957.

The timing (**ti**) and population sizes (**Pi**) at inflection suggest a peak at 1.5 times the inflection population (**Pi). **But taking a higher peak or a Gauss curve gives a better fit. The Hubbert values are as follows:

ti Pi 1.5Pi 1.65Pi best fit Pm

World 1990 5.3 8 8.7 7.5

Less Developed 1990 4.1 6.1 6.8 6.4

Industrialized 1957 0.9 1.3 1.5 1.2

The Hubbert curves have been drawn separately for the two subgroups and the World. The sum of the Developed and Less Developed is identical with the World curve up to 2050 and diverges only slightly thereafter.

The1998 United Nations forecasts of population in 2050 are classed as medium, medium-low and low cases, of which the medium-low is supported by the Hubbert study. The UN forecasts for Less Developed Countries are 6.5 – **6.7** – 7.7 compared with 6.2 from the Hubbert method. For the Developed Countries the numbers are 0.9 – **1** – 1.15 compared with 1 from Hubbert. In other words, the Hubbert technique agrees with the UN forecast for 2050 in respect of the Developed Countries and is within 8% of the UN forecast for the other countries. In fact, the UN has successively reduced its forecasts since 1990, which was the inflection point when population growth reached a maximum of 90 million per year. It suggests that its forecasts are coming into line with the results of Hubbert modelling.

**Figure 15:**

Conclusions

Hubbert’s modelling technique has been variously applauded and criticised, but the constraints to its application have not been widely appreciated. It works well only where applied to a natural domain, unaffected by political or significant economic interference; to areas having a large number of fields; and to areas of unfettered activity. Hubbert himself worked primarily on the US-48, which had the necessary characteristics to be well modelled by a single cycle.

We may note in passing that many other natural phenomena follow a bell-shaped curve as illustrated for example by the number of mad cows in the United Kingdom.

But the application of multiple Hubbert curves, which was not developed by Hubbert, has much wider application, and has proved to be an exceedingly valuable modelling tool not confined to the oil industry, as population studies confirm.

Hubbert modelling gives fair results only when the past data series has not been disturbed by economic and political factors and when the inflection point has been passed. Good results are achieved only when the past data series has passed the peak and when there is one single cycle for discovery.

The procedure of relating the discovery curve with the subsequent production curve after a time shift of a certain number of years is an indispensable tool to obtain a reliable forecast on production. The discovery curve itself, which is commonly volatile, needs to be smoothed. The time-lag is of varying duration. It is 33 years for the USA; 11 years for the United Kingdom; 17 years for the Former Soviet Union; and about 30 years for the world as a whole, although affected by the impact of the Middle East swing producers. This shift gives a reliable trend for coming production except in the unlikely event that a major new cycle of discovery arises.

There is no doubt that Hubbert modelling is a valuable tool but like all tools, it needs to be used properly for the right job. It is important to understand its strengths and weaknesses and to know when to apply single or multi-curve approaches. Knowing backdated annual discovery is a must, but it has to be the annual discovery based mean not proved reserves. Hubbert’s model only from production data is an insufficient tool.

References:

-Al-Fattal S.M. & Startzman R.A. 1999 “Analysis of Worldwide Natural Gas Production” SPE 57463

-Attanasi E.D. & Root D.H. 1994 “The enigma of oil and gas field growth” AAPG 78/3 March, table 1

-Bartlett A.A 1999 ,”An Analysis of U.S. and World Oil Production Patterns Using Hubbert-Style Curves.” *Mathematical Geology.*

-Hubbert M.K. 1974 “on the Nature of Growth” http://www.technocracy.org/articles/hub-gro.html

-Laherrere J. 1999 “World oil supply -What goes up must come down: when will it peak?” Oil and Gas Journal Feb.1 p 57-64 and in Letters of OGJ March 1, March 15 & March 29

-McCabe P.J. 1998 “Energy resources- Cornucopia or empty barrel?” AAPG bul V82 n°11 Nov. p2110-2134

-Perrodon A., Laherrere J., Campbell C.J. 1998 “The world’s non-conventional oil and

gas” Petroleum Economist, March, report 113p