King Hubbert in his famous paper of 1956 ("Nuclear energy and the fossil fuels," API drilling and production practice) describes the future production curve of the US 48 states as a symmetrical (or bell-shaped) curve without giving any equation. As everything which goes up must come down (except if the velocity is high enough to escape the earth), every curve has to come down one day and a symmetrical curve works well most of the time.
Later, he defined what is now known as the Hubbert curve as the derivative of the logistic curve.
The logistic curve was introduced by the Belgian mathematician, Verhulst, in 1845, as a law of population growth, and it was used extensively by the biometrician, Raymond Pearl . It is now called the Pearl-Reed curve, showing population p growing over time t (years) from zero to a maximum U in any geographical area:
p = U/(l+exp(a-b×t)) or p = U/(l+exp(b×(t-tm))
...where tm is the year in which the curve shows an inflexion point or where the derivative shows a maximum (peak time), or
ln((U-p)/p) = -b(t-tm)
It is easy to study this function by linear regression to find the values b and tm.
The logistic curve for cumulative production CP is:
CP = U/(l+exp(-b(t-tm))
The following graph displays the logistic curve (model of cumulative production) and its derivative: the Hubbert curve (model for annual production) with a scale multiplied by 10 (pink). The integration of the Hubbert (green) coincides with the logistic curve (blue).
The equation for the Hubbert curve for the annual production P (being delta CP/delta t) is simple when related to peak annual production Pm occurring at year tm:
P = 2Pm/(l+COSH(-b(t-tm)))
...where the constant b is equal to 4Pm/U, but also to 5/c, c being the half width of the curve on the time axis from peak time to the time when production has fallen to a very low level (0.01 Pm), as ln(100) = 5±.
So, a quick way to compute the ultimate of a Hubbert curve is:
U = 0.8Pm×c
The area below the Hubbert curve is equal to 80% of the triangle containing the Hubbert curve (area = 2c×Pm/2 = Pm×c)
It is interesting to compare the Gaussian bell-shaped curve with a parabola (stone thrown in the air).
The normal curve is generally called the Gaussian curve, although it is also known as the Laplace-Gauss curve or the de Moivre-Laplace-Gauss law, as Laplace, who was influenced by de Moivres work of 1718, discovered the law in 1780 but did not publish it until 1812. Gauss issued the present equation in a more generalized manner as early as 1821 - and it has thus remarkably taken more than a century to properly formalize it. The normal law represents the probability of randomness, as the sum of a very large number of small independent causes, giving the probability (p) In this distribution, the mode (most likely equals the peak of probability), the median (equal number on either side) and the mean (equal area below the curve on both sides) are the same. As applied to oil production, a bell-shape curve is:
P = Pm×exp(-(t-tm)2/2s2)
A comparison between a Hubbert curve and a Gauss curve with the same peak shows that the difference is quite small, while a Hubbert curve is close to a parabola on the upper half part of the curve.
Hubbert predicted the cumulative production of oil over time in the United States, correctly forecasting a peak of annual production in the early 1970s. He was criticised at the time on the grounds that an individual oilfields production profile is assymmetrical whereas his profile was (almost) symmetrical. But in fact it occurs that the present oil production profile for the US 48 States is strongly symmetrical. This symmetry occurs since the sum of skewed asymmetrical curves from a large number of individual oilfields tends to be symmetrical, because the sum of a skewed probability distribution of a large number (>30) of events trends towards a symmetrical normal distribution.
This relates to the Central Limit Theorem, a well-known tenet of statistics, that explains how the distribution of the mean from a skewed distribution tends to become normal as the sampling size increases. This symmetry reflects also the correlation with the discovery curve [Ivanhoe, King Hubbert Center letter 97/1], which is almost symmetrical because of the cyclic status of exploration and the law of diminishing returns.
The Hubbert curve works very well for the US 48 States production which is based on the sum of more than 25,000 oilfields. It is not perfectly smooth because of political factors, displaying three anomalies from a perfect curve, of which the first anomaly from the model was during the depression of the 30s and the last two being shoulders at the same level, at around 7 Mb/d: the first at the end of the 1950s, possibly reflecting the end of proration; and the second in the early 1980s, possibly reflecting a sharp increase in oil price.
The production of the FSU is very similar to the US (48) production, being a country with many basins and fields and a continuous effort in exploration, but with a steeper rise and decline. The Hubbert model agrees very well except for the last four years where economic and logistic problems have prevented the FSU from displaying a normal decline.
The FSU production curve correlates very well with the 15 years shifted (see Multicycles below) discovery curve, as it is for the world outside the swing producers (the 5 largest producers of the Middle East which do not produce at full capacity).
If we model an annual discovery curve and a similar 10 year shifted annual production curve, we obtain the following remaining reserves (cumulative discovery minus cumulative production) and the "famous" (but meaningless) R/P ratio.
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 done with several cycles. We have modeled every production country in the world and every one can be modeled with at most three or four cycles (thus the term "multi-Hubbert"), just as a sound can be modeled with a few harmonics (Fourier analysis). All the modeling was done with correlation of annual production and annual discovery shifted by some years to obtain the best fit. It is obvious that there are exploration cycles followed by production cycles.
France is a good example of two symmetric cycles with similar slopes. The Netherlands displays first two cycles with the first one onshore being much smoother than the second one which is offshore (everyone knows that offshore reserves are produced faster than onshore) and now there is a third cycle beginning.
Notice the good correlation between the oil production curves (dark) and (both shifted by 7 years) the discovery curves (green).
We have accordingly looked for other cyclical events in nature to see if they can be modeled with multi-Hubbert cycles
We have shown in "Parabolic fractal distributions in Nature" that urban agglomerations gather in the same way as galaxies or oil reserves.
Populations of countries change with time, most of the time growing, but past history have shown that civilizations, as stars or dinosaurus or humans, rise and decline, in number or in power, as did the Inca, Mayan, Greek, and Roman civilizations. When the fertility rate of a country is less than 2.1 children per woman, the country's population will diminish: it takes some time to peak because of the pyramid of age, but the population will peak and decline. This rise, peak and decline can be modeled with several curves.
Bourgeois-Pichat  said that the industrial population will peak soon with a symmetrical curve (see http://www.ined.fr/publicat/pop_et_soc/pes321/pes3212.html where the French and European populations are forecasted by the European Demographic Observatory until 2050, peaking around 2020-2030) and that the developing countries (which try to follow the industrial countries) will follow the same pattern 40 years later. I modeled the world population with three Hubbert curves: the industrial population, the developing countries' population and the basic population (which does not care for civilization and comfort) as shown:
We have found a graph of the cod landings in the North Atlantic displaying several symmetric cycles. It is easy to model it with three Hubbert cycles; a fourth one is possible if fishermen are reasonable!
The Multi-Hubbert model is the best model for studying the evolution over time of production activity (human or other physical activity) being the sum of a large number of independent events, absent significant constraints from economics or regulation. It is simple and up to now I have not found any event which cannot be modeled with Hubbert cycles.
I will be glad to receive any contradictory example and I will respond accordingly.
In nature, the size of an object cannot be negative. The law of normal distribution has no difficulty with this constraint when the standard deviation is small compared with the mean, but when the standard deviation increases, a zero limit occurs. In order to avoid negative values, it is necessary to skew the bell-shape to the left, by applying the normal law to the logarithm of the size. It is called the Lognormal Law, where:
The Lognormal Law corresponds with the notion of proportional effect: dy = dx/x . Considering, for example, the wealth of a country, it is as easy for someone with a billion dollars to earn a million dollars, as it is for someone with a million dollars to earn a thousand dollars.
The Lognormal Law can be well applied as a measure of the probability that the reserves in an oilfield are greater than a certain volume. It is in fact the recommended probabilistic approach, where the minimum value is taken for a probability around 85-95% and represents what are termed Proven (Proved) Reserves, or P; the mode (the inflexion point) or the median (50%) or the mean (the expected value or the area below the curve) is taken as Proven + Probable, or 2P; and the maximum value (around 5-15%) is taken as Proven + Probable + Possible, or 3P . The SPE/WPC  has defined, for the probabilistic approach, proved as 90% probability, proved + probable as 50% probability and proved + probable + possible as 10% probability.
In a lognormal distribution, where the minimum case has a 95% probability and the maximum case has a 5% probability,
the mean is equal to (mini + mode + maxi)/3
It is interesting to compare the lognormal distribution which works very well for the distribution of the probability of existence of reserves for a single field, to the parabolic (and linear) fractal distribtuion which is used to model the distribution of reserves in a basin. Lognormal distribution works well when the mean (expected value) has a meaning, i.e., the reserves of the field, but it does not work well when the mean has no physical meaning - the "mean reserves" of a basin is only a statistical value and does not represent any physical field. Furthermore when undiscovered fields are studied in a lognormal distribution model of a basin, if the same law is kept, it means that large fields should be as numberous as small fields, which is not true. Lognormal distribution is too pessimistic on small objects (and too optimistic on large objects). Linear fractal (Pareto-Mandelbrot) distribution is too optimistic on small objects (going sometimes to a divergent series).
In the following example (see Parabolic Fractal Distributions in Nature), lognormal, linear and parabolic fractal are fitted in order to match best the 200 largest objects of a distribution, starting with the largest being 1,000, the number of objects larger than 10 is 850 for lognormal distribution, 5,000 for parabolic fractal and 40,000 for linear fractal!
Most studies of reserves distribution are based on a lognormal distribution (see above). The distribution of probability for the reserves of a single field to be greater than a certain size follows a lognormal distribution, where the expected value ("mean") has a physical meaning (i.e., the reserves of the one field). But the distribution of the reserves of a large number of fields in a basin or a country (frequency of sizes) does not follow a lognormal distribution, as the "mean" has not a physical "meaning", but a statistical one. Pareto recognized long ago the 80/20 rule. 80% of the wealth is held by 20% of the people, 80% of health expenditures are spent for 20% of the patients, in Europe 80% of the subsidies go to 20% of the farmers. Nature is self-similar! Mandelbrot rediscovered the Zipf law (as did others such as Pareto, Hausdorff, ..) and very cleverly found a good name for it, the "fractal". But Mandelbrot stayed with a linear fractal, despite the fact that his best example was the oil reserves distribution based on only 10 data points. From the Petroconsultants' database of the world outside North America for oil and gas fields, I have displayed several thousand points on a log-log size-rank (rank of decreasing size) which is not linear but curved and close to a parabola. When the slope of a linear distribution is close to the horizontal, the quantity of reserves is infinite, which is impossible! Self-similarity (when restricted to a certain domain) rules Nature!
Another fascinating fact is what I call the King effect. Some distributions display an anomalous number one object, just as a new King, to be sure not to be challenged, kills the barons and takes their wealth in order to be above the crowd. The King effect is found, for urban agglomerations in France (Paris) and in the UK (London) but not in the USA, and for some oil distributions, such as the Ghawar field in the ME or the ME in the world!
As highlighted above, a note to the French Academy of Sciences on this matter was published in 1996, "Distribution of parabolic fractals in Nature.
Anyone interested in the subject is welcome to send me e-mail, jean.laherrere at wanadoo.fr.