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Page added on September 8, 2015
Why Hubbert’s Peak Oil Theory Fails
There has been a growing interest in the topic of peak oil theory for the past few decades. The interest emerged after the 1956 M.K. Hubbert peak oil theory successfully forecasted the peak year of crude oil production in the United States. Hubbert used a quantitative technique -Logistic Growth Curve or Bell-Shaped Curve (Fig.1)- to model the conventional crude oil production profile of the lower 48 states in the United States (excluding Alaska). He predicted that crude oil production would peak in the late 1960s or early 1970s. According to Heinberg (2003), many economists and oil companies dismissed his prediction at that time. However, when United States oil production peaked in 1970, his theory proved correct. Thereafter, many industry observers used the Hubbert Curve to forecast the oil production peak year in different countries.
Fig.1. Hubbert’s 1956 prediction of world petroleum production rates (Hubbert,1956).
Although Hubbert peak theory garnered a lot of credibility after its success in predicting the peak of U.S. oil production, it started to fail after 1995 when the actual crude oil production decline rate was totally different from Hubbert’s prediction. Fig.2 shows the deviations of the actual crude oil production rate from what Hubbert predicted in his theory. The main reason for such discrepancy in the actual and predicted rate of crude oil production reduction according to Cambridge Energy Resource Associates (IHS CERA) (2007) is the fact that Hubbert did not consider the likely continuous resource growth, application of new and advanced technology, fundamental commercial factors, and the serious affects of geopolitics on production.
Fig.2. United States oil production excluding Alaska and Hubbert high estimate for the US (Laherrère, 2003).
II. CAUSES OF FAILURE:
1- Resource Growth
In his reasoning, Hubbert did not incorporate the continuous growth in resources. The methodology he used was based on imputing decline curves against currently proven reserves and asserting that the game is over. According to Heinberg (2003), Hubbert used an estimated ultimate recoverable reserve ( EURR ) between 150 to 200 billion barrels of oil. Based on this value, he estimated the U.S. oil production to peak at 3 billion bbl/year ( 8.2 million bbl/day ) anytime between 1965 to 1972 as shown in Fig.3. Hubbert theory succeeded in predicting the peak time, but there were few deviations in the actual production rate form what he predicted.
Fig.3. Hubbert’s Fig. 21 shows his 1956 forecast for USA oil production (Hubbert, 1956).
First of all, U.S. production rate peaked at 10 million bbl/day which was approximately 1.4 million bbl/day higher than what Hubbert predicted as shown in Fig.4. In addition, six years after peaking in November 1970, the U.S. production rate encountered its first deviation from Hubbert’s prediction. Oil production rose to 9.14 million bbl/day forming a secondary peak in the period between 1977 to 1986 as shown in Fig.4.
Fig. 4. Actual monthly USA oil production ( Data Source: EIA )
Furthermore, the second deviation from Hubbert’s prediction is the incline formed since late 2008 until the present time. In 2008, the U.S. produced five million bbl/day and just a year after that in 2009, U.S. production rate begun to rise until it averaged 8.3 million bbl/day in 2014.
Both deviations were due to new discovery and production of oil from the Prudhoe Bay, Kuparak oil fields in the north of Alaska and production from Gulf of Mexico along with unconventional crude oil coming from the Bakken oil play in North Dakota respectively. These deviations -rise in production- prove that the usefulness of Hubbert’s theory is limited because it requires a knowledge of totality of oil reserves in existence which is in a continuos growth leading to increase in production. Fig.5 shows the growth in the world oil reserves.
Fig. 5. World proved oil and gas reserves ( Data Source: EIA )
Although much of the easy oil -mostly located onshore and offshore in shallow water- was discovered and drilled, there are still many potential untapped reserves in different parts of the world. Therefore, predicting production peak and future trend without taking such resource growth into consideration leads to a misleading prediction.
2- Application of New and Advanced Technology
Similarly, Hubbert’s peak theory did not take into account the application of new and advanced technology that made the process of extracting oil cheaper and more efficient than it used to be. The application of such technology resulted in production increase. For instance, oil that 60 years ago may have been difficult and uneconomical to be extracted, could currently be recovered due to higher accuracy of surveys to locate the oil, advanced drilling technology and improved drilling methods.
Furthermore, the application of improved and new technology such as horizontal drilling and hydraulic fracturing have led to the production of shale oil in the United States and other countries that for years have been uneconomical to extract. This leap forward in oil-field technology -used for the drilling and extraction of once uneconomical to produce oil- is the main reason for rising the U.S. oil production rate in 2009. It clearly shows the critical role technology and innovation play in the increase of oil production which was not considered in Hubbert’s peak theory.
3- Economic Factors
Hubbert’s method failed to incorporate economic factors that directly influence productive capacity. The theory ignores the impact of two major drivers of production such as demand and price dynamics. It also fails to comprehend the economic factors that can direct the producing company and government national strategies to control the production rate of crude oil. All these factors play a crucial role in determining the production rate of any country and the world in general.
4- Geopolitical Impact
Hubbert peak theory also did not take into consideration the impact of geopolitics on oil production. For instance, a major war in the most oil-rich part of the world -the Middle East- for example, would definitely skew the Hubbert’s peak theory’s graph dramatically changing the rate of production. This is also one of the examples of Hubbert’s peak theory failure to acknowledge the impact of external variables on oil production rate.
III. CONCLUSION
Despite the remarkable contribution, M. King Hubbert’s peak theory fails because it did not incorporate the impact of resource growth, technology advancement and external variables such as geopolitical and economic events on production. The Hubbert’s model may work well under the assumption that everything else other than currently proven reserves remains equal, but of course in real life they never do. The restrictions we face today are more technological and economical than the amount of oil in the ground. We are limited by how inventive we are to find new resource and extract what is available.
70 Comments on "Why Hubbert’s Peak Oil Theory Fails"
Makati1 on Tue, 8th Sep 2015 7:28 pm
Timing…timing…timing. He could never have seen the drastic measures taken to pretend all is well and nothing has peaked. Such things as frantic fraking, ‘liquids’, ‘moon shine'(biofuels), banks covering oil debts at almost zero interest, etc. Real oil peaked about on his schedule both in the Us and the world. We just don’t want to admit it.
That last paragraph is pure unadulterated bullshit. The rats are packing their bags and about to leave the sinking oily ship. EROEI is ignored here also, but what can you expect from an oil pimp getting desperate?
Robert Tegir on Tue, 8th Sep 2015 7:34 pm
Hubbert’s prediction was for the lower 48 states. Never included Alaska. The peak of conventional oil was likely 2005. Since then, ‘liquids’ of all sort have been added in, not true conventional oil. Yes, fracking has changed things, but fracked oil declines are horrendously high rates.
The annual decline rate of all the world’s oil fields soon won’t be met with ‘new’ sources of oil soon. Most major projects now are getting canceled or slow rolled.
It was those who ‘extended’ Hubbert’s theory to ‘worldwide’ who did not factor in the short life of frack’ed oil…..just delaying the peak…but the peak is still maybe just 5-10 years out.
BC on Tue, 8th Sep 2015 7:39 pm
Per capita. Period. Said. Done. Print it.
Davy on Tue, 8th Sep 2015 7:41 pm
Oil pro is a oil industry promotional site of course MK Hubbert analysis failed.
Don S on Tue, 8th Sep 2015 7:52 pm
How wonderful to be living on a planet with an infinite source of oil! Thank you, Oil Pro, for opening my eyes!
Boat on Tue, 8th Sep 2015 8:53 pm
BC,
If you don’t use oil why would the price of oil affect you. Ethiopians have 6 cars per 1,000. Per capita they make less than $300 income per year.
This is why per capita is just a buzz word. Meaningless except to those who use a lot of oil.
dissident on Tue, 8th Sep 2015 8:58 pm
There is no symmetry between the left and right side of the all time oil production curve. There will be no gradual decline in demand to mirror the increase of demand after the mid-1800s. In the real world, the future tail of the bell curve is shifted towards the peak giving a plateau and not a peak. This shift is due to ever increasing demand prompting more and more scraping of the bottom of every barrel including developing non-conventional sources that were deemed uneconomical in the past. The end result of this shift is that there will be a cataclysmic drop in production in the near future instead of a gradual ramp-down.
Hawkcreek on Tue, 8th Sep 2015 10:28 pm
Hubbert’s Peak Theory did not fail. His prediction of the timing for the peak may be incorrect, but only an idiot will claim that there will not be a peak in production at some time. I will bet that the peak will end up being much closer to Hubbert’s prediction than it will be to the infinite time span that this article seems to claim.
GregT on Tue, 8th Sep 2015 11:06 pm
“If you don’t use oil why would the price of oil affect you.”
Obviously it wouldn’t, now would it. They didn’t use oil before, and they still won’t use oil in the future. They aren’t changing the per capita figure one iota.
For those of us who do use oil, for pretty much every single thing in our day to day lives, per capita means everything. You would be one of those people Boat.
JV153 on Wed, 9th Sep 2015 1:03 am
It’s the price marker that reinvigorated drilling for oil, so Hubbert was wrong – it’s pretty obvious marginal reserves would be drilled at high oil prices.
However, I suspect Hubbert might have omitted this on purpose.
GregT on Wed, 9th Sep 2015 1:08 am
“I suspect Hubbert might have omitted this on purpose.”
Hubbert did not ‘omit this’. His analysis was for the lower 49 conventional reserves. He was very clear that oil sands and tight oil would be exploited, that they could push the peak out, but that they would not be a game changer.
GregT on Wed, 9th Sep 2015 1:14 am
Page 24 of his report here:
http://www.hubbertpeak.com/hubbert/1956/1956.pdf
charmcitysking on Wed, 9th Sep 2015 1:51 am
It’s not a theory.
apneaman on Wed, 9th Sep 2015 2:24 am
GregT, another one trying to debunk something they have never read. Same as AGW, evolution, etc. The flip side of that are the ones who never read any Adam Smith, except the few cherry picked quotes handed to them. That’s how it works now, hardly anyone reads source material, but everyone’s an expert. Just go straight to the debunking source that confirms your pre existing bias. Remember “Cliff Notes”? Well the internet is the Cliff Notes of Cliff Notes. There is a website with short comeback answers to every position there is. My favorite is “Answers in Genesis” it’s for Christians. You know the folks who say they believe in the bible, yet never made the time to read it, but need a comeback answer to defend their beliefs. Apes love their stories, even when they have never actually read them.
GregT on Wed, 9th Sep 2015 2:47 am
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GregT on Wed, 9th Sep 2015 3:04 am
Hmmm,
I wonder who might be paying their tab? My guess? It’s more than likely not the Union of Concerned Scientists.
http://www.ucsusa.org
yoananda on Wed, 9th Sep 2015 3:45 am
yes hubbert was wrong, but technology and finance will ultimately fails and, his theory will prove right a few year later. Maybe THIS year !
comicrelief on Wed, 9th Sep 2015 4:15 am
Hubbert needs no apologists. Conventional peaked in 2005 the rest is smoke and mirrors, just the thrashings of a dying beast.
Makati1 on Wed, 9th Sep 2015 6:14 am
comicrelief, you got it in one!. NET oil energy peaked long ago. THAT is the number that counts, not how many barrels of some flammable liquid is produced.
rockman on Wed, 9th Sep 2015 6:21 am
And a reminder again to all that haven’t read Hubbert’s original work; he specifically states that his projections are based on known (and rather mature) trends. He clearly states that his analysis does not include the shales, DW or any other trend yet to be exploited. His projection of the KNOWN TRENDS has been proven to be very accurate. The confusion comes as a result of some folks mischaracterizing (often intentionally IMHO) to push their own agendas.
OFT on Wed, 9th Sep 2015 9:19 am
Just reading that paper (1956) and note that Hubbert’s assessment (page 27) says:
‘If the world should continue to be dependent upon the fossil fuels as its primary source of industrial energy… On the basis of the present estimates of the ultimate reserves of petroleum and natural gas, it appears that the culmination of world production of these products should occur with about half a century’
In my book that suggests a peak about 2006, which would tie in to a number of the statistical charts regularly seen on these pages.
Another Northwest Resident on Wed, 9th Sep 2015 9:33 am
“M. King Hubbert’s peak theory fails because it did not incorporate the impact of resource growth”
Oil grows?
I had no idea.
I wonder what kind of fertilizer it needs to grow. I’d imagine a healthy dose of bullshit will do the job.
BobInget on Wed, 9th Sep 2015 9:39 am
Comments here are more informative then ‘Oil Pro’s’ hand-out.
But, not as amusing.
Nony on Wed, 9th Sep 2015 10:41 am
Rockman:
1. No, no he didn’t. Not in 1956. I read that thing cover to cover and a lot more carefully than you. Mudlogger.
2. But the cornie argument has LONG been that new developments WILL occur. It’s conceding the argument if you have that caveat.
3. Read the part of the 1956 paper where he talks about multiple peaks in Indiana. He concedes that new developments cause new peaks but says HE DOUBTS THAT WILL HAPPEN AGAIN.
—
His whole approach was FUBAR. And he got his ass kicked. Just butt-slammed. By the growth in oil and natural gas. Face facts, old man.
Makati1 on Wed, 9th Sep 2015 11:32 am
What growth? Are you talking gross or net, Nony? There is HUGE difference in net energy provided at the consumer end in the last 20-30 years and it has all been down hill. Negative. Shrinking. Contracting. De-growthing. Whatever you want to call it. It’s ending.
GregT on Wed, 9th Sep 2015 12:40 pm
Read it again Nony,
Specifically pages 19, 20, and 24.
http://www.hubbertpeak.com/hubbert/1956/1956.pdf
Nony on Wed, 9th Sep 2015 12:49 pm
I already read them. We discussed them.
Lather, rinse, repeat.
Nony on Wed, 9th Sep 2015 12:59 pm
Maki: gross, obviously. Read the 1956 paper. It is about reserve estimation and depletion. Energy used to extract it is irrelevant (good or bad). The Hubbert thesis is that there is so much oil in the ground and when you use it up it’s gone (for millions of years). It doesn’t matter if you have to use a lot or a little energy to get it out (for that problem). Just how much is in the dirt and how fast you take it out.
P.s. The fundamental issue is how do you estimate how much is in the ground. Hubbert’s work relies on earlier papers by Weeks and Platt that characterize the recoverable oil in all the sedimentary basins. But those papers are NOT oil in place. So changes in what is recoverable affect the outcome. To give him credit*, Hubbert does discuss this concept wrt EOR improvements. But still missed several other improvements.
P.s. It’s interesting how Rock simultaneously says nothing ever changes in the oil business and then says Hubbert can’t be blamed for not anticipating new developments.
GregT on Wed, 9th Sep 2015 1:09 pm
Hubbert called the peak in continental US conventional oil production in or around 1973. He was also correct in his call for the peak in global conventional oil production in or around 2006.
All the rest of your diatribe is a complete waste of time and space Nony. You are being willfully ignorant of reality. As of ~2008, the world has entered a new era. An era of declining cheap affordable oil. The economic fallout is plain for all to see. Your constant idiotic cheerleading is beyond annoying. You are a complete fool Nony.
Dubya on Wed, 9th Sep 2015 4:14 pm
Ah yes, the old “how much oil is there” discussion. Well, evidence is there is more than enough to thaw the arctic ice cap, screw up the Antarctic circulation and ice cover, slow the Gulf Stream and northern jet streams.
There is certainly enough to push the population of humans over 10 billion if we try really hard.
We have managed to kill half the wild animals on earth including making hundreds of them extinct in the past few decades, so that has freed up lots of primary productivity.
We have used megatons of oil killing off 90% of the large/commercially valuable fish, so I am so happy to hear, Nony, that we have enough oil to acidify and deoxygenate the ocean to eliminate most of the other contaminating life & allow H2S bacteria to take us to a lovely purple Canfield ocean.
Ah yes, petroleum, the lifeblood of the economy, the source of infinite growth on our finite planet. I assume that Rex & Steve & Tony and you & me are saving up to emigrate to a new planet now that we have made our money here.
Dubya on Wed, 9th Sep 2015 4:28 pm
Sorry, a bit short sighted of me not to point out that now we have infinite oil all the 100 million or so refugees from Syria can join us here in North America, buy big trucks & McMansions located at least 10 meters above current sea level & invite another billion or so rice farmers from Bangladesh and the Indus & Mekong deltas to join us in our Limitless Petroleum Nirvana.
I have a business proposition; let’s build lots of petroleum powered refrigeration units and stick them along the arctic coast – this will prevent the permafrost from melting so that we can keep releasing all the fossil CO2.
shortonoil on Wed, 9th Sep 2015 4:34 pm
Between 1960, and 1999 the EIA reported accumulated world production of 744.14 Gb. That figure deviates from the accumulated production provided by the logistic curve that Hubbert proposed by 0.06 Gb, or 0.0001%.
Around the year 2000 world production began deviating from Hubbert’s logic curve. That was the year that the EIA decided to start including bitumen production, NGLs, high test camel pea, and anything else that might resemble a liquid hydrocarbon. They knew that the Peak was immanent. From 2000 to 2005 their reported accumulated production of 128.08 Gb deviated from the curve by 4.33 Gb, or 3.4%.
It is likely that the petroleum industry realized that if it became generally known that the Peak was about to occur, the impact on their stock prices would be extreme. Obviously, we should fire the EIA, dump Hubbert’s projections, and hire oilpro.com to give us reliable, unbiased information about petroleum production. Hubbert apparently didn’t have a clue!
http://www.thehillsgroup.org/
Boat on Wed, 9th Sep 2015 5:02 pm
short,
We won’t have a clue when peak oil will hit until it happens and then look back in hind site.
Oil loved the Peak oil talk. If it wasn’t for the geopolitics in the middle east the $100 oil would have never happened. Fracking/tar sands, would have been a tech for the future. Turns out the shortage was conflict, not lack of oil.
Boat on Wed, 9th Sep 2015 5:12 pm
Duby,
Why do you have this need to provide for every human. If you want oil just go buy it. Get an education, develop a skill the market needs earn an income and spend it as you choose. This is how the working world operates.
Or you can choose to live like the Ethiopians..6 cars per 1,000 people, The average earns less than $300 per year. Not much health care, few bills, no commute. Very few taxes, very few computers in fact one modern refrigerator uses about the same electricity as 8 Ethiopians use in a year. There are always choices.
shortonoil on Wed, 9th Sep 2015 5:23 pm
“We won’t have a clue when peak oil will hit until it happens and then look back in hind site. “
Obviously, we should fire the EIA, dump Hubbert’s projections, and hire Boat to give us reliable, unbiased information about petroleum production.
In 1970 a barrel of crude delivered 5.3 million BTU to the economy. In 2015 it is delivering 2.7 million. The shortages have already begun!
http://www.thehillsgroup.org/
Don Stewart on Wed, 9th Sep 2015 5:32 pm
See this interview with Graham Turner, a physicist who works on technology. He became interested in the Limits to Growth model, and put together an update.
Particularly notice, at about the 25 minute mark, how many of his words are reminiscent of BW Hill’s model.
http://www.resilience.org/stories/2015-09-09/limits-to-growth-an-update
Don Stewart
marmico on Wed, 9th Sep 2015 5:57 pm
Obviously, we should…dump Hubbert’s projections
Absolutely.
1. Cumulative world oil production through 2015 is greater than Hubbert’s ultimate potential production of 1250 billion barrels.
2. 2015 annual world production is 2 times higher than Hubbert’s 12.5 billions of barrels per year at the 2000 estimated peak.
GregT on Wed, 9th Sep 2015 6:22 pm
Thanks for the link Don.
Sounds like Turner has come to the same conclusions as most of us here have.
jjhman on Wed, 9th Sep 2015 7:29 pm
Question #1
Am I the only one that noticed that the author’s #1 reference was Richard Heinberg?
Heinberg must be just about the #1 doomer on the whole planet.
Question #2
Why beat up on Hubbert for work done in 1956? That’s about as useful as beating up Newton for missing the Lorenz transform.
Boat on Wed, 9th Sep 2015 7:58 pm
jjhman,
Some of the doomers live and spread disinformation. Follow the conversation and jump in.
ghung on Wed, 9th Sep 2015 8:22 pm
Boat said: “Some of the doomers live and spread disinformation.”
Specific examples?
jjhman said: “Heinberg must be just about the #1 doomer on the whole planet.”
More doomer than Guy McPherson? Than James Hansen?
Comparisons? Enquiring minds want to know!
GregT on Wed, 9th Sep 2015 8:22 pm
Boat,
If you feel that “some of the doomers live and spread disinformation”, by all means provide credible links that dispute that “disinformation”.
I for one, am all ears.
Davy on Wed, 9th Sep 2015 8:33 pm
Boat when you open your mouth it reeks of a distorted world view. You remind me of the ugly Americans that give our country a bad name when traveling the world. I am not saying you are one just you have those tendencies. The world is tired of the traditional conservative American flag waiving bullshit. Wake up and get a life please.
apneaman on Wed, 9th Sep 2015 8:36 pm
jjhman if you think Richard Heinberg is anywhere near the #1 doomer on the whole planet then you have not been paying attention. Not even close. Doomer lite at best. Try conservation biologists and professor emeritus Guy McPherson – he says humanity will be extinct by 2030. Or how about the late Frank Fenner, microbiologist who lead the scientific team that eradicated smallpox.
Frank Fenner sees no hope for humans
FRANK Fenner doesn’t engage in the skirmishes of the climate wars. To him, the evidence of global warming is in. Our fate is sealed.
“We’re going to become extinct,” the eminent scientist says. “Whatever we do now is too late.”
http://www.theaustralian.com.au/higher-education/frank-fenner-sees-no-hope-for-humans/story-e6frgcjx-1225880091722
Or wildlife biologist Neil Dawe
Web of life unravelling, wildlife biologist says
“Wildlife biologist Neil Dawe says he wouldn’t be surprised if the generation after him witnesses the extinction of humanity.”
“It’s happening very quickly,” he says.
“Economic growth is the biggest destroyer of the ecology,” he says. “Those people who think you can have a growing economy and a healthy environment are wrong. “If we don’t reduce our numbers, nature will do it for us.”
He isn’t hopeful humans will rise to the challenge and save themselves.
“Everything is worse and we’re still doing the same things,” he says. “Because ecosystems are so resilient, they don’t exact immediate punishment on the stupid.”
https://web.archive.org/web/20150419201929/http://www.oceansidestar.com/news/web-of-life-unravelling-wildlife-biologist-says-1.605499
Notice the pattern? All biology people. You know, the ones who know more about how life works than anyone else. There are a growing number of their colleagues sharing a similar dire message, but what the fuck do they know compared to a couple of intellectual GIANTS like you and Boat?
peterev on Wed, 9th Sep 2015 8:37 pm
Note that Hubbert in 1956 predicted the world would peak in conventional oil production in the year 2000. The actual peak for conventionally drilled wells peaked in 2005 according to the Exxon Mobile 2015 Energy Outlook: A view to 2040. Between 1956 and 2005, we had two oil embargoes that curtailed oil imports to the USA. All in all, not too shabby a prediction.
peakyeast on Wed, 9th Sep 2015 8:38 pm
Hubbert was not incorrect. The writer has obviously not read Hubberts report and the conditions to the prediction – or perhaps he fails to understand what he reads…
I think the author should start caring about other people and just STOP reading and writing – It seems to me that he is causing more harm than good thru sheer stupidity – to himself and others.
It would be the kind thing to do, please.
marmico on Wed, 9th Sep 2015 8:41 pm
All in all, not too shabby a prediction.
That’s a pile of crap. According to Hubbert, not a single drop of oil should be produced in 2015 yet 79+ million barrels per day are supplied.
peakyeast on Wed, 9th Sep 2015 8:45 pm
We do have a nice recent easily understood example of a species that grossly exceeds carrying capacity and it does show the possible suddeness of a collapse brought on by deprivation of an essential substance.
enter Sct. Matthews island.
First year: 6000 deers
Third year: 40 deers
GregT on Wed, 9th Sep 2015 8:45 pm
According to marmico,
He thinks that Hubbert predicted that not a single barrel of oil would be produced in 2015. Of course this is simply more marmico stupidity, and in no way reflects anything that Hubbert said or predicted. As usual marmi is full of it.
shortonoil on Wed, 9th Sep 2015 8:50 pm
“Question #2
Why beat up on Hubbert for work done in 1956? That’s about as useful as beating up Newton for missing the Lorenz transform.”
You are actually more correct in your observation than you may realize. One of the complaints regarding the use of the normal logistic function is that it puts maximum production (Peak) at 2001. Conventional crude Peaked in 2005. This has been used repeatedly to demonstrate that Hubbert’s assumption that production followed a probability distribution described by a logistic curve is incorrect.
In fact the distribution is a skewed logistic curve:
http://www.thehillsgroup.org/depletion2_013.htm
A skewed logistic curve has no explicit mathematical equation to describe it. It can only be solved using Quantile Statistics that at the time of Hubbert had not even been invented. Perez published his first text on the subject in 1979. Also the equations must be solved numerically, and all the computer power in the world at the time that Hubbert published his paper was not powerful enough to derive the graph above. We solved it using software that we wrote for a lap top!
To make matters even more difficult, there is no way to determine the amount that the distribution is skewed until after Peak has been reached. Criticizing Hubbert for not getting it exactly correct is as ridiculous as criticizing the Egyptians for not putting electric lights in the dark corridors of their pyramids.
Some people are so pathetically simple its a wonder that they make it through the day?
http://www.thehillsgroup.org/