[sjn]

Assuming you have faith in the BP data, and that SA, Mexico and Russia don't pass their peaks. While these models are interesting, they don't really reflect reality. It would be interesting to see what happens in different scenarios.

[pup55]

$this->bbcode_second_pass_quote('', 'd')ifferent scenarios.

[khebab]

A few other figures from pup55 simulations:

I put a few demand projections assuming different growth (2%, 2.5% and 3%). This figure represents how same amount of oil (1, Gb) is consumed without geologic or technological contrainst (we extract what we need). In this case we have 30 years of oil remaining. The curve is the Verhulst model (we extract what we can!) which shows that demand will not be satisfy from 2007.

Figure 5

On observation we can make from the different Verhulst based curves is that the projected PO value (maximum production rate) is quite insensitive to the value of Q-inf. The second figure shows a linear fit of the PO maximum production value under the Verhulst based prediction (red line) versus the corresponding value of Q-inf (URR). We can see that to go from 34 Gb to 38 Gb we would have to double the URR! 38 Gb being the projected demand for 2014 under a 2% demand growth per year. The situation is worst than I thought!

Figure 6

[pup55]

khebab, these are excellent graphs!

The second graph is the most interesting. There is probably some way to derive this linear relationship mathematically from the equations, but interesting to have stumbled upon it empirically.

I will have to think some more about this.

Thanks for your hard work.

[pup55]

Think about this:

You can't increase Q-inf to "infinity" because the earth is a sphere with a finite volume.

But you can increase Q-inf to some theoretical maximum, which might be computable.

You should then be able to arrive at some kind of estimate of maximum volume annually pumpable given a theoretical maximum URR or OOIP.

Then, it should be easy to compute the "ultimate peak year" by using some exponential demand growth estimate, a la Bartlett. As I recall, Bartlett's calculation was that if the earth was a hollow sphere full of oil, there would be a maximum of about 350 years of consumption left if consumption grows at the current rate. By limiting this further, for example, by assuming that you can't drill farther down than 10000 meters, and you can't drill in water deeper than X meters, you should be able to arrive at a refined estimate of maximum volume of oil available and therefore define a "maximum peak".

What this is saying is that not only is there a finite total volume that can be pumped, there is also a finite maximum rate at which it can be pumped. This goes back to Smiley's "peak oil derived" model which represented a volume of pressurized gas inside a vessel. At a certain point, the drilling of additional holes in the vessel did not result in increased flow rate.

So, in essence, not only is oil depletion inevitable, "peak" is inevitable.

In any case, it looks like it will happen a heck of a lot faster than 350 years.

[khebab]

$this->bbcode_second_pass_quote('pup55', 'k')hebab, these are excellent graphs!

The second graph is the most interesting. There is probably some way to derive this linear relationship mathematically from the equations, but interesting to have stumbled upon it empirically.

I will have to think some more about this.

Thanks for your hard work.

[khebab]

$this->bbcode_second_pass_quote('pup55', 'T')hink about this:

You can't increase Q-inf to "infinity" because the earth is a sphere with a finite volume.

But you can increase Q-inf to some theoretical maximum, which might be computable.

You should then be able to arrive at some kind of estimate of maximum volume annually pumpable given a theoretical maximum URR or OOIP.

Then, it should be easy to compute the "ultimate peak year" by using some exponential demand growth estimate, a la Bartlett. As I recall, Bartlett's calculation was that if the earth was a hollow sphere full of oil, there would be a maximum of about 350 years of consumption left if consumption grows at the current rate. By limiting this further, for example, by assuming that you can't drill farther down than 10000 meters, and you can't drill in water deeper than X meters, you should be able to arrive at a refined estimate of maximum volume of oil available and therefore define a "maximum peak".

What this is saying is that not only is there a finite total volume that can be pumped, there is also a finite maximum rate at which it can be pumped. This goes back to Smiley's "peak oil derived" model which represented a volume of pressurized gas inside a vessel. At a certain point, the drilling of additional holes in the vessel did not result in increased flow rate.

So, in essence, not only is oil depletion inevitable, "peak" is inevitable.

In any case, it looks like it will happen a heck of a lot faster than 350 years.

[khebab]

One last graph where the production is divided by the projected world population (ref: Table 5.1 from Adam Szirmai, www.dynamicsofdevelopment.com, 2004).


Figure 7

[Shiraz]

Khebab, with regard to the last chart -

You are mistaken in your application of the 2% growth curve. 2% is generally taken to be the expected rate of growth for oil consumption, whereas your chart has used per capita oil consumption as it's vertical axis. The business as usual assumption should be 0% growth, which by eye can be observed to be the trend since 1980.

Interestingly, your chart would thus imply that business as usual might be achievable until 2020?

I imagine this outcome occurs mathematically because the single verhulst model has been used. To all modellers - why use the single verhulst to describe global oil production when we know this idea has been contravened in the past (political disturbances)? Surely using 2 cycles is a minimum conceptually?

Lastly, the early graph showing that differing Qinfs leads to a similar timing of peak with differeing post peak curve is a mistake (even though the conclusion might not be). The effect shown above is a natural artefact of using least square fitting with verhulst curves. In my opinion, to fairly evaluate the likely difference in peak date with differing Qinfs, you need to control for post peak curve - ie. hold the curves to be symetrical, for example, or else constrain them to be equally skewed. If you let two variables fit, you cant infer much about either of them... All this of course is on top of the many difficulties of interpreting the meaning of fitting verhulsts to production curves... [one thing i haven't seen here, that i find most interesting, is Laherreres RMSE/Qinf charts. The vertical axis is the RMSE of the verhulst fit, and the horizontal axis is the Qinf used to generate the fit. The result is a *heavily* skewed u-shaped curve, demonstrating that the fit achieved by least squares fitting of a verhulst is both biased and terribly dubious. Worse still, the skewing becomes many times worse with multi-cycle verhulst fitting, which calls into question the whole meaning of multi-cycle verhulst fits (I think it can still be justified). Anyway, enough rambling.]

Thanks to all the people who put work in here. Enjoyable to see all the ideas. Rest assured that any criticism is intended only to be constructive.

[khebab]

Thank you for your sharp comments:
$this->bbcode_second_pass_quote('Shiraz', 'K')hebab, with regard to the last chart -

You are mistaken in your application of the 2% growth curve. 2% is generally taken to be the expected rate of growth for oil consumption, whereas your chart has used per capita oil consumption as it's vertical axis. The business as usual assumption should be 0% growth, which by eye can be observed to be the trend since 1980.

[nth]

wow, this is very interesting. no matter how much oil they discover, they won't satisfy demand projected by EIA in the long term.

[pup55]

$this->bbcode_second_pass_quote('', 't')he fit achieved by least squares fitting of a verhulst is both biased and terribly dubious

[khebab]

I was wondering, in order to validate the model, if we could use a kind of cross validation technique. I explain myself:

We could use fields or country data which shows advanced depletion (ex: the US). Then cut artificially the dataset in half, were the left side is the "training set" and the right side is the "test set". Of course, the cutoff time has to be before the peak. We train the model based on the training set and compute the error on the prediction using the test set. The cutoff value could be changed and the impact on the prediction accuracy assessed.

This way, we can have a objective methodology to validate the approach and try to evaluate its reliability.

[pup55]

I have bumped to the top the Egypt thread, which has just the sort of test you are talking about.

It would be interesting to repeat this exercise to see if we are any better than we were at the time.

[khebab]

$this->bbcode_second_pass_quote('pup55', 'I') have bumped to the top the Egypt thread, which has just the sort of test you are talking about.

It would be interesting to repeat this exercise to see if we are any better than we were at the time.

[khebab]

I tried to produce the Verhulst curve using Matlab with your parameters, here's the code:

$this->bbcode_second_pass_code('', '
n= 0.149454128;
qinf= 2461.82;
t_half= 106.8467633;
k= 0.039628337;
t= 1900:2050;
P= (qinf / n * k) .* (2^n - 1) .* exp((t - 1900 - t_half) * k) ./ (1 + (2^n - 1) .* exp((t - 1900 - t_half) * k)).^(1/n);
')

The resulting curve seems correct but the peak is not the same (2020 at 38.83 Gb) . is it the correct form for P?

[pup55]

It's going to take me some time to go through it.

Meanwhile, try increasing your range from 2050 to 2085 and see what happens.

[pup55]

I think I see what you did now.

$this->bbcode_second_pass_code('', '(qinf / n * k) ')

This term should be Qinf/(n*tau) and since tau=1/k this should be

Qinf/(n/k) which is the same as (Qinf*(k/n)). so I think that one is OK.

this term

$this->bbcode_second_pass_code('', 'exp((t - 1900 - t_half) * k)')

should be:

(exp((t - 1900 - t_half)) /x

but since X is some constant and is present in both the numerator and denominator, use of k should not change the calculation.


You used the final version of Roper's equation 6.

When I did it, I used the second part of equation 5, which basically calculates the reserves for a given year Q(t) and then I set up the spreadsheet to subtract it from the reserves from the previous year to give the annual production.

But, it should give about the same results.

[Shiraz]

Ok. Question time.

Pup55, what did you do?

I have been unable to replicate your fit.

I have...

Used data set 1900..2008 as per page 1 this post

Constrained Model 2008 = data 2008 (== 32.58 )

Constrained Model Sum 1900..2008 to 1273

Minimized sqaure difference

When I try this, i get a rather absurd fit, given by:

n = 24.2988
Qinf = 21585.6
T1/2 = 317.11
k = 0.0506

Which yields a RMSD (1900..2008) of 1.693

Importantly, when I input your parameters as per page 1, this yields a RMSD of 1.983, indicating that the absurd fit I obtained appears to be a better fit over the test data. I guess this suggests you may have constrained the fit in some other way?

So... have you constrained this fit in some other way, that I am missing?

edit: when i put a bracket immediately after the 8 above it turned into a smiley! inserted a space.

[khebab]

I think there is a mistake on Equation (6) given by Ropert (Depletion Theory):



I used Maple 9 to calculate P(t) and I obtain the following:
$this->bbcode_second_pass_code('', 'P := Q_inf*(2^n-1)*exp((t-t_half)/tau)/((1+(2^n-1)*exp((t-t_half)/tau))^(1/n+1)*n*tau)')
The denominator exposant is in fact (1/n+1)