Egypt depletion curve

by **pup55** » Wed 27 Oct 2004, 08:18:01

The data in

http://www.dieoff.org/42Countries/42Countries.htm

gives the production data through 1997 and estimated depletion curves for 42 countries.

Using this data, and the model we arrived at and refined in the previous thread (verhulst curve) with the fine tuning suggested by Soft_landing, presented below is the actual and predicted depletion curve for Egypt (smooth curve, easy example). Egypt is post-peak, so we are using the model to predict the downslope.

Also, we are able to use the current BP report data to "check" the last 6 years of data to see (relatively objectively) how the model did.

Notes:

a. n=6.8, qinf= 20.14, t(1/2)=47.00446303, k=0.175914413

b. Peak prediction was 1990 or 1991 actual was 1993.

c. Predicted 2003 production was .28 gb, actual was .273 gb.,

Maybe SL would be nice to us today and graph this out for us. If so, we will be able to see that the last six years of predictions "blind" are pretty good today.

Gotta try it with a more complicated case.

Units are gb/yr

Code: Select all: Predicted Actual Recent 1960 -- 0.023 1961 0.018444074 0.027 1962 0.021819692 0.031 1963 0.025775996 0.041 1964 0.030398302 0.046 1965 0.035778817 0.047 1966 0.042014922 0.046 1967 0.049206393 0.046 1968 0.057451287 0.08 1969 0.066840324 0.124 1970 0.077449632 0.17 1971 0.08933195 0.151 1972 0.102506664 0.128 1973 0.116949392 0.093 1974 0.132582281 0.084 1975 0.149266515 0.108 1976 0.166798731 0.119 1977 0.184912925 0.151 1978 0.203288905 0.175 1979 0.221567399 0.192 1980 0.239370757 0.215 1981 0.256326988 0.252 1982 0.27209411 0.243 1983 0.28638156 0.265 1984 0.298965988 0.299 1985 0.309699807 0.325 1986 0.318512233 0.296 1987 0.325403719 0.334 1988 0.330435561 0.319 1989 0.333716723 0.323 1990 0.335389875 0.33 1991 0.335618187 0.329 1992 0.334573904 0.332 1993 0.332429215 0.345 1994 0.329349508 0.339 1995 0.325488818 0.339 1996 0.320987145 0.329 1997 0.315969224 0.321 1998 0.310544372 -- 0.3127068 1999 0.304807075 -- 0.3020201 2000 0.298838033 -- 0.28513052 2001 0.292705461 -- 0.2768145 2002 0.286466485 -- 0.274767 2003 0.28016854 -- 0.273914 2004 0.273850689 2005 0.267544849 2006 0.261276877 2007 0.255067537 2008 0.248933329 2009 0.242887211 2010 0.236939208 2011 0.231096923 2012 0.225365974 2013 0.219750354 2014 0.214252729 2015 0.208874692 2016 0.203616963 2017 0.198479566 2018 0.193461961 2019 0.188563165 2020 0.183781842 2021 0.179116384 2022 0.174564968 2023 0.170125614 2024 0.16579622 2025 0.1615746 2026 0.157458513 2027 0.153445677 2028 0.149533798 2029 0.145720575 2030 0.142003717 2031 0.138380947 2032 0.134850015 2033 0.131408701 2034 0.128054817 2035 0.124786212 2036 0.121600778 2037 0.118496446 2038 0.115471189 2039 0.112523026 2040 0.109650017 2041 0.106850269 2042 0.10412193 2043 0.101463194 2044 0.098872296 2045 0.096347515 2046 0.093887171 2047 0.091489627 2048 0.089153285 2049 0.086876586 2050 0.084658011

by **Soft_Landing** » Wed 27 Oct 2004, 15:38:50

This is good, but from memory, we were hoping to find out how well this model would predict the peak?

Perhaps I can offer you three data sets - all of which have passed peak in reality. I wont give years because that would give too much away. Also, I have made all the quantities into an index (start at 1).

Code: Select all: Time Set1 Set2 Set3 1 1.00 1.00 1.00 2 0.91 2.00 1.06 3 0.85 2.50 1.11 4 0.82 2.40 1.17 5 0.75 1.70 1.23 6 0.79 1.50 1.28 7 0.81 1.30 1.34 8 0.88 1.20 1.40 9 1.01 2.00 1.42 10 1.13 0.80 1.48 11 1.31 0.50 1.58 12 1.32 0.40 1.64 13 1.32 7.50 1.62 14 1.32 39.00 1.76 15 1.32 51.20 1.95 16 1.30 38.00 2.00 17 1.32 38.00 2.02 18 1.32 35.00 2.17 19 1.32 28.00 2.35 20 1.40 57.00 2.15 21 1.78 56.00 2.30 22 1.98 79.00 2.62 23 1.98 87.00 2.67 24 1.98 95.00 2.77 25 1.98 118.00 2.70 26 1.97 115.10 2.88 27 2.15 115.00 3.05 28 2.47 118.00 3.05 29 2.68 134.00 2.75 30 2.98 150.00 3.00

If the task seems too great (as I suspect), perhaps you could suggest why? What further information would help you most?

by **ohanian** » Thu 28 Oct 2004, 02:18:18

Soft_Landing wrote:

This is good, but from memory, we were hoping to find out how well this model would predict the peak?

Perhaps I can offer you three data sets - all of which have passed peak in reality. I wont give years because that would give too much away. Also, I have made all the quantities into an index (start at 1).

Code: Select all
Time Set1 Set2 Set3 1 1.00 1.00 1.00 2 0.91 2.00 1.06 3 0.85 2.50 1.11 4 0.82 2.40 1.17 5 0.75 1.70 1.23 6 0.79 1.50 1.28 7 0.81 1.30 1.34 8 0.88 1.20 1.40 9 1.01 2.00 1.42 10 1.13 0.80 1.48 11 1.31 0.50 1.58 12 1.32 0.40 1.64 13 1.32 7.50 1.62 14 1.32 39.00 1.76 15 1.32 51.20 1.95 16 1.30 38.00 2.00 17 1.32 38.00 2.02 18 1.32 35.00 2.17 19 1.32 28.00 2.35 20 1.40 57.00 2.15 21 1.78 56.00 2.30 22 1.98 79.00 2.62 23 1.98 87.00 2.67 24 1.98 95.00 2.77 25 1.98 118.00 2.70 26 1.97 115.10 2.88 27 2.15 115.00 3.05 28 2.47 118.00 3.05 29 2.68 134.00 2.75 30 2.98 150.00 3.00

If the task seems too great (as I suspect), perhaps you could suggest why? What further information would help you most?

======
Set 1

Model: q(x)= 0.5 * F * (1 - tanh( abs(H - x)/(2.0 * k)))

F = 96.6726 Q_infinity
H = 106.174 T_peak
k = 21.3254

Model: q(x)= 0.25 * k * F * ( 1 - (tanh(0.5 * k * (x - H)))**2 )

F = 2300.55 Q_infinity
H = 107.402 T_peak
k = 0.0475257

============

by **ohanian** » Thu 28 Oct 2004, 02:46:48

======
Set 2

Model: q(x)= 0.5 * F * (1 - tanh( abs(H - x)/(2.0 * k)))

F = 304.142
H = 29.6591
k = 6.25117

Model: q(x)= 0.25 * k * F * ( 1 - (tanh(0.5 * k * (x - H)))**2 )

F = 3500
H = 32.5711
k = 0.175367

============

Set 3

Model: q(x)= 0.5 * F * (1 - tanh( abs(H - x)/(2.0 * k)))

F = 6.09899
H = 27.7847
k = 16.1334

Model: q(x)= 0.25 * k * F * ( 1 - (tanh(0.5 * k * (x - H)))**2 )

F = 204.945
H = 38.9406
k = 0.0650496

by **pup55** » Thu 28 Oct 2004, 09:37:36

Hmmm...

I spent the most time on model 2.
I still do not have excel solutions, but I am not sure it would help.

You have to make two assumptions:

a. you know more or less what the shape of the curve is going to be (so you can adjust k and n)

b. you know more or less how close you are to the peak and/or how much more stuff is still in the ground. This is obviously a potentially self-deluding assumption for the peakers who think we are right at it. You don't need to know exactly, but helpful to know if there are 2 times, 3 times, or 10 times much more remaining or whatever.

There are multiple solutions for error=0.

What I did was use various multiples of "actual cumulative production" (q-inf multiplier), got the curve pretty close by adjusting k and n, and then used "goal seek" to give error=0 by adjusting t-50.

Code: Select all: qinf factor t-50 n 2 30 0.05 0.2 3 34 2 0.2 4 38 3 0.2 10 45 3 0.2 20 48 3 0.2 30 51 3 0.2 100 53 3 0.3 200 55 6 0.3 300 55 6 0.3

By q-inf factor I mean I multiplied cumulative production by that much to get q-inf. So for a factor of 4, Q-inf was 5508 for model 2.

A graph of the first two columns gives gives an interesting curve.

Even in the early stages, where the calculation is most sensitive to changes in Q-inf, a doubling of Q-inf only moves the peak 8 years. So maybe that's close enough of an estimate to make people feel better, or worse, as the case may be.

I think maybe the inflexion point on the curve may be critical. If you are above that on the left arm, the curve is already pretty well defined so easy to deduce the shape, and you know within an order of magnitude where you are on Q-inf, so you can do a halfway job of estimating the peak. Or else, the "flatter" the data, like in model 1, the worse the estimate is going to be anyway.

I don't know what to think about this. You will love my model for Qatar, though.

Code: Select all

by **Soft_Landing** » Thu 28 Oct 2004, 11:47:26

Thanks for your models Ohanian. I'll post the graphs shortly. I'm hoping pup55 might come up with something a little more solid before I do...

Pup55, can you give us something, whether it be a model, predictions for peak dates, or whatever? Just so we can have a quantifiable measure of how close you were. The whole point of the blind test is so you cant go back later and say, "yeah, I can see how that could be right." Even maximums and minimums for possible peak dates would be cool.

by **pup55** » Thu 28 Oct 2004, 13:56:42

I'm in a sporting mood, after your encouragement:
I think you can sort this out for easy graphing

Code: Select all: Set 1 Set 2 Set 3 n 0.5 2 1 Q-inf 200 4100 200 T-50 40.8 34.25 38 k 0.1 0.2 0.0682 peak 43 32 39 actual pred actual pred actual pred time 1 1 1 1 0.91 0.32 2 1.75 1.06 0.97 2 0.85 0.35 2.5 2.13 1.11 1.02 3 0.82 0.39 2.4 2.60 1.17 1.08 4 0.75 0.42 1.7 3.17 1.23 1.15 5 0.79 0.47 1.5 3.86 1.28 1.21 6 0.81 0.51 1.3 4.70 1.34 1.28 7 0.88 0.57 1.2 5.72 1.4 1.35 8 1.01 0.62 2 6.95 1.42 1.42 9 1.13 0.69 0.8 8.44 1.48 1.50 10 1.31 0.75 0.5 10.24 1.58 1.57 11 1.32 0.83 0.4 12.41 1.64 1.65 12 1.32 0.91 7.5 15.00 1.62 1.73 13 1.32 1.00 39 18.09 1.76 1.82 14 1.32 1.09 51.2 21.77 1.95 1.90 15 1.3 1.20 38 26.11 2 1.99 16 1.32 1.31 38 31.21 2.02 2.08 17 1.32 1.43 35 37.13 2.17 2.17 18 1.32 1.56 28 43.95 2.35 2.26 19 1.4 1.70 57 51.71 2.15 2.35 20 1.78 1.85 56 60.41 2.3 2.43 21 1.98 2.02 79 70.01 2.62 2.52 22 1.98 2.19 87 80.38 2.67 2.61 23 1.98 2.37 95 91.34 2.77 2.69 24 1.98 2.56 118 102.60 2.7 2.78 25 1.97 2.77 115.1 113.82 2.88 2.86 26 2.15 2.98 115 124.57 3.05 2.93 27 2.47 3.20 118 134.41 3.05 3.00 28 2.68 3.43 134 142.91 2.75 3.07 29 2.98 3.67 150 149.70 3 3.14 30 3.91 154.50 3.19 31 4.15 157.15 3.24 32 4.38 157.65 3.29 33 4.62 156.09 3.33 34 4.85 152.70 3.36 35 5.06 147.76 3.38 36 5.26 141.57 3.40 37 5.44 134.48 3.40 38 5.60 126.77 3.40 39 5.73 118.72 3.40 40 5.83 110.56 3.38 41 5.90 102.47 3.36 42 5.92 94.58 3.33 43 5.91 87.01 3.29 44 5.86 79.81 3.24 45 5.77 73.03 3.19 46 5.64 66.70 3.14 47 5.47 60.82 3.07 48 5.27 55.38 3.00 49 5.03 50.37 2.93 50 4.77 45.77 2.86 51 4.50 41.56 2.78 52 4.20 37.71 2.69 53 3.90 34.20 2.61 54 3.59 31.01 2.52 55 3.28 28.10 2.43 56 2.98 25.46 2.35 57 2.69 23.06 2.26 58 2.41 20.88 2.17 59 2.15 18.91 2.08 60 1.90 17.12 1.99 61 1.68 15.50 1.90 62 1.47 14.03 1.82 63 1.28 12.70 1.73 64 1.11 11.49 1.65 65 0.96 10.40 1.57 66 0.82 9.41 1.50 67 0.71 8.52 1.42 68 0.60 7.71 1.35 69 0.51 6.98 1.28 70 0.43 6.31 1.21 71 0.37 5.71 1.15 72 0.31 5.17 1.08 73 0.26 4.68 1.02 74 0.22 4.23 0.97 75 0.18 3.83 0.91

by **Soft_Landing** » Fri 29 Oct 2004, 00:20:29

Ok, drum roll.....

Did you guys try to guess which countries the data were from? You were staring at Bahrain, Congo, and the US lower 48. Enough chit chat, here are the results.

What do you think?

by **pup55** » Fri 29 Oct 2004, 10:28:02

Hmmm...

If we had any sort of inkling what Q-inf was in any of these cases, we'd have been much better.

Also, maybe a step back to think about it... what kind of accuracy is "acceptable"? Is getting within 10 years of the peak "acceptable" to validate numerical depletion modeling?

Maybe a goal would be to refine the method to consistently get within 4 years of the peak. That's one presidential term, and enough time to take steps, if steps can be taken.

by **pup55** » Fri 29 Oct 2004, 11:12:59

I take back what I just said after reading the article below on historical production.

Most of these predictions suggest peak within 10 years of y2000, depending on their estimate of Q-inf. Maybe that's all the accuracy anybody can expect from this type of analysis.

For a dry-run blind exercise, though, maybe a different standard is in order.

by **dmtu** » Fri 29 Oct 2004, 11:54:15

Regardless of the accuracy you guys are, to me anyway, pretty damned amazing. Thanks for the contribution.

by **Soft_Landing** » Fri 29 Oct 2004, 16:42:22

Hmmm...

If we had any sort of inkling what Q-inf was in any of these cases, we'd have been much better.

My thoughts exactly. I wrote yesterday to BP to ask them to send me an old version of Statistical review, pre-90's if possible. I don't know if they'll do it (I don't even know when they first made electronic copies of these reports), but if they do, I'll be able to provide old estimates of reserves, which should help guide Q-inf projections. I do have a BP SR from 1999, so that may be old enough to produce some interesting results.

Also, I've realised that I can get infinite possible real life depletion curves by summing production from regions or groups of countries. So there is no longer a lack of raw data to perform tests on.

The added benefit of grouping countries together for depletion modeling purposes is that we can take advantage of the central limit tendency. The central limit theorem points out that when you sum independent distributions, random variation is more likely than not to be decreased. This can be grasped conceptually by imagining that "political interference" (eg. war), large anomylous discoveries, and other deviations to normal production are likely to effect different regions at different times. Thus, the overall curve will be smoother when more precincts are included.

Of course, world oil production does not even become close to the ideal candidate for central limit tendency, because there are other kinds of production interruption that effect the world "precinct". Examples are world recession and international cartel interference. Nonetheless, the effects of central limit tendency are probably responsible for the very stable curve between 1950 and 1970. You wouldn't expect a smaller region to show such a stable trend.

So, perhaps we can try another blind set next week? As for today, well, it's the weekend - no plotting or data for me today.

Do you guys have any other data you'd want in a blind test situation?

by **Cool Hand Linc** » Fri 29 Oct 2004, 20:08:45

This is really amazing to me as well. I will continue to keep up with your postings. Keep it up!

I am wanting to know about North America in paticular. Mexico, US, and Canada.

If I have understood correctly. The technology being used to extract the lighter crudes should cause an eventual rapid decline here rather than a slower decline. When would this occur? Any ideas.

by **smiley** » Sat 30 Oct 2004, 08:49:48

hi. I didn't visit this part of the forum for a while and I have to say that I'm impressed by the amount of efford you have put into this. Kudos.

If I have understood correctly. The technology being used to extract the lighter crudes should cause an eventual rapid decline here rather than a slower decline. When would this occur? Any ideas.

mexico appears to be declining already if you surf to "www.pemex.com" go to investor relations and "operations" you can get all the production data. However the production has a double peak which complicates things.

If you're looking to a high technology profile I think Norway is a good example (single peak). Here are the production data. The second column is the first derivative, (three point average smoothing).

I'm afraid that these countries are very hard to predict since the peak occurs very abrubt.

Code: Select all: year production slope 1971 6 1972 33 13 1973 32 1 1974 35 78.5 1975 189 122 1976 279 49 1977 287 38.5 1978 356 60 1979 407 86 1980 528 52.5 1981 512 2 1982 532 74.5 1983 661 110 1984 752 81 1985 823 77.5 1986 907 115.5 1987 1054 144.5 1988 1196 257 1989 1568 260.5 1990 1717 193.5 1991 1955 250.5 1992 2218 211 1993 2377 237.5 1994 2693 263 1995 2903 270 1996 3233 188.5 1997 3280 -47 1998 3139 -70.5 1999 3139 102 2000 3343 138.5 2001 3416 -7 2002 3329 -78 2003 3260

I've been busy with modelling, but I gave it up from pure frustration. Now that I see that you're busy with I like to give it another try. I'll try to share some of my thoughts, maybe you have some ideas.

First of all I think that a good model should eliminate the need for accurate reserve data, since this is the main problem. Hubbert's curve requires either the position of the peak or a good reserve estimate.

Every time I get the same problem. If you have a model which describes the production of a country which has peaked and eliminate the peak from the data the model goes beserk. The models only seem to work when you have the data of the last 3 years before the peak, when the curve actually levels out.

If you look at the production graphs there appears to be a regime where the production rises almost linearly (In the case of Norway 1989-1995). In Congo that linear regime seems to span from 1978-1998, with the peak occurring shortly after.

The models that are available do not have such a linear regime. So if your data doesnt include the actual peak the model either peaks much too low or far too high. Depending on the type of curve or the fitting method that you use (least squares, Newton etc) the inflection point will end up at the start or end of this linear regime. Somehow we have to incorporate this linear part in the equations.

by **pup55** » Mon 01 Nov 2004, 11:19:42

I will have to blow the dust off of my calculus book and review to make an intelligent comment on this except to say that I think you are on to something.

If you calculate the slope as a percent of the previous year's production. you end up with nearly a straight line, per the data below. When the line crosses zero, you can figure peak has occurred. If you can deduce the equation of that line, you might be able to apply it to the production data to get a suitable peak estimate. It might not be linear, might be logarithmic or some trig function. Also, it might only apply to the regime near the peak, but worthy of some exploration.

Another problem is that the data is so noisy, especially the year between 73 and 74 when they tripled production. If you throw out that, maybe linear regression can give you a decent equation.

Code: Select all: 1976 17.56272401 1977 13.41463415 1978 16.85393258 1979 21.13022113 1980 9.943181818 1981 0.390625 1982 14.0037594 1983 16.64145234 1984 10.7712766 1985 9.416767922 1986 12.73428886 1987 13.70967742 1988 21.48829431 1989 16.61352041 1990 11.26965638 1991 12.81329923 1992 9.513074842 1993 9.991586033 1994 9.766060156 1995 9.30072339 1996 5.830497989 1997 -1.432926829 1998 -2.245938197 1999 3.249442498 2000 4.142985343 2001 -0.204918033 2002 -2.34304596

by **Soft_Landing** » Tue 02 Nov 2004, 00:58:27

I need to make a correction because I think I've made a mistake. It's been a while since I took statistics courses, and the old knowledge is coming back in fits and starts.

I wrote:The added benefit of grouping countries together for depletion modeling purposes is that we can take advantage of the central limit tendency. The central limit theorem points out that when you sum independent distributions, random variation is more likely than not to be decreased. This can be grasped conceptually by imagining that "political interference" (eg. war), large anomylous discoveries, and other deviations to normal production are likely to effect different regions at different times. Thus, the overall curve will be smoother when more precincts are included.

The effect described does exist, but it's simply a result of a larger sample size. The central limit theorem refers to the tendecny for the distribution of samples from a distribution to approach the normal (gaussian) distribution, regardless of the shape of the distribution from which those samples were sourced.

Apologies for the error.

by **seb** » Tue 02 Nov 2004, 01:29:38

Yes I confirm this theorem known as "central limit theorem" and usually taught to 3rd year undergraduate students in mathematics. More precisely, here is the simplest form of this theorem.

Let consider independant and identicaly distributed random variables X1, X2, X3,.... They have to be in the so-called L^2 space, but just don't care about it...Let m be their common mean and sigma^2 their common variance. Then the following variable
Yn= (X1+ ... + Xn -nm)/(sigma*sqrt(n))
goes to the reduced normal law (mean=0 and variance=1) as n goes to infinity.

In other word this means that big sample of random variables behaves like a normal random variables. BUT these variables must have the same mean and variance. This is obviously not the case if you consider depletion curves, the mean and variance are different. But there are more elaborate central limit theorem which can deal with that if I remember well. I am not a probabilist, I'd need to look a bit more carefully in probability theory books. Anyway, the size of the sample, about 100 countries, might be too small to find a normal distribution (in other worda a Bell curve). A limit is a limit, and you need a probabilist estimate of the convergence with a so small sample. Such estimates exist.

Am I clear?

Egypt depletion curve

Egypt depletion curve

really

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