[pup55]

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

$this->bbcode_second_pass_code('', '

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
')

[Soft_Landing]

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).

$this->bbcode_second_pass_code('', '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?

[ohanian]

$this->bbcode_second_pass_quote('Soft_Landing', '[')img]http://members.optusnet.com.au/emieluk/pupegypt.jpg[/img]

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).

$this->bbcode_second_pass_code('', '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?

[ohanian]

======
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

[pup55]

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.

$this->bbcode_second_pass_code('', '
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.






$this->bbcode_second_pass_code('', '')

[Soft_Landing]

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.

[pup55]

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



')

[Soft_Landing]

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?

[pup55]

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.

[pup55]

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.

[dmtu]

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

[Soft_Landing]

$this->bbcode_second_pass_quote('', 'H')mmm...

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

[Cool Hand Linc]

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.

[smiley]

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.

$this->bbcode_second_pass_quote('', 'I')f 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.

[pup55]

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.

$this->bbcode_second_pass_code('', '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')

[Soft_Landing]

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.

$this->bbcode_second_pass_quote('I', 'T')he 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.

[seb]

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.

$this->bbcode_second_pass_quote('', ' ')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.