[khebab]

Motivation:

There is a lot of debate about why a sum of individual oil field productions should produce a symmetric curve. Intuitively, we feel that there is some degree of connection with the Central Limit Theorem:
$this->bbcode_second_pass_quote('', 'a')ny sum of many independent identically distributed random variables will tend to be distributed according to a particular "attractor distribution". The most important and famous result is called simply The Central Limit Theorem which states that if the sum of the variables has a finite variance, then it will be approximately normally distributed

[EnergySpin]

I think that a physically more plausible model for the production of each well, is that it follows a bi-exponential model
f(t) = A1 Exp[-k1 t]+A2 Exp[-k2 t] i.e. a two comparmental model.
This arises as a solution of the following system:

V1--->V2--->f(t)

Where URR = V1(0), V2(0) = 0, and the following differential laws are assumed:
V1'[t]=-k1 V1[t],
V2'[t]=-k2 V2[t]+K1 V1[t]
and f[t]=k2 V2[t].
Solving the system of differential equations leads to:
V1[t] =V1(0) Exp[-k1 t]
V2[t] = (k1*V1(0)) * (Exp[-k1 t] - Exp[-k2 t])/(k2-k1)

and f[t] = (k2*k1*V1(0)) * (Exp[-k1 t] - Exp[-k2 t])/(k2-k1)
The rationale for this model is simple:
The reservoir has a large volume of oil (V1 or URR) which is not directly accessible. Instead one drains a smaller potential compartment (V2) which is fed by V1.
Is it possible to run the simulation assuming this model?

[khebab]

EnergySpin, thanks for the good comment!
$this->bbcode_second_pass_quote('EnergySpin', 'I')s it possible to run the simulation assuming this model?

[WebHubbleTelescope]

$this->bbcode_second_pass_quote('khebab', '
')2- the resulting curve is slightly assymetric (skewness= 0.43) and a gamma function is maybe more appropriate

[EnergySpin]

$this->bbcode_second_pass_quote('WebHubbleTelescope', '
')The gamma function occurs when I use the oil shock model and set all 4 rates to the same value and give it a delta function stimulus or another exponential function as input. In the former you get a 4th order gamma and the latter a 5th order gamma. I know this because it is a great way to check the numerical integration accuracy for my model:

http://mobjectivist.blogspot.com/2005/1 ... ution.html

Khebab, what you have discovered is the sampling version of doing convolution. I thought you did radar stuff, DSP and that. You should really be doing a convolution. If you wanted, you could put it in the frequency domain and perform FFTs and just multiply the results and then do an inverse FFT.

[EnergySpin]

$this->bbcode_second_pass_quote('khebab', 'E')nergySpin, thanks for the good comment!
$this->bbcode_second_pass_quote('EnergySpin', 'I')s it possible to run the simulation assuming this model?

[khebab]

$this->bbcode_second_pass_quote('WebHubbleTelescope', 'K')hebab, what you have discovered is the sampling version of doing convolution. I thought you did radar stuff, DSP and that. You should really be doing a convolution. If you wanted, you could put it in the frequency domain and perform FFTs and just multiply the results and then do an inverse FFT.

[nero]

$this->bbcode_second_pass_quote('', '-') alpha and beta are distributed between in an angle domain such as the resulting slopes are between 2% and 15%,
- the URR is uniformly distributed between 5 and 20,
- the starting year t between 1 and 21

[khebab]

$this->bbcode_second_pass_quote('EnergySpin', 'O')f course the resultant expressions tend to involve generalized hypergeometric (Lauricella) or Meiger functions, which are not that easy work to with.

[WebHubbleTelescope]

$this->bbcode_second_pass_quote('EnergySpin', '
')With all due respect, a garden variety FFT analysis is not going to do much with this messy time series.
Why not apply Bayesian spectrum analysis directly (which will allow us to incorporate prior information)?

[khebab]

$this->bbcode_second_pass_quote('nero', 'T')he starting year should not be uniformly distributed. The discovery is dependent on the exploratory effort which is dependent on the demand(price). The demand is not a constant but develops over time and is dependent in part on the previous supply and previous discovery. That I believe is at the heart of why the central limit theorem is not applicable. The discoveries are not independent random events.

[turmoil]

$this->bbcode_second_pass_quote('khebab', 'A')greed but the goal here is not to find a realistic model about oil production. I'm trying to understand the basic mechanisms that makes some production curves symmetric (US, Norway,...) and others very asymmetric. The reasons why we can get symmetric curve are not obvious in part because of the things you are mentioning.

[khebab]

$this->bbcode_second_pass_quote('turmoil', 'k')hebab, have you considered the differences between deep water and regular wells, how quickly each was exploited, the secondary and/or tertiary techniques that were used in them, etc? thanks.

[WebHubbleTelescope]

$this->bbcode_second_pass_quote('khebab', '')$this->bbcode_second_pass_quote('nero', 'T')he starting year should not be uniformly distributed. The discovery is dependent on the exploratory effort which is dependent on the demand(price). The demand is not a constant but develops over time and is dependent in part on the previous supply and previous discovery. That I believe is at the heart of why the central limit theorem is not applicable. The discoveries are not independent random events.

[WebHubbleTelescope]

$this->bbcode_second_pass_quote('nero', '')$this->bbcode_second_pass_quote('', '-') alpha and beta are distributed between in an angle domain such as the resulting slopes are between 2% and 15%,
- the URR is uniformly distributed between 5 and 20,
- the starting year t between 1 and 21

[khebab]

I explored a little bit the model by gradually increasing the level of randomness in the different variables.

Case 1: trivial case

All fields are identicals in shape (Isoscele triangles) and surface (slope= 5% and URR= 15), only the starting year is a gaussian N(11, 4.33). This ideal case stricly verifies the conditions of the Central Limit Theorem (see Illustration of the central limit theorem). ).

[align=center]

[/align]
The resulting curve is fairly gaussian with maybe heavier tails. Also, the skewness value is decreasing with the number of points in the simulation (see figure in the lower left corner).

Case 2: shape fixed, URR gaussian

The URR is distributed according to a gaussian N(15, 2.89)

[align=center]


Parameters distributions: from left to right and top to bottom: field upslope distribution (in %), URR distribution (log-log representation), field starting year and peak year distribution.
[/align]

Case 3: Isoscele triangles with a gaussian slope, URR gaussian

same as case 2 but we randomly choose the triangle slopes N(8.5, 2.31) (in %). The right tail is getting heavier and the skewness has increased.

[align=center]

[/align]

Case 4: random triangles with gaussian slopes, URR gaussian

Not much difference with case 3 but the skewness value has decreased.
[align=center]
[/align]

Case 5: random triangles with gaussian slopes, URR gaussian but big field produced first

Because big fields have a higher discovery cross-section and are economically more valuable, there is a good chance that they will be produced first and conversly the small fields will be produced last. Same as case 4 but we rank fields according to their size and the biggest are produced first. It has a good influence on the result where the production is much more gaussian especially in the tails.

[align=center][/align]

Case 6: random triangles with gaussian slopes, URR lognormal but big field produced first

The distribution of the URR values is clearly not normal and is probably more like a lognormal distribution where big fields have a very low probably of occurence. The lognormal model is the same that have been used in a previous thread (A Statistical Model for the Simulation of Oil Production). Curiously, the impact of this modification is not very strong and the curve is just slightly more skewed.

[align=center]

[/align]

Some conclusions

1- the production profile is well modeled by a gaussian when most of the underlying parameters are also gaussian distributed.
2- the logistic model do not seem to be a particularly better model especially for the tails
3- the distribution of the URR values do not seem to have a big impact on the resulting curve
4- when fields are produced by order of size, the production curve is getting more gaussian.

[pup55]

All I can think of is that there is no compelling reason to believe that the shape of the distribution would be a lot different if n=15000 or n=20,000.

In other words, you have selected a sufficiently large number for N that within this order of magnitude, the shape of the sum of the curves will start to take on the shape of the individual curves (in this case, gaussian) pretty much no matter what.

I am thinking about the argument of new discoveries and /or reserves growth, and how the discovery of X number of new fields, including a number of super-giants, might influence the shape of the right hand side of the curve.

It looks as though even if X, the new fields, were large in number compared to the known existing fields (which for you is 10,000), the sumnation curve will be more or less the same shape unless the time distribution or the size distribution of the new discoveries is grossly different from the distribution of the currently known discoveries.

What I am saying is that it is possible to have new discoveries to avoid the problem of the right hand side of the curve. These discoveries would have to be sufficiently large and occur in a sufficiently different time sequence to substantially alter the curve shape. Per your post the other day, you might be able to estimate the probability of such an event happening. Example: The discovery of 10 more Ghawar-sized fields, at or about the present time, making the curve a lot broader and flatter than it is in the above examples. (example 6 might be the closest to this).

You would then be able to make a determination kind of like we were talking about the other day, such as that there is y probability that future discoveries would be sufficient to get us out of these problems.

Once you know what y is, no problem to have the conversation with Michael Lynch, who will tell you that we should go on living like we are because "we will probably discover sufficient oil to avoid any problems related to the peak", etc. etc.

[JohnDenver]

khebab,
I know this is slightly off-topic from the issue you're considering, but I'll say it anyway. When I saw Stuart's recent Gaussian, which matches world production so beautifully over different orders of magnitude, the thing which intrigued me most was that it was made of linear segments whose slopes seem to be determined by economic/political factors, as Stuart pointed out.

This makes me think that the shape of the rising side of the Gaussian is determined by demand, not discovery/resources/geology. As a thought experiment, consider a parallel world where there is no constraint on the resource. In this world, there is a fortuitous discovery of a monstrous pool of oil which is (say) 500Gb in size, and is located on the surface like a lake. What would be the production curve in those circumstances? I think it's clear that production in this world will still be constrained -- not by the ability to mobilize supply, but rather by cartel activity which acts to prevent too much oil from flooding the market, thereby crashing prices and bankrupting all the players who invested in infrastructure prior to discovery of the lake.

Clearly, this effect has been a chronic, dominant feature of oil production on the rising side of the curve. The main goal of Rockefeller's strategy was to cartelize, and thereby constrain supply and stabilize prices. The same can be said for OPEC. So my theory is that the slope of the left side of the Gaussian is determined by cartel activity, which in turn is determined by demand/price. The lake will not be produced, even if it is discovered, because there is no way for the market to consume it. Producing it full-blast would be an act of suicide for the producer. It reminds me of something Mike Lynch said in his thread (I'm paraphrasing): predicting supply essentially boils down to predicting demand. I believe that is 100% true on the upside of the curve.

If I'm correct, it makes your question a lot more difficult to answer. Why should the rising and falling sides of the Gaussian be symmetric if the rising side is determined by economics while the falling side is determined by geology?

[khebab]

$this->bbcode_second_pass_quote('pup55', 'I')n other words, you have selected a sufficiently large number for N that within this order of magnitude, the shape of the sum of the curves will start to take on the shape of the individual curves (in this case, gaussian) pretty much no matter what.

[khebab]

$this->bbcode_second_pass_quote('JohnDenver', ' ')Clearly, this effect has been a chronic, dominant feature of oil production on the rising side of the curve. The main goal of Rockefeller's strategy was to cartelize, and thereby constrain supply and stabilize prices. The same can be said for OPEC. So my theory is that the slope of the left side of the Gaussian is determined by cartel activity, which in turn is determined by demand/price. The lake will not be produced, even if it is discovered, because there is no way for the market to consume it. Producing it full-blast would be an act of suicide for the producer. It reminds me of something Mike Lynch said in his thread (I'm paraphrasing): predicting supply essentially boils down to predicting demand. I believe that is 100% true on the upside of the curve.