Posted: Thu Jan 12, 2006 12:01 am Post subject: Convergence of the sum of many oil field productions
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:
Quote:
any 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
However, individual oil field productions are not random variables and are rather deterministic therefore the Central Limit Theorem cannot be invoked directly. The objectives are the following:
1- look at the conditions required to converge toward a particular "attractor distribution"
2- test the validity of the logisitic and gaussian distributions
Methodology:
I made the following assumptions:
- production profiles of individual fields follow a triangular distribution function
- the URR, the left side slope (angle beta) , the right side slope (angle gamma) and the time of production start are random variables
three relevant properties for an oil field are the surface of the triangle (URR) and the slopes on the left and right side of the peak (beta and gamma angles) which represents the production growth an decline rates respectively.
I use basic properties of triangles to derive a, b and c from beta, gamma and the URR: The law of sines and cosines:
Consequently, the oil field production profile is the following:
Code:
f(x|alpha,beta,t,URR)= (x-t)/(a*c*cos(beta)), if x >= t and x <= t + c*cos(beta)
f(x|alpha,beta,t,URR)= (t + a - x)/(a*(a-c*cos(beta))), if x > t + c*cos(beta) and x <= t + a
For now, I assume uniform distributions for the different random variables:
- alpha and beta are distributed within 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
Results: Erratum: for the subplot at the bottom left corner of each figure, the y axis label should be "Prod. / Cum. Prod." instead of "Prod."
Result with 10 oil fields
Result with 100 oil fields
Result with 1,000 oil fields
Result with 10,000 oil fields
Discussion:
1- the total production seems not to converge completely toward a gaussian or a logistic distribution in particular on the tails. However, these two models seem valid
2- the resulting curve is slightly asymmetric (skewness= 0.43) and a gamma function is maybe more appropriate
3- this is a simple model and there are many improvements possible in particular on the probability distribution function of the different variables
Edit (01/12/2006): corrections of a few english mistakes! sorry about that, english is not my first language (especially late at night). _________________ ______________________________________
http://GraphOilogy.blogspot.com
Last edited by khebab on Tue Aug 05, 2008 12:32 pm; edited 3 times in total
Posted: Thu Jan 12, 2006 6:58 am Post subject: Re: Convergence of the sum of many oil field productions
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? _________________ "Nuclear power has long been to the Left what embryonic-stem-cell research is to the Right--irredeemably wrong and a signifier of moral weakness."Esquire Magazine,12/05
The genetic code is commaless and so are my posts.
Posted: Thu Jan 12, 2006 7:03 am Post subject: Re: Convergence of the sum of many oil field productions
EnergySpin, thanks for the good comment!
EnergySpin wrote:
Is it possible to run the simulation assuming this model?
Yes, I can. I will also post the code in R language later. I chose a triangular distribution because it's the simplest unimodal distribution that you can think of and is a good approximation of the envelope of a real oil field production curve. The objective here is that we are trying to find and equivalent of the Central Limit Theorem for the summation of arbitrary curves. _________________ ______________________________________
http://GraphOilogy.blogspot.com
Posted: Thu Jan 12, 2006 7:09 am Post subject: Re: Convergence of the sum of many oil field productions
khebab wrote:
2- the resulting curve is slightly assymetric (skewness= 0.43) and a gamma function is maybe more appropriate
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:
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.
With that, I believe you are on the right track.
Try convolving a 4th order gamma and a segment of a quadratic. It is indistigushable from a gaussian on the up-slope.
Give up the ghost on the logistic and gaussian, the match is only empirical and the truth lies in modelling the stochastic process, with real initial conditions and solving the differential equations. And a lot of this amounts to doing convolutions.
Posted: Thu Jan 12, 2006 7:19 am Post subject: Re: Convergence of the sum of many oil field productions
WebHubbleTelescope wrote:
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:
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.
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)?
There is also a couple of rather interesting pieces of work in exponential signal analysis in (biological) time series analysis which might of some relevance to this. One of it involved regularized algorithms for the numerical inversion of Laplace Transforms, and the other one involved MCMC methods.
I could provide references if anyone is interested ... just PM me. _________________ "Nuclear power has long been to the Left what embryonic-stem-cell research is to the Right--irredeemably wrong and a signifier of moral weakness."Esquire Magazine,12/05
The genetic code is commaless and so are my posts.
Posted: Thu Jan 12, 2006 7:26 am Post subject: Re: Convergence of the sum of many oil field productions
khebab wrote:
EnergySpin, thanks for the good comment!
EnergySpin wrote:
Is it possible to run the simulation assuming this model?
Yes, I can. I will also post the code in R language later. I chose a triangular distribution because it's the simplest unimodal distribution that you can think of and is a good approximation of the envelope of a real oil field production curve. The objective here is that we are trying to find and equivalent of the Central Limit Theorem for the summation of arbitrary curves.
Maybe I'm wrong, but I thought that such a limiting form does not exist as far as the exponential decays are concerned.
But for at least certain distributiions with support on the real positive hemi-axis one can derive closed form expressions for the sum and the product of many such variables. Of course the resultant expressions tend to involve generalized hypergeometric (Lauricella) or Meiger functions, which are not that easy work to with. _________________ "Nuclear power has long been to the Left what embryonic-stem-cell research is to the Right--irredeemably wrong and a signifier of moral weakness."Esquire Magazine,12/05
The genetic code is commaless and so are my posts.
Posted: Thu Jan 12, 2006 8:04 am Post subject: Re: Convergence of the sum of many oil field productions
WebHubbleTelescope wrote:
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.
You'right, but I thought that presenting the problem in the time domain was more accessible for a larger audience . Working in the frequency domain is mainly a computational improvement.
WebHubbleTelescope wrote:
Give up the ghost on the logistic and gaussian, the match is only empirical and the truth lies in modelling the stochastic process, with real initial conditions and solving the differential equations. And a lot of this amounts to doing convolutions.
Again, you'right, but as a Physic guy, I try to keep in touch with tangible quantities that are directly observables (i.e. the production curve). The gaussian model is the easiest, most common function and is a central pdf in statistical analysis. no more than that. I also believe that a good model should use stochastic PDE as you are suggesting. As you know very well , there is a lot of resistance in the community in evolving from the gaussian/logistic model mainly for historical reason. If you want to convince people you have to go gradually with simple intuitive experiments that they can recreate themself if possible. _________________ ______________________________________
http://GraphOilogy.blogspot.com
Joined: May 22, 2004 Posts: 1424 Location: Ottawa, Ontario
Posted: Thu Jan 12, 2006 11:54 am Post subject: Re: Convergence of the sum of many oil field productions
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
The 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. _________________ Biofuels: The "What else we got to burn?" answer to peak oil.
Posted: Thu Jan 12, 2006 8:17 pm Post subject: Re: Convergence of the sum of many oil field productions
EnergySpin wrote:
Of course the resultant expressions tend to involve generalized hypergeometric (Lauricella) or Meiger functions, which are not that easy work to with.
You got it! that's why I started that thread. However, as pointed out by WHT displacing the problem in the Fourier domain should make thinks a lot more easier. _________________ ______________________________________
http://GraphOilogy.blogspot.com
Posted: Thu Jan 12, 2006 8:20 pm Post subject: Re: Convergence of the sum of many oil field productions
EnergySpin wrote:
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)?
Convolution of two functions in the time domain = multiplication of two functions in the frequency domain.
So if you want to do a convolution, its a nice trick to do the FFT, multiply and then the inverse FFT.
Posted: Thu Jan 12, 2006 8:25 pm Post subject: Re: Convergence of the sum of many oil field productions
nero wrote:
The 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.
Agreed 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. _________________ ______________________________________
http://GraphOilogy.blogspot.com
Joined: Aug 13, 2004 Posts: 1185 Location: Richmond, VA, Pale Blue Dot
Posted: Thu Jan 12, 2006 8:37 pm Post subject: Re: Convergence of the sum of many oil field productions
khebab wrote:
Agreed 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, 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. _________________ "If you are a real seeker after truth, it's necessary that at least once in your life you doubt all things as far as possible"-Rene Descartes
"When you have excluded the impossible, whatever remains however improbable must be the truth"-Sherlock Holmes
Posted: Thu Jan 12, 2006 9:37 pm Post subject: Re: Convergence of the sum of many oil field productions
turmoil wrote:
khebab, 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.
not yet, maybe one day! my guess is that the application of EOR will affect the growth rate (angle beta) and the decline rate (angle gamma) probably by producing a longer growth period and then a steep fall (c>>b). _________________ ______________________________________
http://GraphOilogy.blogspot.com
Posted: Thu Jan 12, 2006 11:04 pm Post subject: Re: Convergence of the sum of many oil field productions
khebab wrote:
nero wrote:
The 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.
Agreed 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.
I have been able to come up with a similarity between Norway and UK even though the shapes of the curves are very different. It appears that both regions spiked up their extraction rate circa 1992, to make up for declining supplies.
Posted: Thu Jan 12, 2006 11:38 pm Post subject: Re: Convergence of the sum of many oil field productions
nero wrote:
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
The 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.
I would largely agree with this statement. However, I think the size of the discoveries are largely uncorrelated from year to year.
If I could, I would rephrase it this way: The sampling rate for establishing discoveries is not an independent random event. Much like a boom and bust cycle, this is likely super-linear and perhaps quadratic over time until the sampling starts firing blanks. The sampling distribution for a quasi-infinite pool is much closer to being a set of independent random events. You should be able to see this if you look at an unsorted creaming curve for the entire world since the 1800's. I haven't seen one yet, but it should be linear, with random-walk excursions above and below the line, if the size of discoveries are uncorrelated. It will bend over with a negative inflection if the big ones get discovered earlier than the rest. You will see this negative inflection with finite regions such as the middle east, but this is misleading because the reason the region got subsequently explored heavily was because of the initial big hit.
Bottom-line is that we see many artifacts because of a finite sample space. I think you have to open up the sample space to see the real random effects.
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. Working in the frequency domain is mainly a computational improvement.
my guess is that the application of EOR will affect the growth rate (angle beta) and the decline rate (angle gamma) probably by producing a longer growth period and then a steep fall (c>>b).
