My assumptions are the following (the symbol "~" means "distributed according to the law"):
Each oil field is modeled according to a logistic curve:
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P(t)= kxURR/2(1 - tanh^2(k(t - t_half)))
with random parameters (URR, k) according to the following laws:
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URR ~ Exponential(10)
k ~ Uniform(1) + 0.05
The exponential law for the URR (mean = 10 Gb) results in less frequent big fields and a lot of small fields. For each field I assume a first year production rate equals to 10% of the peak production of the logistic curve:
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P(t0)= 0.1 x URR / (4 x k)
In order to preserve causality, I have to estimate the time between peak production time and t0.
The time of discovery is a mixture of a gaussian and a uniform distribution controlled by the parameter beta:
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t_discovery ~ beta x Gaussian(25,10) + (1 - beta) x Uniform(50)
I assume a systematic oil field developpment time of 5 years +/- 6 months:
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t_developpment ~ Gaussian(5, 0.5)
Some preliminary results for two values of beta (1.0 (just gaussian) and 0.5 (half gaussian and half uniform)):
Figure 1
Figure 2
Two observations:
- the first figure demonstrates that a sum of logistic curves give a monopeak curve which seems to be asymetrical (skewed to the left) and looks like a Gamma distribution.
- the second figure is similar to the UK production where discoveries led to two successive production peaks.
From this first simple test, I already bumped into a few critical questions:
1- is there a relation between the URR and k?
2- is there a relationship between early discoveries and the URR (big fields discovered and exploited first)
3- what is the relationship between starting production level (first year) and the maximum output (I assumed 10%)?
4- What is the observed oil field size distribution? (I assumed an exponential)
Of course, this a very crude statistical model where there are no economical constraints.
Any suggestions is welcomed!
