[khebab]

A lot of very interesting comments! thanks!

I checked on the Ricatti. It looks like you did an excellent job reproducing the math in some of your supporting posts. Weird how the URR seems really low (~1600) which causes the curve to do a nose-dive between 2010-2030. However, if you push up the URR to 2500, the curve flips upward crazily. You said something about adding another control function to suppress this, but this non-linear stuff sometimes has a mind of its own.

I agree, the shock function is hard to handle and is poorly justified by Guseo. Guseo finds an URR around 1,600 Gb and he claims that his approach does not require prior knowledge of the true URR value. It's not entirely true because I found that the final result for the URR will strongly depend on the choosen initial value - which reflects what you think the URR should be! - as you can see on the graph.

I don't have any non-linearities in my formulation apart from the forcing function (which relates to the discovery curve). This makes it well-behaved for all parameter inputs, so that you don't get those unexpected swings.

I have to find time to seriously go through your work at mobjectivist.com.

there is a really good reference book out there
Decision Analysis for Petroleum Exploration, 1975, Newendorp, Planning Press, Colorado, 668 pp.
This is a classic study in Bayesian logic applied to oil and gas problems, the chapter titled "Probabilities of Outcomes of Multiwell Drilling Programs" addresses some of the issues you probably need to think about when simulating..starts on page 327.

Thanks for the reference!

That means that a single WHT-Hubbert curve will only fit in a constant-price domain. It follows that the linearized derivative of the WH curve will also only give you a straight line to an accurate URR in a constant-price domain. When prices shift, URR shifts too! But that's sort of obvious: if you are estimating URR based on past production only in a constant-price domain, you are making the assumption that prices will not change. If prices change, more oil becomes "economical" to produce, so URR increases.

Guseo tried to build a price-based shock function but with not much success! one piece is missing here, you will have to model demand in order to infer the impact of high prices. If demand is strong, the probability of higher prices increases which creates an economical incentive to explore more and to implement thrid/fourth generation EOR techniques. But EOR is not applicable on all fields and will have not the same success on all fields. Projected Increased in URR are mainly projected oil displacement from the P50 reserves to the P90 prior to any real implementation.

(and by the way , most librarians think that I'm crazy when they realize that a MD is poking into the geology section of the library Rolling Eyes)

B) Modelling oil reservoir discovery = EXPLORATION. I found particularly enlightening the comments that rockdock made about log-normal distributions. Are these the terms "creaming curves" are understood? If we had access to discovery data we could estimate such distributions from start by using non-parametric kernel based methods and not rely on fixed parametric assumptions

Agreed. It seems to work. I stumbled on an article of Laherrere on that matter.

A) Modelling of the physical processes that describe indivindual well behaviour.

We need to define what level of granularity (oil field or wells) is really necessary being careful to to go too low because reservoir modeling is a very very complex science! WHT used rather simple models which gave reasonnable results.

C) Modelling the economy i.e. supply and demand . My understanding is that markets are modelled using Stochastic Differential Equations (e.g. Black Sholes formula), but I have absolutely no technical experience in either deploying these mathematical tools (although I'm a fast learner!) or even understand the econometrical context ... Any volunteers?

me neither! this is quite a difficult problem. I`m not aware of any research paper on that issue.

For that I propose we use the open source R (http://www.R-project.org) and (win)BUGS from http://www.mrc-bsu.cam.ac.uk/bugs/welcome.shtml
The second can be dowloaded for free after registration .... and there is also a library that allows R and BUGS to communicate. This will at least guarantee a conformity of tools .... and allow people to test predictions/assumptions on their computers.
However such tools require data, data, data (old and new) and here I rest my case . I hope that other people may contribute ...

I agree, I've already started my learning curve on the R language.

F) Getting the data: rockdock/shakespear1/taskforce_unity any ideas?

I think it should be the first step. Collect the different data and build a data repository which can be easily consulted and updated.

[WebHubbleTelescope]

Khebab's posting of the ASPO discovery curve got me motivated to try an experiment.

Looking at what EnergySpin wrote on Compartmental analysis explains the tact I took on the experiment. I added 3 latencies corresponding to (1) a "fallow" period (2) a "construction" period and (3) a "maturation" period to get the ASPO data shifted enough on the time scale to make sense. This essentially provided a good "active" discovery window that matched my previous triangular and Welch forcing function.

Everything lines up (oil shock perturbations included) if I choose 8-year average latencies for each phase, excepting for the initial ~15 year <i>1/e</i> extraction rate.


As usual, I included the explanation and source code here:
http://mobjectivist.blogspot.com/2005/1 ... -data.html

The fit scares me a bit, but the outcome scares me even more. Can we trust the ASPO data for estimated oil discoveries? Is it way underestimated? Or, are we cooked for sure?

[EnergySpin]

Lots of info here ....
I will try to provide input to a few issues but I feel we need to set up a timetable and a work - plan after we cool down

We need to define what level of granularity (oil field or wells) is really necessary being careful to to go too low because reservoir modeling is a very very complex science! WHT used rather simple models which gave reasonnable results.

I am suggesting a simple model as well ... one that models a region within a basin as the sum of 2 or 3 exponential terms. Consider the following simple model (and the geologists here should comment on its conceptual accuracy).
A well or a set of production wells (P) empty an oil field; the volume of the oilfield is partitioned in the following compartments: V1 (the one directly drained by the pumps), a compartment V2 that communicates with V1 and a compartment V3 that communicates with neither V1 nor V2. Under these circumstances the URR of this particular oil field is bounded from above by the sum of the volumes of the two compartments. Any interventions that establish a communication between V3 and one of the V1 or V2 will lead to an increase in the URR.
In graphical terms the connectivity of this model looks like this:

MODEL1 V2--->V1---->P if the communication between V2 and V1 is unidirectional OR

MODEL2 V2<-->V1----P if V1 and V2 communicate in a bidirectional fashion.
Let's examine the first case (which can be solved analytically)
In order to add function to this model we assume that under the influence of pumping V2 drains into V1 according to the following law:
(Eq1) dV2/dt = -k21*V2(t)
Coservation of mass then mandates that the dynamics of V1 evolve as:
(Eq2) dV1/dt = k21*V2(t)-k1*V1(t)
Boundary conditions are V2(0)=V2o, V1(0)=V1o
Direct integration of Eq1 gives the evolution of the volume in V2 compartment as V2(t)=V2o*exp(-k21*t)
Substituting and integrating equation 2 (which I did with Mathematica cause I'm too old for this ) gives the general solution:

A1*Exp(-k1*t)+A2*exp(-k2*t),

where A1 = (k1*V1o-k21*V1o-k21*V2o)/(k1-k21) and

A2=k21*V2o/(k1-k21)

Side Note: Writing Equations in ASCII sucksssss


The instantaneous production by this system is simply P(t)= k1*V1(t), and the cumulative production from time t=0 to time t=T is simply the integral of k1*V1(t) in the interval [0,T] which is simply given by:

A1*(1-exp(-k1*t))+A2*(k1/k2)*(1-exp(-k2*t))

What are the mathematical properties of the instantaneous production curves ? Does it have any peaks so to speak?
Let's see: P'(t)=0=>k1*V1'(t)=0 which after a lot of mathematical mambo jambo gives us that there is a single peak at time t=log(-k1*A1/(A2*k21))/(k1-k2).
In general the curve is not symmetric ... and the decline in production post peak is slower than the rate of rise pre-peak.
For those of you who have the time .... one can play with the formulas and arrive at expressions for the cumulative production at the peak , the pear production rate etc.

Conclusion
For an oil field whose dynamics of drainage can be described by a rate time-invariant two-compartmental system there will be a single peak!
Note the sentence in italics ... the rates should not change for this to occur. Does this reflect the real world? This would mean that whoever is controlling that particular oil-field is content pumping with the same equipment, is not trying to offset production by "opening up" the pumps or drilling more wells etc

The interesting feature is that with this expressions at hand and production data during a period when no changes in management of the resource occured one could estimate the constants, volumes etc and hence have an idea about the URR. In addition if one knew the geologic constants k21, V1o, V2o and a detailed production history for the oil field one could detect whether declining production is due to someone shutting down the pumps deliberately or due to depletion. (Left as an exercise to derive such expressions )

What if there is a different management policy? One that actually puts more holes in the ground or pumping full throttle? Actually ... the system of differential equations becomes more complicated ... and analytical solutions in general do not exist (however the last time I seriously thought about ODEs in their analytic form was 14 years ago so feel free to correct me).
A particular case is one where the rate constant k1 increases linearly with time i.e. k1(t)=a+b*t. Then the system of ODE's has a rather complicated solution but the general features are the following:
a) the date of the peak is shifted in the future .. but not by far
b) the production at the peak is higher (obviously)
c) the curve becomes more symmetric (and the "symmetry" is determined by the relation a/b)
d) the rate of decline is always smaller than the rate of ascent .

In reality one has to "play" with the version that allows bidirectional communication between the two compartments (I think) and some of the conclusions might be different.
Rockdock what do you think about all this? Does it sound reasonable as an approximate model?

[EnergySpin]

The fit scares me a bit, but the outcome scares me even more. Can we trust the ASPO data for estimated oil discoveries? Is it way underestimated? Or, are we cooked for sure?

I am not sure I'd trust ASPO data . In reality ASPO data includes a hidden assumption: URR is fixed and will not change. But URR is a derived figure determined by how much oil is down there and how much we are expecting to extract. "How much is down there" is fixed by geology ... how much we are expecting to extract is determined by technology.
And the downslope is going to be determined by both geology+economy.

[EnergySpin]

Let's play with the model a little bit to see if we can gain an understanding about North Sea or the Mexican fields.
As I said, contigent on the accuracy of the approximation ... sharp declines post peak do not show up when I explored the resulting curve with Mathematica (I admit that the exploration was limited though). So is there a mechanism that can account for such sharp drops?
One fairly obvious mechanism would be a change in the relative size of the V2 (which is the "mother" compartment feeding everything else) and V3 (a compartment that cannot be drained).
If somehow (?Earthquakes ) a volume of material was transferred from V2->V3, at the time the peak occured .... then the resulting drop post peak will be steeper than it would have been. Obviously this will lead to a reduction in the URR achieved. If such a mechanism were to operate in the ascent phase (i.e. pre-peak), then the most likely outcome would be an acute drop followed by a slow rise to a smaller peak.
Another mechanism would involve a reduction in the rate constant between V2 and V1 i.e. the "mother" compartment empties slowly in the smaller V1 compartment. If such a change were to happen at the peak, then the decline would be steep, and production would go to a lower value . If the change in this rate constant was such, that the inflow rate in V1 was much slower than the outflow rate i.e. k2 << k1 balancing the diffrence in size of the two compartments then extraction of the resource attains a plateau since dV1(t)/dt =0 =>V1(t) approx constant => P(t)=k1*V1(t) approx constant.
However both scenarios argue against a further sharp decline in the future .....A hypothetical discontinuity would have short-lived effects in the rate of decline.
Comments are welcome

[SilentE]

Everything lines up (oil shock perturbations included) if I choose 8-year average latencies for each phase, excepting for the initial ~15 year <i>1/e</i> extraction rate.


...

The fit scares me a bit, but the outcome scares me even more. Can we trust the ASPO data for estimated oil discoveries? Is it way underestimated? Or, are we cooked for sure?

Good match! Very interesting...

A few questions:

1. Is that a "backdated" discovery datebase from ASPO? Cause that's a potential source of error - it would systematically undercount more recent discoveries.

2. What kind of average is the 36 year off-set for production based on - is it weighted? I know there always appears to be a lag, but the size of the lag varies substantially, and the times to develop non-OPEC fields can vary a great dael - and may be decreasing on average. Also, OPEC producers aren't even trying to maximize production. But doesn't a production-lag model (however long it is) assume that they will do so? If nothing else, there reserve-production ratios a likely to be much lower than for non-OPEC. As the market share of OPEC rises, that effect will become stronger.

3. The projection for future discoveries looks a little low. The extrapolation line seems to come off the troughs in discovery, instead of a mean value between the troughs and peaks. It looks like the problem is that the extrapolation comes off a three-year moving average, the last value of which happens to be centered in a discovery trough (coincidentally missing the higher discovery rates of 2000-2001).

4. I'm skeptical because as soon as the historical data run out, the model takes the curve in a totally different direction. The fact that your model seems to be swamping the data - and is even in disagreement with the most pessismistic recent estimates from Campbell - is suspicious.

5. The older post mentions four pre-2000 shocks: 1973, 1979, 1984, and 1991. Which did you drop? And while 1973 and 1979 are obvious, the effects of the Gulf war were more transient, and the 1984 "end of recession" seems truly odd. The recession ended in 1982 - it lasted 3 quarters. What about the 1981 decision to end US domestic price controls (which had strangled domestic E&P), or the 1985 Saudi decision to abandon high prices and finally open their taps?

6. An unexplained fifth "reverse shock" has been added in 2001. Why? What does it represent?

[WebHubbleTelescope]

The "reverse" shock is actually a sudden increase in extraction rate in the last few years.

And as Khebab has pointed out, the ASPO data only shows conventional oil strikes and the BP data shows all production sources. Which means this curve will extend a few more years beyond that shown, if I go back and make a deeper suppressive shock in the late 70's.


Eyeballing the area under the curves, it will extend the peak onset by a few more years. But then again, the ASPO data are estimates of discovery and could be high or low.

[WebHubbleTelescope]

Given the fit to the curve was so good assuming that conventional sources matched all sources, this indicates that the extraction rate of conventional sources likely decreased linearly over the last 30 years with the linearly increasing non-conventional sources making up the gap in demand.

Code: Select all

ConstantRate(t) = ConventionalRate*(1-t/T) + NonconventionalRate*t/T

where t=time from when we started using nonconventional sources, and the rate is defined as proportion extracted of reservoir volume per year. I would consider this an economic argument governing substitutability of resources. The invisible hand of the oil industry at work trying to extend the plateau, so to speak.

If I stick in a suppressive extraction rate that decreased linearly by about 15% between 1974 and 1991, it gives a good eyeball match if you imagine the conventional source curve hovering below that of all sources.


As you can see, this extends the peak to 2008. Yes, about three more years due to the use of non-conventional sources. Whew! Now we can rest easy.

[WebHubbleTelescope]

A groundwater class, eh?

I took a college class in limnology once. I learned that eventually all lakes will eventually fill up completely with muck.

My answer to everything is "I don't know anything but the mean". So for all the information you desire with respect to specific oil field characteristics, such as porocity, etc., I suggest we replace it with some empirically observed macro parameters. For example, humans are greedy creatures; they will extract an amount proportional to the amount of resources available. If you have a big oil field, they will extract at a high rate. If you have a small oil field, they extract correspondingly less. We don't know the exceptions to this rule (such as Ghawar probably didn't get extracted at that high a rate), therefore we can say absolutely nothing about the standard deviation. Like I said, lacking all available information, use the mean. Convolve enough of these means together, and we can approach a kind of central limit theorem for the production curves.

On another level, I think all I want to achieve with this exercise is to replace the logistic curve (i.e. the Hubbert curve) with something that reaches a canonical level of representation. My intuition tells me that knowing the details in oil extraction won't help in any of this. If you want to try it, go ahead, but for me, a simple model with some effective data regression techniques will hopefully take us beyond the logistic curve.

[EnergySpin]

A groundwater class, eh?

I took a college class in limnology once. I learned that eventually all lakes will eventually fill up completely with muck.

My answer to everything is "I don't know anything but the mean". So for all the information you desire with respect to specific oil field characteristics, such as porocity, etc., I suggest we replace it with some empirically observed macro parameters. For example, humans are greedy creatures; they will extract an amount proportional to the amount of resources available. If you have a big oil field, they will extract at a high rate. If you have a small oil field, they extract correspondingly less. We don't know the exceptions to this rule (such as Ghawar probably didn't get extracted at that high a rate), therefore we can say absolutely nothing about the standard deviation. Like I said, lacking all available information, use the mean. Convolve enough of these means together, and we can approach a kind of central limit theorem for the production curves.

On another level, I think all I want to achieve with this exercise is to replace the logistic curve (i.e. the Hubbert curve) with something that reaches a canonical level of representation. My intuition tells me that knowing the details in oil extraction won't help in any of this. If you want to try it, go ahead, but for me, a simple model with some effective data regression techniques will hopefully take us beyond the logistic curve.

Keep it to the basics ... A well is draining a relative small area of the reservoir which is much larger. The rest of the reservoir exchanges matter with the area near the well. A two compartment approximation (i.e. 2 exponential terms) could provide a pretty good approximation for our purposes. Deffeyes is assuming a decline/ascent proportional to the amount of the oil still left in the ground (which gives you a uni-exponential curve). Problem with Deffeyes is that a uni-exponential curve has no maxima but at the beginning ... so he must be relying on a combo of logistic+exponential decay post peak.
However at least in his model , there are parameters with definite physical interpretation and not mambo-jambo stuff like the Ropper/logistic/Verhulst approximation.

I have to disagree WHT , if one had detailed mathematical models of matter fluxes inside an oil field one could derive macroscopic approximations via means of Neural Networks and use them in "macroscopic" applications for prediction. In fact there is a fair number of consulting companies which do this kind of work for the industry.
ElijahJones the model you are proposing (or something similar) is applied in convex hull/finite volume models of reservoirs ... problem is you need 3 detailed and different sets of data and a cluster to get results out of it. We neither have the data nor the computing power ... so realism suggests we stick to approximate models that can be fitted using production data only.

[EnergySpin]

The idea of estimating reserves from well data seems tricky. At best its an inverse problem that may not be well defined, so maybe you get a bounded parameter space instead of a crisp estimate. Use some simple interval arithmetic to get upper and lower bounds on reserves.

Point A: correct . I was thinking more of a Bayesian type of solution that bounds the estimate.
Point b: Good luck with your thesis . I finished mine before I found po.com!
Take care

[WebHubbleTelescope]

If you don't even know the standard deviation how can you judge your results, you need a standard error for confidence intervals?

Isn't this one of the lessons from information theory and maximum entropy principles? All the parameters that I incorporate into my model which happen to use the mean of some observable, have a standard deviation which equals the mean. That happens to give you the exponential function. In general, it is a safe bet that works for all kinds of analyses where you need to maximize the uncertainty because of a lack of complete information.

I suppose the confidence intervals in the solution space can be determined by systematically varying the means of the parameters. That sounds like a good idea and would both provide a good way to do data regression and get the sensitivity error bars at the same time.

[WebHubbleTelescope]

I have to disagree WHT , if one had detailed mathematical models of matter fluxes inside an oil field one could derive macroscopic approximations via means of Neural Networks and use them in "macroscopic" applications for prediction. In fact there is a fair number of consulting companies which do this kind of work for the industry.

ElijahJones the model you are proposing (or something similar) is applied in convex hull/finite volume models of reservoirs ... problem is you need 3 detailed and different sets of data and a cluster to get results out of it. We neither have the data nor the computing power ... so realism suggests we stick to approximate models that can be fitted using production data only.

So then I would suggest we have two classifications for the means. The first class comes about from estimates provided from industry experts and historical data. Such as:
How long does a discovery lay fallow on average before the decision to extract?
How long on average does it take to build a production rig?
How long on average before the rig is pumping to maturity?

The second class comes about from the NN "helper" applications that fuse the available information. These can solidify the assumptions that I have been making. Such as:
Is the rate of extraction proportional to that remaining in the reservoir?

And then we have the complete unknowns, such as how much is actually still underneath the ground?

Agreed that people do this kind of work in the industry. But "consulting companies" = "proprietary knowledge", and I thought that is why we are doing this here -- to let the amateurs finally get a chance to unlock the secrets. The cathedral versus the bazaar. That is partly what facinates me about this work.

[EnergySpin]

Such as:
Is the rate of extraction proportional to that remaining in the reservoir?

And then we have the complete unknowns, such as how much is actually still underneath the ground?

.

As I said in a previous post we should try to decompose it in three or 4 different modelling exercises and then fuse them together.
Let's start with what I call macroscopic reservoir modelling:
The simple conceptual model of oil production in a space with a hole : the fluxes of material from the reservoir to the outside world is then (at first approximation) proportional to the amount that is still remaining:
Conservation of mass requires that the rate of decline in the material = rate of production, hence:
P(t) = dV/dt = -K*V(t)
Integration gives us: V(t)=Vo*exp(-k*t)
Notice two things about this :
a) the production attains a maximum at the beginning i.e. P(t) = k*Vo*exp(-k*t) and there is no "peak".
b) the constant k mixes both geologic factors AND economical factors with no clear way of separation among the two
Therefore a simple exponential curve does not reproduce the phenomenon .
The second approach conceptualises the oil field as two separate compartments:
One (small) compartment around the well and another one communicating with the rest of the reservoir (i.e. ? source rock)
The rate of transfer from the big compartment to the small one is driven by "geology" (porosity and all that), the second rate by the people operating the pumps. In this model there is a clear separation between geology, economy and the volumes of material that can be extracted appears as a parameter to be estimated from production data.
Material fluxes modelling in these complex systems (i.e. from BIG to small reservoir) need an accurate understanding of the specifics of the oil field, however a first approximation would be to treat those as constants.
The mathematical basis of this assumption is also sound in the following sense:
If the rate of transfer of material from BIG->small is an unknwon function, of time one can always expand it in Taylor series (contigent upon certain continuity assumptions) and keep only the first, second etc terms.
At first approximation only the first term is important and one ends up getting the formulas I analysed in a previous post in this thread.
If one has access to production data from period where the oil field was pumped at a stable manner (i.e. the rate constant describing the transfer from the small compartment to the real world was fixed) then one could use these data to obtain estimates (in the Bayesian sense) of the both geological parameters (Vo, rate of transfer BIG->small). Then at latter times he could use these probabilities to find out whether changes in the administration of field occured. Again this is only possible with the Bayesian perspective.
This is the way we use mathematics to arrive at dosage regimes of medications for example i.e. estimate the volumes that a drug is distributed or to time the administration of antidotes in case of poisoning by taking measurements (think of them as production data) and plugging them to equations. The principles are the same!
By the way, have you checked your PMs? I sent you a message about reservoir modelling and estimation via such linear transfer models

[Taskforce_Unity]

[VERY GOOD QUESTION! I have no data on the distribution of the discovery for small fields (0-100 tbpd). Simmons is just saying that there are 4,000+ fields that have been discovered up to the year 2000 which are providing 53% of the world production!. That'it, that's all! In statistics, when we have no knowledge, you usually assume a uniform distribution which gives about 700 discoveries per decades. However, since discoveries have been declining since the 80s I assumed a uniform loss of 100 new fields per decades.

Here is a graph from IHS Energy about the distribution of oil discoveries



This comes from this presentation

Global Oil Supply Issues: Recent Trends and Future Possibilities

As to the timedate of these discoveries.. i have no data.

As for American info i suggest this website:

Michigan oil field data

I am willing to copy data into a spreadsheet to save you guys some time(whatever program you would like, ill download it )

My dad might also be willing to help. He is pensionated now and has a full life of mathematical (and statistical history) at three Dutch Universities. Ill try to summarize what's in this thread for him and ask him to take a look at it. Hopefully he can give some comments/directions.

As to data access, i only can look into public data. The exception to this are journals (through my university) like this one: Quarterly oil supply OECD

Another suggestions of mine is to involve the people from the oildrum in this project. They know a lot of things and are great analysts.

[EnergySpin]

So do you have the data in the forrm of oil field volume, size (i.e. area in square meters etc) and counts?
That would be helpful. Excel would do fine ...

[Taskforce_Unity]

So do you have the data in the forrm of oil field volume, size (i.e. area in square meters etc) and counts?
That would be helpful. Excel would do fine ...

pfft no data publicly available that i know off.

At the DTI UK oil & gas database you can get the numbers of OIIP, recoverable reserves, Cumulative production, Water injection per well, oil production per well, wellhead pressure, days on production, field depth. Oil production and so on.

However i cannot find size (area in square meters) and i don't know what you mean with counts.

DTI data

Just specify what you would like to have in your spreadsheet, ill try to spend some time on it.

Another suggestion of mine is to meet on team/peak speak. Might be helpful to make some agreements on who does what in a easy manner.

Im now going to bed, ill try to find specific field by field data on size, production, wells tomorrow.

[EnergySpin]

Yes we will do that tomorrow. It is getting late here and I am hunting on literature for a paper.
I ey balled DTI. Very interesting ... the production profile of those fields does not really much the Michigan fields you sent me.
There is greater heterogeneity in the upslope part of the production of North Sea oil fields, but once decline starts it seems to proceed at an exponential decline (fast initially, slower then) .
US fields peaked rapidly (within a couple of years from the day they were first drilled) and tapered of slowly. Peak date was at less than 50% of the URR .... typically at 30-40s. This is the only feature that is shared between the two areas.

I wonder whether the fields in North Sea were fragmented or somehow "different" than the US fields.
I will try some bi-exponential fittings at the Michigan oil fields .... and report back. But modelling North Sea is going to be a bitch .... it is as if people were simultaneously draining more than one compartments. Once the smaller ones went out ... production is sustained by a core compartment which now falls according to the US experience.
But at least for the North Sea ... I find it difficult to believe that a nice clean "Hubbert" curve can yield results. But of course I just started playing with the data