Chris,

I applaud your effort. I appreciate that you're coming from a math background and not petroleum engineering. I think that allows a useful different perspective. And I suspect you probably understand part of your characterization of a "typical field decline curve" has limitations but I understand modeling needs to start at a simple base and then allow more complexity to enter. Ignoring gas fields for the moment, oil fields tend to fall into two general reservoir drives: water drive and pressure depletion. The build up to peak production results from the time span to bring all field wells online so I skip that section. Your basic model most closely fits that of a pressure depletion drive. As oil is produced the natural gas saturating the oil comes out of solution and aids in the production. But as pressure is drawn down there is less energy to drive the process. But this model is often complicated by the reinjection of produced NG back into the reservoir to maintain reservoir pressure as high as possible. This would alter you basic curve significantly if the pressure maintenance effort is very effective.

In strong water drive reservoirs the oil production rates drops very slowly initially as the buoyancy force of the water pushing the oil up the well bore remains fairly constant. But when the water level approaches the perforations in a producing well the oil rate can drop very quickly. But after the oil/water ration reaches a very low level (let’s say less than 20% oil) the oil decline rate may become very low. I’ve worked with fields that have recovered 100’s of millions of bbl of oil but produced 70% of their cumulative production at oil rates of less than 20% of the total fluid produced. Many of these fields took over 50 years to recover the bulk of their production. Many of these fields are still producing today even though the “oil cut” is down to 1% or 2%.

In applying a model to current global production and looking forward one must look at the drive mechanism of the remaining super fields. Ghawar in Saudi Arabia is a good place to start. Being a strong water drive reservoir its production rate remained fairly level during the early part of its life. But its total production profile is complicated by various secondary recovery efforts including late life horizontal drilling. I can imagine such analysis could be beyond you capabilities. In time perhaps you can have someone with reservoir engineer expertise to assist in this regard. Good luck with your future effort and don’t let the geobabble get you down. Reservoir dynamics are governed by physical laws which lend themselves well to strict math rules.

Interesting post, thanks!

It reminds me what I've tried to do a few years ago:

Convergence of the sum of many oil field productions

I also observed a Gamma distribution:

Some statistical models are also well established in geology such a Lognormal distribution for the distribution of field size. Once the triangle parameters are random variables, the skweness of the resulting production profile almost disappears:

see here for the details.

It's gonna take me some time to digest your article but from a first look it seems that there are no stochastic aspects in your model, the triangle parameters are not random variables, right?

I believe the premise missed in this analysis is that field sizes have nothing to do with the shape of the underlying curve. Reading the paper, I would argue that assuming a monotonically declining size of field after the initial discovery is a restrictive constraint. What the distribution in field sizes actually leads to is noise on the curves. i want to point this out because alternative models such as dispersive discovery do not make any a priori assumptions on field size distributions.

What if we were robots, did not have finite lifespans, and did not have evolved wet-ware that strongly favored immediate consumption over delayed reward....

Assume the oil technology we had would be the same as we humans now have. We wouldn't sell the oil - our objective would be to maximize the total amount of oil produced over time (could be a decade, could be a thousand years, whatever time period would maximize Qt)

At what point in the initial ramp up in production, would we use technology to 'slow' down the flow, so as to maximize long term sum of oil extraction?

(Said differently, the modern human system is an economic one. To produce oil we have to buy equipment, labor etc. with money. We need to get a return on this money relatively quickly (time value of money, etc.), so we are eager and willing to let production ramp as fast as it can (barring a complete economic depression.) How much does a positive market discount rate borrow from future production?) (I guess this is more of a geology question than a math question but Im curious if anyone has insight as to the magnitude of this number - is it 1% or 50%?)

If you integrate that triangle or sum the cumulative area from left to right you get a pseudo-logistic spline curve. That is a convex quadratic seamlessly joined at an inflection point to a concave quadratic. There's your sigmoid or S-curve which we all agree is the hallmark of increasing returns followed by decreasing returns.

If indeed all-liquids is on a 2005-2008 plateau as suggested on other threads the triangle would seem to have the apex sliced off.

Nice post

No matter how bad a day I have The comment section always cheer me up. Nice info.

David Strahan is a character that deserves a closer look. Why do we keep seeing his name? Who is he?

certainly "ccpo" has asked himself this(these) question(s) every night before beddy byes.

Dudley, Thanks for this contribution, the most important point being IMO your 35:65 split - to which I will return in a moment.

Rockman makes some good points about engineering - and this brings me to human intervention, technology and economics. A couple of pointers as to how this can modify field production profiles.

When the oil industry moved off shore it became preoccupied with maximising flow rates and cash flow as early as possible - often at the expense of ultimate recovery. 20 odd years ago a company working off shore would install the platform having invested in said platform and often a pipeline too. They would then start drilling wells and bring these on sequentially as fast as they could drill them - always keeping an eye on reservoir pressure to keep this above bubble point. At some point they would have to drill some water (or gas) injection wells for pressure maintenance.

What happened next was that companies started to "pre-drill" wells before the platform was built / installed. So maybe 6 to 10 wells would already be there by the time the steel jacket arrived. I suspect this was made partly possible by low rig rates during the recession that affected the world oil industry from 1980 to 1998. When the platform arrived all they had to do was to hook up the pre-drilled wells and the field would hit "plateau" / peak immediately - pretty like your wedge triangle.

So one thing I'd like to see done here is you playing around with a number of different field production profile shapes - including ones with great long drawn out plateaus like the Saudi supergiants where natural peaks have never been reached owing to conservative field management practices.

Another factor to consider is to compare and contrast field management behavior between onshore and offshore environments. Offshore, fields are inclined to have a more finite life span that onshore - again see what Rockman says. In the north sea for example, the cost of maintaining infrastructure will eventually get too high and it will be decommissioned with recoverable oil still in the ground.

And so to your 35:65 split. This makes intuitive sense to me and helps bridge the gap between those who see a 50:50 split and 2 trillion bbls URR for Earth and those who see much higher reserves - IEA et al. My feeling is that we are in peak oil territory - but according to your model we may still have the best part of 2 trillion bbls to produce. In essence, the post peak decline will be shallow as we (Mankind) do everything we can to wring the last drop out of old fields.

A real world example of this is Ghawar where we have tended to only think about dry oil production and ignore the 10s of billions of barrels that may one day be produced from behind the flood front - that is if the Saudis are in any fit state to make the massive investment to bring this about - which is doubtful for a range of complex reasons.

One data set I'd like to see is the cumulative number of producing oil fields 1900 - 2008 together with the cumulative number of decommissioned fields.

The CERA depletion study showed that around 40% (?) of current production comes from new fields where decline is dominated by expending natural reservoir energy.

Global oil production will end up being dominated by a gigantic stack of old fields producing at 90%+ water cut. We do quite desperately need eroei data on strippers - Nate?

There is always the possibility that it is natural reservoir energy that is key to high eroei and that once that is spent globally it is game over for the human race:-(

Does Westexas or Nate or someone have a graph showing the actual curves of former peak areas like Texas, Oklahoma, eastern US and such to use for comparison. To me, the triangle is very front end loaded and that the 35:65 ratio seems too optimistic. I understand ROCKMAN's point about offshore well production peaking almost immediately, but from the North Sea example, I think you still see a less front end loaded triangle and mainly a steeper, shorter time span of the same look, more like Khebab's graphs. I still have alot to learn about how long secondary and tertiary recovery techniques extend field life, but it seems to me that we will tend to sacrifice the tail of production to save the peak if we at all can (atleast until global trade breaks down)

It rises quickly to it's peak at time t=1 and decreases slowly until no oil is produced at time t=6. The idea is that the natural pressure of the oil field causes rapid production initially, after which decline is more gradual. Before and after the peak the curve is linear, so it looks like a triangle.

If the decline was the result of pressure wouldn't that part of the curve be convex, due to Bernoulli's equation--k*sqrt(P)=Q
(where P is pressure and Q is volumetric flow)?

I would guess that the rate of production in the growth
phase is proportional to the number of wells dug in a field (which is economic) and the cost per well goes down
when more wells are dug. Once the number of wells are fixed it is the extraction phase which would be governed by pressure.

But most models seem to be convex in the growth phase and concave/negative exponential in decline which is different to what I would expect.

At any rate, oil production is an economic activity which is why the deterministic R/P model should work. Economics is about the maximizing production rate not maximizing the resource.

This paper looks a little bit like an opening for Fuzzy Logic (which is an improvement over probability theory?).
http://en.wikipedia.org/wiki/Fuzzy_logic

Chris,

Khebab is right. The central limit theorem basically gives you this Bell-curve, even without requiring each oil field to conform to any types of production profile shape. I have pointed out this fact at my own site back in 2006.

http://www.1stmillionat33.com/2006/07/peak-oil/

There is no magic to it. It's just math on random processes.

Regards,

A couple of comments. One overall the rate at which we bring new fields online is limited by the number of drilling rigs. Offered without proof but overall worldwide then number of active rigs in any given year on average is fairly constant within a reasonable range.

Baker Hughes rig counts from 1975 -2008
http://investor.shareholder.com/common/download/download.cfm?companyid=B...

We see the absolute number change with economic conditions but over the entire production period the number of active rigs is fairly consistent. Certainly details matter but the important point seems to be that world wide production rates at least of the period considered are geared by the number of active rigs and previous production.

If one assumes that in any given year the best fields will be drilled as soon as possible then the wells these rigs drill are biased with both production rates higher earlier on and the lifetime of the well lasting longer earlier on since we always select the largest known field first given the limited number of rigs.

All the data I've seen about actual oil well lifetimes performance etc point to signifcant cherry picking of the best known fields first for actual development next discovery although its fairly random and dispersive seems to lead production by a significant amount. So early on you had a large number of choices to exploit at any given time. As long as discovery can lead production then selection pressure to produce the best/largest fields first seems probable. Well production data points towards systematic declines in productivity increases in drilling depth etc.

Now how to model this ?

I suggest the current paper is pretty close but with some modifications.

First the giant and super giant fields can and should be treated as a different group. The production levels from these fields is significant and accounts for 40-60% of overall production the production profiles of these fields span decades and seem to have a trapezoidal shape consistent in field drilling programs can maintain production levels close to that bottle necked by above ground infrastructure for decades.

Inclusion of production from these fields in global estimates will swamp the production of smaller fields. For most of the production history they can for all practical purposes be considered constant.

Next assuming that the current paper is correct for small fields we need only include the effect of a limit on the rate we can exploit the fields. I think its correct that the bias is towards larger fields first but the peak itself is probably not sharp but a long plateau limited by the number of active rigs.

So a distribution of triangles like this is probably accurate.

2 4 8 16 32 64 64 64 64 32 16 4 2

This is the number of wells drilled or fields exploited the actual triangle dimensions are weighted to the earlier side of the distribution say with a linear weighting with the last fields drilled 50% smaller then the first fields.

What this results in is a distortion of the curve into a sort of shark fin like distribution proably it reaches plateau at 36% but production continues close to this rate well past 50% depletion on finally declining later in the sequence say when we hit the final 16 or 4 fields. The only change is that above ground factors throttle the rate at which new fields can be developed but on the same hand it also allows significant selection for the best fields first as discovery significantly leads development.

The final drop of in production is then driven by the lack of replacement fields and is shaped by the lifetimes of the final fields added. So when replacement stops if the fields have a average lifetime of 10 years production declines rapidly as no fields come on line to replace the previously developed fields and average field lifetime also drops. So assume that not only is the curve weighted best first but also that field lifetimes decline from left to right say from 30 - 5 years.

The move from onshore to offshore advanced in technology etc etc support these sorts of trends.

Advanced recovery and stripper wells adds a small amount with a profile quite similar to the super giants but its quite small and can be treated as a constant with new depleted fields moving to stripper status at a fairly constant rate.

The net result in my opinion is that the final decline curve is heavily influenced by the final decline of the small fields once replacement is no longer possible and should be quite steep. If the super giants decline at the same time then replacement can be assumed to be over for sure as any new fields could only be assigned to offset super giant declines. Without new regions to drill then you see a initial sharp decline from the small fields reaching end of life over a period of the same order of magnitude as the average field lifetime. I think right now thats about 10 years or so.