109 comments on The Derivation of "Logistic-shaped" Discovery
Comments can no longer be added to this story.
| Show without comments | PDF version
109 comments on The Derivation of "Logistic-shaped" Discovery
Comments can no longer be added to this story.
| Show without comments | PDF version
Search The Oil Drum with Google
Support The Oil Drum
Recently on TOD:World
TOD:Campfire
- Thanksgiving Open Campfire Thread
- How Relocalization Worked
- How to Set Up and Run a Bicycle Repair Company
TOD:Europe
- Unique Times -- and the Future
- Peak Gold, Easier to Model than Peak Oil? - Part I
- Carbon Capture and Storage
TOD:Canada
- In this house, we obey the laws of thermodynamics!
- The Round-Up: October 24, 2008
- Compressed Air Energy Storage - How viable is it?
TOD:Australia/NZ
- The Bullroarer - Friday 27th November 2009
- International Energy Agency calls 'Peak' on OECD Oil Demand
- Australian Senate: Peak Oil motion defeated 31:6
TOD:Net Energy
Blogroll
Energy Sites
- The Coming Global Oil Crisis
- Die Off
- Dry Dipstick
- Energy Bulletin
- From the Wilderness
- Life After the Oil Crash
- Peak Oil Crisis
- Peak Oil News and Message Boards
- Powerswitch
- Rigzone
- Matthew Simmons
- Wolf at the Door
Environment & Sustainability Sites
- The Daily Green
- EcoGeek
- Eco Street
- Green Car Congress
- Green Options
- green.alltop.com
- Gristmill
- RealClimate
- Sustainablog
- Treehugger
- WorldChanging
Blogs
- Casaubon's Book
- Cleantech Blog
- Clusterf
k Nation (Jim Kunstler) - The Cost of Energy
- David Strahan
- Early Warning
- The Energy Blog
- European Tribune
- GraphOilology
- Health After Oil
- jeffvail.net
- Mobjectivist
- Peak Energy (Australia)
- Peak Energy (USA)
- R-Squared
- Resource Insights
Finance & Economics Blogs
- The Big Picture
- Calculated Risk
- The Crash Course
- Ecological Economics
- Econbrowser
- Environmental Economics
- Infectious Greed
- The Mess That Greenspan Made
- Mish's Global Economic Trend Analysis
Organizations
Peak Oil Primers
Beware email scams!
Beware email scams claiming to be from this site. We do not have any job openings. If anyone contacts you about a job at The Oil Drum, do not reply to them, and definitely do not give them any personal information or send them money. Read more here.
“No civilization can survive the physical destruction of its resource base.”
—Bruce Sterling
User login
Contact
- Content: editors at theoildrum dot com
- Tech support: support at theoildrum dot com
Personnel
- Editors: Nate Hagens, Gail the Actuary, Prof. Goose
- DrumBeat Editor: Leanan
- Contributors: ace, Engineer-Poet, Heading Out, jeffvail, JoulesBurn, Sam Foucher, Robert Rapier
- TOD:Campfire: Glenn, Jason Bradford
- TOD:Europe: Chris Vernon, Euan Mearns, Francois Cellier, Jerome a Paris, Luís de Sousa, Rembrandt, Rune Likvern, Ugo Bardi
- TOD:Canada: benk, Libelle
- TOD:ANZ: Big Gav, Phil Hart, aeldric
- Emeritus: Stuart Staniford
- Technician: Super G
License
This work is licensed under a Creative Commons Attribution-Share Alike 3.0 United States License.
GAIA Host Collective
I'm glad to see that you are still active on that problem.
If I understand correctly you have improved your Dispersive model using a Bayesian approach where the prior on L is specified. You then derive a generalization of the logistic curve (s-curve).
I'm having troubles following changes in your notations, last time you gave:
D(t) = kt6*(1-exp(-Dd/kt6))
It would be helpful to provide a short table linking your model variables to real world quantities:
L0: average depth?
k= Dd: URR
x: the current search depth
etc.
The dispersive model is supposed to model the discovery curve + reserve growth. The way I see it, true reserve growth (i.e. free of political/economical influences and accounting artifacts) is:
1. improved recovery methods applied on fields over time
2. knowledge growth: knowledge of the fields increases with time (i.e. field delineation)
Can you explain what is your interpretation of reserve growth? It seems that in your model, reserve growth comes only from the increase in the search depth which is in fact new smaller discoveries at greater depth over time.
One issue is how to estimate the various parameters from real world datasets. In particular, discovery data is contaminated by backdated reserve growth that is difficult to remove without complete reserve growth history. Reserve growth should be a time-dependent diffusion of the initial discovery volumes. A quick and dirty solution is to first remove reserve growth using a heuristic reserve growth model (e.g. Arrington) and then convolve with the same time dependent reserve growth function which in the case of the Lower-48 gives:
The red curve is now the correct curve where reserve growth is dispersed in time and not instantaneous. IMO, the fit between your model and this new curve is quite remarkable:
Any chance you could graph this method for world production?
Here is a quick trial using the ASPO discovery curve and Russia reserve growth model (Verma et al.):
Here is the fit using the new dispersive model:
Khebab,
I'm a college student and while I am going into Calculus 3 next semester I find this stuff exceedingly difficult to understand, what books, classes or websites might I look at to better understand this statistical modeling. It's tricky :D
Thanks,
Crews
The absolute classic and considered one of the great mathematical texts of the last century is "An Introduction to Probability Theory and Its Applications" by William Feller.
If you want to really get hooked on understanding how the physical world works, I would suggest taking a course on Statistical Mechanics.
Khebab
In my opinion the original discoveries plus a small fraction of the reserve growth is probably a good estimate for the amount of easy to extract oil left.
Just eyeballing the graph to integrate I get.
About 1.7 trillion barrels total.
Original = 1 trillion.
Assume a 20% growth in reserves is easy oil.
1.7*.20 = 340
This gives about 1340 billion barrels of "easy oil".
Lets give this rough estimate a 10% error term or range and its
1206 - 1340.
Given we are I think close to 1100 GB extracted now.
Then we could be at about 80-90% depleted in "easy oil".
This simply little calculation should be enough to make you wonder if we are going to keep production close to the highest levels ever achieved for much longer. Even if you increase the easy oil levels its not hard to see that production rates will probably begin to fall off soon.
The fact that the easy vs hard concept predicts a peak at around 70-78% of total URR inline with WT 60% of URR peak prediction is interesting. The two different approaches are not giving hugely different answers. In fact in my opinion the logistic is telling us a lot about how much easy oil we have left. In fact this easy oil approach does not tell us a lot about peak itself just when decline is certain peak could have been back at 1000 which given a 1700 total is 60% of URR perfectly in line with the logistic.
What the easy oil approach says is that decline is probably certain by 70% of URR or 10% past what was probably peak production.
Time will tell of course but I don't hold high hopes for us getting the next 700GB or 1000GB or whatever number we claim to still have in reserves out of the ground at anywhere near the rate we extracted the first half.
First of all, as everyone realizes this blogspot and scoop blogger software is crap for doing any kind of mathematical markup. Therefore the equations become more ad hoc than I would like, and I resort to using snapshot gifs of markup from various technical equation processing SW apps.
So yes, L0 and Dd both refer to URR give or take a scale factor to convert from some earth volume to cumulative number of barrels. Everything here is dimensionally sound and the time advance of discovery volume follows the URR linearly.
Now here is how the reserve growth comes in. The dispersion in discovery rates gives the source of the reserve growth (not the dispersion in volumes necessarily). The high rates over certain parts of the volume provide the initial fast growth and the slower rates over other parts of the volume give the long tails in the out years. The whole set of rates accelerates over time in terms of mean, but the dispersion stays as the variance of the mean so the slow rates are always there (think dispersion of wavelets, Khebab). I think it is pretty obvious, but no one really understands how to backdate all the discovery curves to reflect this property properly as you indicate. On top of this, the power-law growth rates give much higher reserve growth than the exponential law growth, since the power-law family is stronger initially but weakens in comparison to the exponential as time increases. This turns the strong symmetry in the exponential dispersive/Logistic into the asymmetry of the power-law dispersive. The worst (or best in terms in terms of reserve growth) is fractional power-law growth; this is a diffusion-limited growth that gives incredibly long reserve growth tails. And the diffusion is what you want to see -- my feeling is that the dispersive effects on top of strong technological acceleration outweigh the diffusional aspects on any one particular reservoir. In other words, the statistics rule on an aggregate of reservoirs and the "micro"-diffusion likely applies better to individual reservoirs.
I agree totally with your Arrington approach, but wish we did not have to do this, and suggest that someone place the reserve growth discoveries in the correct places on the timeline.
My problem is how to retrieve the reserve growth function from the available data itself and without access to complete reserve growth history.
Assuming that you have a complete discovery curve for a particular country (e.g. Lower 48) contaminated by backdated reserve growth, I was thinking about the following approach:
1. choose a suitable parametric form for the reserve growth factor function (RGF): RGF(t)= at^b
2. choose values for a and b.
3. remove backdated reserve growth from the original discovery curve.
4. simulate a reserve growth history from the RGF function and the new discovery curve in 3.
5. add simulated reserve growth history to the discovery curve in 3.
5. fit the dispersive model on the new discovery curve.
6. Apply the shock model
7. compare the reserve history generated by the Shock model and available proven reserve history (after anomalous increases removed)
8. go back to 2 and reiterate
9. the best agreement in step 7 gives the more likely parameter values for a and b.
This approach is similar to what I've tried to do with Ghawar (http://www.theoildrum.com/node/2945). I'm also wondering how the (k,n) values for the dispersive values would compare to the (a,b) values.
I think the fundamental distinction between the (k,n) tuple and the (a,b) tuple is that (a,b) always starts at the initial discovery point for a particular region, but (k,n) predates all those points. It is essentially the difference between comparing a(t-t0)^b and k(t-0)^n. The t0 point is based on the discovery time, but t=0 is the time from the start of the search. So the care we must apply is to get the t0 bias correct. For example, if we use a later t0 value for the (k,n) tuple, the reserve growth function will look concave up (2ndderivative positive) whereas we know that reserve growth from the point of the particular discovery is concave down.
Otherwise I think the same principles apply and dispersion looks like a kind of diffusion. The big question that needs to be answered is how strong this search rate is after the initial discovery. Dispersion is us looking for "the stuff", while real diffusion is "the stuff" creeping toward us.