Welcome to the age of diminishing returns

Wednesday, September 18, 2013

Mineral resources and the limits to growth.

This is a shortened version of a talk I gave in Dresden on September 5, 2013. Thanks to Professor Antonio Hurtado for organizing the interesting conference there.



So, ladies and gentleman, let me start with this recent book of mine. It is titled "The Plundered Planet." You can surely notice that it is not titled "The Developed Planet" or "The Improved Planet." Myself and the coauthors of the book chose to emphasize the concept of "Plundering"; of the fact that we are exploiting the resources of our planet as if they were free for us for the taking; that is, without thinking to the consequences. And the main consequence, for what we are concerned here is called "depletion," even though we have to keep in mind the problem of pollution as well. 

Now, there have been many studies on the question of depletion, but "The Plundered Planet" has a specific origin, and I can show it to you. Here it is.  


It is the rather famous study that was published in 1972 with the title "The Limits to Growth". It was one of the first studies that attempted to quantify depletion and its effects on the world's economic system. It was a complex study based on the best available data at the time and that used the most sophisticated computers available to study how the interaction of various factors would affect parameters such as industrial production, agricultural production, population and the like. Here are the main results of the 1972 study, the run that was called the "base case" (or "standard run"). The calculations were redone in 2004, finding similar results. 



As you can see, the results were not exactly pleasant to behold. In 1972, the study saw a slowdown of the world's main economic parameters that would take place within the first two decades of the 21st century. I am sure that you are comparing, in your minds, these curves with the present economic situation and you may wonder whether these old calculations may be turning out to be incredibly good. But I would also like to say that these curves are not - and never were - to be taken as specific predictions. No one can predict the future, what we can do is to study tendencies and where these tendencies are leading us. So, the main result of the Limits to Growth study was to show that the economic system was headed towards a collapse at some moment in the future owing to the combined effect of depletion, pollution, and overpopulation. Maybe the economic problems we are seeing nowadays are a prelude to the collapse seen by this model, maybe not - maybe the predicted collapse is still far away in the future. We can't say right now.

In any case, the results of the study can be seen at least worrisome. And a reasonable reaction when the book came out in 1972 would have been to study the problem more in depth - nobody wants the economy to collapse, of course. But, as you surely know, the Limits to Growth study was not well received. It was strongly criticized, accused of having made "mistakes" of all kinds and at times to be part of a worldwide conspiracy to take control of the world and to exterminate most of humankind. Of course, most of this criticism had political origins. It was mostly a gut reaction: people didn't like these results and sought to find ways to demonstrate that the model was wrong (or the data, or the approach, or something else). If they couldn't do that, they resorted to demonizing the authors - that's nothing now; I described it in a book of mine "Revisiting the limits to growth".

Nevertheless, there was a basic criticism of the "Limits" study that made sense. Why should one believe in this model? What are exactly the factors that generate the expected collapse? Here, I must say, the answer often given in the early times by the authors and by their supporters wasn't so good. What the creators of the models said was that the model made sense according to their views and they could show a scheme that was this (from the 1972 Italian edition of the book):



Now, I don't know what do you think of it; to me it looks more or less like the map of the subway of Tokyo, complete with signs in kanji characters. Not easy to navigate, to say the least. So, why did the authors created this spaghetti model? What was the logic in it? It turns out that the Limits to Growth model has an internal logic and that it can be explained in thermodynamic terms. However, it takes some work to describe the whole story. So, let me start with the ultimate origin of these models:


If you have studied engineering, you surely recognize this object. It is called "governor" and it is a device developed in 19th century to regulate the speed of steam engines. It turns with the engine, and the arms open or close depending on speed. In so doing, the governor closes or opens the valve that sends steam into the engine. It is interesting because it is the first self-regulating device of this kind and, at its time, it generated a lot of interest. James Clerk Maxwell himself studied the behavior of the governor and, in 1868, he came up with a set of equations describing it. Here is a page from his original article


I am showing to you these equations just to let you note how these systems can be described by a set of correlated differential equations. It is an approach that is still used and today we can solve this kind of equations in real time and control much more complex systems than steam engines. For instance, drones. 



You see here that a drone can be controlled so perfectly that it can hold a glass without spilling the content. And you can have drones playing table tennis with each other and much more. Of course they are also machines designed for killing people, but let's not go into that. The point is that if you can solve a set of differential equations, you can describe - and also control - the behavior of quite complex systems.

The work of Maxwell so impressed Norbert Wiener, that it led him to develop the concept of "cybernetics"


We don't use so much the term cybernetics today. But the ideas that started from the governor study by Maxwell were extremely fecund and gave rise to a whole new field of science. When you use these equations for controlling mechanical system, you use the term "control theory." But when you use the equations for study the behavior of socio-economic systems, you use the term "system dynamics"

System dynamics is something that was developed mainly by Jay Wright Forrester in the 1950s and 1960s, when there started to exist computers powerful enough to solve sets of coupled differential equations in reasonable times. That generated a lot of studies, including "The Limits to Growth" of 1972 and today the field is alive and well in many areas.

A point I think is important to make is that these equations describe real world systems and real world systems must obey the laws of thermodynamics. So, system dynamics must be consistent with thermodynamics. It does. Let me show you a common example of a system described by system dynamics: practitioners in this field are fond of using a bathub as an example:

On the right you have a representation of the real system, a bathtub partly filled with water. On the left, its representation using system dynamics. These models are called "stock and flow", because you use boxes to represent stocks (the quantity of water in the tub) and you use double edged arrows to indicate flows. The little butterfly like things indicate valves and single edged arrows indicate relationship.

Note that I used a graphic convention that I like to use for my "mind sized" models. That is, I have stocks flowing "down", following the dissipation of thermodynamic potential. In this case what moves the model is the gravitational potential; it is what makes water flow down, of course. Ultimately, the process is driven by an increase in entropy and I usually ask to my students where is that entropy increases in this system. They usually can't give the right answer. It is not that easy, indeed - I leave that to you as a little exercise

The model on the left is not simply a drawing of box and arrows, it is made with a software called "Vensim" which actually turns the model "alive" by building the equations and solving them in real time. And, as you may imagine, it is not so difficult to make a model that describes a bathtub being filled from one side and emptied from the other. But, of course, you can do much more with these models. So, let me show a model made with Vensim that describes the operation of a governor and of the steam engine.


Before we go on, let me introduce a disclaimer. This is just a model that I put together for this presentation. It seems to work, in the sense that it describes a behavior that I think is correct for a governor (you can see the results plotted inside the boxes). But it doesn't claim to be a complete model and surely not the only possible way to make a system dynamics model of a governor. This said, you can give a look to it and notice a few things. The main one is that we have two "stocks" of energy: one for the large wheel of the steam energy, the other for the small wheel which is the governor. In order to provide some visual sense of this difference in size, I made the two boxes of different size, but that doesn't change the equations underlying the model. Note the "feedback", the arrows that connect flows and stock sizes. The concept of feedback is fundamental in these models.

Of course, this is also a model that is compatible with thermodynamics. Only, in this case we don't have a gravitational potential that moves the system, but a potential based on temperature differences. The steam engine works because you have this temperature difference and you know the work of Carnot and the others who described it. So, I used the same convention here as before; thermodynamic potential are dissipated going "down" in the model's graphical representation

Now, let me show you another simple model, the simplest version I can think of a model that describes the exploitation of non renewable resources:

It is, again, a model based on thermodynamics and, this time, driven by chemical potentials. The idea is that the "resources" stock as a high chemical potential in the sense that it may be thought as, for instance, crude oil, which spontaneously combines with oxygen to create energy. This energy is used by human beings to create what I can call "capital" - the sum of everything you can do with oil; from industries to bureaucracies.

On the right, you can see the results that the model provides in terms of the behavior as a function of time of the stock of the resources, their production, and the capital stock. You may easily notice how similar these curves are to those provided by the more complex model of "The Limits to Growth." So, we are probably doing something right, even with this simple model.

But the point is that the model works! When you apply it to real world cases, you see that its results can fit the historical data. Let me show you an example:


This is the case of whaling in 19th century, when whale oil was used as fuel for lamps, before it became common to use kerosene. I am showing to you this image because it is the first attempt I made to use the model and I was surprised to see that it worked - and it worked remarkably well. You see, here you have two stocks: one is whales, the other is the capital of the whaling industry that can be measured by means of a proxy that is the total tonnage of the whaling fleet. And, as I said, the model describes very well how the industry grew on the profit of killing whales, but they killed way too many of them. Whales are, of course, a renewable resource; in principle. But, of course, if too many whales are killed, then they don't have enough time to reproduce and they behave as a non-renewable resource. Biologists have determined that at the end of this fishing cycle, there were only about 50 females of the species being hunted at that time. Non renewable, indeed!

So, that is, of course, one of the several cases where we found that the model can work. Together with my co-workers, we found that it can work also for petroleum extraction, as we describe in a paper published in 2009 (Bardi and Lavacchi). But let me skip that - the important thing is that the model works in some cases but, as you would expect, not in all. And that is good - because what you don't want is a "fit-all" model that doesn't tell you anything about the system you are studying. Let's say that the model reproduces what's called the "Hubbert model" of resource exploitation, which is a purely empirical model that was proposed more than 50 years ago and that remains a basic one in this kind of studies: it is the model that proposes that extraction goes through a "bell-shaped" curve and that the peak of the curve, the "Hubbert peak" is the origin of the concept of "peak oil" which you've surely heard about. Here is the original Hubbert model and you see that it has described reasonably well the production of crude oil in the 48 US lower states.




Now, let's move on a little. What I have presented to you is a very simple model that reproduces some of the key elements of the model used for "The Limits to Growth" study but it is of course a very simplified version. You may have noted that the curves for industrial production of the Limits to Growth tend to be skewed forward and this simple model can't reproduce that. So, we must move of one step forward and let me show you how it can be doing while maintaining the basic idea of a "thermodynamic cascade" that goes from higher potentials to lower potentials. Here is what I've called the "Seneca model"


You see that I added a third stock to the system. In this case I called it "pollution"; but you might also call it, for instance, "bureaucracy" or may be even "war". It is any stock that draws resource from the "Capital" (aka, "the economy") stock. And the result is that the capital stock and production collapse rather rapidly; this is what I called "the Seneca effect"; from the roman philosopher Lucius Anneaus Seneca who noted that "Fortune is slow, but ruin is rapid".

For this model, I can't show you specific historical cases - we are still working on this idea, but it is not easy to make quantitative fittings because the model is complicated. But there are cases of simple systems where you see this specific behavior, highly forward skewed curves - caviar fishing is an example. But let me not go into that right now.

What I would like to say is that you can move onward with this idea of cascading thermodynamic potentials and build up something that may be considered as a simplified version of the five main stocks taken into account in the "Limits to Growth" calculations. Here it is


Now, another disclaimer: I am not saying that this model is equivalent to that of the Limits to Growth, nor that it is the only way to arrange stocks and flows in order to produce similar results to the one obtained by the Limits to Growth model. It is here just to show to you the logic of the model. And I think you can agree, now, that there is one. The "Limits" model is not just randomly arranged spaghetti, it is something that has a deep logic based on thermodynamics. It describes the dissipation of a cascade of thermodynamic potentials.

In the end, all these model, no matter how you arrange their elements, tend to generate similar basic results: the bell shaped curve; the one that Hubbert had already proposed in 1956


The curve may be skewed forward or not, but that changes little on the fact that the downside slope is not so pleasant for those who live it.

Don't expect this curve to be a physical law; after all it depend on human choices and human choices may be changed. But, in normal conditions, human beings tend to follow rather predictable patterns, for instance exploiting the "easy" resources (those which are at the highest thermodynamic potential) and then move down to the more difficult ones. That generates the curve.

Now, I could show you many examples of the tendency of real world systems to follow the bell shape curve. Let me show you just one; a recent graph recently made by Jean Laherrere.



These are data for the world's oil production. As you can see, there are irregularities and oscillations. But note how, from 2004 to 2013, we have been following the curve: we move on a predictable path. Already in 2004 we could have predicted what would have been today's oil production. But, of course, there are other elements in this system. In the figure on the right, you can see also the appearance of the so-called "non-conventional" oil resources, which are following their own curve and which are keeping the production of combustible liquids (a concept slightly different from that of "crude oil) rather stable or slightly increasing. But, you see, the picture is clear and the predictive ability of these models is rather good even though, of course, approximate.

Now, there is another important point I'd like to make. You see, these models are ultimately based on thermodynamics and there is an embedded thermodynamic parameter in the models that is called EROI (or EROEI) which is the energy return for the energy invested. It is basically the decline in this parameter that makes, for instance, the extraction of oil gradually producing less energy and, ultimately, becoming pointless when the value of the EROEI goes below one. Let me show you an illustration of this concept:



You see? The data you usually read for petroleum production are just that: how much petroleum is being produced in terms of volume. There is already a problem with the fact that not all petroleums are the same in the sense of energy per unit volume, but the real question is the NET energy you get by subtracting the energy invested from the energy produced. And that, as you see, goes down rapidly as you move to more expensive and difficult resources. For EROEIs under about 20, the problem is significant and below about 10 it becomes serious. And, as you see, there are many energy resources that have this kind of low EROEI. So, don't get impressed by the fact that oil production continues, slowly, to grow. Net energy is the problem and many things that are happening today in the world seem to be related to the fact that we are producing less and less net energy. In other words, we are paying more to produce the same. This appears in terms of high prices in the world market.

Here is an illustration of how prices and production have varied during the past decades from the blog "Early Warning" kept by Stuart Staniford.



And you see that, although we are able to manage a slightly growing production, we can do so only at increasingly high prices. This is an effect of increasing energy investments in extracting difficult resources - energy costs money, after all.
So, let me show you some data for resources that are not petroleum. Of course, in this case you can't speak in terms of EROEI; because you are not producing energy. But the problem is the same, since you are using fossil fuels to produce most of the commodities that enter the industrial system, and that is valid also for agriculture. Here are some data.



Food production worldwide is still increasing, but the high costs of fossil fuels are causing this increase in prices. And that's a big problem because we all know that the food demand is highly anelastic - in plain words you need to eat or you die. Several recent events in the world, such as wars and revolutions in North Africa and Middle East have been related to these increases in food prices.

Now, let me go to the general question of mineral production. Here, we have the same behavior: most mineral commodities are still growing in terms of extracted quantities, as you can see here (from a paper by Krausmann et al, 2009 http://dx.doi.org/10.1016/j.ecolecon.2009.05.007)



These data go up to 2005 - more recent data show signs of plateauing production, but we don't see clear evidence of a peak, yet. This is bad, because we are creating a climate disaster. As you seee from the most recent data, CO2 are still increasing in a nearly exponential manner

 

But the system is clearly under strain. Here are some data relative to the average price index for aluminum, copper, gold, iron ore, lead, nickel, silver, tin and zinc (adapted from a graphic reported by Bertram et al., Resource Policy, 36(2011)315)



So, you see, there has been this remarkable "bump" in the prices of everything and that correlates well with what I was arguing before: energy costs more and, at the same time, energy requirements are increasing because of ore depletion. At present, we are still able to keep production stable or even slowly increasing, but this is costing to society tremendous sacrifices in terms of reducing social services, health care, pensions and all the rest. And, in addition, we risk to destroy the planetary ecosystem because of climate change.

Now I can summarize what I've been saying and get to the take-home point which, I think can be expressed in a single sentence "Mining takes energy"


Of course, many people say that we are so smart that we can invent new ways of mining that don't require so much energy. Fine, but look at that giant wheel, above, it used to extract coal in the mine of Garzweiler in Germany. Think of how much energy you need to make that wheel; do you think you could use an i-pad, instead?

In the end, energy is the key of everything and if we want to keep mining, and we need to keep mining, we need to be able to keep producing energy.  And we need to obtain that energy without fossil fuels. That's the concept of the "Energy Transition"



Here, I use the German term "Energiewende" which stands for "Energy Transition". And I have also slightly modified the words by Stanley Jevons, he was talking about coal, but the general concept of energy is the same. We need to go through the transition, otherwise, as Jevons said long ago, we'll be forced to return to the "laborious poverty" of older times.

That doesn't mean that the times of low cost mineral commodities will ever return but we should be able to maintain a reasonable flux of mineral commodities into the industrial system and keep it going. But we'll have to adapt to less opulent and wasteful life as the society of "developed" countries has been accustomed so far. I think it is not impossible, if we don't ask too much:


h/t ms. Ruza Jankovich - the car shown here is an old Fiat "500" that was produced in the 1960s and it would move people around without the need of SUVs

____________________________________________


Acknowledgement:

The Club of Rome team

Daphne Davies
Ian Johnson
Linda Schenk
Alexander Stefes
Jos├ęphine von Mitschke-Collande
Karl Wagner


And the coauthors of the book "Plundering the Planet"

Philippe Bihouix
Colin Campbell
Stefano Caporali
Partick Dery
Luis De Souza
Michael Dittmar
Ian Dunlop
Toufic El Asmar
Rolf Jakobi
Jutta Gutberlet
Rui Rosa
Iorg Schindler
Emilia Suomalainen
Marco Pagani
Karl Wagner
Werner Zittel





20 comments:

  1. That's the short version? Pretty comprehensive treatment.

    One does wonder where all the copper windings for the Wind Turbines will come from.

    RE
    http://doomsteaddiner.net

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    1. Yes, in Dresden I spoke for about 45 minutes, but in this report I am mentioning only about 2/3 of the slides I presented. There is this thing that a spoken presentation turns out to be very, very long when transformed into a written text. And, today, I worked like a dog for a good 3 hours to write this post!

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    2. Copper's an interesting one.

      Chris Clugston, in his 2012 book "Scarcity", reports that global peak copper extraction year is likely to be 2030, supply peaking in 2040.

      27 years to global reserve exhaustion at pre-recession consumption and growth rates.

      Add in the depeltion profiles of the other 88 "essential" non-renewable natural resources which Clugston catalogues and its a very grim picture indeed.

      "Renewables" aren't going to be a magic panacea.

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    3. 27 years sounds like a long time, but then it's much much easier to extract the first half of the resource (copper) than the last half. I wonder if 27 years is the projection for theoretically extracting ALL copper, or if it's a projection of when it'll be uneconomical to extract on any large scale. and of course pairing with peak fossil-fuels, when that'll make it that much harder to extract.

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  2. I've seen that Limits to Growth graph many times, but only just now realised that towards 2100, when everything else is declining, births are rising. How come?

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    1. That's a weak point of the model, indeed. It works on the basis of historical data showing that there is a sort of "peak" in the number of children that families tend to have as a function of income. That's also called the "demographic transition". The model assumes that as income drops, we go backwards along the transition and births increase. But that's just a supposition. I think that births will remain low and the population peak will be much, much earlier. That doesn't change the main features of the model, though.

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    2. Let me add also that the latest version of the "standard run" (2004) sees population peaking earlier and at lower numbers. That is the probable reason for the result that collapse is shifted of about a decade onward. But the overall result is unchanged

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    3. I saw a graph today of the birth and death rates in Greece over the last few years.

      Death rate up, birth rate down.

      Wish I'd taken note of the link...

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  3. Ugo
    I take it that your global numbers indicate, roughly, 'global boundary conditions', over time, given certain assumptions about 'inter-dependencies', but this is for a globe where there are large areas and populations where fossil fuel-dependency has barely reached much more than a ‘threshold presence’?

    I understand your example in the comment above about 'population transition' as a dependency that does not change the main features of the model, but I have still some difficulty with the huge disparities - perhaps ranging two orders of magnitude - in usage of resources that are a fact across the globe.

    'Dependencies' might be different for 'critical uses' of resources in different regions. Which is what we see even now before world 'growth' has actually peaked (i.e. we have yet to see a peak in total industrial activity and capital stocks). Sudden 'phase changes' have occurred historically even in advanced urban/industrial economies, for example in the USSR where change 'suddenly' shrank that economy. And more recently, if differently, we see the ability of relatively low-income countries to compete, sometimes to their advantage, for the global resource-pool.

    To return to my original question: the models show change in rates of use, but cannot define the effects of global rate-changes on critical thresholds in regions that currently show very 'low-end' resource use? Syria, Egypt etc. are very vulnerable net ‘importers of everything’ as low-medium resource consumers, who have experienced very rapid population increase since 2000 with little prospect of increasing their future access to adequate resources. But there are other vast populations with different trajectories?

    Thanks for that hard work!
    Phil H

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    1. Yes, the model is highly "aggregated" as economists say. It can't tell how exactly the decline will spread over the different regions of the world. Of course, the poorest regions will suffer first and that's exactly what's happening. Just as the poor in "rich" regions will suffer first. So, I think the value of the model would have been to warn us about the future - if someone had listened!

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  4. Does the model above take into account the (invariably negative except perhaps for agriculture in the northern latitudes or arctic drilling ) effects of climate change on infrastructure, industrial production, agricultural production, resource extraction, transportation, decreased economic efficiency due to displacement and migration of large numbers of people, and etc. or not? And if not how significant are such effects likely to be, and can they and their increase over time be included in the model? Intuitively they would seem to be quite significant and ever more so. For instance if the EROEI for a particular source of petroleum is X it might only be 70% of X if either the rigs or the pipelines or other infrastructure is in need of constant repair or replacement. Another easy example is the coal mines in Queensland that were flooded. It will take quite a bit more energy to get them back into production than it would have taken to simply continue the ongoing extraction without disruption had they never been flooded .

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    1. Max, the more parameters you enter in the model, the more you risk to go off track with it, because you accumulate uncertainties. So, the model aggregates everything under the banner of thermodynamics, but it doesn't pretend to tell you where and how exactly collapse starts. It is a general view: it is like telling you that traffic at rush hour will be heavier than at 4 a.m.. That's generally true, but nobody can tell you if you'll be run over by a truck at a specific hour.

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    2. Thanks Ugo and yes I understand. The early model included pollution as a general category. (in 1972 climate change was of course already known and understood by some but the extent and speed and magnitude of impact of its effects were not) But climate change is now probably the biggest long term (or even short and medium term) effect or component fitting under the rubric of "pollution" and probably far outweighing the effects of all other pollution components. The estimates I have seen is that is already shaving a couple of percentage points off yearly world GDP and I suspect that percentage will continue to grow even bigger. Maybe it would complicate the model further and unduly but it seems to me like a pretty important variable. Population is going to keep increasing and so is climate change and both are pretty important. But population is included while climate change per se (other than under pollution effects) does not seem to be. That probably made sense in 1972 but it seems to me to make less sense today. On the other hand I have no idea about how it could be included appropriately (including at least three climate scenarios) but without unduly overloading the model with more uncertainties.

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    3. We could disaggregate climate change from the other forms of pollution. Everything can be done but, as I was saying, the more parameters we include, the more we introduce uncertainties

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  5. Ugo, do you know if yeast populations in wine and beer follow the Seneca Cliff, whereby alcohol is the pollution? Also just a point about using the term pollution for all capital destruction is that the Seneca cliff model seems to produce an energy curve that won't go over a hubbert curve, but I suspect that using debt (destroying wealth or economic capital to create income) can create a similar shaped curved to the Seneca curve but goes for a time above the hubbert curve.

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    1. Well, if I remember correctly, yeasts or bacteria are among the few experimental cases where you have a biological system that tends to follow the simple version of the Lotka-Volterra model. That is, the curve tends to be approximately symmetric. This is told from memory - I should look it over; maybe if you have a moment you can look at the literature. About the energy curve that "won't go over a hubbert curve", well, that's the result of stretching a bit the parameters to show more clearly the asymmetry. You can set the model for a significant overlap of the two curves.

      And, in the end, not everything can be modeled!

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    2. First, I find your model fascinating and very helpful in trying to show in a simplistic manner how a civilization can fall quickly. I am not trying to saying you should try to model everything nor be everything to everyone, but was trying to say not all capital destruction is the same and you seem to be talking about a very specific type of pollution (one that prematurely inhibits extraction). Or am I wrong? Can a pollution that speeds extraction for a greater loss in the future be modeled under the parameters you have chosen? As far as the curve of yeast, it looks by eyeballing the graph available like it does follow the Seneca cliff whereby pollution leaves resources short of extracting a hubbert curve with some cellular recycling slightly slowing the fall down the cliff. Though, I can't get to the original article data that in German and locked behind paywalls, so I was hoping you would know. Citation is in this article with a graph of if http://www.oilcrash.com/articles/energy.htm

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    3. Actually, I take back what I said about being short of the Hubbert Curve. I don't really know that, but there might be a cliff. It would be interesting to see if the cellular recycling causes the yeast population to go over a predicted hubbert linerization for the experiment, if there really is a cliff in the data set and it isn't some eyeballing a graph issue. It could give insite to what we will do. Mainly I want to know is: are we grabbing energy faster than one would expect based on EROI and capital feedbacks (my understanding of what hubbert linerization is showing)? Is shale ( unconventionals) an energy tranche that should be added onto the baseline hubbert curve projections or just a way of recycling our earned capital back into energy?

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    4. I am still working at all that. There are many ways to introduce further parameters to model the cliff - but I am still trying to produce models that are both simple and elegant. Nobody likes spaghetti models, of course, but sometimes there is no other way. Give a look to this post by Tom Fiddaman: http://blog.metasd.com/2013/09/are-there-limits/

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  6. I don't have my copies available to me right now, but I remember a very compelling graph in either the original LTG or one of the updates that showed the increase in energy required to mill an ore as the quality of the ore decreased. I suspect that phenomenon can probably in itself generate a Seneca Cliff.

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