Saturday, January 24, 2015

The shale oil "miracle": how growth may falsely signal abundance

Originally published on "Cassandra's Legacy" on Tuesday, February 24, 2015


Oil production (all liquids in barrels per day) in the US and Canada. (From Ron Patterson's blog). Does this rapid growth indicate that the resources are abundant and that all the worries about peak oil are misplaced? Maybe not....


Sometimes, we use a simple metric to evaluate complex systems. For instance, a war is a complex affair where millions of people fight, struggle. suffer, and kill each other. However, in the end, the final result is seen in terms of a yes/no question: either you win or you lose. Not for nothing, General McArthur said once that "there is no substitute for victory".

Now, think of the economy: it is an immense and complex system where millions of people work, produce, buy, sell, and make or lose money. In the end, eventually, we think that the final result can be described in terms of a simple yes/no question: either you grow, or you don't. And what McArthur said about war can be applied to the economy, as well: "there is no substitute for growth".

But complex systems have ways to behave and to surprise you that can't be reduced to a simple yes/no judgement. Both victory and growth may well create more problems than they solve. Victory may falsely signal a military might that doesn't really exist (think of the outcome of some recent wars....), while growth may signal an abundance which is just not there.

Take a look at the figure at the beginning of this post (from Ron Patterson's blog). It shows the oil production (barrels/day) in the US and Canada. The data are in thousand barrels per day for "crude oil + condensate" and the rapid growth for the past few years is mostly due to tight oil (also known as "shale oil") and oil from tar sands. If you follow the debate in this field, you know that this growth trend has been hailed as a great result and as the definitive demonstration that all worries about oil depletion and peak oil were misplaced.

Fine. But let me show you another graph, the US landings of North Atlantic Cod, up to 1980 (data from Faostat).

Doesn't it look similar to the data for oil in the US/Canada? We can imagine what was being said at the time; "new fishing technologies dispel all worries about overfishing" and things like that. It is what was said, indeed (see Hamilton et al. (2003)).

Now, look at the cod landings data up to 2012 and see what happened after the great burst of growth.

I don't think this requires more than a couple of comments. The first is to note how overexploitation leads to collapse: people don't realize that by pushing for growth at all costs, they are destroying the very resource that creates growth. This can happen with fisheries just as with oil fields. Then, note also that we have here another case of a "Seneca Cliff," a production curve where the decline is much faster than growth. As the ancient Roman philosopher said, "The road to ruin is rapid". And this is exactly what we could expect to happen with tight oil

Thursday, January 22, 2015

Sandeels: another Seneca cliff


Originally published on Cassandra's Legacy on Thursday, January 22, 2015




Once you start looking for "Seneca Cliffs" in the exploitation of natural resources, you find them all over the scientific literature. This is my latest find of a production curve where decline is much more rapid than growth: the landings ofsandeels. If you don't know what a sandeel is, here is one: 



In the report (2007), where I found the curve shown above, the authors discuss the causes for the collapse of the fishery, especially in view of climate change. They don't seem to arrive to any definitive conclusion and they don't use the dreaded term "overfishing". But from the fact that trawlerwere used in this fishery, I think it is clear that the fish stock was being destroyed in a process similar to the one that led to the collapse of the whole UK fishing industry. The more resources were aggressively thrown at trying to maintain production, the more the fish stock was depleted. The end result was the rapid collapse observed.

So, as in several other cases, we have a classic example of the "Seneca Collapse", that is a production curve where decline is much more rapid than growth. Below, you can see the Seneca curve as shown in a simulation carried out by system dynamics that takes into account the increased capital expenditure in fishing equipment (the model is described here). 



As Seneca said, "the road to ruin is rapid", indeed.

Monday, January 19, 2015

A Seneca cliff in the making: African elephants on the brink of extinction

Originally published on "Cassandra's legacy" on Monday, January 19, 2015

The graph above refers to effects of the illegal hunting of African elephants. It is taken from a recent paper by Wittemyer et al.



Once you have given a name to a phenomenon and understood its causes, you can use it as a guide to understanding many other things. So, the concept of the "Seneca Cliff" tells us that the overexploitation of natural resources often leads to an abrupt decline that, often, takes people by surprise. In the case of biological resources, such as fisheries, the decline may be so fast and uncontrollable that it leads to the extinction or to the near extinction of the species being exploited. It has happened, for instance, for whales in 19th century and for the Atlantic cod.

If you keep in mind these historical examples, you can examine other cases and identify possible Seneca cliffs in the making. One such case is the ivory trade from the hunting of African elephants. If you look at the plots (from a recent paper), above, you see that the seized ivory mass has shown a considerable increase starting around 2008. It peaked in 2011, then declined. We can probably take these numbers as a "proxy" for the number of African elephants being killed - which is also visible as the red line in the upper box. 

This is very worrisome, because if killings decline, it may very well be because there are fewer elephants left to kill - just as the landings of the fishing industry tend to decline when the fish stocks are depleted. Considering how abruptly these things go (the "Seneca effect") then we may well be seeing a similar trend in progress for African elephants: that is, the prelude of an abrupt crash in their numbers. Considering that elephants are big and reproduce slowly, that may very well lead to their extinction.

On this subject, the authors of the paper seem to be very worried, too. The title, by itself, says it all: "Illegal killing for ivory drives global decline in African elephants". In the text, we can read, among other things, that:


The population [of African elephants] was subjected to unsustainable rates of illegal killing between 2009 and 2012, escalating from a mean of 0.6% (SD = 0.4%) between 1998 and 2008 to a high of 8% in 2011 (Fig. 1). Annual illegal killing of elephants in the Samburu population during 2009 to 2012 exceeded those of all previous years of monitoring (1998–2008) with an estimated aggregate of 20.8% of the known elephants illegally killed during that 4-yperiod. ... Illegal killing rates were strongly correlated with black market ivory prices in the Samburu ecosystem. ... As a result of this illegal killing, the population currently suffers from few prime-aged males, strongly skewed sex ratios, and social disruption in the form of some collapsed families and increased numbers of orphans (immature elephants without a parent)

Are we going to lose the elephants forever? Right now, we can't say for sure; but when it will be clear that it is happening, it will probably be too late to do something about it. Doesn't that sound familiar? 



Wednesday, January 14, 2015

Seneca's pyramids: how fast did the Mayan civilization fall?


Originally published on Cassandra's legacy on Wednesday, January 14, 2015



Monument building cycle of the Mayan civilization. From "Sylvanus G. Morley and George W. Brainerd, The Ancient Maya, Third Edition (Stanford University Press, 1956), page 66.". Courtesy of Diego Mantilla.



Once you give a name to a phenomenon, you can focus your attention on it and learn more and more about it. So, the "Seneca Cliff" idea turns out to be a fruitful one. It tells us that, in several cases, the cycle of exploitation of a natural resource follows a forward skewed curve, where decline is much faster than growth. This is consistent with what the Roman philosopher Lucius Annaeus Seneca wrote: "increases are of sluggish growth, but the way to ruin is rapid." With some mathematical tricks, the result is the following curve:


This curve describes the behavior of several complex systems, including entire civilizations which experienced an abrupt collapse after a long period of relatively slow growth. In my first post on the seneca cliff, I already discussed the collapse of the Mayan Civilization (*)



Here, you can see the the Seneca behavior, although the data for the Maya population density seem to be rather qualitative and uncertain. However, the data that I received recently from Diego Mantilla (see at the beginning of this post) are clear: if you take monument building as a proxy for the wealth of the Mayan civilization, then the collapse was abrupt, surely faster than growth.

Something similar can be said for the ancient Egyptians, although the data for pyramid building are more sparse and uncertain than those for the Maya. Finally, also the Roman civilization appears to have collapsed faster than it grew.

So, the Mayans didn't do better than other civilizations in human history. As other civilizations did, they moved toward their demise by dragging their feet, trying to avoid the unavoidable. They didn't succeed and they didn't realize that opposing the collapse in this way is a classic example of "pushing the levers in the wrong direction". It can only postpone collapse, but in the end makes it more rapid.

Will we do any better than the Mayans? One would hope so, but........





(*) Dunning, N., D. Rue, T. Beach, A. Covich, A. Traverse, 1998, "Human - Environment Interactions in a Tropical Watershed: the Paleoecology of Laguna Tamarindito, Guatemala," Journal of Field Archaeology 25 (1998):139-151.

Thursday, January 8, 2015

The Seneca Cliff of Energy Production


Published on Cassandra's Legacy on Thursday, January 8, 2015


The graph above was created by Gail Tverberg on her blog "Our Finite World". It is, clearly, another case of what I called the "Seneca Cliff" (from the Roman philospher who said "the road to ruin is rapid). The Seneca Cliff takes this shape, when generated by a system dynamics model:


Gail's forecast of the future of energy production is not the result of a the same model I developed, but the reasons behind the steep decline are the same. Gail explains it in a post of hers as:



All parts of our economy are interconnected. If parts of the economy is becoming increasingly inefficient, more than the cost of production in these parts of the economy are affected; other parts of the economy are affected as well, including wages, debt levels, and interest rates.

Wages are especially being crowded out, because the total amount of goods and services available for purchase in the world economy is growing more slowly. This is not intuitively obvious, unless a person stops to realize that if the world economy is growing more slowly, or actually shrinking, it is producing less. Each worker gets a share of this shrinking output, so it is reasonable to expect inflation-adjusted wages to be stagnating or declining, since a stagnating or declining collection of goods and services is all a person can expect.

At some point, something has to “give”. 


Which is a good description of the mathematical model at the basis of the Seneca cliff idea. The burden on the economy of increasing costs becomes more and more heavy in times of diminishing returns (or, as Gail says, increasing inefficiency, which is the same). At some point, something "gives" and the whole thing comes down. Seneca rules.

Tuesday, January 6, 2015

Seneca again: the collapse of the UK fishing industry


Originally published on Cassandra's legacy on Jan 6 2015


Image from a 2010 article by Thurstan, Brockington, and Roberts. It describes the cycle of the UK fishing industry, which collapsed because ofoverfishing in the late 1970s.


The two graphs above (from a 2010 article by Thurstan et al.) speak by themselves. We have here a real life example of the overexploitation of natural resources; that is, of the tendency of people of destroying their own sources of wealth. Other classic examples can be found with the 19th century whaling industry and with the Canadian cod fishery.

Overexploitation typically generates the "Hubbert curve," the name given to a bell-shaped production cycle best known for the case of crude oil, but affecting all the resources which can be exploited faster than they can reform by natural processes. This behavior can be explained by means of mathematical models, but, qualitatively, it is the result of the falling profits generated by the diminishing resource stock. In the long run, lower profits discourage investments and the result is a general production decline. A particular case of this mechanism is when the industry initially reacts to diminishing returns by aggressively increasing the amount of capital invested. In this case, the stocks of the resource are depleted very fast and the result is a crash of the production rate; we still have a bell shaped curve, but skewed forward. The rapid decline that occurs after the peak is what I called the "Seneca Cliff." 

There are several historical examples of the Seneca cliff; in the case of fisheries, it is especially evident in the case of the Canadian cod fishery and for the Caspian Sturgeon; but it is evident also in the case of the UK fishing industry. Note, in the figure above, the steep decline of the landings of the late 1970s, it is significantly steeper than the growth of the left side of the curve. This is the essence of the Seneca mechanism. And we can see very well what causes it: the start of the decline in production corresponds to a rapid growth of investments. The result is the increase of what the authors of the paper call "fishing power" - an estimate of the efficiency and size of the fishing fleet.

The results were disastrous; a textbook example of how to "push the levers in the wrong directions", that is, of a case when the attempt to solve a problem worsens it considerably. In this case, the more efficient the fishing fleet was, the more rapidly the fish stock was destroyed. This is a classic mechanism for falling down the Seneca cliff: the more efficient you are at exploiting a non renewable (or slowly renewable) resource, the faster you deplete it. And the faster you get into trouble.

This case, as others, is such a staggering disaster that one wonders how it was possible at all. How could it be that nobody in the fishing industry or in the government realized what was happening? In their article on this subject, Thurstan and his colleagues don't comment on this point, but we can cite an article by Hamilton et al. on the Canadian Atlantic Cod fishery, where they say "Some say they saw trouble coming, but felt powerless to halt it."That seems to be not describing not just the fishing industry, but our entire civilization.

Friday, December 19, 2014

Peak pyramids: the way to ruin is rapid


Originally published on Cassandra's Legacy on Friday, December 19, 2014


The graph above is a little exercise in cliodynamics, the attempt of quantitatively modeling historical data. Here, the size of the great Egyptian pyramids is plotted as a function of their approximate building date, taken as the last year of the reign of the Pharaoh associated to them. The data are fitted with a simple Gaussian, which approximates the cycle of the Hubbert model of resource depletion.


The great Egyptian pyramids built during the 3rd millennium BCE are the embodiment of the power and of the wealth of the Egyptian civilization of the time. But why did the Egyptians stop building them? Not lack of interest, apparently, since they kept building pyramids for a long time. But they never built again pyramids on such a giant scale.

Probably, we will never have sufficient data to understand the economics of the Egyptian pyramid building cycle of the 3rd and 4th Egyptian dynasties. But we can try at least to examine the quantitative data we have. So, I went toWikipedia and I found data for the size of pyramids and their approximate dates. The result is the graph above. Here, I show only the data for completed pyramids as a function of the last year of the reign of the Pharaoh associated for each one.

As you can see, it is possible to fit the data with a Gaussian curve, which approximates the Hubbert curve, known to describe the depletion of a limited, non renewable resource. This suggests that the Egyptians had run out of resources, possibly in the form of the fertile soil necessary to sustain the large workforce needed to build pyramids. Or, perhaps, in an age of increasing warring activity, they were forced to funnel more and more resources into the military sector, taking them away from pyramid building.

Another phenomenon we can note in the graph is the rapid collapse of the size of the pyramids at the end of the cycle. The last pyramid of this cycle, the one associated to Pharaoh Menkaure, is even smaller than the first one of the cycle, the "stepped pyramid" of Pharaoh Djoser. Perhaps, this rapid decline is a manifestation of the "Seneca Effect", a term that I coined to describe economic cycles in which decline is faster than growth. Unfortunately, however, the data are too scattered and uncertain to be sure on this point. But surely there was no "plateau" nor a slow decline after the construction of the largest pyramids andit is suggestive to think that even pyramid building may be described with Seneca's words "increases are of sluggish growth, but the way to ruin is rapid."





Monday, December 15, 2014

Seneca cliffs of the third kind: how technological progress can generate a faster collapse

Originally published on Cassandra's Legacy on Monday, December 15, 2014



The image above (from Wikipedia) shows the collapse of the North Atlantic cod stocks. The fishery disaster of the early 1990s was the result of a combination of greed, incompetence, and government support for both. Unfortunately, it is just one of the many examples of how human beings tend to worsen the problems they try to solve. The philosopher Lucius Anneus Seneca had understood this problem already some 2000 years ago, when he said, "It would be some consolation for the feebleness of our selves and our works if all things should perish as slowly as they come into being; but as it is, increases are of sluggish growth, but the way to ruin is rapid."


The collapse of the North Atlantic cod fishery industry gives us a good example of the abrupt collapse in the production of resources - even resources which are theoretically renewable. The shape of the production curve landings shows some similarity with the "Seneca curve", a general term that I proposed to apply to all cases in which we observe a rapid decline of the production of a non renewable, or slowly renewable, resource. Here is the typical shape of the Seneca Curve:


The similarity with the cod landings curve is only approximate, but clearly, in both cases we have a very rapid decline after a slow growth that, for the cod fishery, had lasted for more than a century. What caused this behavior?

The Seneca curve is a special case of the "Hubbert Curve" which describes the exploitation of a non renewable (or slowly renewable) resource in a free market environment. The Hubbert curve is "bell shaped" and symmetric (and it is the origin of the well known concept of "peak oil). The Seneca curve is similar, but it is skewed forward. In general, the forward skewness can be explained in terms of the attempt of producers to keep producing at all costs a disappearing resource.

There are several mechanisms which can affect the curve. In my first note on this subject, I noted how the Seneca behavior could be generated by growing pollution and, later on, how it could be the result of the application of more capital resources to production as a consequence of increasing market prices. However, in the case of the cod fishery, neither factor seems to be fundamental. Pollution in the form of climate change may have played a role, but it doesn't explain the upward spike of the 1960s in fish landings. Also, we have no evidence of cod prices increasing sharply during this phase of the production cycle. Instead, there is clear evidence that the spike and the subsequent collapse was generated by technological improvements.

The effect of new and better fishing technologies is clearly described by Hamilton et al. (2003)

Fishing changed as new technology for catching cod and shrimp developed, and boats became larger. A handful of fishermen shifted to trawling or “dragger” gear. The federal government played a decisive role introducing newtechnology and providing financial resources to fishermen who were willing to take the risk of investing in new gear and larger boats. 
 ...

Fishermen in open boats and some long-liners continued to fish cod, lobster and seal inshore. Meanwhile draggers  and other long-liners moved onto the open ocean, pursuing cod and shrimp nearly year round. At the height of the boom, dragger captains made $350,000–600,000 a year from cod alone. ... The federal government helped finance boat improvements, providing grants covering 30–40% of their cost.
....
By the late 1980s, some fishermen recognized signs of decline. Open boats and long-liners could rarely reach their quotas. To find the remaining cod, fishermen traveled farther north, deployed more gear and intensified their efforts. A few began shifting to alternative species such as crab. Cheating fisheries regulation—by selling unreported catches at night, lining nets with small mesh and dumping bycatch at sea—was said to be commonplace. Large illegal catches on top of too-high legal quotas drew down the resource. Some say they saw trouble coming, but felt powerless to halt it. 

So, we don't really need complicated models (but see below) to understand how human greed and incompetence - and help from the government - generated the cod disaster. Cods were killed faster than they could reproduce and the result was their destruction. Note also that in the case of whaling in the 19th century, the collapse of the fishery was not so abrupt as it was for cods, most likely because, in the 19th century, fishing technology could not "progress" could not be so radical as it was in the 20th century.

The Seneca collapse of the Atlantic cod fishery is just one of the many cases in which humans "push the levers in the wrong directions", directly generating the problem they try to avoid. If there is some hope that, someday, the cod fishery may recover, the situation is even clearer with fully non-renewable resources, such as oil and most minerals. Also here, technological progress is touted as the way to solve the depletion problems. Nobody seems to worry about the fact that the faster you extract it, the faster you deplete it: that's the whole concept of the Seneca curve.

So take care: there is a Seneca cliff ahead also for oil!


____________________________________

A simple dynamic model to describe how technological progress can generate the collapse of the production of a slowly renewable resource; such as in the case of fisheries. 


by Ugo Bardi

Note: this is not a formal academic paper, just a short note to sketch how a dynamic model describing overfishing can be built. See also a similar modeldescribing the effect of prices on the production of a non renewable resource


The basics of a system dynamics model describing the exploitation of a non renewable resource in a free market are described in detail in a 2009 paper byBardi and Lavacchi. According to the model developed in that paper, it is assumed that the non renewable resource (R) exists in the form of an initial stock of fixed extent. The resource stock is gradually transformed into a stock of capital (C) which in turn gradually declines. The behavior of the two stocks as a function of time is described by two coupled differential equations. 
R' = - k1*C*R C' = k2*C*R - k3*C, 
where R' and C' indicate the flow of the stocks as a function of time (R' is what we call "production"), while the "ks" are constants. This is a "bare bones" model which nevertheless can reproduce the "bell shaped" Hubbert curve and fit some historical cases. Adding a third stock (pollution) to the system, generates the "Seneca Curve", that is a skewed forward production curve, with decline faster than growth.  

The two stock system (i.e. without taking pollution into account) can also produce a Seneca curve if the equations above are slightly modified. In particular, we can write:  
R' = - k1*k3*C*R C' = ko*k2*C*R - (k3+k4)*C. 
Here, "k3" explicitly indicates the fraction of capital reinvested in production, while k4 which is proportional to capital depreciation (or any other non productive use). Then, we assume that production is proportional to the amount of capital invested, that is to k3*C. Note how the ratio of R' to the flow of capital into resource creation describes the net energy production (EROI), which turns out to be equal to k1*R. Note also that "ko" is a factor that defines the efficiency of the transformation of resources into capital; it can be seen as related to technological efficiency. 
The model described above is valid for a completely non-renewable resource. Dealing with a fishery, which is theoretically renewable, we should add a growth factor to R', in the form of k5*R. Here is the model as implemented using the Vensim (TM) software for system dynamics. The "ks" have been given explicit names. I am also using the convention of "mind sized models" with higher free energy stocks appearing above lower free energy stocks





If the constants remain constant during the run, the model is the same as the well known "Lotka-Volterra" one. If the reproduction rate is set at zero, the model generates the symmetric Hubbert curve. 

In order to simulate technological progress, the "production efficiency" constant is supposed to double stepwise around mid-cycle. A possible result is the following, which qualitatively reproduces the behavior of the North Atlantic cod fishery. 




Among other things, this result confirms the conclusions of an early paper of mine (2003) on this subject, based on a different method of modeling.

Let me stress again that this is not an academic paper. I am just showing the results of tests performed with simple assumptions for the constants. Nevertheless, these calculations show that the Seneca cliff is a general behavior that occurs when producers stretch out their system allocating increasing fractions of capital to production. Should someone volunteer to give me a hand to make better models, I'd be happy to collaborate!











Sunday, December 7, 2014

Fossil fuels: are we on the edge of the Seneca cliff?


Originally published on "Cassandra's Legacy" on Sunday, December 7, 2014




"It would be some consolation for the feebleness of our selves and our works if all things should perish as slowly as they come into being; but as it is, increases are of sluggish growth, but the way to ruin is rapid." Lucius Anneaus Seneca, Letters to Lucilius, n. 91

This observation by Seneca seems to be valid for many modern cases, including the production of a nonrenewable resource such as crude oil. Are we on the edge of the "Seneca cliff?"



It is a well known tenet of people working in system dynamics that there exist plenty of cases of solutions worsening the problem. Often, people appear to be perfectly able to understand what the problem is, but, just as often, they tend to act on it in the wrong way. It is a concept also expressed as "pushing the lever in the wrong direction."

With fossil fuels, we all understand that we have a depletion problem, but the solution, so far, has been to drill more, to drill deeper, and to keep drilling. Squeezing out some fuel by all possible sources, no matter how difficult and expensive, could offset the decline of conventional fields and keep production growing for the past few years. But is it a real solution? That is, won't we pay the present growth with a faster decline in the future?

This question can be described in terms of the "Seneca Cliff", a concept that I proposed a few years ago to describe how the production of a non renewable resource may show a rapid decline after passing its production peak. A behavior that can be shown graphically as follows:



It is not just a theoretical model: there are several historical cases where the production of a resource collapsed after having reached a peak. For instance, here are the data for the Caspian sturgeon, a case that I termed "peak caviar".




Do we risk to see something like this in the case of the world production of oil and gas? In my opinion, yes. There are some similarities; both fossil fuels and caviar are non-replaceable resources; and in both cases prices went rapidly up at and after the peak. So, if Caspian sturgeon showed such a clear Seneca cliff, oil and gas could do the same. But let me go into some details.

In the first version of my Seneca model, the fast decline of production was interpreted in terms of growing pollution that places an extra burden on the productive system and reduces the amount of resources available for the development of new resources. However, I found that the Seneca behavior is rather robust in these systems and it appears every time people try to "stretch out" a system to force it to produce more and faster than it would naturally do.

So, in the case of the Caspian sturgeon, above, growing pollution is unlikely to be the cause of the rapid collapse of production (even though it may have contributed to the problem). Rather, the main factor in the collapse is likely to have been the effect of the growing prices of a rare and non replaceable resource (caviar). High prices enticed producers to invest more and more resources in raking out of the sea as much fish as possible. It worked, for a while, but, in the end, you can't fish sturgeon which isn't there. It ended up in disaster: a classic case of a Seneca Cliff.

Can this phenomenon be modeled? Yes. Below, I describe the model for this case in some detail. The essence of the idea is that producers need to reinvest a fraction of their profits in developing new resources in order to keep producing. However, the yield of the new investments declines as time goes by because the most profitable resources (e.g. oilfields) are exploited first. As a result, less and less capital is available for new investments. Eventually production reaches a maximum, then it declines. If we assume that companies re-invest a constant fraction of their profits in new resources, the model leads to the symmetric bell shaped curve known as the "Hubbert Curve."

However, as I describe in detail below, decline can be postponed if high prices provide extra capital for new productivedevelopments. Unfortunately, growth is obtained at the cost of a fast burning out of capital resources. The final result is not any more the symmetric Hubbert curve, but a classic Seneca curve: decline is more rapid than growth.

Is this what we are facing for fossil fuels? Of course, we are only dealing with qualitative models, but, on the other hand, qualitative models are often robust and give us an idea of what to expect, even though they can't tell us much in terms of predicting events on a precise time scale. The ongoing collapse of oil prices may be a symptom that we are running out of the capital resources necessary to keep developing new fields. So, what we can say is that there are some good chances of rough times ahead - actually very rough. The Seneca cliff may well be part of our near term future.


_______________________________________________________

The Seneca curve as the result of increasing fractions of profits allocated to the production of a non renewable resource

by Ugo Bardi - 07 Dec 2014


Note: this is not a formal scientific paper; it is more a rough "back of the envelope" calculation designed to show how increasing capex fractions can affect the production rate of a non renewable resource. If someone could give me a hand to make a more refined and publishable study, I would be happy to collaborate!


The basics of a system dynamics model describing the exploitation of a non renewable resource in a free market are described in detail in a 2009 paper byBardi and Lavacchi. According to the model developed in that paper, it is assumed that the non renewable resource (R) exists in the form of an initial stock of fixed extent. The resource stock is gradually transformed into a stock of capital (C) which in turn gradually declines. The behavior of the two stocks as a function of time is described by two coupled differential equations.

R' = - k1*C*R
C' = k2*C*R - k3*C,

where R' and C' indicate the flow of the stocks as a function of time (R' is what we call "production"), while the "ks" are constants. This is a "bare bones" model which nevertheless can reproduce the "bell shaped" Hubbert curve and fit some historical cases. Adding a third stock (pollution) to the system, generates the "Seneca Curve", that is a skewed forward production curve, with decline faster than growth. 

The two stock system (i.e. without taking pollution into account) can also produce a Seneca curve if the equations above are slightly modified. In particular, we can write: 

R' = - k1*k3*C*R
C' = ko*k2*C*R - (k3+k4)*C.

Here, "k3" explicitly indicates the fraction of capital reinvested in production, while k4 which is proportional to capital depreciation (or any other non productive use). Then, we assume that production is proportional to the amount of capital invested, that is to k3*C. Note how the ratio of R' to the flow of capital into resource creation describes the net energy production (EROI), which turns out to be equal to k1*R. Note also that "ko" is a factor that defines the efficiency of the transformation of resources into capital; it can be seen as related to technological efficiency. These points will not be examined in detail here.

Here is the model as implemented using the Vensim (TM) software for system dynamics. The "ks" have been given explicit names. I am also using the convention of "mind sized models" with higher free energy stocks appearing above lower free energy stocks




If the k's are kept constant over the production cycle, the shape of the curves generated by this model is exactly the same as with the simplified version, that isa symmetric, bell shaped production curve. Here are the results of a typical run:




Things change if we allow "k3" to vary over the simulation cycle. The characteristic that makes "k3" (productive investment fraction) somewhat different than the other parameters of the model, is that it is wholly dependent on human choice. That is, while the other ks are constrained by physical and technological factors, the fraction of the available capital re-invested into production can be chosen almost at will (of course, there remains the limit of the total amount of available capital!). 

Higher prices will lead to higher profits for producers and to the tendency to increase the fraction reinvested in new developments. It is also known that in the region near the production peak prices tend to be higher - as in the historical cases of whale oil and caviar and whale oil. In the case of caviar, the price rise was nearly exponential, in the case of whale oil, more like a logistic curve. Assuming that the fraction of reinvested capital varies in proportion to prices, some modeling may be attempted. Let me show here the results obtained for an exponential increase of the fraction of reinvested Capex.
  


I have also tried other functions for the rising trend of k3. The results are qualitatively the same for a linear increase and for a logistic one: the Seneca behavior appears to be robust, as long as we assume a significant increase of the fraction of the reinvested capex

Let me stress once more that these are not supposed to be complete results. These are just tests performed with arbitrary assumptions for the constants. Nevertheless, these calculations show that the Seneca cliff is a general behavior that occurs when producers stretch out their system allocating increasing fractions of capital to production.