Cassandra has moved. Ugo Bardi publishes now on a new site called "The Seneca Effect."

Sunday, December 7, 2014

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

"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. oil fields) 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 productive developments. 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 by Bardi 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 is a 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. 


  1. Traducción al español 1
    "Sería un consuelo para la debilidad de nuestro ser y nuestras obras si todas las cosas se perdieran mucho más lentamente, de lo que vienen a ser; pero como ocurre que los aumentos son de crecimiento lento, el camino a la ruina es rápido." Lucius Anneaus Séneca, Cartas a Lucilio, n. 91.

    Esta observación por Séneca parece ser válida para muchos casos modernos, incluyendo la producción de un recurso no renovable como el petróleo crudo. ¿Estamos en el borde del "precipicio Seneca?" (Vínculo)

    Es un principio bien conocido de las personas que trabajan en la dinámica de sistemas que existen un montón de casos de soluciones que empeoran el problema. A menudo, las personas parecen ser perfectamente capaces de entender cuál es el problema, pero, con la misma frecuencia, tienden a actuar sobre el en el camino equivocado. Es un concepto que también se expresa como "empujar la palanca en la dirección equivocada". (Vínculo)

    Con los combustibles fósiles, todos entendemos que tenemos un problema de agotamiento, pero la solución, hasta ahora, ha sido para perforar más, para perforar más profundo, y para mantener la perforación. Exprimir un poco de combustible de todas las fuentes posibles, no importa lo difícil y costoso, podría compensar el declive de los campos convencionales y mantener la producción creciente de los últimos años. Pero, ¿es una solución real? Es decir, ¿no iremos pagar el actual crecimiento con un descenso más rápido en el futuro?

    Esta pregunta puede ser descrita en términos del “Precipicio de Seneca", (Vínculo) un concepto que me propuse hace unos años para describir cómo la producción de un recurso no renovable puede mostrar un rápido descenso después de pasar su pico de producción. No es sólo un modelo teórico: hay varios casos históricos en los que la producción de un recurso colapsó después de haber alcanzado su punto máximo. Por ejemplo, aquí están los datos para el esturión del Mar Caspio, un caso que yo denominé "pico del caviar” (vínculo).

    ¿Nos arriesgamos a ver algo como esto en el caso de la producción mundial de petróleo y gas? En mi opinión, sí. Hay algunas similitudes; tanto los combustibles fósiles y el caviar son recursos no reemplazables; y en ambos casos los precios subieron rápidamente en y después del pico. Así que, si Caspio esturión mostró una clara acantilado tales Seneca, petróleo y gas podrían hacer lo mismo. Pero déjame entrar en algunos detalles.

    En la primera versión de mi modelo Séneca, (Vínculo) el rápido declive de la producción se interpreta en términos de la creciente contaminación, lo que supone una carga adicional en el sistema productivo y reduce la cantidad de recursos disponibles para el desarrollo de nuevos recursos. Sin embargo, me encontré con que el comportamiento Seneca es bastante robusto en estos sistemas y que aparece cada vez que la gente trata de "estirar" un sistema para forzarlo a producir más y más rápido de lo que sería natural hacer.

    Así, en el caso del esturión del Caspio, arriba, la contaminación puede no ser la causa de la rápida caída de la producción. Más bien, lo que pasa es que los altos precios de un recurso raro y no reemplazable (caviar) atrajeron a los productores a invertir cada vez más recursos en rastrillar del mar tanto como sea posible. Funcionó, por un tiempo, pero, al final, no se puede pescar esturiones que no están allí. Se terminó en desastre: un caso clásico de un Precipicio Séneca .

    ¿Se puede modelar este fenómeno? Si. A continuación, describo el modelo para este caso en detalle. La esencia de la idea es que los productores deben reinvertir una parte de sus beneficios en el desarrollo de nuevos recursos con el fin de seguir produciendo. Sin embargo, el rendimiento de las nuevas inversiones disminuye a medida que pasa el tiempo, porque los recursos más rentables (por ejemplo, campos de petróleo) son explotados en primer lugar. Como resultado, menos y menos capital está disponible para nuevas inversiones. Eventualmente la

  2. Traducción al español 2
    producción alcanza un máximo, entonces se disminuye. Si suponemos que las empresas vuelvan a invertir una fracción constante de sus ganancias en nuevos recursos, el modelo lleva a la curva en forma de campana simétrica conocida como la "curva de Hubbert".

    Sin embargo, como lo describo en detalle más adelante, la disminución puede posponerse si los precios altos proporcionar capital adicional para los nuevos desarrollos productivos . Desafortunadamente, se obtiene el crecimiento en el costo de una quema acelerada de recursos de capital. El resultado final no es más la curva de Hubbert simétrico, sino una clásica curva Seneca: el descenso es más rápido que el crecimiento.

    ¿Es esto lo que nos enfrentamos a los combustibles fósiles? Por supuesto, sólo se trata de modelos cualitativos, pero, por otro lado, los modelos cualitativos son a menudo robustosa y nos dan una idea de qué esperar, a pesar de que no nos pueden decir mucho en términos de predecir acontecimientos en una precisa escala de tiempo. El actual colapso de los precios del petróleo puede ser un síntoma de que nos estamos quedando sin los recursos de capital necesarios para mantener el desarrollo de nuevos campos. Por lo tanto, lo que podemos decir es que hay algunas buenas posibilidades de tiempos difíciles por delante - en realidad duros . El precipicio Seneca bien puede ser parte de nuestro futuro a corto plazo.


    La curva de Seneca como resultado del aumento de fracciones de beneficios asignados a la producción de un recurso no renovable
    por Ugo Bardi - 07 de diciembre 2014

    Nota: esto no es un trabajo científico formal; es más unos deslavazados cálculos escritos sobre una servilleta de papel para mostrar cómo el aumento de las fracciones de gasto de capital pueden afectar la tasa de producción de un recurso no renovable.¡ Si alguien me puede dar una mano para hacer un estudio más refinado y publicable, yo estaría encantado de colaborar!
    Los fundamentos de un modelo de dinámica de sistemas que describe la explotación de un recurso no renovable en un mercado libre se describen en detalle en un documento de 2009 (Vínculo) by Bardi y Lavacchi. Este documento proporciona una descripción teórica del modelo de Hubbert y de la curva de producción "forma de campana". En el modelo, se supone que existe el recurso no renovable (R) en forma de una reserva inicial de la extensión fija. El stock de recursos se transforma gradualmente en una reserva de capital (C) que a su vez disminuye gradualmente. El comportamiento de las dos poblaciones como una función del tiempo se describe mediante dos ecuaciones diferenciales acopladas.

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

    donde R 'y C' indica el flujo de las acciones en función del tiempo (R 'es lo que llamamos la "producción"), mientras que los "ks" son constantes. Se trata de un modelo de "Bare Bones", que sin embargo puede reproducir la curva de Hubbert y encajar algunos casos históricos (Vínculo presentan). Adición de un tercio de valores (la contaminación) al sistema, genera la "Curva Seneca" (Vínculo), que es una curva de producción hacia adelante sesgada, con el declive más rápido que el crecimiento.

    El sistema de dos valores también puede producir una curva Seneca si las ecuaciones anteriores están ligeramente modificados. En particular, se puede escribir:

  3. Traducción al español 3

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

    Aquí, "K3" aquí indica explícitamente la fracción de capital reinvertido en la producción, mientras que k4 que es proporcional a la depreciación del capital (o cualquier otro uso no productivo). Entonces, se supone que la producción es proporcional a la cantidad de capital invertido, que es C * k3. Tenga en cuenta también que "ko" es un factor que define la eficiencia de la transformación de los recursos en capital; se puede observar en relación con la eficiencia tecnológica, pero este punto no se examinará aquí.

    Aquí es el modelo tal como se aplica utilizando el software Vensim (TM) para la dinámica del sistema. Los "ks" se han dado nombres explícitos. También estoy usando la convención de "modelos de tamaño de la mente" (Vínculo) presentan con las reservas de energía gratis más altas que aparecen por encima de las reservas de energía gratis más bajas


    Si se mantiene constante durante todo el ciclo de producción de la k, la forma de las curvas generadas por este modelo es exactamente el mismo que con la versión simplificada, que es una simétrica, curva de producción en forma de campana. Estos son los resultados de un ensayo normal:

    Las cosas cambian si permitimos "K3" para variar el ciclo de simulación. La característica que hace "K3" (fracción de la inversión productiva) un poco diferente a los otros parámetros del modelo, es que es totalmente dependiente de la elección humana. Es decir, mientras que los otros ks se ven limitadas por factores físicos y tecnológicos, la fracción del capital disponible reinvertidos en la producción se puede elegir casi a voluntad (¡ por supuesto, sigue siendo el límite de la cantidad total de capital disponible!).

    Los precios más altos darán lugar a mayores beneficios para los productores como para la tendencia a aumentar la fracción reinvertidos en nuevos desarrollos. También se sabe que en la región cerca de la producción pico de precios tienden a ser mayores - como en los casos históricos de aceite de ballena y caviar y aceite de ballena. En el caso de caviar, la subida de precios era casi exponencial, en el caso del aceite de ballena, más parecido a una curva logística. Suponiendo que la fracción de capital reinvertido varía en proporción a los precios, algunos modelos se pueden intentar. Aquí, déjame mostrarte sólo los resultados obtenidos con un aumento exponencial.
    Traducción al español 3

    También he intentado otras funciones de la tendencia al alza de K3. Los resultados son cualitativamente el mismo para un aumento lineal de una logística uno: reglas Seneca.

    Quiero subrayar una vez más que estos no se supone que son los resultados completos. Estos son sólo pruebas realizadas con supuestos algo arbitrarios para las constantes. Sin embargo, estos cálculos muestran que el comportamiento Seneca se produce cuando se supone que los productores se extienden a su sistema de asignación creciente fracciones del capital a la producción.

  4. Thanks for these translations, Anselmo. But are you publishing them somewhere?

    1. No, I only publish them in your blog.

    2. You are doing a lot of work. It would deserve to appear on a specific, Spanish language, blog. Why don't you set up one?

    3. I can not guarantee continuity in my effort, so do not see a good idea to create a blog. On the other hand, I ´m guided by the desire to learn and to cooperate in the dissemination of knowledge about problems that afflict our world. And. i think, the best way is to support authors of prestige.

      Usually I upload translations and I write posts in the Crash Oil forum. I especially like translate Posts by John Michael Greer.

    4. Or... Ask Antonio (Turiel), he will probably be interested on publishing it (normally, we translate and publish him in the Italian blog. This time would be the other way around :-))

    5. Yes, but it is too bad. Placed in the comments, the translations won't have any visibility. Why should people look for translations in spanish to be found in a blog in English? You should place them in a repository, somewhere. It is what we do with the italian version of this blog: we place translations in a site of their own. We could easily make a version in Spanish. Would you be interested in putting your translations in there?

    6. Many readers of AMT (Dr. Antonio Turiel) also read this blog (like me). And in the linked forum there is the topic rised by Mr. Mindundi, with another translation., and also with comments by Anselmo.

      Anselmo is doing a big, great job translating your blog, as well as JMG's Archdruid report, also in that forum.

      From Spain, best regards to all of your for your great effort in the diffussion of Peak Oil (and everything), and for any translation.


    7. Estimated prof. Bardi, told him that are translated into Spanish some of his articles and discuss in a forum that is born in the shadow of Dr. Antonio Turiel blog.

      If you are interested in seeing the comments generated, the link to one of your items is as follows:

      And for now, the forum administrator, is bringing translations of several authors in this thread:

      A greeting and thanks for your extensive work

    8. Dear professor Bardi:

      I will be glad to upload translations of your posts where you prefer

    9. Well, it would be easy to make a "mirror" blog in Spanish. How would you translate "Resource Crisis" in Spanish? Crisis de recursos would be all right?

    10. Anselmo, would you please write me at Thanks

  5. Kopits presented some interesting data about capex values and oil production for the listed oil majors (pages 40-42) -
    That might be useful for the model, but not quite enough to quantify k3...if that is necessary after all.

    1. Very interesting link, thanks! Too bad that the capex data are only for a limited period. But they do show an evident increase, could be exponential

    2. Ugo
      What do you think of these people and their model of depletion? Are they saying the same thing as you, in their case using perhaps the same kind of data as Kopits?
      Quote: "Optimistic estimates place the world's total petroleum reserve at 4,300 billion barrels. Of that quantity the ETP model predicts that it will be possible to extract 1,760.5 billion barrels. This constitutes 40.9% of the total reserve."


    3. Sorry about that - I missed out the link!


    4. Looks like the correct approach, but I'll have to work on that to understand the details

    5. Yes, it seems that data availability is a problem for capex data of oil industry. I thought that it might be possible to compile correct values from annual reports for shareholders of respective companies (tedious task, I know), but even BP has the last downloadable report only from the year 2005. I would like to help you with the paper, but it seems that it will be a problem to find any suitable data, even for other nonrenewables.

    6. Ok friends, oil is nearing below 60 dollars/barrel. Not too much time for modelling Seneca cliff, hurry up! ;-)


  6. Regarding the oil shale plays, it would seem the capex is driven by external finance rather than reinvestment of profits. I guess this also extends the peak beyond what would be capable using only reinvestment of profits.

    I guess the system really is on its last legs when oil is produced non-profitably to keep the system running simply via central banks magically printing money by fiat. Obviously such slight of hand is not sustainable.

    1. This might explain why, recently, oil prices have fallen even though production costs are high. I was going to ask Prof. Bardi if he can explain the recent price declines so near the presumed Seneca Peak. Looks like it may be from something external to the simple models he presented.

    2. I suspect a combination of ongoing demand destruction (aka economic stagnation) and good old-fashioned market manipulation (by OPEC, mainly, mebby others).

      Not to put words in Prof. Bardi's mouth, but the models presented here are doubtless intended to be illustrative, not definitive. In any case, of course, "definitive" modeling is only possible after (usually long after) the modeled event(s) have run to completion and all significant factors have been (retrospectively) identified. This is the core difficulty with, for example, the global climate-change models; unforeseen-but-necessary consequences, both as to type and rate of event, keep arising. Thus the ever-moving target of "probable consequence and schedule."

  7. Might be an interesting atom of information:

  8. "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. "

    Whenever I complain to my grandmother about something in society that doesn't make sense, she tells me, "Everything makes sense to the people doing it, you just have to ask who and when does it help?"

    From that perspective, I have come to accept that market failures on finite resources are what us programmers call a feature, not a bug. In the sturgeon example (and bluefin tuna is another good one) the collective industry gets richer and richer as the price rises, since profits rise faster than expenses, but even on the down slope it causes superior outcomes for some players. At first, these are the legacy players that happen to be able to hold out the longest and find a high price, lower competition environment; but eventually a lot of resources are spent even when total output is low because it becomes a lottery.

    If the goal of producers was to contribute to society then you are right they are doing the wrong thing, but if it is the hope of getting rich then they are doing the absolutely correct thing. After all, the producers are able to get external financing and can restructure debt when things go wrong, so it is a heads we win, tails you lose situation with their investors.

    Traditional fishing villages in Japan rarely catch bluefin these days, but any ship that does catch a few is a ticket to retirement, so a myriad of boats still set out, backed by small investor pools playing a physical equivalent of the lotto.

    Random positive reinforcement is *the* strongest operant conditioner. The only factor that seems to break the cycle is complete and sustained demand destruction. As long as demand remains, even if it is just among the luxury class, these dynamics seem to hold.

    This seems to make the Seneca Cliff not only plausible, but inevitable.

    1. Your words remind me of a poem I wrote a while back: Fail big for the win.

  9. Mikkel's astute comments (for which thanks!) resonate perfectly with Garrett Hardin's observations on the "tragedy of the commons" so many years ago. Hence the necessity and inevitability of "demand destruction" aka collapse/depopulation so eloquently posited by the "doom-sayers."

    The only way forward/out of this morass, so far as I can see, is a truly massive (and highly improbable) change/elevation of collective human consciousness. Ironically, such an event seems most likely as a late (likely too late) consequence of that "demand-destruction."



Ugo Bardi is a member of the Club of Rome, faculty member of the University of Florence, and the author of "Extracted" (Chelsea Green 2014), "The Seneca Effect" (Springer 2017), and Before the Collapse (Springer 2019)