###### Also published at Resilience.org.

The “energy cliff” is a central concept in ecological economics, and it’s based on a very simple ratio. But for me this principle was a slippery thing to grasp, and I eventually realized some of the most common graphs used to illustrate the Energy Cliff were leaving me with a misleading mental image.

This column takes a closer look at Energy Return on Energy Invested (ERoEI, EROEI or simply EROI) and the Energy Cliff, concluding with the question of how and whether the Energy Cliff might be experienced as a historical phenomenon.

#### The Energy Cliff as a mathematical function

Below are two frequently used versions of the Energy Cliff graph, based on the pioneering work of Charles Hall. They illustrate the relationship between Energy Return on Energy Invested and the percentage of energy production that is “surplus”, i.e., not needed by the energy sector for its own work and therefore available for use by the rest of society.

In each case the EROEI is shown on the horizontal axis with lowest values at the right. The apparent suddenness of the drop-off in surplus energy depends on the relative scales of the axes and maximum value shown for EROEI, but in each case the drop-off becomes nearly perpendicular as EROEI falls below 10 – thus the name “Energy Cliff”.

Simple enough, eh? But after seeing this graph presented in several books and essays, I still found the concept hard to master. I kept asking myself, “How does that work again?” or “Why does energy supply drop off so suddenly?”

The problem, I realized, is that the impression these graphics leave in my mind is at odds with the intent. As these examples show, the “Energy for society” or “Profit energy” dominates the graphic visually, and the “Energy used to procure energy” or “Cost energy” seems like such a small sliver that it couldn’t possibly be that important. Mathematically naïve as that impression may have been, it nevertheless made it difficult for me to retain a clear understanding of the Energy Cliff.

The solution for me was to play with the graph until I felt I understood it clearly, using imagery that reinforced the understanding.

It was most helpful, I found, to present the graph not as an unbroken continuum between the two variables, but as a bar chart showing discrete values of Energy Return on Energy Invested: 1, 2, 3, 4, etc up to 50.

Visualizing the numbers this way minimizes the tendency to see the surplus energy, or Net energy output, as one massive block. Just as importantly, it allowed me to easily focus on the relationship between specific values of Energy input and Net energy output.

For example, at the far right end of the graph is the ERoEI value 1. This corresponds to a bare break-even scenario. An oil well with this ERoEI would not be worth drilling: we would use up one barrel of oil to drill and operate the well, and it would spit out exactly one barrel in return, leaving us with no surplus energy for our efforts.

An ERoEI of 2 corresponds to a Net energy output of 50%. To return to our Proverbial Oil Corp., we burn one barrel of oil to drill and operate a well, and the well spits out two barrels, leaving us with a net gain of 1 barrel or 50% of the Total energy output.

Our oil wells with ERoEI of 3 give us 3 barrels total for every one we invest, for a net energy gain of 2 barrels or 66.6%, wells with ERoEI of 4 give us a net energy output equal to 75% of their total energy output, wells with ERoEI of 5 give us a net energy output equal to 80% of their total energy output, and so on.

We can also see clearly that the Energy input and Net energy output percentages change very slowly for ERoEI values above 20 – at which point Energy input is 5% and Net energy output is 95% of Total energy output).

There is another simple tweak to this chart that can vividly illustrate the sudden drop-off: animation. (And since most of us use supercomputers capable of guiding a moon mission for our morning reading, why not throw in some animation?)

By focusing attention on just a narrow range of ERoEI values at a time, this moving bar graph illustrates the fact that Net energy output changes slowly throughout most of the range, and then drops off suddenly and swiftly.

The animated graph relies on the element of time as a key facet of the presentation. That begs the question: can the Energy Cliff chart be read as a function of time?

#### The Energy Cliff as a historical phenomenon

It is easy to look at the Energy Cliff graphic as a chronological progression, given the convention of viewing timelines with past on the left and future on the right. That would be a mistake – there is no element of time in the chart – but it might be a useful mistake if made consciously.

It’s true that ERoEI rates have been declining slowly for the past 50 years, and many new energy technologies today have ERoEI rates of 10 or lower. And in fact, the Energy Cliff chart is sometimes presented as evidence that an impending energy crisis is mathematically inevitable. While that would be an unwarranted extrapolation from a graph of a simple exponential curve, it isn’t hard to cherry-pick data that graphs to a shape similar to the Energy Cliff.

Consider the following table of ERoEI rates over time.

This table starts with EROEI rates before the industrial age, and finishes with rates that could plausibly represent the collapse of industrial society. When graphed these numbers show a drop-off much like the Energy Cliff, with the addition of a steep slope going up at the outset of industrial civilization. The values are roughly scaled chronologically, to represent the length of time during which very high EROEI prevailed – basically, the 20th century.

The numbers cherry-picked for this chart include, crucially, an EROEI for photovoltaic panels in Spain as calculated by Charles Hall and Pedro Prieto, which was the subject of spirited discussion recently on Resilience. At 2.45, this EROEI is far below the level needed to support a highly complex economy. If this number is correct and turns out to be representative of photovoltaics more generally, then the scenario suggested in the above chart is plausible. As high EROEI petroleum sources are depleted, we turn to bottom-of-the-barrel resources like tar sands, and then to solar panels which are even less energy-efficient. Complex industrial society soon collapses, and the vast majority of us must return to the fields.

For a very different picture, we could use the EROEI for solar panel installations presented by Ugo Bardi in Resilience, from a study by Bhandari et al. In this view, photovoltaics in Spain have an EROEI of 11–12, safely out of the drop-off zone of the Energy Cliff. In this scenario we’d have no need for last-ditch fossil fuels from tar sands, solar panels would produce enough surplus energy to create more solar panels and keep industrial society rolling cleanly along, and the Energy Cliff would be a mathematical function but not a historical reality.

These two charts are equally over-simplified, ignoring other renewable resource energy technologies with widely varying EROEI rates such as hydro-electric generation. It’s unknown how long we might stretch out the dwindling supply of high-EROEI fossil fuels, or whether there will be a collective decision to clamp down on carbon emissions and leave fossil fuels in the ground. And I’m unqualified to make any judgment on whether the Hall/Prieto or the Bhandari assessment of photovoltaics is most realistic.

In presenting these two different charts I merely want to illustrate that while the Energy Cliff graph of a mathematical function is simple and direct, extrapolating from this simple function to forecast historical trends is fraught with uncertainty.

*Top graphic: “The Fool” in the Rider-Waite Tarot deck dances gayly at the edge of a precipice.*