# EROEI limits - basics

Completed 2010-07-20

This note addresses the issue of EROEI (energy returned on energy invested) and its influence on the world economy. A quantity *e* is defined as the efficiency of the entire economy in converting (non-renewable) energy resources into useful energy. It is found that in the limit when EROEI approaches 1*/e*, then there can be no other economic activity than energy extraction itself. For a typical efficiency 0.2, the lower limit of EROEI = 5.

Definitions:

Ep - the flow of primary energy into the stores

Eu - the flow of useful energy into the economy

V - the stock of available primary energy

f - the flow of useful energy into the energy extraction process

R - EROEI of the energy extraction process

e - efficiency of converting primary to useful energy

d - the fraction of the useful energy flux going to energy extraction

a - the inverse timescale with which the primary energy storage is emptied

We make the further simplification that the reservoirs in the ground are infinite - a simplification that can be alleviated later on.

Then the following relations hold:

(1) Ep = R f

(2) Eu = a e V

(3) f = d Eu

(4) d/dt(V) = Ep - (1/e)Eu

Then we get:

(5) Ep = R d a e V

(6) d/dt(V) = a V ( d e R -1 )

In order to have d/dt(V) >= 0, then

(7) d >= 1/(e R)

Since the storage capacity is for all practical purposes finite, a useful assumption is:

d/dt(V) = 0

hence (8) d = 1/(e R)

Interpretation: As EROEI (R) plummets, the relative use of extraction energy has to increase accordingly, and/or the efficiency e has to increase (asymptotically towards unity).

For a society who wants to use more energy (assuming R constant), must increase refinery capacity (*d*) and/or its economic activity (*a*) (and/or efficiency, *e*). Only increasing *a *implies that refinery capacity etc is adjusted accordingly - in propostion with the rest of the economy - through the constant *d*. Increasing *d* is just adding extra capacity or exploiting new fields etc., to the cost of the remaining economy.

The delivery of useful energy during steady state is:

(9) a V (e - 1/R)

i.e., as EROEI decreases, there is less available energy for the economy (not surprisingly).

We also immediately see that there is a lower limit on R, namely R>= 1/e

As R decreases, it is seen from (9) that in order to maintain output, a (or e or V) can be increased (i.e. speeding up the economy, as a fraction *d* of it feeds directly back to extraction). But increasing extraction in turn increases depletion, decreasing EROEI at an accelerated rate. Sooner or later the rate of change of a cannot compete with the change in R. If the time evolution of R can be determined, then one can predict more precisely when a becomes overwhelmed by R.

### Some numbers

From the Global Energy Assessment (GEA) Chapter 1, we find that roughly speaking, in 2008 Oil comprised 30%, coal 25, gas 20 and bio 10% of global primary energy. Neglecting the remainder, renormalizing yields 35, 30, 24, 11.

For the well-head EROEI we pick from Hall and Murphy's (2010) figures from Wikipedia for coal 80, oil 40 gas 40 and bio 20 (wood is 24, Baobab) we arrive at EROEI = 47 as an estimate for the total mix. From GEA we see that the ratio of useful to primary energy is *u *= 169 EJ / 496 EJ = 34%

Hence, the part going to extraction based on the EROEI defined here, *R*, is *P*/*R*, where P is primary energy. *e = U/P* = .34, as found above. Thus, extraction energy to useful, is *P/R / eP* = 1/(*eR*) = *d* = 0.06.

I.e., roughly 5% of today's final energy goes back to extraction, which is not a big amount.

However, if one assumes that the GDP is proportional to the use of useful energy, *GDP* = *g*(1-*d*), we see that in order to have positive growth, d[GDP] / dt > 0, [] denoting the relative quantity, choosing units so that g=1,

d(*d*)/dt /(1-d)< - d[*GDP*]/dt/GNP = - C

i.e., it has to *decrease*, while from the EROEI dependency it must *increase* as EROEI degrades with time.

For a future running on wind and photo-voltaics, R =O(10), the fraction of extraction energy has to be about *d* = 30%. Factoring in that the efficiency, *e*, may be increased may help:

[e]+[R] > (1-d) C, where [] means time derivative of the relative quantity.

Hence, the relative growth in *e* must outpace the decrease in *R* accordingly. For a 2% increase in BNP annualy, we get

[e] > 0.02 (1-d) - [R]

i.e. [e] > 4% per year, if [R] is set to -2%y-1, i.e. 1.2 units per year from the present level of 34%.

Effiency of American passenger cars increased linearly from 34% in 1985 to 43% in 2004, i.e. 0.47 per year, or 1.3% relative increase per year. It turns out that 4% per year of [*e*] is too optimistic.

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The major part of this article was written July 2010. The article by T. Murphy appeared October 2011 *)

The 'Some numbers' part was added February 2013

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Further reading:

*) Tom Murphy (Do the Math), 2011-10-28. http://physics.ucsd.edu/do-the-math/2011/10/the-energy-trap/#more-452

http://www.theoildrum.com/node/8625?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+theoildrum+%28The+Oil+Drum%29&utm_content=FaceBook

The article by Dale, Krumdieck and Bogder (2011) provides a much more in-depth treatment of the themes presented here.

On the linkage between GDP growth and energy use

Dave Murphy, 2013: Implications of declining EROEI

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