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.

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Further reading:

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.

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