Short answers, pulled from the story.
EROI is the ratio of usable energy delivered from an energy resource to the amount of energy used to obtain that resource. It is calculated by dividing energy output by energy input. An EROI of 1:1 means the source is a net energy sink and can no longer supply usable energy.
An energy source must have an EROI of at least 3:1 to be considered viable as a prominent fuel. Murphy and Hall's 2010 extended methodology sets 5 as the minimum for sustainability, while Hall's own methodology places 12 to 13 as the threshold for supporting technological progress and complex society.
Charles A. S. Hall, a systems ecology and biophysical economics professor at the State University of New York, is credited with popularizing the energy analysis field. His method drew on biological research from an Ecosystems Marine Biological Laboratory and gained wide attention when a 1984 paper appeared on the cover of the journal Science.
A 2015 review found mean harmonized EROIs for photovoltaic modules ranging from 8.7 to 34.2, with energy payback times of 1.0 to 4.1 years. Wind turbines averaged an EROI of 19.8 in 2018 data, while Vestas reports 31 for its V150 model. Hydropower averages about 110 over a 100-year operational life.
The ratio for fossil fuel discovery in the United States fell from about 1000:1 in 1919 to roughly 5:1 in the 2010s. Originally, one barrel of oil could find, extract, and process about 100 barrels; that figure has declined steadily as easier reserves were depleted.
Thomas Homer-Dixon argues in "The Upside of Down" that declining EROI was one of the reasons for the collapse of the Western Roman Empire in the fifth century CE. Ecological damage including deforestation and soil fertility loss in southern Spain, southern Italy, Sicily, and north Africa began reducing agricultural output in the 2nd century. Rome's population, which peaked at 1.5 million under Trajan, fell to only 15,000 by 1084.