This relationship starts with the heat content of oil and natural gas. Since these fuels are used to create heat, and can be substituted for each other in some applications (electricity generation, industrial heating, greenhouses, etc), the heat content is very important. A barrel of oil produces 6 times the heat energy of an MMBtu of gas. (A GJ is about the same as an MMBtu).
Historically however, the ratio has gone up to as much as 10:1. This tended to be in times when oil was very expensive and the prospects for the natural gas market were not good. There is a very good, but long and scholerly, study done by Rice University in 2007.
A key quote from this study is as follows.
"Importantly, we conclude that U.S. natural gas and crude oil prices remain linked in their long term movement...... One implication of this finding is that, if international crude oil prices remain high, U.S. natural gas prices are unlikely to collapse substantially over the long term.
According to our analysis, a $70 per barrrel WTI average price is likely to promote a long run equilibrium natural gas price at the Henry Hub of around $9.40 per MMBtu. Furthermore, our analysis also shows that factors such as weather shocks and changes in storage can lead to substantial deviations from this long run price ratio.
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According to this study, the approximate price relationship should be about 7.5:1, which looks like this.
Oil Natural Gas
$60 $ 8.00
$70 $ 9.33
$80 $ 10.67
$90 $ 12.00
$100 $ 13.33
$110 $ 14.67
$120 $ 16.00