[Peakoil] The Formula: to be memorised!

[W & C Steensby]

Hello folks,

(This post has become a bit long, but honestly I think it's worth reading through. Apologies if it's too much.)

Ever notice how, when a new reserve of a resource is discovered, it's usually accompanied by press releases announcing that this resource will last for some large number of years at current rates of consumption? It's always at current rates of consumption, and yet we know that our economic system cannot survive unless it grows at an exponential rate, i.e. unless it consumes more resources this year than it did last year, even if the extra yearly amount of resource consumed is not very great. ANY growth rate greater than unity is exponential, even a hundredth of a percent. For decades Prof Al Bartlett has been warning about the behaviour, features and consequences of exponential growth, particularly growth in population and in resources consumption.

Here's a formula which lets you calculate how long a given amount of resource will last given various rates of exponentially-increasing consumption. Some of you already know this formula, but if you don't, it's really handy to know or at least know about. Most people don't know of it — or don't seem to know of it, especially politicians and their advisers. Maybe they prefer to ignore it.

It looks a bit complicated but it's not.

The Formula: T = (1/k) ln (kR/r0 + 1)

where
T = number of time periods until the resource is all gone.
k = rate of growth of consumption per time period.
R = quantity of the resource we have now.
r0 = rate of consumption of the resource at time zero, i.e. now.

ln = the natural logarithm function. On a scientific calculator it's usually on the key next to the one with the log function.

Here's an astonishing worked example based on the one in Prof Bartlett's book. Suppose planet Earth were full of oil, a great big spherical oil tank floating in space. How long would the oil last at (a) today's rate of consumption held constant, and (b) at an annually-increasing rate of consumption?

1. We need to know k. From 1880 to 1970 world oil consumption grew at about 7.04% per annum. This is a doubling every 9.8 years, and let's keep doubling at this rate. We'll make our time period a day, not a year. Hence k=7.04% per year. Divide by 365.25 to find k = 1.9274 x 10^-4 per day.

2. We need to know r0. In 2011 the world consumed 88 million barrels daily, which is 13.99 million or near enough to 14 x 10^6 cubic metres daily.

3. We need to know R. This is the volume of the oil tank, i.e. the volume of the Earth, which is 1.083 x 10^21 cubic metres.

(a) The first calculation is very simple: divide R by r0 and we get
      1.083 x 10^21 / 14 x 10^6 = days or 2.118 x 10^11 years.
That's 211,800 million years. Think about this length of time. It's inconceivably long. It's over 15 times the age of the universe.

(b) The second calculation (I won't type it all out) produces 121,538 days or just over 333 years. That's ALL. So short. Ponder this.

Wherever possible always double-check estimates of how long a resource will last. You may be surprised.

Regards,
Walter