Limit on geothermal energy that could be extracted before the earth's magnetic field collapsed?

Question

This is more of a theoretical thought-experiment question.

Basically, how much geothermal energy can we extract before the loss of the magnetic field makes it a terribly bad idea?

Will the geothermal energy inside the earth now, which is lost through "natural processes", last until the end of the Sun (say, another 5billion years?) And if so, how much excess is leftover?

a) do we know how much thermal energy there is? Call this: A

b) do we know at what point the core would stop producing a magnetic field? (ie: is it when the iron in the core is no longer molten?) Call this: B

c) do we know the rate that heat is radiated out naturally? Call the annual rate: C

So I guess: A-B - (5.0 * 10^9 * C) is the amount of heat for "safe" geothermal extraction?

also: Considering such extreme timescales -- are there any other factors adding heat into the Earth? Maybe tidal interactions with Sun/Moon/etc add anything over billions of years?

Thanks!

Answer 1

The amount of energy we can capture as geothermal is only a very tiny fraction of the amount that leaks naturally through the earth's crust. In addition, the distance from the earth's surface to the generator is so very long that the time constant for heat movement at one end to cause heat change at the other is very long.

To get an idea of the calculation for this last point, note that we can measure the climate for going back 80,000 years by looking at the temperatures in the Ural superdeep borehole which goes down 12km. When digging such a hole, the temperatures at the top of the hole is the current temperature. Deeper temperatures correspond to averages of temperatures over longer periods of time.

The temperature T at depth z at time t is given by: $$T(z,t) = \int_{-\infty}^{t}\frac{z\;\exp(-z^2/4\alpha(t-\tau))\;f(\tau)\;d\tau}{\sqrt{4\pi\alpha(t-\tau)^3}}$$ where $f(\tau)$ is the surface temperature and $\alpha$ is the thermal diffusivity. From this, we see that the temperature change at the surface is exponentially damped at a depth $z$ by $\sqrt{t}$.

From the borehole paper above, we have that it takes about 80,000 years to change the temperature of 12km of rock through conduction. The distance to the earth's center is around 6400 km, maybe 6000 km to the magnetic core. So the earliest time we can expect to see a (just barely measurable) change to the earth's magnetic core is around $$80,000\;\textrm{years}\;\times \frac{6000^2}{12^2} = 20\; \textrm{billion years}.$$

In actual fact, the earth's core will cool down a lot sooner than this. The reason for the discrepancy is that I've ignored convection. But in any case, there's nothing to worry about.

Answer 2

Pretty much an echo of Carl. We can really only mine heat from the top few kilometers. The impedence from the deep earth will be dominated by the vast distances to the deep. There really is a huge amount of heat down there, the cores temperature is similar to the surface tempoerature of the sun. Also we still have radioactive decay going on. I have read, that a limit that will hit earlier (with or without geothermal heat mining) is that mantle convection will decrease and no longer support plate tectonics. With plate tectonics, the recycling of important chemical elements stops, and import ones (for life) would get trapped in sediment. But that is a few billion years off. Solar evolution (its growing more luminous at about 10% per billion years at the moment) will cause a runaway greenhouse long before then.

I wouldn't say geothermal extract can't exceed the current rate of geothermal flux. It is heat mining, i.e. using up heat that took millions of years to accumulate in say a hundred years (at a given locality). It really should be considered more as an energy store, then as a renewable source, as any aggressive rate of exploitation would greatly exceed the rate of recharge.