Digging depth and heat

I have a question about the underground heat at different distances from the core of the Earth.

I was wondering if there was a depth at which you can dig that the decrease in heat (as a result of being lower) and the increase in heat due to the depth that you dug that the loss of heat was the same as the gain in heat.

Here are the assumptions that I am using:

• The lower an object is, the cooler the air around it and therefore the cooler the object can become. Cold air tends to sink below heated air. (I'm thinking like a basement is cooler than the top floor during summer without air conditioning).
• The deeper you dig towards the centre of the earth, the hotter your surroundings become. (While there may be some initial decrease in temperature, I'm assuming as you dig towards the mantle or core the hotter the ground around you is).

I am holding constant the heat from the sun and atmosphere. Is there a defined depth at which the loss of heat (from falling cool air) = gain in heat (from the heat of the earth)? Does the shape of the hole dug affect the depth at which this occurs?

• Your first assumption is only valid for constant pressure, which you will not have moving vertically above or below the surface. May 30, 2015 at 13:31

The heat equilibrium depth will vary for different locations due to:

1. Differences in geothermal gradient at different locations (heat emitted by rock). It is generally accepted that the global geothermal gradient is between ${25 ^o C/km}$ & ${30 ^o C/km}$. At some locations the gradient can vary between ${15.4 ^o C/km}$ & ${102.6 ^o C/km}$.
2. The temperature of the air will increase with depth due to auto-compression of the air (also known as lapse rate). [Environmental Engineering of South African Mines, 1989, pp 403-404]. The increase in dry bulb temperature of air due to auto-compression is ${9.66 ^o C/km}$, if there is no change to the humidity of the air.
3. Moisture content of the air. The greater the humidity, the greater the heat in the air.
4. Ventilation of the hole being dug and how much heat the moving air is removing from the bottom of the hole.
5. Temperature of the air entering the hole from the Earth's surface

6. The presence or absence of other sources of heat such as machinery, electrical equipment, warm water springs entering the hole and warm bodies.

7. The age of the hole and its exposure to ventilating air. New holes expose fresh warm rock to air in the atmosphere and the air eventually carries away the heat via ventilation. New holes, or new extensions to holes, are warmer/hotter than old holes.

The shape of the hole will be determined more by its function and geotechnical conditions, such as ground stability, than temperature.