From the ground...down

If you’ve ever lived in a house with a basement, or spent time in a cave, you are probably aware that the temperature underground remains fairly constant regardless of what the air temperature is doing.  Many times the basement is the warmest part of the house in the winter, and the coolest in the summer.  Old time root cellars took advantage of this uniform temperature before air conditioners and central heat were common. 

  

Obviously, fluctuations in the air temperature will penetrate some distance into the ground.  How deep will they go?  While this question is fascinating all by itself, it has some practical implications, too.

As one example, if you were building a pipeline to carry water, and you wanted to bury it deep enough so that it would NEVER freeze, how deep should you go?  Clearly, if you buried it one hundred feet underground, it would never freeze, but if your pipeline were twenty miles long, that is a lot of dirt and rock to move.  You could save a lot of digging if you just stuck it an inch or two below the surface, but if you lived in a cold climate, the pipeline might freeze and break during the first cold spell.  So, you’d like to be deep enough, but not way too deep.

We can look at the effect in a concrete and mathematical way by making some simplifying assumptions. Then, later on we’ll remember that we made those assumptions and not try to get too exact when we go to use the results.  First, imagine that the daily variation in temperature right at the surface of the ground looks like a cosine wave:

Obviously, it doesn’t behave like that exactly, but it does tend to go up and down most days sort of like a cosine wave does.  Then, we’ll also assume that the ground has constant and uniform material properties like density, heat capacity, and thermal conductivity.  

With those two assumptions, we can solve the transient heat equation to get this result:

where

x—distance from the surface

t—time

α—thermal diffusivity (thermal conductivity divided by the product of heat capacity and density)

ω—frequency of the sinusoidal temperature fluctuations at the surface

Now, if you stare at that equation for a minute, you can see that the temperature at any depth varies just like the temperature at the surface, except that:

(a) it is shrunk by the decaying exponential term with  the shrinkage getting more severe  the deeper you go, and

(b) the phase is shifted with increasing depth. 

If you set  x=0, you are right back to the surface temperature, which is what you’d expect.

  

This animation shows the idealized ground temperature (along the x-axis) as a function of depth (y-axis, each line represents 5 cm) assuming a sinusoidal variation in surface temperature between 5 deg C and 15 deg C over a span of time of five 24 hr days. 

 It is easy to see that by about 20 cm deep, the temperature is hardly changing at all from the average temperature of 10 deg C.  If you look at just the exponential term in the equation above: 

you can find that the variation at 20 cm deep is 2.2% of the total (α=0.1 mm^2/s ).

Now, let’s pretend that 2.2% is pretty close to zero—that is, we’ll declare that as the depth at which surface temperature fluctuations don’t matter, and call it the “penetration depth”.  

For the animation, I used α=0.1 mm^2/s  (properties of soil) as the thermal diffusivity of the ground.  But you might imagine that different kinds of ground have different properties.  If I’d used the properties for sand, α=0.22 mm^2/s, then the penetration depth would be about 30 cm instead of 20 cm.  And using α=1.37 mm^2/s, (properties of granite) gives a penetration depth of 74 cm.  While it would be pretty hard to dig in solid granite, it isn’t hard to imagine that some kinds of dirt might have a high percentage of granite rocks or a high percentage of sand mixed in with the soil, and that would make a difference in the temperature distribution.

Next post we’ll talk about how deep you should bury a pipe to avoid freezing.