Page 192 - Physical Principles of Sedimentary Basin Analysis

174 Heat flow
√
exp(−ˆz/ 2). It follows that the amplitude is reduced by the factor 1/e ≈ 0.36 at the depth
√ √
ˆ z = 2 ≈ 1.41 and by the factor 0.01 at the depth ˆz =− 2ln(0.01) ≈ 6.5. The depth
√
ˆ z = 2 is called the skin depth. Another point is that the subsurface lags behind the sur-
√
face. The depth z reaches its temperature maximum at time ˆ t =ˆz/ 2 after the surface,
√
and the temperature at the depth ˆz = 2π ≈ 4.44 is in opposite phase to the surface
temperature. The temperature is at its maximum at this depth when the temperature at the
surface is at its minimum.
The temperature solution is plotted in Figures 6.35a, and b, which show how the tem-
perature variations decay with depth. The figures also show how the phase changes with
the depth.
Some examples of the characteristic time and length for different periods are shown in
2 −1
the following table when the thermal diffusivity is κ = 10 −6 m s :

l 0 Skin depth
Period [m] [m]
24 hours 0.1 0.2
1 month 0.6 0.9
1 year 2.2 3.2
10 years 7.1 10.0
100 years 22 32
1000 years 71 100
4
10 years 224 316
5
10 years 708 1002
6
10 years 2240 3168
4 3 2 1 0 4 5 6 7 8
0 0

2 2

4 4
z [−] z [−]
^ ^
6 6

8 8
(a) (b)
10 10
−1 −0.5 0 0.5 1.0 −1 −0.5 0 0.5 1
^ ^
T [−] T [−]
Figure 6.35. The figures show the temperature into the subsurface for time steps ˆ t n = nπ/4 through
one period, where n is an integer from 0 to 8.