Page 174 - Physical Principles of Sedimentary Basin Analysis

156 Heat flow

0
1 day
10 days
2
30 days
70 days
depth [m] 4 150 days

310 days
6

8

10
0 2 4 6 8 10
temperature [°C]
Figure 6.25. The heating of the ground after the surface temperature has suddenly increased by
2 −1
◦
10 C. The thermal diffusivity of the ground is κ = 10 −6 m s .
which is seen by inserting the definition erfc = 1 − erf into equation (6.202). The temper-
ature solution is plotted in Figure 6.25 for heating 10 m into the ground after the surface
◦
temperature has increased by 10 C.
The initial temperature of the subsurface is the constant T 0 , which is a sufficiently good
approximation for shallow depths. A typical thermal gradient could be 35 C/km, which
◦
◦
implies that the temperature increase over 10 m is only 0.35 C. However, we could have
added the stationary temperature solution T (z) = Az + T 0 to the transient solution (6.202)
where A is the thermal gradient, because the temperature equation is linear. Any linear
combination of temperature solutions that together fulfill the boundary conditions is also a
solution.
The depth into the ground where the temperature has changed by 10% of the difference
T s − T 0 is called the thermal boundary layer. The use of 10% is arbitrary – it is just a
suitable number that serves as a measure for a noticeable change in the ground temperature.
The temperature for heating of the infinite half-space gives

T (z, t) − T 0
= erfc(η) = 0.1, (6.206)
T s − T 0
where Figure 6.24 gives that erfc(η) = 0.1for η = 1.16. The equation for the thermal
boundary layer as a function of time is therefore
√
z T (t) = 2.32 κt. (6.207)

The depth where the temperature has increased by 10% of T s − T 0 (the difference between
the new surface temperature and the initial temperature) is proportional to the square
root of t.