Page 175 - Physical Principles of Sedimentary Basin Analysis

6.14 Instantaneous heating or cooling of semi-infinite half-space 157

Fourier’s law gives that the surface heat flow is

∂T
q =− λ
∂z z=0
∂ ∂η
=−λ(T s − T 0 ) (erfc(η)) η=0
∂η ∂z
2 −η 2 1

= λ(T s − T 0 ) √ e √
π η=0 2 κt
λ(T s − T 0 )
= √ . (6.208)
πκt
The differentiation of the temperature solution (6.202) with respect to z simplifies by use
of the chain rule. The surface heat flow becomes infinite at t = 0 and it then decreases as
√
1/ t. The heat flow can be used to compute the energy E that has passed into the ground
through an area A as a function of time. We have
t t

E(t) = A q(t )dt = Aλ(T s − T 0 ) (6.209)
0 πκ
√
which shows that the energy transferred to the ground increases as t. (Recall that heat
flow has units of energy per unit area and unit time.)
The temperature solution (6.202) gives the temperature of the subsurface at time t in
response to a step-increase in the surface temperature at t = 0. We can also consider
the step rise in temperature as an event that happened a time t ago. We know that the
temperature equation (6.201) is linear and we can therefore add step functions for changes
in the surface temperature. If for instance the temperature increased by a step T 2 at time t 2
and then decreased with the step −T 2 at a later time t 1 the temperature into the ground is

z z
T (z) = T 2 erfc √ − erfc √ . (6.210)
2 κt 2 2 κt 1
This is the temperature transient of the subsurface from a rectangular pulse in the surface
temperature from t 2 to t 1 with the size T 2 . We can continue in this way, by adding pulses,
and then generate the thermal response of the subsurface to a piecewise constant surface
temperature. The temperature becomes
N
z z
T (z) = T n erfc √ − erfc √ (6.211)
2 κt n 2 κt n−1
n=1
when the temperature is T n in the interval from t n to t n−1 . The surface temperature is first
changed a time t N ago and it is then changed at the steps t n until the present time t 0 = 0.
Expression (6.211) has been used in the reconstruction of the past surface temperature from
borehole temperature measurements. Observations of the temperature at a large number of
vertical positions along a borehole makes it possible to obtain the optimal piecewise surface
temperature as demonstrated by Beltrami et al. (1997) and Beltrami (2001).