Page 176 - Physical Principles of Sedimentary Basin Analysis

158 Heat flow

Note 6.7 It is normally a good idea to make the temperature equation dimensionless by
scaling time and distance by characteristic quantities before it is solved. But, there is not
obvious length scale for an infinitely deep ground. We have already seen that the char-
2
acteristic time t 0 is related to the characteristic length l 0 by t 0 = l /κ (or alternatively
√ 0
l 0 = κt 0 ). It now turns out that the temperature for the semi-infinite half-space can be
expressed as a function of the dimensionless ratio
z
η = √ (6.212)
2 κt
where it is anticipated that a factor 1/2 simplifies the matter. The dimensionless tempera-
ture as a function of η is
T (z, t) − T 0
= (η) or T (z, t) = T 0 + (T s − T 0 ) (η) (6.213)
T s − T 0
where the function measures the heating of the half-space from the initial temperature
T 0 to the temperature T s using the unit interval (the interval from 0 to 1). The boundary
conditions for heating of the semi-infinite half-space can therefore be written

(η = 0) = 1 and (η →∞) = 0 (6.214)
because z = 0 implies η = 0 and z =∞ implies η =∞. Differentiation of T (z, t) (6.213)
leads to
∂T z
1

=− · √ · (6.215)
∂t 2 κt 2t
∂T 1

= · √ (6.216)
∂z 2 κt
2
∂ T 1

= · . (6.217)
∂z 2 4κt
Insertion of these parts into the temperature equation (6.201) yields

(η) =−2η (η). (6.218)

Letting f = gives
df
=−2η dη (6.219)
f
which is integrated to
2

(η) = c 1 exp(−η ) (6.220)
where c 1 is an integration constant. One more integration then yields the temperature
solution
η 2
e −x (6.221)
(η) = c 1 dx + c 2
0