Page 195 - Physical Principles of Sedimentary Basin Analysis

6.19 Variable surface temperature 177

The last factor, the exponential of a complex number, is written as a real and an imaginary
part using that e iφ = cos φ + i sin φ. We only need the real part, and the temperature
solution is therefore

b b
ˆ
T (ˆz, ˆ t) = c exp − ˆ z cos bˆ t − ˆ z . (6.280)
2 2
The integration constants b and c are found by comparing T (ˆz = 0, ˆ t) = c cos(bˆ t) with
ˆ
the boundary condition (6.267), and we then get that b = 1 and c = T .

6.19 Variable surface temperature

The previous section showed how the temperature varies with depth when there is a peri-
odic variation in the surface temperature, and Section 6.14 showed the thermal response of
the subsurface to a piecewise constant surface temperature history. We will now see how
we can obtain the temperature at a depth at a specific time for any variation in the surface
temperature. We recall that a linear combination of solutions of the temperature equation
is also a solution. The temperature can therefore be written as a sum of two parts:
T (z, t) = T s (z) + T (z, t) (6.281)

where T s (z) is the stationary solution and T (z, t) is the transient solution. Note 6.11
shows that the transient part can be written as
t 2
z 2
T (z, t) = T surf t − exp(−μ ) dμ (6.282)
√
z/2 κt 4κμ
when the surface temperature at time t is T surf (t). It is assumed that the surface temperature
is zero until t = 0, before it starts to vary. The transient solution is therefore zero before
t = 0. The thermal transient (6.282) can easily be integrated numerically for any given
2
surface temperature T surf (t), because the factor exp(−μ ) goes rapidly towards zero. The
integration can also be carried out exactly for simple forms of the function T surf .
The simplest application of expression (6.282) is to assume that the surface temperature
makes a step from zero to a constant T surf at t = 0. We then obtain directly the solution
for instantaneous heating or cooling of the semi-infinite half-space. Another application is
a surface temperature that increases linearly with time as T surf (t) = ct from t = 0, where
c is the heating rate. The transient part of the temperature solution then becomes

2
2
z z z z
T (z, t) = ct 1 + erfc( √ ) − √ exp(− ) (6.283)
2κt 2 κt πκt 4κt
where Note 6.12 shows how it follows from the general expression (6.282). The solu-
tion (6.272) for periodic heating of the surface can also be derived starting with
equation (6.282) as shown in Section 2.6 of Carslaw and Jaeger (1959).