Page 133 - Physical Principles of Sedimentary Basin Analysis

6.4 Stationary 1D temperature solutions with heat generation 115

4 3 total

heat generation [μW/m3] 2 K40

1
U238
Th232
U235
0
−4 −3 −2 −1 0
time [Ga]
Figure 6.3. Heat generation in the average oceanic crust in the past, where the present-day isotope
concentrations are given in Table 6.2. Notice that the present-day heat production is dominated by
238 U and 232 Th, but the isotopes 235 U and 40 K were important in the Earth’s early history.

heat production is therefore ignored in models that do not go further back in time than a
few hundred million years. Figure 6.3 shows also that the present-day heat production is
dominated by 238 U and 232 Th, but that the 235 U and 40 K were important in the Earth’s early
history.
Exercise 6.3 What is the ratio C Th /C U when Th and U contribute equally to heat gen-
eration? And what is the ratio C K /C U when K and U have the same contribution to heat
generation?

Exercise 6.4 Assume that C Th /C U ≈ 4 and C K /C U ≈ 1.2 and use equation (6.40)to
derive the estimates (6.41)for C U , C Th and C K as a function of heat production.

Exercise 6.5 Show that the number of radioactive atoms as a function of time (6.43)
follows by integration of equation (6.42).
Solution: From equation (6.43) we get
N t
dN
=− λdt =−λt (6.47)
N
N 0 0
which yields

N −λt
ln =−λt or N = N 0 e . (6.48)
N 0

6.4 Stationary 1D temperature solutions with heat generation

Solutions of the stationary (time-independent) temperature equation can be used to esti-
mate geotherms when a geological environment does not change much over a long time