Page 150 - Physical Principles of Sedimentary Basin Analysis

132 Heat flow

T J C P E O M P Q
0 0
10 Quaternary Quaternary
20 VR−observations
30 Pliocene easy−ro Pliocene
1 40 Utsira 1 Utsira
50
Hordaland Hordaland
60
70 80 Rogaland 2 Rogaland
depth [km] 2 90 100 110 Shetland depth [km] 3 Shetland
3
Viking
Viking
Brent
Brent
4 120 130 Cromer_Knoll 4 Cromer_Knoll
140 Dunlin Dunlin
Statfjord Statfjord
150
5 160 Triassic 5 Triassic
170
6 6
−250 −200 −150 −100 −50 0 0.0 0.5 1.0 1.5 2.0
time [Ma] %Ro [−]
(a) (b)
Figure 6.12. (a) A burial history and its paleo-temperature. (b) VR observations and computed VR
(easy-ro) for the temperature history in (a).
The following two sections marked by ∗ can be considered as extended mathematical
exercises and are not necessary reading.

6.8 Stationary heat flow in a sphere ∗
This section is a first attempt to show that conduction in a sphere is not a good model
for the thermal state of the entire planet Earth. It turns out that heat transfer by conduc-
tion dominates only in the lithosphere, and that convection in the mantle is more efficient
than conduction. Nevertheless, heat flow in a sphere is an interesting exercise that shows
what the temperature at the center of the planet would have been if there had been heat
conduction all the way to the center.
The equation for conservation of heat is now written for a thin shell rather than a box,
see Figure 6.13. We then have that the radial transfer of energy into the shell at radius r
added to the energy generated in the shell is equal to the radial transfer of energy out of the
shell at radius r + dr. This can be written as
2
2
2
4π(r + dr) q(r + dr) − 4πr q(r) = 4πr dr S(r) (6.116)
where q(r) and S(r) are the radial heat flow and heat production per unit volume,
respectively, at radius r. To the first order in dr equation (6.116) becomes
dq
2
2
4π(r + dr) 2 q + dr − 4πr q(r) = 4πr dr S(r) (6.117)
dr
which can be further simplified by expanding the term in parentheses and collecting only
terms to first order in dr. We are then left with
dq(r) 2q(r)
+ = S(r). (6.118)
dr r