Page 239 - Physical Principles of Sedimentary Basin Analysis

7.10 Finite duration stretching and temperature 221

Solution: (b) Use the subsidence (7.21). Subsidence of the simple shear model was studied
in great detail by Weissel and Karner (1989).

Exercise 7.15 Show that the strain rate (7.76) applies for vertical line segments of any
length when it is constant.
l

Solution: A long line l can be divided into small (infinitesimal) segments l i as l = i i
where G is the same for each line segment. We therefore have
d dl

G l i = l i or Gl = . (7.96)
dt dt
i i
Exercise 7.16 Show that expression (7.87) for the strain rate as a function of the stretching
factor follows directly from the definition of strain rate.
Solution: The strain rate is defined as G = (1/l)( l/ t), where l = βl 0 is a small
line segment that has been stretched a factor β from the initial length l 0 . The line seg-
ment is stretched the step l = β l 0 during the time interval t, which then gives
equation (7.87).

Exercise 7.17 Let us consider deposition of sediments during a rift phase. Show that the
net thickness of the synrift formation becomes
1
1
ζ 1 = 1 − ζ 0 (7.97)
lnβ β
where β is the stretching factor and ζ 0 is the thickness of the unstretched formation as (net)
porosity-free sediments. Assume that the deposition rate (as net sediments) and the strain
rate are constant.
Solution: The duration of extension is first divided into time steps t, and a thickness
ζ = ω t is deposited during each time step, where the deposition rate is ω.The β-factor
is β = e Gt as a function of time, where G is the strain rate, and t = 0 is the beginning of
the rift phase. The thickness ζ is therefore reduced to ζ exp(−Gt) after a time span t.
The sum of the thicknesses through the entire rifting phase becomes
t 0 t 0 ω

ζ 1 = e −Gt dζ = ω e −Gt dt = 1 − e −Gt 0 (7.98)
0 0 G
where the time t = t 0 is the end of the formation. The strain rate is G = lnβ/t 0 and the
unstretched formation thickness is ζ 0 = ωt 0 . Replacing the strain rate by the β-factor at
the end of rifting then gives expression (7.97).

7.10 Finite duration stretching and temperature
The velocity field controls the temperature of the mantle when heat conduction is negligible
compared to heat convection by mantle flow. The temperature equation for convective heat
flow is then
∂T
+ v ·∇T = 0 (7.99)
∂t