Page 203 - Physical Principles of Sedimentary Basin Analysis

6.20 Temperature transients from sediment deposition or erosion 185

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condition, with the surface temperature T = 0 in the new layer, after t = 10 years. More
than 1 Ma is needed for the temperature to approach the new thermal state. We can use the
same reasoning as with the thermal boundary layer in Section 6.14 to study the transient.
The depth where the temperature has regained 90% of the of depression T = Ah is
√
z = 0.176 κt, because erfc(0.088) = 0.9. For instance a time span t = 16 Ma is needed
for 90% of the perturbation to have died out at the depth z = 4km.
Deposition or erosion of sediments may bring sedimentary basins away from a thermal
steady state for tens to hundreds of million years. The importance of thermal transients
from deposition or erosion of sediments has been studied by several authors after the pio-
neering work of Benfield (1949) – among others Hutchison (1985), Lucazeau and Douaran
(1985), Karner (1991) and Wangen (1995).
Note 6.13 Dimensionless formulation. It is possible to scale the temperature equa-
tion (6.298) and Benfield’s solution (6.299) using the characteristic length l 1 = κ/v,the
2
characteristic time t 1 = κ/v and characteristic temperature T 1 = Aκ/v. Both the equation
and the solution then become parameterless. The dimensionless deposition rate is 1. It is
then possible to compare time and depth with t 1 and l 1 , respectively, in order to predict the
amount of thermal blanketing.
Note 6.14 Fourier series solution. The Fourier series (6.166) solves the dimensionless
convection–diffusion equation (6.150), when the temperatures at the boundaries are fixed.
The Peclet number now represents the deposition rate. The dimensionless solution (6.166)
can be written with units as
l 0 − z t

T (z, t) = T 0 + (T bot − T 0 ) T ˆ , (6.302)
l 0 t 0
where l 0 is the depth to the base of the lithosphere and T bot is the temperature at the depth
2
l 0 . Time is scaled with characteristic time t 0 = l /κ and the Peclet number becomes Pe =
0
v/v 0 , where v 0 is the characteristic velocity v 0 = κ/l 0 . The solution (6.302)gives the
same results as Benfield’s solution as long as the amount of deposited sediments s = vt is
much less than the length scale l 0 . The dimensionless formulation (6.150) has a stationary
part controlled by the Pe-number, which tells us that Pe
1 gives negligible thermal
blanketing and that the opposite regime, Pe
1, may give strong thermal blanketing. The
regime in between, which gives moderate blanketing, is characterized by a deposition rate
2 −1
v 0 ∼ 250 m Ma −1 when κ = 10 −6 m s and l 0 = 125 km. The time it takes to reach a
steady state with the Fourier solution can be estimated by the half-life of the first term in the
2 2
2
series. Each term in the Fourier series goes to zero as exp(−k n t) where k n = n π +Pe /4
ˆ
for n ≥ 1. The first term, which dies out most slowly, has the half-life
ln2
ˆ t 1/2 = 1 2 . (6.303)
2
π + Pe
4
The condition for strong thermal blanketing is that Pe should be larger than 1 and that
ˆ t should go beyond ˆ t 1/2 . Furthermore, the time must not be so large that the amount of