Page 215 - Physical Principles of Sedimentary Basin Analysis

7.1 Isostatic subsidence 197

Pratt isostasy is less used than Airy isostasy, probably because of the assumption of a
crustal base at the same depth regardless of elevation. There are few observations that
support such an assumption. On the other hand, Airy isostasy can easily be extended with
lateral variations in density.
Exercise 7.1 Basins compact even if there is no sediment infill, because of porosity reduc-
tion from chemical processes. Let a sedimentary basin have the initial thickness S 0 ,the
initial average porosity φ 0 , and the initial water depth w 0 . The average basin density is
= φ w +(1−φ) s , where w is the water density and s is the sediment matrix density.
(a) Assume Airy isostasy and find an expression for the water depth as a function of the
average basin porosity. What is the maximum water depth when the average basin porosity
is zero?
(b) Find the total compaction of the basin as a function of the average basin porosity.
Solution: (a) The Airy isostasy gives that pressure in the ductile mantle
(7.6)
w 0 w + S 0 0 + a 0 m = w w + S + a 1 m
is the same at the same depth:
w 0 + S 0 + a 0 = w + S + a 1 . (7.7)
The basin has the thickness S and the water depth is w when the porosity is φ. The mantle
thickness before and after the change of porosity is a and a 1 , respectively. Elimination of
a and a 1 gives
S − S 0 0 − (S − S 0 ) m
w = w 0 + . (7.8)
m − w
The weight of the basin becomes

ζ 0
S = φ w + (1 − φ) s = (e w + s )ζ 0 (7.9)
1 − φ
where ζ 0 = (1 − φ 0 )S 0 is the net (porosity-free) thickness of the basin, and where e =
φ/(1 − φ) is the void ratio. The same calculation gives that the initial weight of the basin
is S 0 0 = (e 0 w + s )ζ 0 . The basin thickness is S = (1 + e)ζ 0 in terms of the void ratio.
When these are inserted into equation (7.8) we get
w = w 0 + (e 0 − e)ζ 0 . (7.10)
The water depth grows linearly with decreasing void ratio. The maximum increase in water
depth is therefore w = e 0 ζ 0 . Expression (7.10) for the water depth is reasonable because
eζ 0 measures the amount of water in the sedimentary basin with void ratio e. The difference
(e 0 − e)ζ 0 is therefore the amount of water lost from the basin when the void ratio goes
from e to e 0 .
(b) The increase in water depth from compaction of the basin is
w = w 0 + S 0 − S = w 0 + (e 0 − e)ζ 0 (7.11)
using that S 0 = (1 + e 0 )ζ 0 and S = (1 + e)ζ 0 .