Page 216 - Physical Principles of Sedimentary Basin Analysis

198 Subsidence

Exercise 7.2
(a) Equation (7.3) gives the water depth that corresponds to the sediment thickness S.Find
an expression for the uncertainty w in the water depth when there is an uncertainty in
the average (bulk) sediment density s . Express the uncertainty as a relative uncertainty
w/w.
(b) Calculate the relative uncertainty for s = 250 kg m −3 , when s = 2300 kg m −3 and
a = 3300 kg m −3 .
Solution: (a) The water depth becomes w = S ( a − s )/( a − w ) when a sedimentary
basin of thickness S and density s is replaced by water. The uncertainty w caused by
the uncertainty s is then w =−S s /( a − w ), and the relative uncertainty is
w/w =− s /( a − s ), which is independent of the sediment thickness.
(b) The numbers above for the densities give that w/w =−0.25. In other words, if the
average bulk sediment density s = 250 kg m −3 is too high then it implies that the water
depth is 25% too low.
Exercise 7.3 The bulk sediment density is b = φ w + (1 − φ) s , where w is the water
density, s is the sediment matrix density and φ the porosity. Assume that the Athy func-
tion, φ = φ 0 exp(−z/z 0 ), gives the basin porosity as a function of depth, where φ 0 is the
porosity at the basin surface and z 0 is a depth that characterizes the compaction. The bulk
s
density averaged over the basin is ¯ b = (1/s) b dz, where the basin surface is at z = 0
0
and the s is the thickness of the basin.
(a) Show that an uncertainty z 0 in the parameter z 0 implies an uncertainty ¯ b =
−( s − w ) ¯ φ z 0 /z 0 in the basin average of the sediment density, where ¯ φ is the basin
average of the porosity.
(b) Use the result of Exercise 7.2 to compute relative uncertainty in the water depth that
corresponds to the sediments when z 0 /z 0 = 0.2, ¯ φ = 0.15, s = 2300 kg m −3 .

7.2 Thickness of crustal roots

Isostasy can be used to obtain the depth of the crustal roots of the continents. We already
know that the continents are made of thicker, older and also less dense crustal rocks than
the oceanic crust. It is now assumed for simplicity that the continental and the oceanic crust
have the same density c . The oceanic crust has the thickness c and it is covered by water
of depth w, see Figure 7.4. The principle of isostasy requires that the pressures at the same
depth in the mantle beneath the oceanic crust and the continental crust are the same. We
therefore have the following equality:

w w + c c + d m = (d + c + w + h) c (7.12)
where the left-hand side is the pressure in the mantle beneath the oceanic crust and the
right-hand side is the pressure beneath the continent. The height of the continent above the
sea level is h and the depth of the crustal root below the base of the oceanic crust is d (see
Figure 7.4). Rearranging equation (7.12)gives