Page 109 - Physical Principles of Sedimentary Basin Analysis

4.6 Gravitational compaction of a hydrostatic clay layer 91

the effective stress is then given by equation (4.63). The vertical effective stress under
hydrostatic conditions is given by

z

σ = ( − f )g(1 − φ) dz (4.67)
0
where the z-axis is pointing downwards and the surface of the layer is z = 0. The inte-
gral (4.67) is not trivial to carry out in the z-coordinate, because the porosity is not yet
known as a function of z. The trick is now to switch to the Lagrangian coordinate defined
by the position in the layer measured as net (porosity-free) rock above the base of the layer.
This is the so-called ζ-coordinate, and we have that dζ = (1 − φ)dz.(The ζ-coordinate
was introduced in Section 3.17.) The vertical effective stress under hydrostatic conditions
is therefore
∗

σ = ( − f )g(ζ − ζ) (4.68)
where ζ ∗ is the ζ-coordinate for the top of the layer. We then get that the normal
consolidation line (4.63)asafunctionofthe ζ-coordinate is
∗
ζ − ζ
e = e 0 − C v ln (4.69)
ζ 0

where ζ 0 = σ /(( − f )g) is the reference ζ-thickness that corresponds to σ . We would

0 0
like to have the void ratio as a function of z rather than ζ, and a step in that direction is to
have the z-coordinate as a function of ζ too. The z-coordinate is given by equation (3.153):
∗
ζ ∗ dζ ζ
z = = 1 + e(ζ) dζ. (4.70)
ζ 1 − φ(ζ) ζ
This integration is straightforward to carry out and the z-coordinate as a function of the
ζ-coordinate is

∗

ζ − ζ
∗
z = (ζ − ζ) 1 + e 0 + C v − C v ln . (4.71)
ζ 0
Notice that both the void ratio (4.69) and the z-coordinate (4.71) are only functions of the
∗
ζ-depth from the surface ζ − ζ. The void ratio can now be plotted as a function of the
z-coordinate by combining equations (4.69) and (4.71), since both equations are parame-
terized by the ζ-coordinate. Figure 4.3 shows the porosity of a clay layer as a function of
4

depth (z-coordinate) when C v = 0.2, e 0 = 1.5, σ = 1 kPa and ( − f )g = 10 Pa/m.
0
With these data we get that ζ 0 = 1m.

Exercise 4.6 Derive equation (4.71). Hint: ln x =−x + x ln x.
Exercise 4.7 Equation (4.69) for the normal consolidation line and equation (4.71)for the
z-coordinate can be extended with a surface load σ L . The equation for the void ratio is then
ζ − ζ σ L
∗
e = e 0 − C v ln + (4.72)
ζ 0 σ
0