Page 113 - Physical Principles of Sedimentary Basin Analysis

5.1 Porosity as a function of net sediment thickness 95

minerals by the pore fluid flow. The mass in a layer (or computational element) is therefore
conserved.
A challenge with burial history modeling is the reconstruction of the thicknesses of each
layer through the geohistory. This chapter looks at this particular problem.

5.1 Porosity as a function of net sediment thickness
We will show that a convenient way to deal with burial histories is to work with a
Lagrangian vertical coordinate that is constant for each layer boundary through the geohis-
tory. The natural choice for such a Lagrangian coordinate is the height up to a “grain” in
the sedimentary column measured as net (fully compacted) rock. This height, denoted ζ,is
a constant of each grain in the basin during deposition and compaction. See Section 3.17,
where this Lagrangian coordinate is explained further.
In order to make some simple exact burial histories we assume that porosity decreases
exponentially with the net (porosity-free) sediment depth as
(ζ − ζ)

∗
φ = (φ 0 − φ min ) exp − + φ min (5.1)
ζ 0
when expressed in the ζ-coordinate. The parameter φ 0 is the surface porosity, φ min is
∗
the minimum porosity found at great depth, ζ is the current height (or total thickness)
of the basin measured as porosity-free rock, and ζ 0 is a depth that characterizes the
decreasing porosity. The porosity φ 0 − φ min is seen to be reduced to half at the ζ-depth
∗
ζ − ζ = ln2 ζ 0 ≈ 0.69 ζ 0 . The porosity function (5.1) is a modification of Athy’s porosity
function (Athy, 1930), who fitted porosity observations to the real sediment depth with an
exponential function. (See Section 2.1.)
The real depth of the sediments from the basin surface is obtained from the ζ-coordinate
and the porosity function (5.1) using equation (3.153), which is recaptured here:
ζ ∗
dζ

z = . (5.2)
ζ 1 − φ(ζ)
When the porosity function (5.1) is inserted into the integral (5.2), and the integration is
carried out, we get
1
1 − φ
∗
z = (ζ − ζ) + ζ 0 ln (5.3)
1 − φ min 1 − φ 0
as shown in Exercise 5.1.The z-coordinate is as expected only a function of the ζ-depth
∗
from the surface, ζ −ζ, because the porosity function, φ, is only a function of the ζ-depth.
We have now both the z-coordinate (5.3) and the porosity (5.1) as explicit functions of the
ζ-coordinate. It is then possible to plot the porosity as a function of depth as shown in
Figure 5.1. The figure also shows some other porosity–depth trends.
We have seen that each “grain” of the sediments is uniquely identified with a
ζ-coordinate. Equation (5.3) can therefore be used to trace the real depth (relative to the