Page 117 - Physical Principles of Sedimentary Basin Analysis

5.3 Porosity as a function of z 99

The total ζ-thickness of the column is
N−1

∗
ζ = ζ k = ζ i + η i (5.11)
k=1
for every i from 1 to N (see Figure 5.3). The net depths are found in an iterative manner,
starting from the basin surface, where η N = 0. Let us say that η i+1 is found and that we
want to compute the next net thickness ζ i in order to obtain η i = η i+1 + ζ i . This com-
putation can be done by approximating the average porosity of the layer with the porosity
at the center of the layer,

φ i = φ(η i+1 + ζ i /2). (5.12)
¯
We then have

ζ i = 1 − ¯ φ z i = 1 − φ i (η i+1 + ζ i /2) z i (5.13)
which is an equation for the net layer thickness ζ i . This equation is solved numerically
by simple iteration, for nearly any porosity function. Each evaluation of the right-hand
side yields a ζ i that can be inserted back into the right-hand side to obtain an improved
ζ i . This scheme can be initialized with the real layer thickness, and only a few iterations
are needed to obtain an accurate numerical solution. Once all net thicknesses are found
from the surface to the basement, we also have the ζ-coordinates of each horizon. A burial
history simulation can then be done where the final thicknesses will match today’s real
thicknesses. This procedure works with any porosity function as long as it is a function of
the net sediment depth, and the algorithm also remains unchanged if each layer has its own
porosity function. We return in Section 5.4 to calibration of burial histories with erosion
processes.

5.3 Porosity as a function of z

Porosity as a function of z, rather than the ζ-depth, simplifies the process of obtaining the
net amount of porosity in each layer from today’s real thicknesses. We simply get

ζ i = 1 − φ(¯z − z N ) z i , (5.14)
where ¯z = (z i +z i+1 )/2 is the present-day depth to the center of the layer, and where z N is
the present-day position of the basin surface. (The porosity function takes as argument the
real depth relative to the basin surface.) We want the net thickness of each layer, because
they are constant through the burial history. The computation of the real layer thicknesses
from the ζ-thicknesses, at any time in the geohistory, is done with an iterative algorithm
quite similar to the one in the previous section. We start at the basin surface z N , and then
proceed downwards layer by layer. When the depth z i+1 is found, the task is to find the