Page 115 - Physical Principles of Sedimentary Basin Analysis

5.2 Pre-calibration of burial history calculations 97

ζ-coordinates. The parameters in the porosity function (5.1)are φ 0 = 0.5, φ min = 0.03
and ζ 0 = 1350 m. The horizons in plot 5.2a are separated in space by 200 m net sediments,
and show a burial history where deposition of net (porosity-free) sediment is at a constant
rate of 3700 m over 250 Ma. The total amount of net rock as a function of time is therefore
∗
ζ = ωt, where ω = 14.8m Ma −1 . Such simple burial histories, with deposition of net
sediment at a constant rate, are ideal to study the basic behavior of processes related to
deposition, like for instance overpressure build-up. The plot 5.2b has horizons that are
separated by 10 Ma in time, and shows a burial history where the deposition rates are
piecewise constant in intervals of 50 Ma.
Burial histories shown in Figure 5.2 could be generalized by allowing for a separate
porosity function for each layer. The solution (5.3) then has to be modified to an expression
for a layer thickness. The real depths down to the horizons are obtained starting from the
basin surface by computing the real layer thicknesses, layer by layer. Such an algorithm is
straightforward to implement in a computer program.
Exercise 5.1 Show that the integral (5.2) becomes the function (5.3) when the porosity
function (5.1) is inserted.
Solution: A change of integration variable from ζ to u = φ gives du = (u−φ min )(dζ/ζ 0 ),
and the integral (5.2) becomes

du
φ 0
z = . (5.4)
φ (1 − u)(u − φ min )
The function inside the integral is rewritten as
1 a b
= + (5.5)
(1 − u)(u − φ min ) 1 − u u − φ min
where a = b = 1/(1 − φ min ). It is now straightforward to do the integration and the result
is equation (5.3).
Exercise 5.2 Show that equation (5.3)for the z-coordinate can be rewritten as the
following expression for the layer thickness:
1
1 − φ 1
dz = dζ + ζ 0 ln (5.6)
1 − φ min 1 − φ 2
where dz = z 2 − z 1 , dζ = ζ 2 − ζ 1 and where the subscripts 1 and 2 denote the horizon
boundaries of the layer.

5.2 Pre-calibration of burial history calculations
It was shown in the previous section how simple burial histories could be made from
knowledge of the net (porosity-free) thickness of rock in each layer, by assuming that the
porosity is a function of net (porosity-free) depth. But we did not say anything about how
one could obtain the net layer thicknesses from real thicknesses and the porosity function.