where ct is the total stress. As fluid escapes, pore pressure decreases and effective stress
increases, causing porosity to decrease. For sediments that are normally compacting
(sediments at hydrostatic pore pressure), porosity has been observed to decrease
exponentially with depth, as suggested by Athy (1930)
n = noexp[-bz] (8)
where no is initial porosity, b is a constant [L-1], and z is burial depth [L]. For hydrostatic
conditions, the change in effective stress, oe, with depth is given by
(do-, / dz) = (po,- ,of) (l-n) g (9)
where p, is grain density. Combining Equations 8 and 9 results in
(dn / do ) = (-bn) / [(p, f) (1- n) g] (10)
Equation 10 is only applicable for sediments undergoing compaction. For this
investigation, it is assumed that sediments cannot decompact or expand when the
effective stress is reduced. Thus, porosities will remain constant unless the effective
stress exceeds the previous maximum effective stress value. The change in volume (dV)
can be related to (dn) by
dV/V= (dn) / (1 n) (11)
In one dimension, the volume change represents only the change in vertical
thickness and thus the horizontal dimension of the sediment column stays constant. As
porosity decreases with depth, matrix compressibility (a) is also reduced. Based on
Equations 10 and 11, the matrix compressibility is calculated as
a= (bn)/[(p, pf) (1-n)2 g] (12)