Physics for Geologists, Second edition

Stress and strain 99
from which we obtain by integration (remembering that the boundary
conditions are that the sedimentary rock had an initial porosity, and that
the porosity cannot be reduced below zero),
The quantity b is called the scale length [L] (it is more convenient to divide
by a large number than to multiply by a small one), and fo is the porosity at
zero depth (which should be interpreted as the depth at which the sediment
accumulated into the stratigraphic record). The value of b can be obtained
from the data by solving Equation 9.17 for b. Equally, it is the depth at
which the porosity of normally compacted mudrock reaches the value fo/e
(or 0.368 fO). The value of fo is usually about 0.5, so b is usually about the
depth at which about 18 per cent porosity is achieved in normal compaction.
This equation is found to be a good summary of the compaction of
mudrocks, with scale lengths varying from about 500 m to 4 km. When the
scale length is large, the curve approaches a straight line, and it could be that
sandstone compaction, which appears to be linear with depth, follows the
same laws with a very large scale length.
Bulk density (pb) is related to porosity by
where pf and ps refer to the mean mass densities of the pore fluid and of the
solids.
Let us digress briefly to a practical application. In boreholes, it is much
easier to measure electric or acoustic properties of rocks than it is to mea-
sure their porosity, but we are often more interested in the porosity. We
are therefore concerned with the search for formulae relating one property
with another. What then is the relationship between mudrock porosity and
the sonic velocity, or its inverse, called the shale transit time (symbol Atsh)
[L-IT, inverse velocity] in mudrock?
The boundary conditions are that there will be some maximum value
of porosity, fo, when the mudrock first accumulates into the stratigraphic
record. This is usually close to 50 per cent. (Note that we are not concerned
with the superficial porosities of 70 per cent and more in mud that has yet
to accumulate into the stratigraphic record.) And the minimum porosity
at depth will be close to zero. There will be a corresponding transit time,
Ato, near the surface, and a limiting value of the transit time correspond-
ing to zero porosity. This latter is called the matrix transit time, At,,,,;,
or At,, .
When seeking a relationship between two quantities of different dimen-
sions, in this case L3/L3 and L-'T, it may be assumed that a dimensionless
form will be required, and that the dimensionless porosity, f/fo, will
be related to a dimensionless transit time involving Atsh, At0 and At,,.
The boundary conditions are that when f = fo, Atsh = Ato; and when
Copyright 2002 by Richard E. Chapman