Page 71 - Petroleum Geology

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We take the relationship between porosity and sonic transit time in mud-
stones to be:

f= fo(At- 55)/(At0 - 55) z (At - 55)/220. (3.7b)
Equation 3.7b gives meaning to the empirical constants of Magara’s equa-
tions in the form f = mA t + n.
It must be pointed out here that error in the determination of At is not
symmetrical about the mean. In a perfect hole, its true value will be closely
determined by the sonic log, but any geometrical irregularities in the wall of
the borehole lengthen the travel path and so lead to a larger value of At than
the true one. Also, in shallow and in thick mudstones there may be a degree
of undercompaction near the middle, which also increases the transit time.
So we are concerned with the smallest values of At, and with values in thin
mudstones and near the top and bottom of thick mudstones for the deter-
mination of true normal compaction curves. Likewise, we are concerned
with the smaller values of porosity, which represent maximum compaction.
Combining eqs. 3.5a and 3.7a, we obtain:
f/fo = e--z/b = (At - At,,)/(Ato - At,,)
from which we write:

This equation is an improvement on eq. 3.6 in that it satisfies the boundary
conditions of A to at z = 0, and At,, when porosity is eliminated. It remains
to determine the value of the scale length b.
The scale length can be determined by two methods, the choice of which
depends on the nature of the data. First, by setting the dimensionless depth,
z/b, equal to unity and solving eq. 3.8, b is equal to the depth z at which At
is 95-96 ps/ft in normally compacted mudstones (z being measured from
ground level or seafloor). If there is no reliable data around 95-96 ps/ft,
take the deepest level at which At is thought to represent normal compaction
and solve eq. 3.8 for b:

b = z/ln {(At - 55)/(At0 - 55)j (3-9)
the units being the same as those of depth, and the transit time beinginps/ft.
We test this result using the data of Hottmann and Johnson (1965, p. 719,
fig. 2), which has hitherto been accepted as satisfactory evidence that a plot
of the logarithm of the transit time against depth is linear to depths of about
4 km in the US. Gulf Coast region. Inserting into eq. 3.8 the values At, = 165
ps/ft, At,, = 55 ps/ft, and z/b = 1, gives At = 95.5 ps/ft when b = z. This
value is reached at a depth of about 12,500 ft (3800 m) using the trend
of shorter transit times, indicating this value for the scale length. Alternatively,