Physics for Geologists, Second edition

Some dangers of mathematical statistics 135
Figure 13.3 The same data plotted logarithm vs logarithm. When the data of
Table 13.1 are plotted as logarithms, the slope is evidently 2 or 0.5
(depending on which is dependent variable).
finding a formula relating the two? As mentioned earlier, an association
between a velocity, with dimensions LT-I and porosity, which is dimen-
sionless, must be in terms of a dimensionless velocity - in this case, the ratio
of the measured velocity to the acoustic velocity when the porosity is zero.
The boundary conditions are that when f = 0, V = Vo; and there is some
maximum value of f, the value obtaining when the sediment accumulated
into the stratigraphic record. This we labelled fo on page 99.
If you feed your data into a computer, and tell it to compute the lin-
ear correlation coefficients, and so on, it will come up with a formula that
appears quite satisfactory, with very small errors between prediction and
measurement. Try plotting the solids proportion, 1 - f, against the ratio of
the velocity observed to the velocity in the matrix, V/Vo -no, the logarithms
of those quantities - anticipating an equation of the dimensionless form
where Vo is the acoustic velocity when the porosity is zero. Linear regression
of In (1 -f) and In (V/Vo) or log (1 -f) and log (V/Vo) may suggest the values
of b and m - but the warnings above apply here too: any gentle curve may
appear to be linear with statistical significance.
What about the fluid in the pores? The short answer is that the acoustic
velocity in quartz is about four times that in liquids, so if the wavelength is
very short, the path of first arrival will be through pore fluid only over very
short sections close to the points of contact between grains in uncemented
sandstones. And because the path will be in general far from tangential to
the grain, very little of it will be through the pore fluid. If the sandstone is
cemented, virtually none of the path will be through fluid.
Table 13.2 gives 13 measured values of the solid proportion, 1 - f, and
the dimensionless velocity, V/Vo, in sandstone assuming a matrix velocity,
Copyright 2002 by Richard E. Chapman