Since the possible linear combinations of the elastic bulk and shear
moduli (*K* and ) are infinite, it is natural
to ask why (or if) the choice
is special? Is there
perhaps some other combination of these constants that works as well or even
better than the choice made here?
There are some rather esoteric reasons based on recent work
(Berryman *et al.*, 1999) in the
analysis of layered anisotropic elastic media that lead us to believe that the
choice is indeed special, but we will not try to describe
these reasons here.
Instead we will point out some general features of the two types of
plots that make it clear that this choice is generally good, even
though others might be equally good or even better in special
circumstances.
First, in the diagram using the (, )-plane,
it is easy to
see that *any* plot of data using linear combinations of the form
(, ), where *c* is any real constant,
will have precisely the same information
and the display will be identical except for a translation of the
values along the ordinate by the constant value *c*. Thus, for
example taking , plots of (, )will have exactly the same interpretational value as those presented
here.
But, if we now reconsider the data-sorting plot (*e.g.*, Figure 2)
for each of these choices,
we need to analyze plots of the form (,). Is there an optimum choice of the parameter
*c* that makes the plots as straight as possible whenever the only
variable is the fluid saturation? It is not hard to see that the
class of best choices always lies in the middle of the range of
values of taken by the data. So setting
will always guarantee that there are very large positive and negative
values of , and therefore that these data fall
reliably (if somewhat approximately) along a straight line.
But the minimum value of has an absolute minimum
of , based on the physical requirement of positivity of *K*.
So is a physical requirement, and since
is a fairly typical value for
porous rocks, it is expected that an optimum value of will
generally be obtained using this criterion. Thus, plots based on
bulk modulus *K* instead of will not be as effective
in producing the quasi-orthogonality of porosity and saturation that we
have obtained in the data-sorting style of plotting. We conclude that
the choice is not unique (some other choices might be as
good for special data sets), but it is nevertheless an especially
simple choice and is also expected to be quite good for most real
data.

4/28/2000