An important anisotropy parameter for qP-waves is Thomsen's parameter
, defined by
= a-c2c. Formula (avea) for a may be rewritten as
a = <(+2)^2 - ^2+2> + <+2>^2 <1+2>^-1, which can be rearranged into the convenient and illuminating form
a = <+2> -
[<^2+2>
<1+2>
-<+2>^2]
<1+2>^-1.
This formula is very instructive because the term in square brackets
is again in Cauchy-Schwartz form, so this
factor is nonnegative. Furthermore, the magnitude of this term
depends principally on the fluctuations in the Lamé
constant, largely independent of
. Clearly, if
,then this factor vanishes identically, regardless of the behavior of
. Large fluctuations in
will tend to make this term large.
If in addition we now consider the combination
2=
[<+2><1+2> -
1]
-[<^2+2><1+2>
-<+2>^2],
we find that the first bracket on the right hand side is again in
Cauchy-Schwartz form, which makes it always positive
unless , in which case it vanishes.
Similarly, the second term always makes a negative contribution
unless
, in which case it vanishes.
The properties of the isotropic layers
making up the anisotropic rocks depend on mineralogy, porosity,
and fluid content.
We can gain some insight into how the relative sizes of
and
are related to mineralogy by considering the
measured values of these stiffnesses for quartz and calcite.
Simmons and Brace (1965) report measured values of
about 28 to 31 GPa for the shear modulus of fused quartz,
and measured values of about .027 to .032 GPa-1 for the
compressibility, corresponding to values of about 13 to 16 GPa
for
. (These measurements were made for confining pressures
of 0.1 MPa to 10 GPa.) Wilkens et al. (1984) gave ultrasonic
velocity values measured in single crystals of quartz and
calcite that yield values of about 44 GPa for the shear modulus
of quartz, 8.4 GPa for
for quartz, 31 GPa for the shear
modulus of calcite, and 55 GPa for
for calcite. (These
measurements were made without accounting for the anisotropy
in quartz and calcite crystals. See Aleksandrov and Ryzhova (1961)
for measured values of all the elastic stiffnesses, for
single calcite crystals and quartz crystals.) Both the static
and the dynamic measurements show that
values in quartz and
calcite are similar, whereas the
values differ considerably
for the two minerals, with
being larger than
for
calcite and smaller than
for quartz. We also know that
vp/vs ratios in sandstones are close to 1.5, and vp/vs ratios
are about 1.8 to 2.0 for sedimentary rocks containing calcite,
dolomite, or feldspar (e.g., Wilkens et al., 1984;
Castagna et al., 1985).
If
is approximately equal to
, the vp/vs ratio is 1.7.
Rocks having
larger than
have vp/vs ratios larger than
1.7, and rocks having
smaller than
have vp/vs ratios
smaller than 1.7.
Although the stiffnesses of clay
minerals can be found in the literature (e.g., Katahara, 1996),
it is more useful to consider instead the velocities of rocks
containing clays (e.g., Tosaya and Nur, 1982; Han et al., 1986)
because the clays themselves typically contain micropores
and the location of the clays in the microstructure (e.g.,
lining pores, or between grains) is more important than
mineralogy, for elastic properties (Caruso et al., 1985;
Wilkens et al., 1986).
Porosity does not directly affect and
values of the
grains, but since it affects the velocities, it affects the vp/vs
ratio. Clay content and the presence of pores both tend to
increase vp/vs ratios
(Castagna et al., 1985). If randomly-oriented flat cracks are
present, they will also tend to increase the vp/vs ratio,
since vs is more sensitive than vp to flat cracks
(Wilkens et al., 1984).
Fluctuations in in the earth have important implications
for oil and gas exploration. Gassmann's well-known results
(Gassmann, 1951) show that, when isotropic porous elastic media are saturated
with any fluid, the fluid has no mechanical effect on the shear
modulus
, but can have a significant effect on the bulk modulus
, and therefore on
. Thus, observed variations
in
have no direct information about fluid content, while
observed variations in
, especially if they are large variations,
may contain imporant clues about variations in fluid content.
So the observed structure of
in (Pwavep)
strongly suggests that small positive and all negative values of
may be important indicators of significant fluctuations
in fluid content.
If the finely layered medium is composed of only two distinct types of isotropic elastic materials and they appear in the layering sequence with equal spatial frequency, then we find that
2=
(_2-_1)(_2-_1) + (_2-_1)
(_1+2_1)(_2+2_2).
This result agrees with Postma (1955) except for an obvious
typographical error in the denominator of his published formula.
This formula shows clearly that if then Thomsen
parameter
is identically equal to zero as expected.
Also, if
but
, then (Postma)
implies
, as we inferred from (Pwavep).
Now, we can use this formula to deduce the smallest possible value of
the right hand side of (Postma). The shear moduli must not be
equal, so without loss of generality we suppose that .Then, the numerator is seen to become negative by taking
towards negative values and
. The smallest value
can take is determined by the bulk modulus bound
. So we may set
in both the numerator and denominator. This choice also makes the
factor
as small as possible
in the denominator, thus helping to magnify the effect of the
negative numerator as much as possible. The result so far is that
2= 34(_2-_1_2)
(-_1+_2/3-_1_1+2_1).
The parameter may vary from
to plus
infinity.
At
, the second expression in parentheses is
positive; but, this expression is also a monotonically decreasing
function of
and approaches -1 as
.Thus, the smallest value of Thomsen's parameter
is given by
= - 38_2-_1_2
-38.
This result differs by a factor of 2 from the corresponding result of
Postma (1955), which was obtained improperly by allowing
three of the four elastic constants to vanish and also using a
physically motivated but unnecessary restriction that both
and
must be nonnegative.
If we had used the nonnegativity constraint on the
's, the
present result would have changed to
= 1 - _2-_12_2 -14, which is the same inequality as that found by Postma, but his equality differed from that in (restricted) and was in fact improperly obtained.
As a final point about the formula (arearranged), we note that it implies in general that
a <+2>,
so we have an upper bound on in finely layered media
stating that
2 (<+2><1+2>-1) = (<v_p^2><v_p^-2> - 1), where the final form is true for layered media having constant density.
Before concluding this section, we want to note one further
identity for .The general formula can be rearranged to give
=
2[<(+)+2>
<1+2>
-<++2>
<+2>].
This formula is not in Cauchy-Schwartz form, but is nevertheless
probably the simplest form of the result for this anisotropy parameter.
In particular, it is easy to see from this form that if
either or
, then
the right hand side vanishes identically. The first result is
well-known and the second has been known since Postma's (1955) work
to be true for two-constituent periodic layered media
[also see (Postma)].
The present result generalizes Postma's observation in this case.