For isotropic elastic
materials there are two bulk elastic wave speeds
(Ewing *et al.*, 1957; Aki and Richards, 1980), compressional
and shear .Here the Lamé parameters and are the constants that appear in Hooke's law relating stress to strain
in an isotropic material.
The constant gives the dependence of shear stress on shear
strain in the same direction. The constant gives the dependence of
compressional or tensional stress on extensional or dilatational
strains in orthogonal directions.
For a porous system with porosity (void volume fraction)
in the range , the overall density of the rock or sediment
is just the volume weighted density given by
,where , , are the densities of the
constituent solid, liquid and gas, respectively, and *S*
is the liquid saturation, *i.e.* the fraction of liquid-filled
void space in the range (Domenico, 1974).
When liquid and gas are distributed uniformly in all pores and cracks,
Gassmann's equations say that, for quasistatic isotropic elasticity and low
frequency wave propagation, the shear modulus will be mechanically
independent of the properties of any fluids present in the pores,
while the overall bulk modulus of the rock or
sediment including the fluid depends in a known way on porosity
and elastic properties of the fluid and dry rock or sediment
(Gassmann, 1951).
Thus, in the Gassmann model, the Lamé parameter
is elastically *dependent* on fluid properties,
while is not.
The density also depends on saturation.
At low liquid saturations, the fluid bulk modulus is dominated by
the gas, and therefore the effect of the liquid on is negligible
until full saturation is approached. This means that both seismic
velocities *v*_{p} and *v*_{s} will decrease with increasing fluid
saturation (Domenico, 1974; Wyllie *et al.*, 1956;
Wyllie *et al.*, 1958) due to the ``density effect,''
*i.e.*, the only quantity changing is the density which increases in the
denominators of both *v*_{p}^{2} and *v*_{s}^{2}. As full saturation is
approached, the shear velocity continues its downward trend, while the
compressional velocity suddenly (over a very narrow range of change of
saturation) shoots up to its full saturation value.
An example (Murphy, 1982; 1984) of this behavior is shown in Figure 1a.
This is the expected (ideal Gassmann)
behavior of porous rocks at low frequencies (sonic and below).

10/25/1999