For isotropic elastic materials there are two bulk elastic wave speeds (Aki and Richards, 1980), compressional and shear .Here is the overall density, and 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

= (1-)_s + [S_l + (1-S)_g],
where , , are the densities of the
constituent solid, liquid and gas, respectively. *S*
is the liquid saturation, *i.e.*, the fraction of liquid-filled
void space in the range [see 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 *K* ()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; Berryman, 1999).
Thus, in the Gassmann model, the Lamé parameter
is elastically *dependent* on fluid properties,
while is not.
The density also depends on saturation, as shown in
equation (rho).
At low liquid saturations, the bulk modulus of the fluid mixture
is dominated by the gas, and therefore the effect of the liquid
on is negligible
until the porous medium approaches full saturation. This means that both
velocities *v*_{p} and *v*_{s} will decrease with increasing fluid
saturation (Domenico, 1974) due to the ``density effect,''
wherein the only quantity changing is the density, which increases in the
denominators of both *v*_{p}^{2} and *v*_{s}^{2}. As the medium approaches
full saturation, the shear velocity continues its downward trend, while the
compressional velocity suddenly (over a very narrow range of
saturation values) shoots up to its full saturation value.
A well-known example of this behavior was provided by Murphy (1984).
Figure 1 shows how plots of these data for sandstones
will appear in several choices of display, with Figure 1(a)
being one of the more common choices.
This is the expected (ideal Gassmann-Domenico)
behavior of partially saturated porous media.
The Gassmann-Domenico relations hold for frequencies low enough (sonic
and below) that the solid frame and fluid will move in phase,
in response to applied stress or displacement. The fluid pressure
must be (at least approximately) uniform throughout the porous medium,
from which assumption follows the homogeneous saturation requirement.

4/28/2000