Gassmann's equation (Gassmann, 1951) for fluid substitution states that

K = K_dr + ^2(-)/K_m + /K_f,
where *K*_{m} is the bulk modulus of the solid mineral,
*K*_{dr} is the bulk modulus of the drained porous frame,
is the Biot-Willis (1957) parameter,
is the porosity, and *K* is the effective bulk modulus
of the undrained fluid-mixture-saturated porous medium,
where, for partial saturation conditions with homogeneous mixing of
liquid and gas,

1/K_f = S_l/K_l + (1-S_l)/K_g.
The saturation level of liquid is *S*_{l}, *K*_{l} is the bulk modulus
of the liquid, and *K*_{g} is the bulk modulus of the gas.
When *S*_{l} is small, (Kf) shows that ,since . As , *K*_{f} remains close to
*K*_{g} until *S*_{l} closely approaches unity. Then, *K*_{f} changes
rapidly (over a small range of saturations) from *K*_{g} to *K*_{l}.

The bulk modulus *K*_{f} contains the only dependence on *S*_{l}
in (KGassmann).
Thus, for porous materials satisfying
Gassmann's homogeneous fluid conditions and for low enough frequencies,
the theory predicts that, if we use seismic velocity data
in a two-dimensional plot with one axis being the saturation *S*_{l}
and the other being the ratio ,then the results will lie along a straight (horizontal) line until
the saturation reaches (around 95% or higher),
where the curve formed by the data will quickly rise to
the value determined by the velocities at full liquid saturation.

On the other hand, if the porous medium contains gas and liquid
mixed in a heterogeneous manner, so that patches of the medium
hold only gas while other patches hold only liquid, then the theory
predicts that, depending to some extent on the spatial distribution
of the patches, the results will deviate from Gassmann's results.
If we consider the most extreme cases of spatial distribution possible,
which are laminated regions of alternating liquid saturation and
gas saturation, then the effective bulk modulus at low frequencies
will be determined by an average of the two extreme values
of (KGassmann): *K*(*S*_{l}=0) = *K*_{dr} and *K*(*S*_{l}=1).
Using saturation as the weighting factor, the harmonic mean and the mean are
the well-known results for these two extremes of behavior.
Of these two, the one that differs most
from (KGassmann) is the mean. But, on our plot,
the results for the mean will again lie along a straight line,
So this time the line goes directly from the
dry (or unsaturated *S*_{l} = 0) value to the fully saturated value (*S*_{l} = 1).
The two straight lines
described are rigorous results of the theory, and form two sides of
a triangle that will contain all data for partially saturated systems,
regardless of the type of saturation present.

10/25/1999