Gassmann's equation (Gassmann, 1951) for fluid substitution states that
K = K_dr + ^2(-)/K_m + /K_f, where Km is the bulk modulus of the solid mineral, Kdr 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 Sl, Kl is the bulk modulus of the liquid, and Kg is the bulk modulus of the gas. When Sl is small, (Kf) shows that ,since . As , Kf remains close to Kg until Sl closely approaches unity. Then, Kf changes rapidly (over a small range of saturations) from Kg to Kl.
The bulk modulus Kf contains the only dependence on Sl 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 Sl 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(Sl=0) = Kdr and K(Sl=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 Sl = 0) value to the fully saturated value (Sl = 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.