Biot's sigmoid velocity versus frequency curves

 

Fig2.7
Figure 9 Calculated velocity versus frequency for sand, silt, and clay. For a sand of 40% porosity and 10 Darcy permeability, critical frequency is in the sonic-ultrasonic range (10-100 kHz). For a silt of 60% porosity and 10 mD permeability, critical frequency is in the high ultrasonic frequency range (10 MHz). In clay with 80% porosity and permeability of the order of 0.5 mD, critical frequency is in the GHz range. Figure and caption taken from Marion 1990, Figure 2.7.

Figure (9) displays some theoretical velocity-frequency curves for different materials (sand, silt, and clay). The shape of these curves is sigmoid: the velocity is constant for low frequencies until it reaches a transitional region where it strongly increases to reach eventually another constant-velocity level corresponding to the high frequencies. As explained in the section on diffracted waves, using diffraction theory we can predict the low and high levels of velocity for low and high frequencies. Moreover, in the preceding section we arrived at a condition that locates the limit between low frequencies and high frequencies. If we apply this condition to the three curves by picking the frequency at which the sigmoid pattern occurs, we can estimate the radius of the grains. For the clay, the estimate of the radius is 0.1 $\mu$m. The silt and the sand give us approximately 10 $\mu$m and 2 mm, respectively. The silt and clay samples can be considered as suspensions because of their high porosity (60% and 80%). Therefore, the estimation of the grain size gives some realistic results. Unfortunately, since the sand with 40 percent porosity cannot be considered a suspension, condition (8) is no longer valid. However, the condition gives interesting predictions of the grain size.