Hydraulic Conductivity Procedure

The general procedure is as follows:

1. Drill a bore hole.
2. Do a down hole survey with a source that generates SH-waves.
3. Invert body wave dispersion and decay with CAINV3 8.2.7. This will provide stiffness, C1, and damping, C2 values under a Kelvin-Voigt model (Equation 4).
4. Select a relevant frequency and porosity and compute Kelvin-Voigt damping ratios using KD4kvmb 8.3.1. The coupled damping ratio is the most likely one. The program will also return the corresponding hydraulic conductivities, $KD$ in $m/s$.

   Note this solution should agree with figure like Figure 34 when that figure is computed for the same relevant frequency.

Figure 36: Prompt for input in KD4kvmb.m run

Why use the Kelvin-Voigt (KV) constitutive representation for a soil? The problem with the KV model is that frame and fluid masses are lumped together as one. The KVMB representation frees the two masses to move which leads to an estimate of permeability. Engineering practice has been to use the KV representation, as in resonant column analysis. The CAINV3 joint inversion of wave dispersion and decay to values of stiffness and damping follows that same KV practice. However, the KVMB representation can also be used to map up to two (coupled and uncoupled) cases of equivalent KV damping. It is also possible that there will be only one result of KV damping if one is at the peak of the curve. Consider drawing a horizontal line to intersect a curve like those in Figure 34.

An example of running KD4kvmb is shown to illustrate the final step of the procedure 8.3.1.1. The results of the run shown in Figure 36 are:

SOLUTION (+/- 95 Percent Confidence)
Freq=12(Hz) Resonator_L=1.33(m)
Damping Ratios: Peak=0.030293 Wave=0.018850 (+/-0.01450)
Coupled (b_case): DR=0.018850 KD=0.01224(+/-0.0117m/s)
UnCoupled (a_case): DR=0.018849 KD=0.09169(+/-0.0881m/s)
Porosity: 0.250 (+/-0.038)
Relaxation Time Tr=C2/C1=0.50 msec

The notation is as follows:

• Peak damping ratio (DR) is the theoretical maximum for the case at hand.
• Wave damping ratio is the result of the CAINV3 joint inversion of a down-hole SH-wave survey.
• KD is the estimate of hydraulic conductivity ($m/s$). There are two of these unless the wave equals the peak damping ratio.
• Relaxation time is in milliseconds. It is analogous to the time it takes a sponge to recover its shape when squeezed under water and then released. The more permeable the sponge, the quicker the water can enter the sponges pores.