Hydraulic Conductivity from Seismic Damping

The use of the Kelvin-Voigt (KV) representation has long been a standard in geotechnical engineering and soil dynamics. In the vibrator perspective, it consists of a mass, spring, and dashpot with the spring and dashpot in parallel configuration. Figure 65-A shows both the vibrator and wave perspectives of the KV model. The KV model is limited by the single mass, making it unable to represent multi-phase media like a water saturated sand. In the case of a water saturated sand, there are two masses (solid frame, pore fluid).

Figure 65: (A). Kelvin-Voigt (KV) representation for both vibrator and wave assemblage. (B) Kelvin-Voigt-Maxwell-Biot (KVMB) representation.
\includegraphics[scale=0.7]{FigurePP}

Building on the work of others, it became apparent that the ideas of Maxwell and Biot could be combined to produce an alternative model with two masses that would produce vibratory behavior close to from the KV model (Michaels (13)). This alternative representation is the Kelvin-Voigt-Maxwell-Biot (KVMB) model shown in 65-B. The two masses represent the solid frame and the fluid (typically water, but could be any fluid with a viscosity). The dashpot can be represented by a collection of conduits for the fluid to move relative to the solid frame. While the model based on parallel tubular pores within a solid mass is unlikely to be a true image of a water saturated soil, it is capable of capturing the behavior of real granular water saturated media when excited by seismic forces.

Some questions can not be addressed by the KV model. Consider, for example, this question:

Should seismic damping increase or decrease with an increase in hydraulic conductivity?
The problem is that there is no place for permeability in the KV model. It turns out that neither is likely to be the only answer. Once one introduces a mechanism to capture permeability or hydraulic conductivity for a specific fluid into the model, the question can be addressed. The KVMB model, on the other hand, introduces permeability and fluid viscosity through the dashpot placed between the two masses. The result is the ability to capture both coupled and uncoupled possible motions. This results in behavior that answers the question with both alternatives being true.

The dashpot provides viscous friction. Friction can be low when the soil is very tight (low permeability) and the motion of fluid and frame are largely coupled. That is, they move together. Low levels of friction are also possible in a highly permeable soil, one with large pore spaces that permit easy flow between frame and fluid. This is the uncoupled case. The KVMB model predicts that there will be an intermediate place between these two extremes where friction is greatest, and this produces a peaked response.

The metric chosen to represent viscous friction is the KV damping ratio. This is possible even though that representation can not explain the friction as the KVMB model does. There is a mapping between the two representations because the KVMB model predicts behavior very close to that of a true KV vibrator. The mathematics are developed in Michaels (13) and involve dropping the real eigenvalue of the KVMB system and relating the two complex KVMB eigenvalues to the only two eigenvalues of the KV system.

With this mapping, the body of KV based measurements can be related hydraulic conductivity. Be these measurements from vibratory experiments (like resonant column) or from shear wave propagation of dispersion and propagation decay, the mapping provides a way to connect them to permeability of the medium (once the fluid viscosity and porosity are known). Knowledge of porosity is required since it splits the mass into the two components, fluid and frame.



Subsections