Revisiting Axis Translation for Unsaturated Soil Testing

Hilf’s (1956) axis translation (AT) (also called pressure plates) is arguably the most widely used technique for controlling matric suction in laboratory unsaturated soil testing. Two assumptions are inherently involved in the AT: cavitation is not important in soil water retention (SWR), and matric suction ${\psi }_{\mathrm{m}}$ controlling a soil’s water content is defined as the difference between pore air pressure $u_{\mathrm{a}}$ and pore water pressure $u_{\mathrm{w}}$, i.e. [Fig. 1(a)]

A more general definition should include both capillarity and adsorption (Lu and Zhang 2019) [see Fig. 1(b) for illustration]where $x$ = statistical distance to a particle surface; and ${\psi }_{\mathrm{ads}}(w)$ = soil sorptive potential (SSP) including electrical double layer, van der Waals, and hydration (Lu and Zhang 2019).

In nature, pore air pressure is controlled by the atmospheric pressure with a mean of 101.3 kPa at sea level. Pore water pressure can be higher or lower than pore air pressure, but cannot be lower than the cavitation pressure of water. The cavitation (saturated vapor) pressure at 25°C (298.2 K) is 3.2 kPa. Thus, capillary pressure ($=-{\psi }_{\mathrm{m}}=u_{\mathrm{w}}-u_{\mathrm{a}}$) reaches its cavitation pressure under a natural condition of ${(u_{\mathrm{w}}-u_{\mathrm{a}})}_{\mathrm{c}}=3.2-101.3=-98.1\mathrm{kPa}$. The negative sign indicates a value lower than the atmospheric pressure. Capillary pressure in soil with a water surface tension $T_{\mathrm{s}}$ ($=72\mathrm{mN}/\mathrm{m}$ at 25°C) and the interface radius $r$ is governed by the Young–Laplace equation [Fig. 1(a)]

Therefore, the largest pore radius for cavitation pressure is: $r_{\mathrm{c}}=-2T_{\mathrm{s}}/{(u_{\mathrm{w}}-u_{\mathrm{a}})}_{\mathrm{c}}=\sim {10}^{-6}\mathrm{m}$. As such, without the SSP, capillary water cannot exist in pores $<\sim {10}^{-6}\mathrm{m}$. All capillary water down to $\sim {10}^{-9}\mathrm{m}$ (the size of water molecule) would be cavitated if there is no SSP.

In the lab, the practical upper limit of the AT is $u_{\mathrm{a}}=\mathrm{1,500}\mathrm{kPa}$, which by Eq. (3) is equivalent to a capillary radius of $\sim {10}^{-7}\mathrm{m}$. Thus the AT will prevent cavitation for capillary water in pores with sizes from $\sim {10}^{-6}$ to ${10}^{-7}\mathrm{m}$, i.e., from μm to sub-μm.

Since all three components of SSP always elevate pore water pressure with a magnitude inversely proportional to distance from particle surface, soil water is less likely to cavitate. Van der Waals and hydration potentials generally only affect pore water pressure within a distance $<10$ water molecules or $3.0\times {10}^{-8}\mathrm{m}$ (e.g., Lu and Zhang 2019), so their effect on pressure increase has little impact on cavitation radius under the AT [Fig. 1(c), lightly shaded area]. Electrical double layer, however, can affect a distance of ${10}^{-7}\mathrm{m}$ (e.g., Lu and Zhang 2019), depending on the thickness of the double layer that is strongly affected by pore water salinity and particle surface electrical potential. As such, it could elevate pore pressure at the same scale with the cavitation radius by the AT, suppressing the cavitation.

Experimental SWR data by the AT and dew point (DP) (measuring matric potential through vapor pressure) for a silty (Palouse) soil [Fig. 1(c)] (Bittelli and Flury 2009) show significant and increasing discrepancies in the soil water content for matric suction $>100\mathrm{kPa}$, implying suppression of cavitation by the AT and a significant role of cavitation in soil water drainage in soils with pore sizes $<{10}^{-6}\mathrm{m}$ in nature.