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Fredlund & Xing model

In general, a relation between relative permeability and the (effective) degree of saturation or the suction can be derived from any soil water retention curve (SWRC) by integrating it (statistical approach). Depending on the complexity of the SWRC, an analytical function may be derived as it's the case e.g. for the Van Genuchten - Mualem model.

In general, all statistical approaches follow the same principle: the integral from a dry or residual state (\(S^{wr}\)) related to the current state (\(S^w\)) is compared with the integral of the fully saturated state (\(S^w=S^{w,sat}\)) and may be expressed as1:

\[\begin{equation*} k^{r,wf}(S^w,p^c) = \frac{k(S^w)}{k^{sat}} = \frac{\int_{S^{wr}}^{S} \frac{d S}{(p^c)^2(S)}}{\int_{S^{wr}}^{S^{w,sat}} \frac{d S}{(p^c)^2(S)}}, \end{equation*}\]

or

\[\begin{equation*} k^{r,wf}(S^w,p^c) = \frac{\int_{\ln(p^c)}^{1\cdot 10^6} \frac{S^w(e^y) - S^w(p^c)}{e^y} S^\prime(e^y) \, dy}{\int_{\ln(p^c_i)}^{1\cdot 10^6} \frac{S^w(e^y) - S^w(p^c_i)}{e^y} S^\prime(e^y) \, dy} , \end{equation*}\]

when the capillary pressure axis is integrated on a logarithmic scale. The integral between the upper end of the capillary pressure range (\(1\cdot 10^6\) kPa) and the current capillary pressure \(p^c\) is compared to the total integral with a lower limit \(p^c_i\). \(y\) is a dummy variable of integration representing the logarithm of integration and \(S^\prime\) is the derivative of the SWRC with respect to \(p^c\).

The SWRC and its derivative according to Fredlund & Xing (1994)1 can be found see here.

  1. D. G. Fredlund, Anqing Xing, and Shangyan Huang. Predicting the permeability function for unsaturated soils using the soil-water characteristic curve. Can. Geotech. J., 31(4):533–546, August 1994. doi:10.1139/t94-062. ↩↩