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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 by integrating it (statistical approach). Depending on the complecity 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 (\(S^r\)) 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 as:

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

or

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

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

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

The input line takes the form:

*Relative permeability = Fredlund-Xing 
k^w_min, k^a_min, h_r, a_f, n_f, m_f, s_i

\(h_r\), \(a_f\), \(n_f\) and \(m_f\) are material parameters regarding the suction at residual water content, air entry value, the rate and the residual water content respectively. Note that the model features a parameter for residual water content and does not require the definition of a residual degree of saturation \(S^{wr}\). The last parameter \(s_i\), which defines the lower integration value for the suction, can be chosen as the air-entry value2. Note that this results in \(k^{r,wf} = 1.0\) for \(S^w \le 1.0\) which might be unfavourable for some applications.

Note that an adapted Nguyen approach with \(m=1.0\) is currently used for the relative permeability of the free air phase.

  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. URL: https://doi.org/10.1139/t94-062, doi:10.1139/t94-062. ↩

  2. Feixia Zhang and Delwyn Fredlund. Examination of the estimation of relative permeability for unsaturated soils. Canadian Geotechnical Journal, 52:150603144238006, June 2015. doi:10.1139/cgj-2015-0043. ↩