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Fredlund & Xing model

Fredlund and Xing 1 2 proposed a relation between the water content \(w\) and the suction \(s\)

\[\begin{equation*} w = \left[ 1-\frac{\textrm{ln}\left(1 + \frac{s}{h_r}\right)}{\textrm{ln}\left(1+ \frac{1\cdot 10^6}{h_r}\right)} \right] \frac{w^{\textrm{sat}}}{\textrm{ln}\left(e+\left(\frac{s}{a_f}\right)^{n_f}\right)^{m_f}}, \end{equation*}\]

wherein \(w^{sat}\) is the water content at full saturation. In numgeo a modified version, which interrelates the degree of saturation \(S^w\) with the suction \(s\) is implemented:

\[\begin{equation*} S^w = \left[ 1- \frac{\textrm{ln}\left(1 + \frac{s}{h_r}\right)}{\textrm{ln}\left(1+ \frac{1\cdot 10^6}{h_r}\right)} \right] \frac{1}{\textrm{ln}\left(e+ \left(\frac{s}{a_f}\right) ^{n_f}\right)^{m_f}}, \end{equation*}\]

wherein \(h_r\), \(a_f\), \(n_f\) 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 input line takes the form:

*Hydraulic = Fredlund-Xing 
h_r, a_f, n_f, m_f

Theory

The contributions to the Jacobian read:

\[\begin{align*} \frac{\partial S^w}{\partial p^c} &= - \left( m_f \cdot n_f \left(\frac{s}{a_f}\right)^{-1 + n_f} \left[ 1- \frac{\textrm{ln}\left(1 + \frac{s}{h_r}\right)}{\textrm{ln}\left(1+ \frac{1\cdot 10^6}{h_r}\right)} \right] \frac{\textrm{ln}\left( e + \left(\frac{s}{a_f}\right)^{n_f} \right) ^{-1 - m_f}}{a_f \left(e+\left(\frac{s}{a_f}\right)^{n_f}\right) }\right) \\- & \frac{1}{\textrm{ln}\left(e + \left(\frac{s}{a_f}\right)^{n_f}\right)^{m_f}} \cdot \frac{1}{h_r \left(1 + \frac{s}{h_r}\right) \textrm{ln}\left(1 + \frac{1\cdot 10^6}{h_r} \right)} \end{align*}\]

  1. Delwyn Fredlund and Anqing Xing. Equations for the soil-water characteristic curve. Canadian Geotechnical Journal - CAN GEOTECH J, 31:521–532, August 1994. doi:10.1139/t94-061. ↩

  2. D. G. Fredlund, H. Rahardjo, and M. D. Fredlund. Unsaturated Soil Mechanics in Engineering Practice. John Wiley & Sons, Inc., 7 2012. doi:10.1002/9781118280492. ↩