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Fredlund & Xing model

The model proposed by Fredlund and Xing (1994) 1 and revised in Fredlund and Xing (2012) 2 provides a continuous mathematical relationship for the soil-water retention curve over the entire capillary pressure range, from zero to an oven-dry state (typically \(10^6\) kPa). A key advantage of this model is that it avoids the slope discontinuity present in models like Brooks & Corey, making it more robust for numerical applications.

Governing Equation

The original model related gravimetric water content (\(w\)) to capillary pressure (\(p^c\)). The version implemented in numgeo is adapted for the degree of saturation (\(S^w\)) and is expressed as:

\[ S^w(s) = C(p^c) \cdot G(p^c) \]

where the equation is a product of a correction function, \(C(p^c)\), and the primary desaturation function, \(G(p^c)\):

  • Correction Function, \(C(p^c)\): This term forces the saturation to zero at a very high capillary pressures (defined as \(10^6\) kPa).

    \[ C(p^c) = \left[ 1 - \frac{\ln\left(1 + \frac{p^c}{h_r}\right)}{\ln\left(1 + \frac{10^6}{h_r}\right)} \right] \]

  • Main Function, \(G(p^c)\): This term describes the shape of the desaturation curve.

    \[ G(p^c) = \frac{1}{\left[ \ln\left(e + \left(\frac{p^c}{a_f}\right)^{n_f}\right) \right]^{m_f}} \]

An important feature of this model is that it accounts for residual saturation through its fitting parameters, primarily \(m_f\), and thus does not require a separate residual degree of saturation, \(S^{r}\), as an input.

The parameters of the Fredlund & Xing model are:

  • \(a_f\) (kPa): A fitting parameter that is related to the air-entry value (AEV) of the soil. It generally corresponds to the capillary pressure at the inflection point of the SWRC.
  • \(n_f\) (-): A dimensionless parameter that controls the slope of the SWRC at the inflection point. It is related to the uniformity of the pore-size distribution.
  • \(m_f\) (-): A dimensionless parameter that controls the curvature at high capillary pressures and is related to the residual water content.
  • \(h_r\) (kPa): A fitting parameter corresponding to the capillary pressure at which the residual water content is reached. It is used in the correction function to ensure the curve reaches zero saturation at high capillary pressure.

Jacobian Contribution

The partial derivative of the degree of saturation with respect to capillary pressure, \(\frac{\partial S^w}{\partial p^c}\), is calculated using the product rule:

\[\frac{\partial S^w}{\partial p^c} = G(p^c) \frac{\partial C}{\partial p^c} + C(p^c) \frac{\partial G}{\partial p^c}\]

The individual derivatives of the component functions are:

\[ \frac{\partial C}{\partial p^c} = - \frac{1}{h_r \left(1 + \frac{p^c}{h_r}\right) \ln\left(1 + \frac{10^6}{h_r} \right)} \]
\[ \frac{\partial G}{\partial p^c} = - \frac{m_f \cdot n_f}{s \cdot \left[e + \left(\frac{p^c}{a_f}\right)^{n_f}\right]} \cdot \left(\frac{p^c}{a_f}\right)^{n_f} \cdot G(p^c) \cdot \left[ \ln\left(e + \left(\frac{p^c}{a_f}\right)^{n_f}\right) \right]^{-1} \]

  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. ↩