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Gallipoli Model

The Gallipoli model1 is an extension of the van Genuchten model2 to account for the influence of the void ratio on the Soil-Water Retention Curve (SWRC).

Governing Equation

The degree of saturation \(S^w\) is given as a function of the capillary pressure \(p^c\) and the void ratio \(e\) by

\[ S^w = \left[ \frac{1}{1 + (\phi e^\psi p^c)^n} \right]^m \]

The parameters are defined as:

  • \(\phi\) is an empirical parameter with units of inverse pressure (e.g., kPa\(^{-1}\)).
  • \(\psi\) and \(m\) are soil constants (dimensionless)
  • \(n > 1\) is an empirical parameter related to the pore-size distribution (dimensionless).

Jacobian Contribution

For the finite element solver, the derivatives of \(S^w\) with respect to \(p^c\) and \(e\) are required. Differentiating the governing equation yields

\[ \frac{\partial S^w}{\partial p^c} = -\,m\,n\,\big(\phi\,e^{\psi}\big)^{n}\,(p^c)^{\,n-1} \left[\,1 + \big(\phi\,e^{\psi}\,p^c\big)^{n}\,\right]^{-(m+1)}, \]
\[ \frac{\partial S^w}{\partial e} = -\,m\,n\,\psi\,\phi\,p^c\,e^{\,\psi-1} \big(\phi\,e^{\psi}\,p^c\big)^{\,n-1} \left[\,1 + \big(\phi\,e^{\psi}\,p^c\big)^{n}\,\right]^{-(m+1)}. \]

Note

We evaluate the Jacobian as the consistent tangent operator within each Newton step to account for the strong material non-linearities of the SWRC.

  1. D. Gallipoli, S. J. Wheeler, and M. Karstunen. Modelling the variation of degree of saturation in a deformable unsaturated soil. Géotechnique, 53(1):105–112, 2003-02. doi:10.1680/geot.2003.53.1.105. ↩

  2. M. Th. van Genuchten. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Science Society of America Journal, 44(5):892–898, 1980. URL: http://doi.wiley.com/10.2136/sssaj1980.03615995004400050002x (visited on 2022-04-07), doi:10.2136/sssaj1980.03615995004400050002x. ↩