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Exponential model

This model provides a simple, single-parameter exponential function to describe the soil-water retention curve, as presented in a study by Abed and Sołowski (2017)1. Its simplicity is advantageous when detailed experimental data is unavailable or for preliminary analyses.

Governing Equation

The model relates the effective degree of saturation, \(S^e\), to capillary pressure, \(p^c\), through an exponential decay function. The version implemented in numgeo for the total degree of saturation, \(S^w\), is:

\[ S^w(s) = S^{wr} + (1-S^{wr}) e^{-\zeta p^c} \]

The parameters in this equation are defined as:

  • \(\zeta\) is a fitting parameter with units of inverse pressure (e.g., kPa\(^{-1}\)) that controls the rate of desaturation. A higher value of \(\zeta\) results in a steeper drop in saturation as suction increases.
  • \(S^{wr}\) (optional) is the residual saturation: The degree of saturation at which a further increase in capillary pressure does not produce a significant additional amount of drainage. This parameter is written as Swr in the input.

Implementation Note: Pressure vs. Head

The formulation used in numgeo is based on capillary pressure (\(p^c\)). This differs from the original formulation in some literature, including 1, which may use pressure head (\(h\)). The pressure-based parameter \(\zeta\) used in numgeo is related to the head-based parameter, \(\zeta_h\), through the unit weight of water, \(\gamma_w\):

\[ \zeta = \frac{\zeta_{h}}{\gamma_w} \]

Jacobian Contribution

The partial derivative of the degree of saturation with respect to capillary pressure, \(\frac{\partial S^w}{\partial p^c}\), is required for the numerical solver and is given by:

\[ \frac{\partial S^w}{\partial p^c} = - \zeta (1-S^{wr}) e^{-\zeta p^c} \]

  1. Ayman A. Abed and Wojciech T. Sołowski. A study on how to couple thermo-hydro-mechanical behaviour of unsaturated soils: physical equations, numerical implementation and examples. Computers and Geotechnics, 92:132–155, 2017. doi:10.1016/j.compgeo.2017.07.021. ↩↩