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Van Genuchten Model

The van Genuchten model (1980)1 is a widely-used mathematical model for the Soil-Water Retention Curve (SWRC). It describes the relationship between the degree of saturation and capillary pressure in a porous medium.

Governing Equation

The numgeo implementation gives the degree of saturation \(S^w\) as a function of the capillary pressure \(p^c\) (defined as positive, e.g., in kPa):

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

where the effective saturation, \(S^e\), is given by:

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

The parameters are defined as:

  • \(\alpha\) is an empirical parameter with units of inverse pressure (e.g., kPa\(^{-1}\)).
  • \(n > 1\) is an empirical parameter related to the pore-size distribution (dimensionless).
  • \(m\) is an exponent, defined as \(m = 1 - 1/n\).
  • \(S^{wr}\) (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.

Pressure vs. Head

It's important to distinguish this pressure-based formulation from the head-based version commonly found in literature. The head-based version uses capillary head \(h_c = p^c/\gamma_w\) and a corresponding parameter \(\alpha_h\) with units of inverse length (e.g., m\(^{-1}\)). The \(\alpha\) parameter used in numgeo is related through the unit weight of water to \(\alpha_h\), \(\gamma_w\):

\[ \alpha = \frac{\alpha_{h}}{\gamma_w} \]

Jacobian Contribution

For the finite element solver, the derivative of the degree of saturation with respect to the capillary pressure \(p^c\) is required. Based on the governing equation used in numgeo, the derivative is:

\[ \frac{\partial S^w}{\partial p^c} = - \alpha m n (1 - S^{wr}) (\alpha p^c)^{n-1} \left[ 1 + (\alpha p^c)^n \right]^{-(m+1)} \]

  1. 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. ↩