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Van Genuchten model

Van Genuchten1 proposed the following relation for the effective degree of saturation:

\[\begin{equation*} S^e = \left( \frac{1}{1+(\alpha s)^n} \right)^{1-1/n}, \end{equation*}\]

where \(\alpha\) and \(n\) represent empirical curve-fitting parameters. Notice that contrary to 1, the suction \(s\) is used instead of the pressure head. Therefore, the parameter \(\alpha\) is related to the original formulation (\(\alpha^o\)) via \(\alpha = \alpha^o/\gamma^w\).

Note that in numgeo, the version using the degree of saturation \(S^w\) is implemented:

\[\begin{equation*} S^w = S^{wr}+ (1-S^{wr}) \left( \frac{1}{1+(\alpha s)^n} \right)^{1-1/n}. \end{equation*}\]

The input line takes the form:

*Hydraulic = van Genuchten [,Swr]
alpha, n

Theory

The exponent ist often expressed as \(m=1-1/n\), which is adopted in numgeo.

The contributions to the Jacobian read:

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

  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. ↩↩