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Van Genuchten - Mualem model

Considering the pore-space as a bundle of capillary tubes, Mualem1 proposed a "geometry-based" model relating the relative permeability to the soil-water-retention curve. In numgeo a modified form of the equation proposed in 2 in terms of the van Genuchten hydraulic model is implemented:

\[\begin{equation*} k^w = (1-k^w_{min}) \sqrt{S^e} \left( 1-\left(1- \left(S^e\right)^{m} \right)^{1/m} \right)^2 + k^w_{min} ~~~\mbox{and}~~~ k^a = (1-k^a_{min}) \sqrt{1-S^e}\left( 1- \left(S^e\right)^{m} \right)^{2/m} + k^a_{min} \end{equation*}\]

The input line for the van Genuchten model takes the following form:

*Relative permeability = van Genuchten 
k^w_min, k^a_min,  n

where \(k^{w}_{min}\) and \(k^{w}_{min}\) are the minimum relative permeability for the pore water and pore air phase, respectively. \(n\) is the van Genuchten parameter, see the hydraulic model and \(m = n/(n-1)\).

Theory

The contributions to the Jacobian read:

\[\begin{align*} \frac{\partial k^w}{\partial S^e} &= (1-k^w_{min}) \left( \frac{ \Big( 1- \zeta^{\frac{1}{m}} \Big)^2}{2 \sqrt{S^e}} + 2\big(S^e\big)^{m-\frac{1}{2}} \zeta^{\frac{1}{m}-1} \Big(1-\zeta^{\frac{1}{m}} \Big) \right), \label{eq:rel_perm_deriv_vanGenuchten_1} \\ \frac{\partial k^a}{\partial S^e} &= (1-k^a_{min}) \left(-2 \sqrt{1-S^e} \big( S^e \big)^{m-1} \zeta^{\frac{2}{m}-1} - \frac{\zeta^{\frac{2}{m}}}{2 \sqrt{1-S^e}} \right), \label{eq:rel_perm_deriv_vanGenuchten_2} \end{align*}\]

with \(\zeta = 1 - (S^e)^m\). Note that at completely dry states (\(S^e=0\)) and fully saturated states (\(S^e = 1\)) contain zero-divide-exceptions. This is prevented at the programming level.

The development of relative permeability \(k^\beta\) with effective degree of saturation \(S^e\) is presented here:


  1. Yechezkel Mualem. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resources Research, 12(3):513–522, 1976. Publisher: American Geophysical Union (AGU). doi:10.1029/wr012i003p00513. ↩

  2. O. Dury, U. Fischer, and R. Schulin. A comparison of relative nonwetting-phase permeability models. Water Resources Research, 35(5):1481–1493, 1999. Publisher: American Geophysical Union (AGU). doi:10.1029/1999wr900019. ↩