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Brooks & Corey model

Governing Equation

Modified version of the original formulation of the Brooks & Corey - Mualem1 model according to Dury et al. (1999)2 :

\[\begin{align*} k^w & = (1-k^w_{min}) \left(S^e\right)^{\frac{2+3\lambda}{\lambda}} + k^w_{min} \\ k^a &= (1-k^a_{min}) \left(1-S^e\right)^2 + k^a_{min} \left(1-\left(S^e\right)^{\frac{2+\lambda}{\lambda}}\right). \end{align*}\]

Therein:

  • \(k^{w}_{min}\) and \(k^{w}_{min}\) are the minimum relative permeability for the pore water and pore air phase, respectively.
  • \(\lambda\) is the pore-size distribution index, see the hydraulic model. This dimensionless exponent reflects the uniformity of the pore sizes. A smaller \(\lambda\) value corresponds to a wider range of pore sizes, while a larger value indicates a more uniform pore structure.

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


Jacobian Contribution

The contributions to the Jacobian read:

\[\begin{align*} \frac{\partial k^w}{\partial S^e} &= (1-k^w_{min}) \frac{2+3\lambda}{\lambda} \Big(S^e\Big)^{\frac{2+3\lambda}{\lambda} - 1}, \\ \frac{\partial k^a}{\partial S^e} &= (1-k^a_{min}) \left( -2 \left(1-S^e\right) \Big( 1 - \left( S^e \right)^\frac{2+\lambda}{\lambda} \Big) - \frac{2+\lambda}{\lambda} \left(1-S^e\right)^2 \left(S^e\right)^{\frac{2+\lambda}{\lambda} - 1} \right). \end{align*}\]

  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. ↩