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Nguyen model

Based on experimental observations, Nguyen1 proposed to relate the relative permeability to the effective degree of saturation using the following simple cubic expressions. Instead of the original expression, we adopted a generalized exponential expression:

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

For \(m=3\) the original relation of Nguyen is recovered. The input line takes the form:\

*Relative permeability = Nguyen 
 k^w_min, k^a_min, m

where \(k^{w}_{min}\) and \(k^{w}_{min}\) are the minimum relative permeability for the pore water and pore air phase, respectively.

Note that an exponential model for the relative permeability analogously to the exponential hydraulic model can be achieved by choosing the Nguyen model with \(m=1\).

Theory

The contributions to the Jacobian read:

\[\begin{align*} \frac{\partial k^w}{\partial S^e} &= m (S^e)^{m-1}, \\ \frac{\partial k^a}{\partial S^e} &= -m (1-S^e)^{m-1}. \end{align*}\]

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


  1. Viet Hoai Nguyen, Adrian P. Sheppard, Mark A. Knackstedt, and W. Val Pinczewski. The effect of displacement rate on imbibition relative permeability and residual saturation. Journal of Petroleum Science and Engineering, 52(1-4):54–70, 6 2006. Reservoir Wettability. URL: http://www.sciencedirect.com/science/article/pii/S0920410506000581, doi:10.1016/j.petrol.2006.03.020. ↩