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Permeability - Relative permeabiliy constitutive model

The coefficient of hydraulic conductivity of unsaturated soil can vary considerably during a transient process as a result of combined changes in void ratio and degree of saturation (or water content) of the soil. At the beginning of a desaturation process, air replaces some of the water in the larger pores. Ongoing desaturation causes water to flow through the smaller pores, which leads to increased tortuosity and consequently a much slower flow. With increasing capillary pressure, the air-water interface is drawn closer to the soil particles, leading to a further decrease in the pore volume occupied by water. As a result, the coefficient of permeability with respect to the water phase decreases rapidly as the space available for water flow is reduced1.

It is convenient to express the hydraulic conductivity at various saturation states as the product of a relative permeability \(k^\beta\) and the hydraulic conductivity under saturated conditions \(\bar{K}^\beta_{ij}\). Therefore, the relative permeability coefficient of the fluid phase \(\beta=\{w,a\}\) is related to the effective degree of saturation \(S^e\) as:

\[\begin{equation*}\label{eq:relative_permeability} K_{ij}^{\beta} = k^\beta \bar{K}^\beta_{ij} ~~~\mbox{and}~~~k^{\beta}=k^\beta(S^{e}) ~~~~;~~\beta=\{w,a\}, \end{equation*}\]

Available relative permeability models in numgeo are:

Model Description
Nguyen Generalised version of the exponential relative permeability model originally proposed by Nguyen et al. (2006)2
Van Genuchten Van Genuchten type relative permeability model according to Dury et al. (1999) 3
Brooks & Corey Modified version of the original formulation of the Brooks & Corey - Mualem4 model according to Dury et al. (1999)3
Fredlund & Xing Relative permeability model according to Fredlund & Xing (1994)5
Logistic (curve) fit Simple logistic fit function

Choice of relative permeability model

When selecting a constitutive model for relative permeability, it is essential to ensure compatibility with the previously chosen soil-water retention curve (SWRC). In particular, the air entry value (AEV) of the SWRC should coincide with the onset of permeability reduction in the relative permeability function. Inconsistencies between these two models may lead to unrealistic simulation results.

The partial derivative of \(k^\beta\) with respect to the unknowns \(u\) (solid displacement) and \(p^\beta\) (pore fluid pressures) with \(\beta=\{w,a\}\) read:

\[\begin{align*}\label{eq:rel_perm_deriv} \frac{\partial k^\beta}{\partial p^\beta} &= \frac{\partial k^\beta}{\partial S^e} \frac{\partial S^e}{\partial p^\beta}, \\ \frac{\partial k^\beta}{\partial u} &= \frac{\partial k^\beta}{\partial S^e} \frac{\partial S^e}{\partial e} \frac{\partial e}{\partial u}. \end{align*}\]

Therein, \(e\) is the void ratio of the porous medium and \(S^e\) is the effective degree of saturation.

  1. Delwyn Fredlund and Anqing Xing. Equations for the soil-water characteristic curve. Canadian Geotechnical Journal - CAN GEOTECH J, 31:521–532, August 1994. doi:10.1139/t94-061. ↩

  2. 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. doi:10.1016/j.petrol.2006.03.020. ↩

  3. 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. ↩↩

  4. 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. ↩

  5. D. G. Fredlund, Anqing Xing, and Shangyan Huang. Predicting the permeability function for unsaturated soils using the soil-water characteristic curve. Can. Geotech. J., 31(4):533–546, August 1994. doi:10.1139/t94-062. ↩