Relative permeability curves and capillary pressure curve
Relative permeability curves and capillary pressure curve are essential tools for describing unsaturated flow in soils or other multiphase flows in geologic media. Relative permeability curves are defined by a set of functional relations typically between liquid saturation and relative permeabilities. However, capillary pressure curve is used to describe soil matric potential or capillary pressure as a function of soil water saturation. It is more frequently called Soil-Water Characteristic Curve (SWCC) for agrohydrologists or ecologists, and has been widely used for estimation of unsaturated soil properties.
In the present study, liquid relative permeabilitykrl is calculated with the van Genuchten-Mualem model (Mualem, 1976; van Genuchten, 1980). Gas relative permeabilitykrg can be calculated with the Corey’s model (Corey, 1954) for Sgr >0 and the relation between krg andkrl for Sgr = 0, and both relative permeabilities are limited to the range from 0 to 1.
\(k_{\text{rl}}=\left\{\par \begin{matrix}{\sqrt{S^{*}}\left(1-\left(1-{[S^{*}]}^{1/\lambda}\right)^{\lambda}\right)}^{2},\text{\ \ \ \ S}_{l}<S_{\text{ls}}\\ 1,\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ S}_{l}\geq S_{\text{ls}}\\ \end{matrix}\right.\ \) (1)
\(k_{\text{rg}}=\left\{\par \begin{matrix}\left(1-\hat{S}\right)^{2}\left(1-{\hat{S}}^{2}\right),\text{\ \ \ \ S}_{\text{gr}}>0\\ 1-k_{\text{rl}},\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ S}_{\text{gr}}=0\\ \end{matrix}\right.\ \) (2)
where S* and \(\hat{S}\) are defined by the following formulations, respectively.
S* = (SlSlr )/( SlsSlr ), \(\hat{S}\) = (SlSlr )/( 1 – SlSgr )
Sl and Sls is liquid and maximum liquid saturation, respectively; Slr andSgr is residual liquid and gaseous saturation, respectively. λ is a parameter representing the distribution characteristics of soil pore sizes, which is m in van Genuchten’s notation.
Since the van Genuchten model (van Genuchten, 1980) is extensively used for describing the relation between matric potential and water saturation. The following function is employed in our study to calculatePcap for SWCCs.
\(P_{\text{cap}}=-P_{0}{({[S^{*}]}^{-1/\lambda}-1)}^{1-\lambda}\)(3)
subject to the restriction –PmaxPcap ≤ 0.