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* = (Sl −Slr )/( Sls −Slr ), \(\hat{S}\) = (Sl −Slr )/( 1 – Sl –Sgr )
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 –Pmax ≤Pcap ≤ 0.