# Unsaturated Hydraulic Conductivity in Composite Porous Media
## 2 Materials and Methods
### Unsaturated Hydraulic Conductivity Measurements  
The mini-disk infiltrometer (MDI) was used to record the \(k_{u}\) values (Figure 2), with a disk radius of 4.5 cm. The Mariotte compartment controls the suction head, and the lower chamber stores the volume of water allowed for infiltration, with a maximum value of 95 mL. The MDI allows regulating the suction from 0.5 to 6 cm to obtain additional information on the distribution of pores after the imposition of air values lower than the suction of the infiltrometer. Suction rates were adjusted for five sets of tests from 1 to 5 cm, and \(k_{u}\) values were measured for each circuit of 5 replicates per type of material. For each suction range, the water content was measured during the execution of the tests every 20 s (because it starts from a known humidity and the water that enters is known). The infiltration rates at the end of each experimental cycle were considered steady-state and were used to find the values of unsaturated hydraulic conductivity, \(k_{u}\). The unsaturated hydraulic conductivity (\(k_{u}\)) depends on factors such as matric suction and soil moisture. This phenomenon occurs because when the soil enters a pendular state, the continuity of the front hydraulic path is hindered, and the water cannot travel as well as when it has a continuous saturated medium. The technique chosen for the study based on [50], designed to estimate infiltration measurements in soil, can adequately correlate with the hydraulic conductivity of the material. The infiltration can be calculated utilizing a fitting equation shown below. \[I=C_{1}\sqrt{t}+C_{2}t \tag{2}\]  
\(C_{1}\) (cm/s) and \(C_{2}\) (cm/s) are adjustment parameters of the equation, but they have physical meaning. \(C_{1}\) is related to the soil absorption/desorption process, and \(C_{2}\) is the hydraulic conductivity. However, the desired hydraulic conductivity is in a partially saturated condition, so this new value must be calculated as \(k_{u}=(C_{1}/A)\), where \(C_{1}\) is the slope of the infiltration curve vs. t\({}^{1/2}\) (Figure 5). \(A\) is the variable that contains the suction values and the radius of the infiltrometer that controls \(k_{u}\). These parameters can be found in [51]. \begin{table}
\begin{tabular}{c c} \hline \multicolumn{3}{c}{**Water Permeability Characteristics**} \\ \hline
**Characteristic** & \(k_{s}\) **(m/s)** \\ \hline Very highly permeable & \(>\)10\({}^{-2}\) \\ Highly permeable & 10\({}^{-2}\)–10\({}^{-4}\) \\ Permeable & 10\({}^{-4}\)–10\({}^{-6}\) \(\star\) \\ Slightly permeable & 10\({}^{-6}\)–10\({}^{-8}\) \\ Very weakly permeable & \(<\)10\({}^{-8}\) \\ \hline \multicolumn{3}{c}{**Hydraulic conductivity**} \\ \hline
**Material** & \(k_{s}\) **(m/s)** \\ \hline Gravel & 10\({}^{-1}\)–10\({}^{-2}\) \\ Coarse sand & \(\cong\)10\({}^{-3}\) \\ Medium-grained sand & 10\({}^{-3}\)–10\({}^{-4}\) \\ Fine-grained sand & 10\({}^{-4}\)–10\({}^{-5}\) \\ Silty sand & 10\({}^{-5}\)–10\({}^{-7}\) \\ Silty clay & 10\({}^{-6}\)–10\({}^{-9}\) \(\star\) \\ Clay & \(<\)10\({}^{-9}\) \\ \hline \end{tabular} \({}^{\dagger}\) There is a theoretical transition between permeable and an “improvious” soil which occurs near a \(k_{s}\) value of 10\({}^{-6}\) m/s.  
\end{table} 
Table 2: Permeability characteristics and hydraulic conductivity [47]. The parameter \(A\) can be calculated from the following equations, depending on the induced suction and the radius of the infiltrometer (\(r_{o}=2.25\) cm). \[A=\frac{11.65(n^{0.1}-1)e^{[2.92(n-1.9)ah_{o}]}}{(\alpha r_{o})^{0.91}}\therefore n \geq 1.9 \tag{3}\]  
\[A=\frac{11.65(n^{0.1}-1)e^{[7.5(n-1.9)ah_{o}]}}{(\alpha r_{o})^{0.91}}\therefore n \leq 1.9 \tag{4}\]  
where \(n\) and \(\alpha\) are called Van Genuchten parameters. According to [51], for granular soil, these variables take the values of 2.68 and 0.145 for the Guamo sand. In this study, the variable \(C_{1}\) shows a value of 0.0241 cm/s (Figure 5).