# Permeability and Stiffness Assessment of Paved and Unpaved Roads with Geocomposite Drainage Layers
## 6 Numerical Simulation of Drainage Performance under Heavy Rainfall  
To improve the drainage conditions of unpaved roads, most agencies choose to build and maintain a 4% roadway crown to enable lateral water flow to drain excess water from the surface course into side drains or ditches. However, this practice requires frequent maintenance, and the effectiveness of the method needs to be quantified. In this study, a numerical simulation was performed to compare the drainage capacity offered by the geocomposite drainage layer with that of a 4% roadway surface crown for unpaved roads. A 2D water infiltration model was programmed in Matlab using [PERSON]'s equation and [PERSON]'s soil water retention characteristic (WRC) model [22; 23]. [PERSON] equation (Equation (6)) is used to predict volumetric water content (\(\theta\)) and matric potential (\(h\)) changes of the unpaved road surface material during transient water flow;  
\[\frac{\partial\theta}{\partial t}=\frac{\partial}{\partial z}\bigg{[}K(h) \bigg{(}\frac{\partial h}{\partial z}+1\bigg{)}\bigg{]} \tag{6}\]  
Liquid water flow in the vertical and horizontal directions is expressed using the well-known Darcy-Buckingham flux equations:  
\[q=-K(h)\bigg{(}\frac{dh}{dz}-\cos\alpha\bigg{)}\ (\text{for vertical direction}) \tag{7}\]  
\[q=-K(h)\bigg{(}\frac{dh}{dx}-\sin\alpha\bigg{)}\ (\text{for horizontal direction}) \tag{8}\]  
where \(q\) is the water flux in cm/s, \(K(h)\) is the unsaturated hydraulic conductivity in cm/s, \(h\) is the matric potential in cm, \(\frac{dh}{dz}\) or \(\frac{dh}{dx}\) is the matric potential gradient, \(\alpha\) is the slope of the soil layer in degrees, and the material is assumed to be isotropic and homogenous. [PERSON]'s water retention characteristic (WRC) model was used to estimate relationships between unsaturated hydraulic conductivity, matric potential, and the water content of the unpaved surface material during transient flow, as shown in Equations (9) and (10):  
\[\theta(h)=\theta_{r}+\frac{\alpha(\theta_{s}-\theta_{r})}{\alpha+|h|^{\beta}} \tag{9}\]  
where \(\theta(h)\) is the volumetric water content at the corresponding matric potential (\(h\)), \(\theta_{r}\) is the residual volumetric water content, \(\theta_{s}\) is the saturated volumetric water content, \(\alpha\) is equal to 1.611 \(\times\) 10\({}^{6}\), and \(\beta\) is equal to 3.96. \[K(h)=K_{s}\frac{A}{A+|h|^{\beta}} \tag{10}\]  
Figure 9: Surface conditions of the unpaved road sections after a heavy rainfall (photos taken on 2 September 2015). where \(K(h)\) is the unsaturated hydraulic conductivity at the corresponding matric potential (\(h\)) in cm/s, \(K_{\mathrm{s}}\) is the saturated hydraulic conductivity in cm/s, \(A\) is equal to 1.175 \(\times\) 10\({}^{6}\), and, \(\beta\) is equal to 4.74. In the above two equations, the coefficient values (\(\alpha\), \(\beta\), and \(A\)) were those suggested by [PERSON] et al. for granular materials based on infiltration experiments [22]. The residual and saturated volumetric water contents (0.034 and 0.27 cm\({}^{3}\)/cm\({}^{3}\)) and the saturated hydraulic conductivity (34.56 cm/day) of the base material were measured in the laboratory using the HPT tests. The saturated hydraulic conductivity of the subgrade of the unpaved road sections was measured using the laboratory rigid-wall compaction mold permeameter test in accordance with ASTM D5856 [24]. The measured saturated hydraulic conductivity of the subgrade is 1.8 \(\times\) 10\({}^{-8}\) cm/day, which can be considered impermeable relative to the surface aggregate material. In the present numerical analysis, the rate of the simulated rainfall event was taken as 12.7 cm/h, which can be considered a violent rain event for the local weather conditions. This rainfall rate is less than the saturated hydraulic conductivity (K\({}_{\mathrm{sat}}\)) of the surface aggregate so surface runoff cannot occur. The rainfall duration was 30 min, and the total simulation time was 90 min. The rainfall direction is assumed to be perpendicular to the ground surface.