## 2 Research Methodology
### Theory and Data Collection
#### 2.1.2 [PERSON] and [PERSON] Method  
The first approximate method for the development of free surface was proposed by [PERSON] and [PERSON] [81]. As an earthen embankment on an impervious base (Figure 2) with no tail water, and with applying Equation (1) to triangle CAB, the discharge per unit width (with \(x\) taken as \(+\) ve to the left) can be achieved:  
Figure 1: Definition sketch for [PERSON]’s solution. \[q\ =\ -k\,y\ \frac{dy}{dx}\ =\ k\,a\ \sin\alpha.\ \tan\alpha \tag{5}\]  
where a denotes the length of seepage surface, and K stands for hydraulic conductivity of the embankment material. Equation (5) can be used to determine the value of 'a'. \[q\int_{asin\alpha}^{h}ydy\ =\ \int_{asin\alpha}^{d}asin\alpha.tan\alpha dx \tag{6}\]  
Or,  
\[\frac{1}{2}(h^{2}-a^{2}sin^{2}\alpha=asin\alpha.tan\alpha(d-acos\alpha) \tag{7}\]  
Or,  
\[(h^{2}-a^{2}sin^{2}\alpha=2 ad\frac{sin^{2}\alpha}{cos\alpha}-2a^{2}sin^{2}\alpha \tag{8}\]  
Or,  
\[a^{2}-\frac{2d}{cos\alpha}a+\frac{h^{2}}{sin^{2}\alpha}=0 \tag{9}\]  
Or,  
\[a=\frac{1}{2}[\frac{2d}{cos\alpha}-\sqrt{\left(\frac{4d^{2}}{cos^{2}\alpha}- \frac{4h^{2}}{sin^{2}\alpha}\right)}] \tag{10}\]  
Therefore,  
\[a=\frac{d}{cos\alpha}[\sqrt{\left(\frac{d^{2}}{cos^{2}\alpha}-\frac{h^{2}}{ sin^{2}\alpha}\right)}] \tag{11}\]  
Remember that the parabolic free surface in this case is tangent to the downstream slope, which is not the case with the [PERSON]'s solution. This plays a critical role for the entrance conditions correction at the upstream slope. As suggested by [PERSON], point D\({}_{0}\) (Figure 3) is better to be taken as the starting point of a seepage line, rather than point D. As can be seen, distance between \(D_{0}\) and D is 0.3 X. In fact, X is OJ. It means that \(D_{0}\)D as a distance depends on upstream slope (\(\beta\)). After that, the actual entrance condition can be determined through sketching in the area DF tangent to the parabolic free surface and normal to the upstream slope. Figure 2: Seepage line determination. In case of a homogenous embankment that is positioned upon an impervious foundation in which the discharge slopes are flatter than 1:1 (see Figure 3), a possible assumption is that the point where the seepage line and the downstream face are intersected is as expressed below:  
\[e=asin\alpha=\frac{h}{3} \tag{12}\]  
where \(e\) stands for the distance between the impervious base and the intersection of the seepage line upon the downstream side, and h signifies the height of water upon the upstream side. The discharge through a unit width of embankment (\(q\)) is obtainable through the use of [PERSON]'s law:  
\[ \begin{array}{ll}q&=\ -\mathrm{k}\ \mathrm{y}\ \frac{dH}{dL}\ \times\mathrm{Area}( \mathrm{mean\ discharge\ area\ of\ vertical\ cross\ section})\\ &=\ -\mathrm{k}\ \left(\frac{c-h}{L}\right)\times\left(\frac{h+\mathrm{z}}{2} \times 1\right)\\ &=\ \mathrm{k}\ \left(\frac{h-e}{L}\right)\times\left(\frac{h+\mathrm{z}}{2} \right)\\ &=\ \frac{h}{2}\left(\frac{h^{2}-e^{2}}{L}\right)\end{array}  \tag{13}\]  
where k stands for hydraulic conductivity of the material of the embankment or permeability coefficient, and L denotes the mean length of the seepage line (the distance from the starting point of the seepage line \(D_{0}\) to the midpoint A of the seepage face BC). Replacing for \(e\) = h/3 in Equation (14) we will obtain:  
\[\begin{array}{ll}q&=\ \frac{k}{2}\left(\frac{h^{2}-\left(\frac{k}{2} \right)^{2}}{L}\right)\\ &\ \ \ \ \ \ \ \ \ \  
The seepage formula is applied to a coding program with respect to the visual basic language using Excel 2014. Based on the scope of study, input data are used to extract the output data that is the seepage rate for homogenous embankment. A total of 24,197 models were taken into consideration in this study.