Earth dam with drain pipe: phreatic coordinates

Given: crest 26.50 m, water level 25.00 m, ground 0.00 m, crest width 6 m, $m_1=3$, $m_2=2$, $k=15\times10^{-4}$ m/s, drain pipe at O, 10 m from the d/s toe. Casagrande's parabola has its focus at O (the end of the drainage).

Step 1: Geometry

$$\begin{aligned} B &= 3(26.5)+6+2(26.5)=138.5\ \text{m}\\ b &= 3\times25 = 75\ \text{m},\quad 0.3b=22.5\ \text{m}\\ d &= B-0.7b-10 = 138.5-52.5-10=76.0\ \text{m} \end{aligned}$$

($d$ = horizontal distance from the corrected entry point A to the drain pipe O.)

Step 2: Parabola constant

$$y_0=\sqrt{h^2+d^2}-d=\sqrt{25^2+76.0^2}-76.0=4.006\ \text{m}$$ $$y=\sqrt{y_0^2+2y_0x}=\sqrt{16.05+8.012\,x}$$

Step 3: Co-ordinates of the phreatic line

$x$ is measured from O towards the u/s side, $y$ above the ground:

At $x=76$ m the height is 25 m, which is the corrected entry point A. Position from the u/s toe: O is $138.5-10=128.5$ m, A is $52.5$ m.

Step 4: Specific discharge

$$q=k\,y_0=15\times10^{-4}\times 4.006=6.009 \times 10^{-3}\ \text{m}^3/\text{s per m}$$

Answer: $q=6.009 \times 10^{-3}$ m³/s per metre ($6.01$ litre/s per m); phreatic line $y=\sqrt{16.05+8.012\,x}$ as tabulated, with $y_0=4.006$ m.