Phreatic line and discharge (crest 300 masl)

Given data and assumptions: homogeneous dam on an impervious foundation, no drain, tail water neglected. Casagrande's basic parabola with focus at the d/s toe.

• Dam height $= 300-278 = 22$ m; water head $h = 297.5-278 = 19.5$ m; $k = 5\times10^{-4}$ cm/s $=5\times10^{-6}$ m/s.

Step 1: Geometry

$$\begin{aligned} B &= 3(22)+4.5+2(22)=114.5\ \text{m}\\ b &= 3\times 19.5 = 58.5\ \text{m},\quad 0.3b=17.55\ \text{m}\\ d &= B-0.7b = 114.5-40.95 = 73.55\ \text{m} \end{aligned}$$

Step 2: Basic parabola

$$y_0=\sqrt{h^2+d^2}-d=\sqrt{19.5^2+73.55^2}-73.55=2.54\ \text{m}$$ $$y=\sqrt{y_0^2+2y_0x}=\sqrt{6.46+5.08\,x}$$

Coordinates ($x$ from the d/s toe, $y$ above the base):

At $x=d$, $y=h=19.5$ m (corrected entry point A, located $0.3b=17.55$ m upstream of the point where the water line meets the u/s face).

Step 3: Corrections

• Entry: the line starts at right angles to the u/s face and joins the parabola smoothly at A.
• Exit: $\alpha=\tan^{-1}(1/2)=26.57^\circ$, so the parabola meets the d/s face at $a+\Delta a=\dfrac{y_0}{1-\cos\alpha}=24.1$ m from the toe. Since $\alpha<30^\circ$ (outside Casagrande's chart), the exit point is found from Schaffernak's formula:

$$a=\frac{d}{\cos\alpha}-\sqrt{\frac{d^2}{\cos^2\alpha}-\frac{h^2}{\sin^2\alpha}}=82.23-\sqrt{6,762-1,901}=12.5\ \text{m}$$

so the line leaves the d/s face about 12.5 m along the slope from the toe (about 5.6 m above the base). (Extrapolating Casagrande's factor 0.36 would give about 15.4 m.)

Step 4: Discharge

$$q=k\,y_0=5\times10^{-6}\times2.54=1.271 \times 10^{-5}\ \text{m}^3/\text{s per m}$$

(Schaffernak's $q=k\,a\sin\alpha\tan\alpha=5\times10^{-6}\times12.5\times0.447\times0.5=1.40\times10^{-5}$ m³/s per m is of the same order.)

Answer: $y_0=2.54$ m, phreatic line $y=\sqrt{6.46+5.08\,x}$, exit point about 12.5 m up the d/s face; discharge $q=1.271 \times 10^{-5}$ m³/s per metre ($\approx1.098$ m³/day per m).