Seepage expressions: no drain vs toe drain

Seepage through a homogeneous earth dam on an impervious base is found by Casagrande's method, which treats the phreatic line as a basic parabola with its focus $F$ at the d/s toe (or at the start of the drain), based on Dupuit's assumption. Notation: $h$ = reservoir depth above the base, $b$ = horizontal projection of the wetted u/s slope, $B$ = base width, $\alpha$ = angle of the d/s face.

Case 1: Homogeneous dam without a drain

       A'  entry corrected 0.3b
   h ~~~~\___  phreatic line (parabola)
        /     \____
       /  dam      \___ a (exit point on d/s face)
   ///////////////////F (toe) \\\
        |<----- d ------>|

• Horizontal distance from the corrected entry to the toe: $d=B-0.7b$.
• Basic parabola: $y_0=\sqrt{h^2+d^2}-d$ and $y=\sqrt{y_0^2+2y_0x}$, with $x$ from $F$.
• Seepage (per unit length): $q=k\,y_0$.
• Exit point on the d/s face (distance $a$ along the slope from the toe):

$$a+\Delta a=\frac{y_0}{1-\cos\alpha},\qquad a=(1-C)(a+\Delta a)$$

where $C=\Delta a/(a+\Delta a)$ is read from Casagrande's chart (0.36 at $30^\circ$, 0.26 at $90^\circ$, 0.10 at $150^\circ$, 0 at $180^\circ$).

• For $\alpha<30^\circ$, Schaffernak's formula is also used:

$$q=k\,a\sin\alpha\tan\alpha,\qquad a=\frac{d}{\cos\alpha}-\sqrt{\frac{d^2}{\cos^2\alpha}-\frac{h^2}{\sin^2\alpha}}$$

Case 2: Dam with a toe drain

   h ~~~~\___  phreatic line
        /     \___
       /  dam      \___
   ///////////////[ toe drain ]\\\
        |<--- d --->|F (start of drain)

• The parabola is drawn with the focus at the u/s end of the toe drain, so $d=B-0.7b-\ell$ ($\ell$ = drain length).
• $y_0=\sqrt{h^2+d^2}-d$, $y=\sqrt{y_0^2+2y_0x}$ and $q=k\,y_0$.
• The line enters the drain nearly horizontally ($\alpha=180^\circ$, $C=0$), so no exit correction is required and the line meets the drain at $a=\dfrac{y_0}{1-\cos180^\circ}=\dfrac{y_0}{2}$ from the focus. The d/s face stays dry above the drain, which improves the stability of the d/s slope.