--- title: "Seepage expressions: no drain vs toe drain" subject: "Hydropower Engineering" chapter: 4 key: "semester_7/hydropower_engineering:seepage-expressions-without-drain-toe-drain" source: https://www.bctengineer.com/q/semester_7/hydropower_engineering/seepage-expressions-without-drain-toe-drain site: BCT Engineering --- # Seepage expressions: no drain vs toe drain Hydropower Engineering · Chapter 4: Headworks of Storage Plants - Asked in: 2073 Shrawan, 2+3 marks - Video: [Seepage though Earthen Dam || Soil Mechanics | Chapter 7 | Most important theory](https://www.youtube.com/watch?v=wmVYDaAoP7s) — Sudip Dhakal - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/hydropower_engineering/seepage-expressions-without-drain-toe-drain - Whole chapter: https://www.bctengineer.com/md/semester_7/hydropower_engineering/chapter-4 ## Question Write with neat sketches the expressions for computing seepage and the phreatic surface in earthen dams for two cases: homogeneous dam without drain and dam with toe drain. ## Short answer For a homogeneous dam without drain, Casagrande’s method uses a basic parabola with focus at the downstream toe: q = k y₀, y₀ = √(h²+d²) – d, and an exit correction is needed. With a toe drain, the focus is at the drain’s upstream end, giving q = k y₀ with no exit correction and the line entering the drain horizontally. ## Answer 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 ```text 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 ```text 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. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In Casagrande's method, the seepage per unit length is A. q = k y0 B. q = k h C. q = k d D. q = k y0^2 Answer: A. q = k y0, where y0 is the focal ordinate of the basic parabola. (Key point: Seepage formula) 2. y0 is computed as A. sqrt(h^2 - d^2) + d B. h - d C. sqrt(h^2 + d^2) - d D. d^2/h Answer: C. y0 = sqrt(h^2 + d^2) - d, with d = B - 0.7b. (Key point: Basic parabola) 3. With a toe drain, the focus of the parabola is at A. The d/s toe B. The u/s end of the drain, so d = B - 0.7b - l C. The crest D. The entry point Answer: B. The focus moves to the start of the drain, reducing d by the drain length. (Key point: Focus with toe drain) 4. Why does a toe drain improve stability? A. It makes the line exit high on the slope B. It increases seepage C. The line enters the drain and the d/s face stays dry D. It removes the need for a core Answer: C. No exit correction is needed (C = 0) and the d/s face remains dry. (Key point: Benefit of toe drain) ## Related in Hydropower Engineering - [Toe-drain earthen dam: discharge and phreatic line](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Aearthen-dam-toe-drain-2069) — Hydropower Engineering, chapter 4 - [Seepage and phreatic line: horizontal filter at toe](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Aseepage-horizontal-filter-derivation) — Hydropower Engineering, chapter 4, asked 3× - [Earth dam with drain pipe: phreatic coordinates](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Aearthen-dam-drain-pipe-phreatic) — Hydropower Engineering, chapter 4 - [Seepage line for 22 m high earthen dam](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Aearthen-dam-seepage-line-22m-45deg) — Hydropower Engineering, chapter 4 - [Seepage discharge through dam body (k = 1×10⁻⁶ m/s)](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Aearthen-dam-seepage-gradient-figure-2072k) — Hydropower Engineering, chapter 4 - [Seepage through dam with central core (45 m dam)](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Aearthen-dam-impervious-core-seepage-2075c) — Hydropower Engineering, chapter 4 ## Related in other subjects - [Steady 2D model: seepage under a dam](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Aseepage-under-dam-2d-model) — Computational Techniques in Civil Engineering, chapter 7, asked 3× - [Earth dam with filter, seepage quantity](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aearth-dam-horizontal-filter-seepage) — Soil Mechanics, chapter 7 - [Stresses in gravity dam with drain holes](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Agravity-dam-stresses-figure) — Hydro Power, chapter 4