---
title: "Earth dam with drain pipe: phreatic coordinates"
subject: "Hydropower Engineering"
chapter: 4
key: "semester_7/hydropower_engineering:earthen-dam-drain-pipe-phreatic"
source: https://www.bctengineer.com/q/semester_7/hydropower_engineering/earthen-dam-drain-pipe-phreatic
site: BCT Engineering
---

# Earth dam with drain pipe: phreatic coordinates

Hydropower Engineering · Chapter 4: Headworks of Storage Plants

- Asked in: 2076 Ashwin, 6 marks
- Video: [[HYDROPOWER #3] Earthen Gravity Dam | Phreatic Line Equation| Numerical Problem | Failure of Dam](https://www.youtube.com/watch?v=TN1xkNzh7fo) — Just Learn Civil
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/hydropower_engineering/earthen-dam-drain-pipe-phreatic
- Whole chapter: https://www.bctengineer.com/md/semester_7/hydropower_engineering/chapter-4

## Question

An earthen dam of homogeneous materials with a drain pipe is shown in the figure. Determine the co-ordinates of the phreatic line and the specific discharge passing through the body of the dam. Coefficient of permeability = $15 \times 10^{-4}$ m/s.
[Figure: dam with U/S water level at 25.00 m, crest level 26.50 m, crest width 6 m, upstream slope m1 = 3, downstream slope m2 = 2, ground level 0.00 m, drain pipe (point O) at horizontal distance 10 m from the downstream toe.]

## Short answer

Focusing at the drain pipe gives y0 = 4.006 m, phreatic line y = sqrt(16.05+8.012x). Specific discharge q = k·y0 = 6.009×10⁻³ m³/s per m.

## Answer

**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:

| x from focus (m) | y (m) |
|---|---|
| 0.0 | 4.01 |
| 2.0 | 5.66 |
| 5.0 | 7.49 |
| 10.0 | 9.81 |
| 20.0 | 13.28 |
| 30.0 | 16.01 |
| 40.0 | 18.35 |
| 50.0 | 20.41 |
| 60.0 | 22.29 |
| 76.0 | 25.00 |

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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Where is the focus of Casagrande's parabola in a dam with a drain pipe at O?
   A. At the drain pipe O
   B. At the u/s toe
   C. At the crest centre
   D. At the corrected entry point
   Answer: A. The parabola's focus is at the end of the drainage, here the drain pipe O. (Key point: Focus of the parabola)

2. How is d computed here?
   A. d = B - 0.7b - 10 = 76 m, from corrected entry point A to O
   B. d = B - 0.3b = 117 m
   C. d = b = 75 m
   D. d = B - 10 = 128.5 m
   Answer: A. d is measured from the corrected entry point to the drain pipe: 138.5 - 52.5 - 10 = 76 m. (Key point: Horizontal distance d)

3. What is the specific discharge and how is it found?
   A. q = k y0 = 15x10^-4 x 4.006 = 6.009 x 10^-3 m3/s per m
   B. q = k h = 15x10^-4 x 25 = 0.0375 m3/s per m
   C. q = k y0^2 = 2.4 x 10^-2 m3/s per m
   D. q = y0 / k = 2670 m3/s per m
   Answer: A. Flow is q = k y0 with y0 = sqrt(25^2 + 76^2) - 76 = 4.006 m. (Key point: Specific discharge)

## 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 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
- [Phreatic line and discharge (k = 5.1×10⁻⁴ cm/s)](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Aearthen-dam-phreatic-line-2082) — Hydropower Engineering, chapter 4, asked 2×
- [Phreatic line and discharge (crest 300 masl)](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Aearthen-dam-phreatic-line-2079-baishakh) — Hydropower Engineering, chapter 4, asked 2×
- [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 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×

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