---
title: "Seepage and phreatic line: horizontal filter at toe"
subject: "Hydropower Engineering"
chapter: 4
key: "semester_7/hydropower_engineering:seepage-horizontal-filter-derivation"
source: https://www.bctengineer.com/q/semester_7/hydropower_engineering/seepage-horizontal-filter-derivation
site: BCT Engineering
---

# Seepage and phreatic line: horizontal filter at toe

Hydropower Engineering · Chapter 4: Headworks of Storage Plants

- Asked in: 2081 Bhadra, 4 marks; 2080 Baishakh, 6 marks; 2075 Ashwin, 6 marks (asked 3 times)
- Video: [Design of Phreatic line in earthen dam with filter](https://www.youtube.com/watch?v=V5pnA13B3r0) — MSR Geo Spatial Solutions
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/hydropower_engineering/seepage-horizontal-filter-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_7/hydropower_engineering/chapter-4

## Question

Derive the expression for seepage flow (specific discharge) through a homogeneous earthen dam section with a horizontal filter (drain) at the toe, and explain how the phreatic line is drawn, with a neat sketch and governing equations.

## Short answer

Casagrande models the phreatic line as a parabola with focus at the filter start. Using Dupuit’s flow q = k y dy/dx and the parabola equation yields q = k(√(d²+h²) – d), where h is water height and d is the horizontal distance from entry point B (0.3Δ upstream of the water face) to the focus. The line meets the filter vertically.

## Answer

For a homogeneous earth dam with a horizontal filter (drain) at the toe, Casagrande's method treats the top flow line (phreatic line) as a **basic parabola** with its focus at the starting point $F$ of the filter. The seepage follows Darcy's law with the Dupuit assumption.

#### Sketch and notation
```text
        B     E
 FRL ~~~*~~~~~*.
        :    /  `.   phreatic line
   h    :   / u/s `-.
        :  /  face    `-._
 -------:-/---------------`|=========  filter
 heel    |<------- d ------>F
        (B is 0.3Δ upstream of E)
```
- $h$ = height of water above the dam base (head), $d$ = horizontal distance from point $B$ to the filter start $F$.
- $E$ is the point where the water surface meets the u/s face, and $\Delta$ is the horizontal projection of the wetted part of the u/s face. The basic parabola starts at $B$, on the water surface a distance $0.3\,\Delta$ horizontally **upstream** of $E$ (towards the reservoir).

#### Equation of the phreatic line
Put the origin at the focus $F$, with $x$ measured positive upstream. A parabola with its focus at the origin and focal length $y_0/2$ (vertex at $x=-y_0/2$) is

$$
y^2 = 2y_0x + y_0^2
$$

where $y_0$ is the ordinate of the parabola at the focus ($x=0$).

#### Seepage discharge (Dupuit)
At any section of the dam the flow per unit width is $q = k\,i\,A$, with the hydraulic gradient $i=dy/dx$ and the flow area $A=y\times1$:

$$
q = k\,y\,\frac{dy}{dx}
$$

Differentiating the parabola: $2y\,dy = 2y_0\,dx$, so $y\,\dfrac{dy}{dx} = y_0$. Therefore

$$
q = k\,y_0
$$

#### Finding $y_0$
The parabola passes through $B$, where $x = d$ and $y = h$:

$$
h^2 = 2y_0d + y_0^2 \;\Rightarrow\; y_0^2 + 2dy_0 - h^2 = 0
$$

$$
y_0 = \sqrt{d^2+h^2} - d
$$

$$
\boxed{q = k\left(\sqrt{d^2+h^2}-d\right)}
$$

#### Drawing the phreatic line
1. Mark $E$, then point $B$ at $0.3\Delta$ upstream of $E$, and measure $d$ and $h$.
2. Calculate $y_0$ and $q$.
3. Find points of the parabola from $y=\sqrt{2y_0x+y_0^2}$ for several values of $x$ and plot them.
4. Entry correction: start the actual phreatic line at $E$, **perpendicular** to the u/s face (an equipotential line), and join it smoothly to the basic parabola a little downstream.
5. Downstream, the parabola meets the horizontal filter vertically at its vertex, a distance $y_0/2$ downstream of $F$. With a horizontal filter no exit correction is needed.

## Check yourself

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

1. In Casagrande's method for a dam with a horizontal toe filter, the phreatic line is taken as a parabola with its focus at:
   A. The corrected entry point B
   B. The top of the dam
   C. The start point F of the filter
   D. The midpoint of the base
   Answer: C. The basic parabola has its focus at F, the start of the horizontal filter. (Key point: Casagrande basic parabola)

2. Using Dupuit's assumption, q = k y dy/dx. For the parabola y^2 = 2y0x + y0^2, what is y dy/dx and hence q?
   A. y dy/dx = h, q = k h
   B. y dy/dx = y0, so q = k y0
   C. y dy/dx = d, q = k d
   D. y dy/dx = 2y0, q = 2k y0
   Answer: B. Differentiating gives 2y dy = 2y0 dx, so y dy/dx = y0 and q = k y0. (Key point: Seepage from Darcy and Dupuit)

3. Substituting the point B (x = d, y = h) into the parabola gives which result for q?
   A. q = k (d - sqrt(d^2 + h^2))
   B. q = k (sqrt(d^2 + h^2) + d)
   C. q = k h^2/d
   D. q = k (sqrt(d^2 + h^2) - d)
   Answer: D. h^2 = 2y0 d + y0^2 gives y0 = sqrt(d^2 + h^2) - d, so q = k (sqrt(d^2 + h^2) - d). (Key point: Solving for y0)

4. How is the phreatic line joined at the upstream face?
   A. Entry point B is 0.3 of the horizontal projection of the wetted u/s face from the water-line intersection, and the line meets the u/s face perpendicularly
   B. Entry is at the water-line intersection and tangent to the face
   C. Entry is at the base heel with a 45° angle
   D. The line is straight from B to F
   Answer: A. B is corrected by 0.3 Δ into the dam, and the line is drawn perpendicular to the u/s face because the face is an equipotential line. (Key point: Entry correction and entry angle)

## 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
- [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 expressions: no drain vs toe drain](https://www.bctengineer.com/question.md?key=semester_7%2Fhydropower_engineering%3Aseepage-expressions-without-drain-toe-drain) — 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
- [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×

## Related in other subjects

- [Top flow line, dam with horizontal filter](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Atop-flow-line-horizontal-filter) — Soil Mechanics, chapter 7
- [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