---
title: "Earth dam flow net and discharge"
subject: "Soil Mechanics"
chapter: 7
key: "semester_4/soil_mechanics:earth-dam-flow-net-compare-discharge"
source: https://www.bctengineer.com/q/semester_4/soil_mechanics/earth-dam-flow-net-compare-discharge
site: BCT Engineering
---

# Earth dam flow net and discharge

Soil Mechanics · Chapter 7: Seepage through Soils

- Asked in: 2077 Chaitra, 6+2 marks
- Video: [Earth Dam - Introduction, types and calculation of seepage through it](https://www.youtube.com/watch?v=mIm13hrBoMs) — Elementary Engineering
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/soil_mechanics/earth-dam-flow-net-compare-discharge
- Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-7

## Question

Draw a flow net diagram for the given earthen dam data and compare the discharge with the theoretical calculation. Top width = 15 m, upstream and downstream slope = 2H:1V, height of dam = 30 m, free board = 5 m, length of drain = 30 m and coefficient of permeability = 40 m/day.

## Short answer

With dam height 30 m, reservoir depth 25 m, slopes 2H:1V, filter 30 m, d=70 m, h=25 m, y0=4.33 m. Theoretical q = k y0 = 40 × 4.33 = 173.2 m³/day per m. A flow net with Nf=3, Nd≈17 gives q ≈ 176 m³/day per m, matching closely.

## Answer

#### Given

Top width 15 m; slopes 2H:1V both sides; height 30 m; free board 5 m, so reservoir depth $h = 25$ m; horizontal toe drain 30 m long; $k = 40$ m/day.

#### Geometry (horizontal drain, Casagrande)

- Base width: $2(2\times30) + 15 = 135$ m.
- The wetted upstream slope has horizontal projection $\Delta = 2\times25 = 50$ m, so A (where the water surface meets the slope) is 50 m from the upstream toe.
- The base parabola starts at B, $0.3\Delta = 15$ m **upstream** of A, i.e. $50 - 15 = 35$ m from the upstream toe.
- The focus F (start of the drain) is at $135 - 30 = 105$ m from the upstream toe.
- Horizontal distance B to F: $d = 105 - 35 = 70$ m.

#### Theoretical (Casagrande) discharge

$$
y_0 = \sqrt{d^2 + h^2} - d = \sqrt{70^2 + 25^2} - 70 = 74.33 - 70 = 4.33\ \text{m}
$$

$$
q = k\,y_0 = 40 \times 4.33 = 173.2\ \text{m}^3/\text{day per m}
$$

#### Flow net

Base parabola ($x$ from F towards the upstream side): $x = \dfrac{y^2 - y_0^2}{2y_0}$

| $y$ (m) | 4.33 | 10 | 15 | 20 | 25 |
|---|---|---|---|---|---|
| $x$ (m) | 0 | 9.38 | 23.81 | 44.02 | 70.00 |

The vertex (where the line meets the drain) is $y_0/2 = 2.17$ m downstream of F.

```text
  B    A            crest 15 m
  .  ▽\_____________
   ·   \·            \
       / ··  parabola  \  2:1
      / entry  ··.      \
     /________________F====\__ drain 30 m
   upstream toe
```

Plot these points, correct the entry from A (perpendicular to the upstream slope), and end the top flow line on the drain at right angles. Draw the flow lines between the top flow line and the base, and equipotentials at equal head drops so that the fields are curvilinear squares. A typical sketch gives $N_f = 3$ flow channels and $N_d \approx 17$ drops; the head drop per field is then $25/17 = 1.47$ m.

$$
q = k\,h\,\frac{N_f}{N_d} = 40 \times 25 \times \frac{3}{17} \approx 176\ \text{m}^3/\text{day per m}
$$

#### Comparison

| Method | $q$ (m³/day per m) |
|---|---|
| Casagrande (theory) | 173.2 |
| Flow net ($N_f = 3$, $N_d \approx 17$, as sketched) | about 176 |

The flow net value depends on how accurately the fields are drawn, but with $N_f/N_d \approx 0.17$ to $0.18$ it agrees with the theoretical value within a few percent.

## Check yourself

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

1. Dam height 30 m and free board 5 m. What is h?
   A. 35 m
   B. 30 m
   C. 25 m
   D. 5 m
   Answer: C. h = 30 - 5 = 25 m. (Key point: Reservoir depth)

2. A is 50 m from the toe, B 15 m further, and F 105 m from the toe. What is d?
   A. 55 m
   B. 40 m
   C. 50 m
   D. 65 m
   Answer: B. B is at 65 m, so d = 105 - 65 = 40 m. (Key point: Distance d)

3. With y0 = 7.17 m and k = 40 m/day, q is about:
   A. 5.6 m3/day per m
   B. 28.7 m3/day per m
   C. 286.8 m3/day per m
   D. 1000 m3/day per m
   Answer: C. q = 40 x 7.17 = 286.8. (Key point: Theoretical discharge)

4. How does the flow net discharge compare?
   A. Agrees within a few percent (N_f/N_d about 0.29)
   B. Ten times larger
   C. Half as large
   D. Cannot be compared
   Answer: A. 40 x 25 x 2/7 is about 286, close to the theory. (Key point: Comparison)

## Related in Soil Mechanics

- [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
- [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 flow type](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aearth-dam-confined-or-unconfined) — Soil Mechanics, chapter 7
- [Flow net under sheet pile, discharge](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Asheet-pile-flow-net-steps-discharge) — Soil Mechanics, chapter 7
- [Flow net discharge, 200 cc/day](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aflow-net-discharge-numerical-200cc) — Soil Mechanics, chapter 7
- [Why a filter on dam downstream](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Afilter-downstream-earth-dam) — Soil Mechanics, chapter 7

## Related in other subjects

- [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 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