---
title: "Darcy's law with neat figure"
subject: "Soil Mechanics"
chapter: 6
key: "semester_4/soil_mechanics:darcy-law-inclined-figure"
source: https://www.bctengineer.com/q/semester_4/soil_mechanics/darcy-law-inclined-figure
site: BCT Engineering
---

# Darcy's law with neat figure

Soil Mechanics · Chapter 6: Principle of Effective Stress, Capillarity and Permeability

- Asked in: 2076 Baisakh, 2 marks
- Video: [Soil Permeability - Darcy's Law](https://www.youtube.com/watch?v=-bYptlimsdI) — 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/darcy-law-inclined-figure
- Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-6

## Question

Write down Darcy's law if Q amount of water flows per unit time through an inclined soil length 'L' of cross section 'A'. Take the hydraulic head difference at the entry and exit points of soil as 'h'. Draw a neat figure and explain each term used in the law.

## Short answer

Darcy’s law for an inclined soil sample: Q = k i A = k (h/L) A, where h is the total head loss between entry and exit, L is the flow path length along the specimen, and A is the cross-sectional area. The gradient i = h/L uses total heads including elevation, so inclination does not appear separately.

## Answer

**Darcy's law:** the discharge velocity of water through a saturated soil is directly proportional to the hydraulic gradient (laminar flow):

$$
v = k i \quad \Rightarrow \quad Q = k\,i\,A = k\,\frac{h}{L}\,A
$$

#### Figure (inclined soil specimen)

```text
  ▽ ---------------------------- H1 = total head at entry
  |                  ^
  |                  | h = H1 - H2 (head loss)
  |                  v
  |              ▽ ------------- H2 = total head at exit
  |              |
 (1)             |
   \\            |
     \\  soil    |
       \\ L, A   |
         \\_____(2)
  z1 = height of (1)   z2 = height of (2)
 ================================== datum
```

The total head at entry (1) and exit (2), measured from a common datum, each equals elevation head + pressure head: $H_1 = z_1 + h_{p1}$, $H_2 = z_2 + h_{p2}$. The water levels in the piezometers at (1) and (2) show these total heads.

#### Terms

- $Q$ = discharge per unit time (m³/s)
- $k$ = coefficient of permeability (m/s), a property of the soil and the water
- $h = H_1 - H_2$ = head loss (difference in total head between entry and exit) (m)
- $L$ = length of the flow path measured along the soil (m)
- $i = h/L$ = hydraulic gradient
- $A$ = cross-sectional area of the soil normal to the flow (m²)
- $v = Q/A$ = discharge velocity

The inclination does not appear separately, because the hydraulic gradient uses the total head (which already includes the elevation head), and $L$ is the length along the inclined sample.

$$
Q = k\,\frac{h}{L}\,A
$$

## Check yourself

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

1. Darcy's law states that, for laminar flow, discharge velocity is:
   A. Proportional to the square of the gradient
   B. Directly proportional to the hydraulic gradient
   C. Inversely proportional to the gradient
   D. Independent of the gradient
   Answer: B. v = k i, so Q = k i A. (Key point: Statement)

2. The head loss h in the law is:
   A. The elevation difference of the sample ends only
   B. The pressure head at the entry
   C. H1 - H2, the difference in total head between entry and exit
   D. The height of the tank
   Answer: C. Total head (elevation + pressure head) is used, so h = H1 - H2. (Key point: Head loss)

3. For an inclined sample, L in i = h/L is:
   A. Horizontal projection
   B. Vertical height
   C. The length along the soil, with no separate angle term
   D. L cos(theta)
   Answer: C. L is measured along the sample; elevation is already included in total head, so inclination does not appear separately. (Key point: Length and inclination)

4. The discharge through the sample is:
   A. Q = k (h/L) A
   B. Q = k (L/h) A
   C. Q = k h L / A
   D. Q = k i / A
   Answer: A. Q = k i A with i = h/L and A normal to flow. (Key point: Discharge formula)

## Related in Soil Mechanics

- [Darcy's law; tests for permeability](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Adarcy-law-tests-for-k) — Soil Mechanics, chapter 6
- [Inclined specimen, total heads, gradient](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Ainclined-specimen-total-head-gradient) — Soil Mechanics, chapter 6
- [Seepage in inclined permeable layer](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Ainclined-permeable-layer-seepage) — Soil Mechanics, chapter 6
- [Constant head permeameter, seepage velocity](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aconstant-head-permeameter-seepage-velocity) — Soil Mechanics, chapter 6
- [Variable head test, two soils](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Avariable-head-two-soils-permeability) — Soil Mechanics, chapter 6
- [Average permeability of layered soil](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Alayered-soil-average-permeability) — Soil Mechanics, chapter 6

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