---
title: "Darcy's law"
subject: "Engineering Geology I"
chapter: 6
key: "semester_3/engineering_geology_i:darcys-law"
source: https://www.bctengineer.com/q/semester_3/engineering_geology_i/darcys-law
site: BCT Engineering
---

# Darcy's law

Engineering Geology I · Chapter 6: Physical Geology

- Asked in: 2066 Jestha (old course), 3+5 marks; 2064 Jestha (old course), 2 marks; 2061 Baisakh (old course), 2 marks; 2063 Baisakh (old course), 2 marks; 2066 Bhadra (old course), 2 marks (asked 5 times)
- 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_3/engineering_geology_i/darcys-law
- Whole chapter: https://www.bctengineer.com/md/semester_3/engineering_geology_i/chapter-6

## Question

State and explain Darcy's law and its applicability (how does water flow under the ground?).

## Short answer

Darcy’s law states that discharge Q = K i A, where K is hydraulic conductivity, i hydraulic gradient, A cross-sectional area. It applies to laminar flow in saturated, homogeneous media with Reynolds number <1–10. Groundwater moves from high to low hydraulic head through pores.

## Answer

**Darcy's law** states that the rate of flow of water through a porous medium is directly proportional to the hydraulic gradient and to the cross-sectional area normal to the flow.

$$
Q = K\,i\,A = K\,\frac{\Delta h}{L}\,A
$$

where $Q$ is the discharge (m$^3$/s), $K$ is the coefficient of permeability (hydraulic conductivity, m/s), $i = \Delta h / L$ is the hydraulic gradient, and $A$ is the total cross-section area (m$^2$). The Darcy velocity is $v = Q/A = K i$. The actual seepage velocity is $v_s = v/n$, where $n$ is the porosity.

```text
   h1 ->  _________ L _________
         |   soil sample (A)   |-> Q
   h2 ->  ---------------------
   head loss  dh = h1 - h2
```

#### How water flows underground
Rain infiltrates, moves down through the unsaturated zone, and reaches the water table. In the saturated zone it moves slowly (cm/day to m/day) through pores and fractures, from high to low hydraulic head, until it comes out at springs, rivers or wells.

#### Applicability
- Valid for **laminar flow** in saturated, homogeneous and isotropic media, with Reynolds number $R_e$ less than about 1 (up to 10).
- Not valid for very coarse gravel or fractured rock with turbulent flow, very high gradients, or very fine clay at very low gradient.

**Example:** $K = 1\times10^{-4}$ m/s, $i = 0.02$, $A = 50$ m$^2$ gives $Q = 1\times10^{-4}\times0.02\times50 = 1\times10^{-4}$ m$^3$/s.

**Answer:** $Q = 1\times10^{-4}$ m$^3$/s, i.e. 0.1 litre/s.

## Check yourself

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

1. According to Darcy's law, discharge is proportional to what?
   A. Square of velocity
   B. Hydraulic gradient and cross-sectional area
   C. Porosity only
   D. Reciprocal of area
   Answer: B. Q = K i A. (Key point: Statement)

2. How is actual seepage velocity related to the Darcy velocity?
   A. v_s = v x n
   B. v_s = v
   C. v_s = v/n, so it is larger than v
   D. v_s = v/K
   Answer: C. Water flows only through pores, so actual velocity is Darcy velocity divided by porosity. (Key point: Seepage velocity)

3. When is Darcy's law valid?
   A. Turbulent flow in coarse gravel
   B. Any flow in fractured rock
   C. Flow at very high gradients
   D. Laminar flow in saturated media with Reynolds number below about 1 (up to 10)
   Answer: D. It fails for turbulent flow. (Key point: Applicability)

4. With K = 1x10^-4 m/s, i = 0.02, A = 50 m^2, what is Q?
   A. 1x10^-4 m^3/s
   B. 1x10^-6 m^3/s
   C. 2.5x10^-3 m^3/s
   D. 1x10^-2 m^3/s
   Answer: A. Q = 1e-4 x 0.02 x 50 = 1e-4 m^3/s = 0.1 litre/s. (Key point: Numerical)

## Related in other subjects

- [Darcy's law with neat figure](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Adarcy-law-inclined-figure) — Soil Mechanics, chapter 6
- [Darcy's law and groundwater movement](https://www.bctengineer.com/question.md?key=semester_4%2Fengineering_geology_ii%3Adarcys-law) — Engineering Geology II, chapter 3, asked 8×
- [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
- [Flow duration curve and its applicability](https://www.bctengineer.com/question.md?key=semester_8%2Fpower_plant_design%3Aflow-duration-curve) — Power Plant Design, chapter 3, asked 3×