---
title: "Darcy's law and groundwater movement"
subject: "Engineering Geology II"
chapter: 3
key: "semester_4/engineering_geology_ii:darcys-law"
source: https://www.bctengineer.com/q/semester_4/engineering_geology_ii/darcys-law
site: BCT Engineering
---

# Darcy's law and groundwater movement

Engineering Geology II · Chapter 3: Hydrogeology

- Asked in: 2081 Chaitra, 1 marks; 2079 Chaitra, 1 marks; 2078 Chaitra, 1 marks; 2077 Chaitra, 2 marks; 2071 Bhadra, 1 marks; 2070 Magh, 2 marks; 2075 Bhadra, 1.5 marks; 2075 Baisakh, 2 marks (asked 8 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_4/engineering_geology_ii/darcys-law
- Whole chapter: https://www.bctengineer.com/md/semester_4/engineering_geology_ii/chapter-3

## Question

State Darcy's law. How does it describe groundwater movement and why is it important? (Include the relation between hydraulic gradient and hydraulic conductivity.)

## Short answer

Darcy's law: Q = K A i (i = Δh/L). Groundwater moves from high to low head; flow rate is proportional to hydraulic gradient and conductivity. For K=2×10⁻⁴ m/s, i=0.1, A=20 m², Q=4×10⁻⁴ m³/s. It is vital for well yield, seepage, and drainage design.

## Answer

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

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

where $Q$ = discharge ($m^3/s$), $K$ = hydraulic conductivity ($m/s$), $A$ = total cross-sectional area ($m^2$), $i = \Delta h / L$ = hydraulic gradient (dimensionless), $\Delta h$ = head loss over flow length $L$.

Discharge velocity $v = Q/A = K\,i$. The actual seepage velocity is $v_s = v/n$, where $n$ is porosity.

#### Relation of gradient and conductivity
Since $v = K\,i$, for a given gradient a higher $K$ gives a larger flow, and for a given flow the gradient is inversely proportional to $K$ ($i = v/K$): a low-$K$ layer needs a steep gradient to pass the same flow. Water moves from high head to low head; the gradient is the driving force, and $K$ (depending on grain size, sorting, fractures and fluid viscosity) is the ease of flow. $K$ is large for gravel (about $10^{-2}$ m/s) and small for clay (about $10^{-9}$ m/s).

#### Describing groundwater movement
Groundwater moves slowly, in laminar flow, from areas of high hydraulic head (recharge) to low head (discharge) in the direction of the gradient. Darcy's law is valid for laminar flow (Reynolds number below about 1-10).

#### Example
$K = 2\times10^{-4}$ m/s, head difference 5 m over 50 m, flow area 20 $m^2$:

$$
\begin{aligned} i &= \frac{5}{50} = 0.1 \\ Q &= 2\times10^{-4}\times 20\times 0.1 = 4\times10^{-4}\ m^3/s \end{aligned}
$$

**Answer:** $Q = 4\times10^{-4}\ m^3/s$ ($\approx 34.6\ m^3$/day).

#### Importance
- Estimating groundwater yield of wells and aquifers.
- Seepage under dams and from reservoirs and canals.
- Design of dewatering and drainage systems.
- Calculating transmissivity ($T = K b$) and aquifer testing.
- Groundwater contamination and flow modelling.

## 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. Hydraulic gradient and area of flow
   B. Porosity only
   C. Grain colour
   D. Depth of well only
   Answer: A. Q = K A i: flow rate is proportional to the hydraulic gradient and cross-sectional area. (Key point: Statement)

2. For a given gradient, what does a higher hydraulic conductivity K give?
   A. Smaller flow
   B. Larger flow
   C. No flow
   D. Turbulent flow only
   Answer: B. K is the ease of flow, so larger K gives larger Q at the same gradient. (Key point: Role of K)

3. For K = 2x10^-4 m/s, head difference 5 m over 50 m, area 20 m^2, what is Q?
   A. 4x10^-3 m^3/s
   B. 2x10^-4 m^3/s
   C. 4x10^-4 m^3/s
   D. 4x10^-5 m^3/s
   Answer: C. i = 5/50 = 0.1, so Q = 2x10^-4 x 20 x 0.1 = 4x10^-4 m^3/s. (Key point: Worked example)

4. Darcy's law is valid for what kind of flow?
   A. Laminar flow
   B. Highly turbulent flow
   C. Free-falling water
   D. Supersonic flow
   Answer: A. It applies to laminar flow, Reynolds number below about 1-10. (Key point: Validity)

## Related in Engineering Geology II

- [Factors and process of groundwater movement](https://www.bctengineer.com/question.md?key=semester_4%2Fengineering_geology_ii%3Agroundwater-movement-factors) — Engineering Geology II, chapter 3, asked 4×

## 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](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_geology_i%3Adarcys-law) — Engineering Geology I, chapter 6, asked 5×
- [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
- [Permeability versus hydraulic conductivity](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_geology_i%3Apermeability-versus-hydraulic-conductivity) — Engineering Geology I, chapter 6