Darcy's Law

Describes the flow of a fluid through a porous medium such as an aquifer.

Written by the Vorzep Team

GeologyHydrogeologyGroundwater Flow

Darcy's Law Calculator Q = −KA · dh/dl

Q = −K · A · dh/dl

Q = flow rate (m³/s) · K = hydraulic conductivity (m/s) · A = cross‑sectional area (m²) · dh/dl = hydraulic gradient (dimensionless)

⟹ Solve ⟷ Q, K, A, dh/dl

K (Hydraulic Conductivity)

m/s

A (Cross‑sectional Area)

m²

dh/dl (Hydraulic Gradient)

—

Q (Flow Rate)

m³/s

Please fix the errors above.

Solve for: Q (Flow Rate) K (Conductivity) A (Area) dh/dl (Gradient)

Presets:

Flow Rate (Q) —

—

K: — A: — dh/dl: — Q: —

✓ Copied!

Flow Rate (Q) —

Low (< 1e-5) Moderate (1e-5–1e-2) High (> 1e-2)

Q = −K·A·(dh/dl) · The negative sign indicates flow in the direction of decreasing hydraulic head.

Q = −KA(dh/dl)

Darcy's Law

Variables

Symbol	Quantity	Unit
Q	Flow rate	m³/s
K	Hydraulic conductivity	m/s
A	Cross-sectional area	m²
dh/dl	Hydraulic gradient

What it means

Darcy’s law is the fundamental equation governing groundwater flow in porous media. It states that the volumetric flow rate Q is proportional to the hydraulic conductivity K, the cross‑sectional area A, and the hydraulic gradient (dh/dl). The negative sign indicates flow in the direction of decreasing head. It assumes laminar flow and is valid for most groundwater systems. This law is used to calculate groundwater flow rates, design wells, assess aquifer yield, and predict contaminant transport. In petroleum engineering, it is applied to oil and gas flow. Understanding Darcy’s law is essential for hydrogeologists, environmental engineers, and civil engineers to manage water resources and to design remediation strategies.

Worked example

Darcy's Law – Two Detailed Examples

Real‑World

Scenario: A hydrogeologist is investigating the flow of groundwater through a sand aquifer. The hydraulic conductivity K is 1.0×10⁻⁴ m/s, the cross‑sectional area A is 10 m², and the hydraulic gradient (dh/dl) is 0.01. Using Darcy's law Q = −KA(dh/dl), they compute the volumetric flow rate Q = −1.0e-4 × 10 × 0.01 = −1.0e-5 m³/s (the negative sign indicates flow direction). This helps them estimate the water supply that can be extracted from the aquifer for a small community.

Parameter	Value
K (m/s)	1.0e-4
A (m²)	10
dh/dl	0.01

1Q = −1.0e-4 × 10 × 0.01 = −1.0e-5 m³/s

2Convert to litres/day: 1.0e-5 × 86400 = 0.864 L/s ≈ 74.6 m³/day

Result 1.0×10⁻⁵ m³/s ✓ Groundwater flow rate

Scenario: A consulting engineer is designing a dewatering system for a construction site. The aquifer has K = 3.0×10⁻³ m/s, A = 15 m², and the hydraulic gradient is 0.005. Using Darcy's law, they calculate Q = −3.0e-3 × 15 × 0.005 = −2.25e-4 m³/s (about 19.4 m³/day). This informs the pump sizing and well placement to keep the excavation dry during construction.

Parameter	Value
K	3.0e-3
A	15
dh/dl	0.005

1Q = −3.0e-3 × 15 × 0.005 = −2.25e-4 m³/s

Result 2.25×10⁻⁴ m³/s ✓ Dewatering flow

Insight: Darcy's law is fundamental to groundwater flow analysis. The negative sign indicates that flow occurs from high to low hydraulic head. The equation is valid for laminar flow in porous media.

Common mistakes

Darcy’s law: Q = −KA(dh/dl) – for saturated flow in porous media.
Negative sign: Indicates flow from high to low hydraulic head – do not drop it.
Hydraulic conductivity K: In m/s – depends on fluid and medium – not constant.
Cross‑sectional area A: Perpendicular to flow direction – use the effective area, not total.
Hydraulic gradient dh/dl: Dimensionless (m/m) – ensure units cancel.

Applications

Darcy's law, Q = −KA(dh/dl), is the fundamental equation for fluid flow through porous media, relating the volumetric flow rate (Q) to the hydraulic conductivity (K), cross‑sectional area (A), and hydraulic gradient (dh/dl). It is used extensively in hydrogeology to predict groundwater flow, to design wells, and to assess contamination migration. By applying Darcy's law, engineers can calculate the quantity of water moving through aquifers, design dewatering systems, and evaluate the performance of drainage and irrigation systems. The negative sign indicates flow from high to low head. Darcy's law is also applied in petroleum engineering for oil and gas production. Understanding this law is essential for water resource management, environmental engineering, and subsurface flow modelling.

Groundwater flow modelling and aquifer characterisation
Design of wells, pumping systems, and drainage networks
Contaminant transport and remediation design
Petroleum reservoir engineering (oil and gas recovery)
Irrigation and soil drainage planning