Darcy's Law Equation, Derivation & Rearrangements

Core idea

Overview

Darcy's Law describes the flow of fluid through a porous medium, typically used to model groundwater movement through aquifers. It states that the discharge rate is directly proportional to the hydraulic conductivity, the cross-sectional area, and the hydraulic gradient.

When to use: Apply this equation when analyzing laminar flow in saturated porous media such as sand, gravel, or silt. It is valid for low Reynolds numbers, typically less than one, where viscous forces dominate over inertial forces.

Why it matters: This principle is fundamental for managing groundwater resources, predicting the migration of underground contaminants, and designing construction projects like dams or landfills. It allows scientists to quantify how much water is moving through the subsurface and at what velocity.

Symbols

Variables

Q = Discharge, K = Hydraulic Conductivity, A = Cross-Sectional Area, i = Hydraulic Gradient (i)

Q

Discharge

m^{3} / d a y

K

Hydraulic Conductivity

m / d a y

A

Cross-Sectional Area

m^{2}

i

Hydraulic Gradient (i)

V a r iab l e

Walkthrough

Derivation

Understanding Darcy's Law

Darcy's law links the discharge of groundwater through a porous medium to the hydraulic conductivity, cross-sectional area, and hydraulic gradient.

Laminar, steady-state flow through a saturated porous medium.
The medium is homogeneous and isotropic.

1

Observe the experimental proportionality:

From Darcy's 1856 sand-column experiments: doubling the area or gradient doubles the flow.

Q \propto A \cdot i

2

Introduce hydraulic conductivity K:

K is the constant of proportionality (m/day). It incorporates both fluid viscosity and the permeability of the medium.

Q = K \cdot A \cdot i

Note: K values: gravel ≈ 100–10,000 m/day; sand ≈ 1–100 m/day; clay ≈ 0.00001–0.01 m/day.

Result

Q = K \cdot A \cdot i

Source: A-Level Geology — Hydrogeology

Free formulas

Rearrangements

Solve for $Q$

Darcy's Law

Q = K \cdot A \cdot i

This rearrangement demonstrates how to express Darcy's Law using the hydraulic gradient symbol, `i`, which simplifies the notation of the equation.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Graph unavailable for this formula.

The graph is a straight line passing through the origin, representing a direct linear relationship between the hydraulic gradient and groundwater discharge. Since the discharge is proportional to the gradient, the slope of the line is determined by the product of the hydraulic conductivity and the cross-sectional area.

Graph type: linear

Why it behaves this way

Intuition

A steady stream of water percolating through a sand-filled cylinder, where the flow speed is determined by the steepness of the pressure drop and the coarseness of the grains.

Q

Volumetric flow rate or discharge

The total volume of water passing through a specific cross-section of the aquifer per unit of time.

K

Hydraulic conductivity

A measure of the 'ease' with which a fluid moves through the pore spaces; it accounts for both the permeability of the rock and the viscosity of the fluid.

A

Cross-sectional area

The 'window' size perpendicular to the flow; a larger area provides more parallel paths for the water to travel through.

dh/dl

Hydraulic gradient

The driving force or 'slope' of the water's energy; it represents the loss of pressure or elevation head over a specific distance.

Signs and relationships

dh/dl: In vector calculus forms, a negative sign is often included to indicate that flow occurs in the direction of decreasing hydraulic head (from high energy to low energy).

Free study cues

Insight

Canonical usage

This equation is used to calculate the volumetric flow rate of a fluid through a porous medium, requiring consistent units for length and time across all variables.

Common confusion

A common error is mixing length units (e.g., K in m/s, A in ft2) or time units (e.g., K in m/day, Q in m3/s) without proper conversion.

Dimension note

The hydraulic gradient (dh/dl) is a dimensionless quantity, representing a ratio of lengths. This is crucial for unit consistency in the equation.

Unit systems

$Q$ m3/s · Represents the volumetric flow rate or discharge, which is the volume of fluid passing through a cross-section per unit time.

$K$ m/s · Hydraulic conductivity, a measure of the ability of a porous medium to transmit water. It has the dimensions of velocity.

$A$ m2 · The cross-sectional area perpendicular to the direction of flow.

$d h / d l$ dimensionless · The hydraulic gradient, which is the change in hydraulic head (dh) over a given distance (dl). It is a dimensionless ratio, e.g., m/m or ft/ft.

One free problem

Practice Problem

A sandy aquifer has a hydraulic conductivity of 15 m/day and a cross-sectional area of 200 m². If the observed hydraulic gradient is 0.005, calculate the total discharge rate (Q).

Hydraulic Conductivity15 m/day

Cross-Sectional Area200 m²

Hydraulic Gradient (i)0.005

Solve for: $Q$

Hint: Multiply the hydraulic conductivity by the area and the gradient.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

K=10 m/day, A=20 m², Gradient=0.05. Flow (Q) = 10 * 20 * 0.05 = 10 m³/day.

Study smarter

Tips

Ensure all units for length and time are consistent across K, A, and Q.
The hydraulic gradient (GRAD) is a dimensionless ratio of vertical head loss over horizontal distance.
Remember that hydraulic conductivity (K) is a property of both the porous medium and the fluid.
Area (A) must be measured perpendicular to the flow direction.

Avoid these traps

Common Mistakes

Using porosity instead of hydraulic conductivity.
Failing to convert units (e.g. area to m² and conductivity to m/day).

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Darcy's law links the discharge of groundwater through a porous medium to the hydraulic conductivity, cross-sectional area, and hydraulic gradient.

Apply this equation when analyzing laminar flow in saturated porous media such as sand, gravel, or silt. It is valid for low Reynolds numbers, typically less than one, where viscous forces dominate over inertial forces.

This principle is fundamental for managing groundwater resources, predicting the migration of underground contaminants, and designing construction projects like dams or landfills. It allows scientists to quantify how much water is moving through the subsurface and at what velocity.

Using porosity instead of hydraulic conductivity. Failing to convert units (e.g. area to m² and conductivity to m/day).