Kozeny-Carman Equation Equation, Derivation & Rearrangements

Core idea

Overview

The Kozeny-Carman equation is a semi-empirical relationship used to estimate the intrinsic permeability of granular porous media like sand and gravel. It relates the flow capacity of the medium to its porosity and the average diameter of the constituent particles, modeling the pores as a network of tortuous channels.

When to use: This equation is best applied to laminar flow conditions in well-sorted, non-cohesive soils or packed beds of uniform particles. It is particularly useful when laboratory permeability tests are unavailable but grain size distribution and porosity data are known.

Why it matters: Accurate permeability estimates are vital for modeling groundwater aquifers, predicting the movement of subsurface contaminants, and optimizing drainage in civil engineering. It provides a theoretical bridge between measurable physical geometry and hydraulic performance.

Symbols

Variables

k = Permeability, \phi = Porosity, d_p = Grain Size

k

Permeability

m^{2}

ϕ

Porosity

V a r iab l e

d_{p}

Grain Size

m

Walkthrough

Derivation

Understanding the Kozeny-Carman Equation

Relates the permeability of a porous medium to its porosity and grain size.

Laminar flow through uniformly packed spherical grains.
No dead-end pores or fractures.

1

Model flow through capillary channels:

The Kozeny-Carman equation treats the pore space as a bundle of tortuous capillary tubes. Permeability increases with grain size squared and porosity cubed.

k = \frac{d _{p}^{2}}{180} \cdot \frac{ϕ ^{3}}{( 1 - ϕ ) ^{2}}

2

Note the key proportionality:

Even small changes in porosity produce large changes in permeability because of the cubic dependence.

k \propto \frac{ϕ ^{3}}{( 1 - ϕ ) ^{2}}

Note: The constant 180 is empirical (sometimes written as 150 depending on the grain packing model).

Result

k \propto \frac{ϕ ^{3}}{( 1 - ϕ ) ^{2}}

Source: University Hydrogeology — Porous Media Flow

Free formulas

Rearrangements

Solve for $k$

Make k the subject

k \propto \frac{ϕ ^{3}}{( 1 - ϕ ) ^{2}}

Simplify the Kozeny-Carman equation to show the proportionality of permeability (k) with respect to porosity ( $ϕ$ ). This involves identifying constant terms and recognizing that $ϵ$ and $ϕ$ both represent porosity.

Difficulty: 2/5

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

Visual intuition

Graph

The graph of permeability (k) versus porosity (ε) follows a power-law curve that increases rapidly as porosity approaches one. Because the term ε³/(1-ε)² dominates the relationship, the curve starts at the origin and features a vertical asymptote at ε = 1, where permeability theoretically approaches infinity.

Graph type: power_law

Why it behaves this way

Intuition

Imagine the porous medium as a complex network of interconnected, tortuous channels, where the overall ease of fluid flow depends on the total volume of these channels, their average width, and how straight or winding

k

Intrinsic permeability of the porous medium

A higher 'k' means the material allows fluid to flow through it more easily. Think of how quickly water drains through coarse sand versus fine clay.

Φ_{s}

Sphericity of the particles

A dimensionless measure of how close a particle's shape is to a perfect sphere. More spherical particles (higher

Φ_{s}

) tend to pack more efficiently, creating less tortuous flow paths.

ϵ

Porosity of the medium

The fraction of the total volume occupied by void space (pores). More empty space (higher

ϵ

) means more pathways for fluid to flow.

d_{p}

Average particle diameter

A characteristic measure of the size of the solid particles. Larger particles (higher

d_{p}

) generally create larger pore spaces and less surface area for fluid friction.

150

Empirical constant

A dimensionless scaling factor derived from experimental observations, accounting for the combined effects of tortuosity and frictional resistance in typical granular media.

Signs and relationships

ε^3: Porosity is cubed because a small increase in the available void space drastically increases both the number and size of interconnected flow paths, leading to a much larger increase in permeability.
(1-ε)^2: This term represents the volume fraction of solids. As the solid fraction increases, the void space decreases, and the flow paths become more constricted and tortuous.
d_p^2: Particle diameter is squared because larger particles create larger pore throats and less surface area per unit volume for frictional resistance.
\Phi_s^2: Sphericity is squared because more spherical particles reduce tortuosity and improve packing efficiency, significantly enhancing the ease of fluid flow through the medium.

Free study cues

Insight

Canonical usage

The Kozeny-Carman equation relates intrinsic permeability (k) to the square of the particle diameter ( $d_{p}^{2}$ ), porosity (ε), and sphericity (Φ_s).

Common confusion

A common mistake is using porosity (ε) or sphericity (Φ_s) as a percentage instead of a dimensionless fraction. Another is mixing units, such as using $d_{p}$ in centimeters and expecting k to be in $m^{2}$ , or attempting to

Unit systems

$k$ L^2 · Intrinsic permeability, representing the ease with which a fluid can flow through a porous medium. Its unit is always length squared.

$Φ_{s}$ dimensionless · Particle sphericity, a measure of how spherical a particle is. It is a ratio of surface areas, ranging from 0 to 1. Use as a fraction.

$ϵ$ dimensionless · Porosity, the fraction of the total volume of a porous medium that is occupied by void space. Use as a fraction (e.g., 0.3 for 30%).

$d_{p}$ L · Average particle diameter. The unit chosen for d_p will determine the unit of k.

$150$ dimensionless · An empirical constant derived from theoretical and experimental work on flow through packed beds.

One free problem

Practice Problem

A sand sample from a coastal aquifer has a porosity of 0.30 and an average grain diameter of 0.2 mm. Assuming a sphericity of 1.0, calculate the intrinsic permeability k in m².

Porosity0.3

Grain Size0.0002 m

Solve for: $k$

Hint: Convert the diameter from 0.2 mm to 0.0002 meters before plugging it into the equation.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Predicting the productiveness of a new oil well from core samples.

Study smarter

Tips

Always convert particle diameter (dp) to meters to ensure the permeability result is in m².
Ensure porosity (phi) is entered as a decimal fraction between 0 and 1, never as a percentage.
Note that sphericity ( $P h i_{s}$ ) is often assumed to be 1.0 for well-rounded grains in simplified textbook problems.
The equation loses accuracy in clay-rich soils due to electrochemical interactions and extremely small pore sizes.

Avoid these traps

Common Mistakes

Applying it to fractured rocks (it only works for granular media).

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Relates the permeability of a porous medium to its porosity and grain size.

This equation is best applied to laminar flow conditions in well-sorted, non-cohesive soils or packed beds of uniform particles. It is particularly useful when laboratory permeability tests are unavailable but grain size distribution and porosity data are known.

Accurate permeability estimates are vital for modeling groundwater aquifers, predicting the movement of subsurface contaminants, and optimizing drainage in civil engineering. It provides a theoretical bridge between measurable physical geometry and hydraulic performance.

Applying it to fractured rocks (it only works for granular media).