This porosity and permeability calculator does two jobs that are often confused with each other. The first is arithmetic and exact: porosity is the fraction of a sample's bulk volume that is void space, and if you know the bulk volume and the solid volume you know the porosity. The second is an estimate: given that porosity and a mean grain size, the Kozeny-Carman relation returns an intrinsic permeability. The first number is a measurement. The second is a model output, and the difference matters more than anything else on this page.
Arb Digital publishes free physics and engineering calculators that each state what they can and cannot do. This one computes porosity, then estimates permeability from it, then converts that permeability into a hydraulic conductivity for the fluid you name. If you already have a conductivity and want flow rates, the Darcy's law calculator takes conductivity as an input and gives discharge, seepage velocity and travel time. If you only need to move between permeability units, the permeability converter handles that.
What This Porosity and Permeability Calculator Does
Porosity, written φ, is defined as void volume divided by bulk volume. You can get it three ways, and all three are offered above. From volumes it is (Vbulk − Vsolid) ÷ Vbulk. From densities it is 1 − (ρbulk ÷ ρgrain), which is the same statement rearranged. Or you can type a porosity you already have and skip straight to the permeability part.
Permeability, written k, is a different property altogether. It measures how readily a fluid moves through the connected pore network under a pressure gradient. It has units of area — square metres in SI, or darcies in petroleum and groundwater practice — and it is a property of the solid skeleton alone. Two rocks with identical porosity can differ in permeability by six orders of magnitude, because permeability depends on pore size, pore connectivity and tortuosity, none of which porosity records.
The calculator reports porosity as the headline figure because that is the number it can compute honestly. The permeability, the darcy value and the hydraulic conductivity are all downstream of one empirical correlation, and are labelled as estimates throughout.
How to Use It
- Pick how you want to supply porosity. Volumes are the most direct if you have a pycnometer measurement. Densities are the usual soil-science route. Direct entry is there when the porosity came from a log or a published table.
- Enter the mean grain diameter. This is the single most influential input on the permeability estimate, because Kozeny-Carman scales with the square of it. A factor of two in grain size is a factor of four in permeability.
- Leave the constant at 180 unless you have a calibration. That value comes from packed uniform spheres. Angular, poorly sorted or platy grains do not obey it well.
- Set the fluid properties if you want a conductivity. The defaults describe fresh water near 20 °C. Warm water, brine or oil all give a different conductivity for exactly the same permeability.
- Read the porosity as a result and the permeability as an order of magnitude. That is the honest way to use the two outputs.
The Formula: How Porosity and Permeability Are Calculated
Porosity is φ = (Vb − Vs) ÷ Vb, equivalently φ = Vvoid ÷ Vb, and in the density form φ = 1 − ρb ÷ ρs. Void ratio, used in soil mechanics rather than hydrogeology, is e = φ ÷ (1 − φ) — void volume over solid volume rather than over total volume. The two are interchangeable but not equal, and mixing them is a common slip.
The Kozeny-Carman relation used here is k = (d² ÷ C) × φ³ ÷ (1 − φ)², with d the mean grain diameter in metres, C the Kozeny-Carman constant (180 by default) and k in square metres. It is derived by treating the pore space as a bundle of tortuous capillaries and applying Poiseuille flow to them, which is why it works tolerably for clean well-sorted sands and badly for fractured rock, clays or anything with a wide grain-size distribution. Engineering LibreTexts' section on diesel particulate filters shows the same Carman-Kozeny relation being used to model pressure drop through a porous medium of assumed parallel channels, which is exactly the idealisation involved.
Converting to hydraulic conductivity uses K = k ρg ÷ μ, where ρ is fluid density, g is 9.80665 m/s² and μ is dynamic viscosity. The groundwater flow section of Physical Geology on Geosciences LibreTexts sets out how hydraulic conductivity enters Darcy's law and why groundwater velocities are so small in ordinary aquifer materials. The United States Geological Survey publishes measured porosity and permeability ranges for common aquifer materials; consult their tables rather than treating a model output as a material property.
Work the defaults through by hand. Bulk volume 100 cm³ and grain volume 65 cm³ give a void volume of 35 cm³, so φ = 35 ÷ 100 = 0.35, or 35 per cent. Void ratio e = 0.35 ÷ 0.65 = 0.5385. For permeability, d = 0.25 mm = 2.5 × 10−4 m, so d² = 6.25 × 10−8 m² and d² ÷ 180 = 3.4722 × 10−10. The porosity group is 0.35³ ÷ 0.65² = 0.042875 ÷ 0.4225 = 0.101479. Multiplying gives k = 3.524 × 10−11 m². Dividing by 9.869233 × 10−13 m² per darcy gives about 35.7 darcy. Then K = 3.524 × 10−11 × 998 × 9.80665 ÷ 0.001002 = 3.44 × 10−4 m/s, which is about 29.7 metres per day — a plausible figure for a clean medium sand.
Why Kozeny-Carman Is a Model and Not a Law
Darcy's law is a law in the ordinary sense: over a wide range of conditions, flux really is proportional to gradient, and the constant of proportionality is a well-defined material property. Kozeny-Carman is not that. It is a correlation built on a picture — uniform spheres, a bundle of capillaries, a fixed tortuosity — and it is calibrated by a constant chosen to make the picture agree with data for some class of materials.
The consequences are practical. Poor sorting makes it over-predict, because fine grains fill the throats between coarse ones and shut down flow faster than the mean diameter suggests. Clay defeats it, because a swelling clay's effective pore size bears no relation to its grain size. Fractures make it meaningless, because nearly all the flow bypasses the model entirely.
Other published correlations exist for the same job — Hazen's relation for clean sands using the effective grain size, Terzaghi's variant, several petroleum-industry forms based on log data. They disagree with each other, sometimes by an order of magnitude, and each carries its own validity range. Naming which correlation produced a number is part of reporting it. This page uses Kozeny-Carman with a user-settable constant, and says so on the face of the tool.
Permeability Versus Hydraulic Conductivity: the Boundary That Matters
Intrinsic permeability k belongs to the porous medium. Hydraulic conductivity K belongs to the medium and the fluid moving through it. The link is K = k ρg ÷ μ, so anything that changes the fluid changes K while leaving k untouched.
This is not a technicality. Water viscosity nearly halves between 5 °C and 35 °C, so the same aquifer has roughly double the conductivity to warm water that it has to cold. A petroleum engineer quotes darcies because the rock property transfers between fluids; a hydrogeologist quotes metres per day because the fluid is always water.
The hydraulic conductivity calculator works the conductivity side directly, including the temperature dependence of water viscosity, and is the right page when the fluid is the thing you are varying. This page is the right one when you are starting from the solid: volumes, densities, grain size. Between them, the permeability converter handles darcy, millidarcy, square metre and square centimetre conversions without any physics attached.
What Porosity Does Not Tell You
Total porosity counts every void, including sealed ones and ones so small that the water in them is bound to mineral surfaces. Effective porosity counts only the connected voids that carry flow, and is always smaller. For a clean sand the two are close; for a vesicular basalt they differ enormously.
Neither figure says anything about pore size, and pore size is what permeability is really about. Halving the grain diameter at constant porosity cuts the Kozeny-Carman permeability to a quarter. That explains why a clay and a gravel with the same void fraction behave as different materials entirely.
Where This Sits Among the Related Tools
Use this page to get porosity and an order-of-magnitude permeability from sample properties. Take the conductivity into the Darcy's law calculator for discharge and travel time, or into the hydraulic gradient calculator if you are working out the driving gradient itself. For flow in pipes rather than porous media, the pipe flow calculator and the Reynolds number calculator are the right tools; Darcy flow is creeping flow, and a Reynolds number above roughly ten already puts you outside its validity.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Treating the permeability output as a measurement — it is a correlation applied to your inputs. Core analysis or a pumping test is what produces a real permeability.
- Confusing porosity with void ratio — porosity divides by total volume, void ratio divides by solid volume. At 35 per cent porosity the void ratio is 0.54, not 0.35.
- Using total porosity where effective porosity is meant — dead-end and clay-bound water does not flow, so seepage velocities computed from total porosity come out too low.
- Applying Kozeny-Carman to clay or fractured rock — the derivation assumes a granular skeleton of similar-sized grains. Outside that, the answer can be wrong by orders of magnitude.
- Quoting a conductivity without naming the fluid and temperature — conductivity is fluid-specific. Permeability is the figure that transfers between fluids.
Related Free Tools From Arb Digital
The Darcy's law calculator turns a conductivity and a gradient into discharge and travel time. The hydraulic conductivity calculator owns the fluid-dependent side of the same physics, and the permeability converter handles units alone. For gradients use the hydraulic gradient calculator, and for open-channel geometry the hydraulic radius calculator. Pipe flow lives in the pipe flow calculator and its flow regime in the Reynolds number calculator. Sample properties come from the density calculator and the soil volume calculator. Everything Arb Digital publishes is listed on the free online tools hub.
Frequently Asked Questions
Porosity is the fraction of a material's volume that is void space. Permeability is how readily fluid moves through those voids under a pressure gradient. They are different properties, and a material can have high porosity and almost no permeability if its pores are tiny or poorly connected.
It is an empirical correlation, not a law. For clean, well-sorted granular materials it usually gets within an order of magnitude. For clays, poorly sorted sediments or fractured rock it can be badly wrong, because the picture of uniform grains and capillary pores it is built on no longer applies.
It computes porosity exactly from the volumes or densities you enter. Permeability is estimated from that porosity and your grain size using the Kozeny-Carman relation with a constant you can change. The porosity is a result; the permeability is a model output.
Intrinsic permeability depends only on the porous medium. Hydraulic conductivity depends on the fluid as well, through K equals k times fluid density times gravity divided by viscosity. Change the fluid or its temperature and the conductivity changes while the permeability does not.
The darcy is the unit of intrinsic permeability used in petroleum and groundwater work. One darcy is 9.869233 times ten to the minus thirteen square metres. A clean medium sand is typically tens of darcies; a tight shale can be measured in nanodarcies.
Use effective porosity whenever you are converting a flux into a real fluid velocity, because only connected pores carry flow. This calculator returns total porosity from the volumes you supply, so reduce it yourself if some of the void space is isolated or clay-bound.
That is the value quoted for a packing of uniform spheres, which is the case the Kozeny-Carman derivation describes. Real sediments deviate, so published constants vary. If you have a measured permeability for your material, back-calculate the constant and use that instead.
The same relation is applied to packed beds, filter media and powders, and the calculator will run for them. The caveats are identical: it is a granular-skeleton model, and you should treat the result as an order-of-magnitude estimate until you have measured the material.
This tool is provided for educational and study use. Porosity is computed from your own inputs; permeability is an empirical Kozeny-Carman estimate that is material-specific and can be wrong by orders of magnitude outside clean granular media. It is not a substitute for core analysis, a pumping test or a qualified hydrogeologist's assessment.
