A permeability that is only the geometry

Darcy’s law defines a permeability and says nothing about what sets it. This collection has an essay on the velocity nobody has, which is what the law is written in terms of, and another on where the law stops, which is where inertia arrives. Neither says how to predict the coefficient.

The standard answer is Kozeny–Carman:

$$k = \frac{\varepsilon^3}{K\,S^2},$$

with $\varepsilon$ the porosity, $S$ the surface area per unit of bulk volume and $K$ a constant near five. It is in every handbook. It is used to infer permeabilities that were never measured, to correct measurements to different porosities, and to convert particle-size data into flow properties.

And it is a correlation rather than a law, because what it asserts is that two numbers determine a third.

The simplest porous medium there is #

Take parallel circular tubes of assorted radii through a solid block. Everything about it is exact. Each tube carries Hagen–Poiseuille flow, so the volume flow through a tube of radius $r$ under a given pressure gradient is proportional to $r^4$, and summing over the tubes gives the block’s permeability

$$k = \frac{\varepsilon\,\langle r^4\rangle}{8\,\langle r^2\rangle}.$$

Its wetted surface per unit bulk volume, counted the same way, is

$$S = \frac{2\,\varepsilon\,\langle r\rangle}{\langle r^2\rangle}.$$

Both are moments of one distribution. And they are different moments. The permeability is a fourth over a second; the surface is a first over a second.

For a bundle of identical tubes every moment is a power of the same radius, the two expressions collapse, and the constant that makes the correlation exact is two — not near two, exactly two, at every porosity.

It is worth pausing on why a bundle is the right model to make this argument with, since it is obviously not a rock.

The point of the counter-example is not to describe a medium; it is to show that the correlation’s two inputs are insufficient, and for that the simplest medium in which both inputs are exactly computable is the best one available. A bundle has an exact permeability, an exact porosity and an exact specific surface, all in closed form, and no modelling assumptions anywhere. If the two inputs fail to determine the third in the simplest case, no amount of complication in a real medium will repair that.

The choice also removes every alternative explanation. There is no tortuosity, no connectivity, no constriction, no dead-end porosity and no surface roughness in a bundle of straight parallel tubes. Whatever the constant is absorbing here, it is absorbing only the pore-size distribution.

Where the constant goes #

Widen the distribution at a fixed median and the two moments move in opposite directions. The permeability rises by five orders, because the widest tubes dominate a fourth moment and a tube twice the radius carries sixteen times the flow. The surface falls, because the narrowest tubes dominate a first moment and there are more of them.

Back out the constant that makes the correlation exact and it falls from two to 0.027 across the range — a factor of seventy-four. And it is not falling in a complicated way: it is exactly two over

$$\frac{\langle r^4\rangle\,\langle r\rangle^2}{\langle r^2\rangle^3},$$

which is one for a uniform bundle and grows without bound as the distribution widens.

So the “constant” is not a material property being measured. It is a bookkeeping term absorbing the shape of the pore-size distribution, and the handbook’s five corresponds to no particular medium.

Two media the correlation cannot tell apart #

The sharpest form is to build the counter-examples. Forty-nine two-size bundles are constructed, each with exactly the target’s porosity and exactly its specific surface — matched to eight figures — and otherwise as different as the two free parameters allow.

Their permeabilities span a factor of eighty-two.

Everything Kozeny–Carman asks for is identical between them. The information it requests is real, exactly measurable, and does not determine the answer — which is the situation this collection has a general statement about: a constraint fixes what lies in its own span, and the permeability does not lie in the span of the porosity and the surface.

The matching itself is worth a paragraph, because a single matched pair would prove very little.

A two-size bundle has three free numbers: the two radii and the fraction of tubes at the larger size. Matching the specific surface uses one of them, and matching the porosity is automatic once the total pore area is set. So two free parameters remain, and the family of media sharing the target’s two measured properties is two-dimensional rather than a single alternative.

That matters because the first matched bundle tried differed from its target by a factor of 1.25, which is agreement rather than a counter-example. Sweeping the remaining two parameters over size ratios from two to thirty and mixing fractions from a fifth of a per cent to eighty per cent gives the factor of eighty-two, and the honest measurement is the family’s extent rather than any member of it — which is the same correction this collection made about insensitivity in general: the freedom is a property of the family swept, and quoting one member is quoting a sample.

Where the flow actually goes #

A two-size bundle makes the mechanism concrete. Take a bundle in which two per cent of the tubes are four times the radius of the rest.

Those two per cent are two per cent of the tubes, seven and a half per cent of the wetted surface, twenty-five per cent of the pore volume — and eighty-four per cent of the flow.

That is the whole difficulty in one figure. A measurement of the surface is dominated by the tubes that carry almost none of the flow; a measurement of the flow is dominated by tubes that are almost none of the surface. Two quantities, weighted at opposite ends of the same distribution.

There is a useful way to see the size of the effect without any of the arithmetic. Permeability goes as radius squared for a single tube — that is Poiseuille’s law, once the number of tubes is fixed by the porosity. So a bundle in which the flow-carrying tubes are four times the radius of the rest has sixteen times the permeability of a bundle of the small ones, and it needs only a couple of per cent of them to dominate.

The specific surface, meanwhile, goes as the reciprocal of the radius. So the same substitution moves the surface downwards by a small amount and the permeability upwards by a large one, and the ratio the correlation forms moves by their product.

Anything that puts a small fraction of the pore volume into large pores raises the permeability enormously and changes the surface hardly at all. That is a fracture in a rock, a channel in a filter cake, a wormhole in an acidised well, a crack in a concrete. All of them are invisible to a porosity-and-surface estimate, and all of them are what actually decides how much fluid the medium passes.

Which half of the correlation works #

It is worth separating the failure from what is not failing.

For a bundle the permeability is exactly proportional to the porosity and the surface exactly proportional to it, so $\varepsilon^3/S^2$ has exactly the right porosity dependence and the constant is left with none. Computed at porosities from 0.05 to 0.8 on one distribution, the constant does not move by one part in 10¹⁵.

So Kozeny–Carman gets the porosity right. What it gets wrong is that a single moment of a distribution cannot stand for two — which is a different criticism from the usual one, and a more specific one.

At a realistic width of the distribution the handbook value with $K = 5$ is out by a factor of seventy-four, and the error grows with the width.

What a permeability is, and is not, an average of #

The general statement worth extracting is about averages, and it has come up in this collection before.

A permeability is a flow-weighted property: what it measures is where the fluid goes, and the fluid goes where the resistance is least. A specific surface is a surface-weighted property: what it measures is where the solid is, and the solid is mostly around the small pores. Two weightings of the same distribution, at opposite ends of it.

That is the same structure as the two averages of a variable-density flow, where a time-weighted mean and a mass-weighted one differ by a factor of two; and as the sound speed of a mixture, which takes a heavy phase’s inertia and a light phase’s springiness. In each, two properties of one system combine by different weights, and a formula built on one cannot predict the other.

Mixtures and distributions do not interpolate. Where two quantities average with different weights, the combination is not between the parents’ values, and a correlation relating them will have a coefficient that is not constant.

What the literature’s scatter is measuring #

The reported Kozeny constant in the literature ranges from about two to well over a hundred, and the usual explanation is tortuosity — the notion that a real pore path is longer than the sample and the constant absorbs the excess.

That is a real effect and it is not what the numbers above are. There is no tortuosity in a bundle of straight parallel tubes: every path is exactly the sample length. The factor of seventy-four here comes entirely from the width of the pore-size distribution, in a medium with no tortuosity at all.

Which suggests a reading of the literature’s scatter. A constant that varies by a factor of fifty across materials is not measuring a geometric detour; it is absorbing a moment ratio, and the moment ratio is a property of the pore-size distribution that the two inputs do not contain. The materials with the largest constants are the ones with the widest distributions, which is what a sedimentary rock, a filter cake and a fibrous mat have in common.

Why the correlation survives anyway #

None of this is a reason to expect Kozeny–Carman to be abandoned, and the reasons it works are worth stating as carefully as the reasons it fails.

Within one material it interpolates well. A packed bed of monodisperse spheres compacted to different porosities has one pore-size distribution shape at every porosity, so the moment ratio is constant and the correlation’s porosity dependence is doing all the work — which this page finds to be exactly right. That is the case Carman fitted, and that is the case in which $K$ really is a constant near five.

Its inputs are the ones that can be measured. Porosity is a weighing, and specific surface is a gas adsorption measurement that a laboratory can do on a gram of powder. A pore-size distribution is harder, and a permeability requires a flow experiment on a sample large enough to be representative. A correlation using the two cheap measurements will always be reached for first.

And it is right about the dependences it does capture. The cube of the porosity, the square of the surface and the dimensional structure are all correct, so it extrapolates sensibly in every direction except the one this page is about.

So the useful conclusion is narrow rather than dismissive. Kozeny–Carman transfers well within a material family and badly between them, the size of the transfer error is the moment ratio, and the moment ratio is the quantity to ask about when a constant fitted on one medium is being applied to another.

What to do about it #

Measure the permeability. It is one experiment, it is not difficult, and it is the only route to the quantity that does not go through a distribution nobody knows.

If it has to be estimated, estimate the moment ratio too. A mercury-intrusion or image-based pore-size distribution supplies exactly the fourth, second and first moments the exact expression needs, and using them is a better calculation than fitting a constant.

And treat a correlated Kozeny constant as a fitted parameter for that material. Within one material at one preparation, the constant is stable and the correlation interpolates well — that is why it survives in practice. Across materials it is not a constant, and transferring one is transferring a moment ratio.

The same arithmetic in three other places #

Once the moment structure is visible it turns up repeatedly, and naming three instances makes it easier to spot a fourth.

A packed bed’s pressure drop is usually estimated from the Ergun equation, whose viscous term is Kozeny–Carman with a particular constant and whose inertial term is a second correlation. Both use a single particle diameter, so a bed of mixed sizes is being described by one number where the flow needs a fourth moment and the surface a first — the same insufficiency, in the correlation almost every chemical engineer reaches for.

A filter’s clogging is the same effect run in time. Deposition narrows the small pores first, because they are where most of the surface is, and the permeability barely responds; then it reaches the large ones, and the permeability collapses. A clogging curve that is flat and then falls off a cliff is the fourth moment finally being touched.

And a fractured rock is the extreme. A fracture aperture of a tenth of a millimetre in a metre of rock is a negligible porosity and a negligible surface, and it can carry more flow than the entire matrix. Every hydrogeological estimate that treats a fractured formation with a matrix permeability is making this error at its largest.

The pattern to watch for is a fourth power. Wherever a transport property goes as a high power of a length and a measurable property goes as a low one, the two will decouple, and a correlation relating them will have a coefficient that quietly carries the distribution’s shape.

What Darcy’s law does and does not promise #

It is worth closing by saying what survives, because the argument has been about a correlation rather than about the law it feeds.

Darcy’s law itself is in excellent shape. It is a statement that the flux is proportional to the pressure gradient, it follows from the Stokes equations by averaging over a representative volume, and the permeability it defines is a well-defined tensor property of the medium — measurable, repeatable, and independent of the fluid. Nothing on this page is a criticism of it.

What is being criticised is a shortcut around measuring that property. The permeability is a functional of the pore geometry, and no two-number summary of that geometry determines it, in exactly the sense this collection has a general statement about: two constraints on a distribution do not fix a third functional of it unless the third lies in their span, and a fourth moment does not lie in the span of a first and a second.

So the practical conclusion is narrow and firm. The permeability is a measurement, not an estimate. Where it must be estimated, the estimate should carry a factor-of-a-few uncertainty rather than a percentage, and the uncertainty should be stated as coming from the pore-size distribution rather than from the correlation’s scatter — because the two have different remedies, and only one of them is available.

What a measured permeability is worth #

Since the recommendation is to measure, it is worth being clear about what a measurement gives and what it does not.

A permeability measured on a core is a property of that core, at the scale of that core, in the direction the flow was driven. It is anisotropic in a layered medium, it is scale-dependent wherever the pore-size distribution has structure larger than the sample, and it is different in a fractured formation depending on whether the sample happened to contain a fracture.

That scale-dependence is the same arithmetic as everything else on this page. A core that misses the large pores measures the matrix; one that catches them measures something dominated by them; and the distribution of measured permeabilities across many cores is not experimental scatter but a measurement of the distribution’s tail.

So the honest form of a permeability report is a distribution rather than a number, with the sample size stated — because a single core’s value is one draw from a quantity whose fourth-moment weighting makes it heavy-tailed by construction. That is not a counsel of perfection: it is why formation evaluation reports permeability logs rather than a permeability, and why a single laboratory value is treated with suspicion by everybody who works with them.

A closing note on what a bundle is not. Real pores are connected sideways, so fluid can leave a narrowing pore and find a wider one, which raises the permeability above what a bundle of the same size distribution would give. That connectivity is a further property the two inputs do not carry, and it moves the answer in the same direction as the moment ratio — so the bundle’s factor of eighty-two is a lower bound on the freedom rather than an estimate of it.

What is not claimed #

A bundle of straight tubes is not a rock. It has no tortuosity, no connectivity, no dead ends and no constrictions, all of which matter in a real medium and all of which the constant is also absorbing. What the bundle establishes is that the constant is already not a constant before any of those are added, which is a lower bound on the problem rather than an account of it.

The specific surface is per unit bulk volume here. Written per unit solid volume the same relation acquires a factor $(1 - \varepsilon)^2$, and the constant near five becomes a constant near five divided by it. That is a bookkeeping convention and both forms are in use; the argument is unaffected, and the convention is stated so that the exactly-two result is checkable.

The moments are of a distribution, not of a sample. They are computed by quadrature over the distribution at forty thousand nodes rather than by sampling, so nothing here depends on a seed or a sample size, and a real measurement of a finite sample would have sampling error on top of everything described.

And Darcy’s law is assumed throughout. Every tube is in Poiseuille flow, so the pore Reynolds number is small and the permeability is a well-defined constant. Where inertia arrives the permeability stops being the whole story and a second coefficient appears, which is a different correction from the one this page is about.