A velocity nobody has

Every solved flow on this site so far has had a body in it — one body, with a shape, and a field computed round it. Water through sand has no such thing. There are bodies everywhere, ten to the twelfth of them in a cubic metre, and nothing can be resolved around any of them: what is left to describe is an average.

That change of question is the whole subject of this rung, and it comes with two prices. The first is paid in the velocity, which stops being anybody’s. The second is paid in the boundary condition, which stops existing.

The velocity that is nobody’s #

Darcy’s law is

$$u = -\frac{k}{\mu}\frac{dp}{dx}$$

and u is the flow rate divided by the total area, solid and fluid together. It is what a flowmeter on the pipe reports, and it is the natural quantity for a mass balance, so it is the right variable for the law to be written in.

It is also a speed nothing in the bed has. The fluid moves through the fraction ε of the area, so its average speed is u/ε — the interstitial velocity — and the two differ by exactly the porosity.

That factor is the easiest correction in the subject and the one most frequently dropped, and it is worth listing what needs it:

• a residence time, which is what a tracer, a reagent or a contaminant actually experiences;
• a pore Reynolds number, which decides whether the linear law applies at all;
• any statement about when something arrives rather than how much of it passes.

For a typical packing ε ≈ 0.4, so the factor is 2.5. Getting it wrong makes a groundwater plume arrive two and a half times too late, which is the sort of error that survives peer review because every number in the calculation is right.

The boundary condition that became a body force #

The second price is structural. In every flow this site has solved, the difficulty and the interest both live at the wall: the no-slip condition is what makes the equations hard and what supplies drag, separation and everything downstream of them.

Averaging over a volume containing many grains throws the walls away. What replaces them is a resistance proportional to velocity — a body force spread through the fluid — and the constant of proportionality is the permeability, a purely geometric property with the dimensions of an area.

The whole content of the model is in that replacement. It is why Darcy’s law is linear, why it is a low-Reynolds statement in exactly the way Stokes drag is, and why it stops working when inertia arrives — which is the next rung.

A permeability is an area, and a small one #

Two things about that curve are worth reading.

It is steep. Permeability goes as ε³/(1 − ε)², so a modest change in packing is a large change in resistance. A filter bed that settles, a soil that compacts, a catalyst that shrinks: each loses flow far faster than the change in porosity suggests, and the exponent is why.

It goes as the square of the grain size. Halving the grain size quarters the permeability, which is why fine sand is a barrier and gravel is a drain, and why grinding a catalyst finer to increase its surface area costs pressure drop by the square.

The tortuosity that turns out not to be a fit #

Two routes to the permeability are available and neither is a solved flow.

A bundle of capillaries. Model the pore space as straight tubes, tilted so that a path through the bed is longer than the bed by a factor τ. Poiseuille’s law inside each tube is exact, and both consequences of the tilt have to be counted: the driving gradient along a tube is smaller by τ because the path is longer, and the component of the velocity that gets a particle through the bed is smaller by τ again. Hence k = εd²/32τ², with τ squared rather than τ.

A hydraulic radius. Take the pore space’s own hydraulic radius, ε/(S(1 − ε)) with S = 6/d the specific surface of spheres, and feed the equivalent tube diameter into the bundle model. That gives Kozeny–Carman, whose familiar form is k = ε³d²/180(1 − ε)².

The two are the same model with the tortuosity left free, so requiring them to agree determines it.

Carman’s 2.5 is not a fit. It is what one model has to say about another for the two to be the same model, and the site’s search finds it to nine figures at every porosity it is asked about.

That is a different kind of claim from a measured constant, and it is worth being clear about what it does and does not establish. It does not establish that a real bed’s path length is 1.58 times its thickness — nobody has measured that, and the tortuosity of a real pore space is not a well-defined quantity. It establishes that if the pore space is idealised as tubes, and if the hydraulic radius is the right equivalent diameter, then 2.5 is forced.

What the permeability says about one grain #

A control volume supplies an independent check, and it is a strong one: the bed’s pressure gradient has to be carried by the grains, so a permeability is a statement about the force on one of them.

$$-\frac{dp}{dx} = nF \quad\Longrightarrow\quad F = \frac{\mu u}{kn}$$

with n the number of particles per unit volume. Comparing that force with Stokes drag on an isolated sphere moving at the interstitial velocity gives a ratio with a closed form: 10(1 − ε)/ε².

That crossing is where the model stops being an approximation and starts being the wrong kind of model. Above ε = 0.916 — computed here at 0.916080 — Kozeny–Carman predicts less drag than a single isolated sphere would feel, and in the limit ε → 1 it predicts none at all. A sphere alone in an infinite fluid feels 3πμdU whatever its neighbours do or do not do.

The reason is structural rather than numerical. A hydraulic radius presumes channels, and a dilute suspension has no channels: the fluid is not threading between grains, it is flowing past them with plenty of room. A model is not merely inaccurate outside its range; it is a statement about something that is not there.

Recording that failure explicitly is the point of computing the drag route at all. The permeability formula would have gone on returning cheerful numbers all the way to ε = 1 and nothing in a plot would have complained.

The same bed, coarser and looser #

Two parameters set everything, and seeing them moved separately is worth more than the algebra.

The pair of them is the practical content of the model. A designer choosing a filter medium is choosing a permeability, and has two levers: the grain size, which is squared, and the packing, which enters as ε³/(1 − ε)². Neither lever is free — finer grains catch more and clog sooner, looser packing passes more and filters worse — and the trade is the same shape as a flowmeter’s between signal and running cost.

What is not being drawn, and why #

There is no pore-scale flow anywhere in this essay. Not in a figure, not in an approximation, not in a sketch. The flow between grains is a real thing with a real solution and this site does not compute it, so nothing here claims a resolution it does not have — the grains in the first figure are drawn precisely to say that.

That restraint has a cost worth naming. Everything above is an averaged statement, and the averaging has hidden all the questions that would matter for a filter, a fuel cell or a catalyst: where the flow actually goes, whether some channels carry most of it, how much of the surface the fluid touches. A real bed has preferential paths, and a bed with preferential paths has a perfectly ordinary permeability and a completely useless residence-time distribution.

The grains are spheres of one size. Real beds are graded, and a small fraction of fines fills the gaps between the coarse grains and collapses the permeability far more than its volume fraction suggests. The specific surface S = 6/d is where that enters and it is where a real bed departs first.

No inertia anywhere. Every statement above belongs to the creeping limit, where the resistance is linear in the velocity and the flow is reversible in the sense the Stokes essay measures. What that costs is the subject of the next rung, and the answer is that it costs nothing at all until a pore Reynolds number of about ten and then everything.

Nothing here is anisotropic. A sedimented or a fibrous bed has a permeability that depends on direction, which makes k a tensor rather than a number, and the whole of this essay is the isotropic case.

Where the linearity comes from, and what it is worth #

Darcy’s law’s real value is that it is linear, which puts groundwater flow, filtration and reservoir engineering into the hands of the same mathematics that solves electrostatics: combine it with mass conservation and the pressure satisfies Laplace’s equation, so everything the potential-flow field knows applies to an aquifer — the superposition that adds two flows together, the images that make a wall out of a reflection, and the whole apparatus of sources and sinks, which in this setting are wells.

That is a large borrowing and it is worth noticing where it comes from. The linearity is not a property of fluids; it is a property of slow flows, the same property that makes creeping flow reversible and Stokes drag proportional to speed. A bed at a higher flow rate loses it, and loses the mathematics with it.

The geometric property that depends on the fluid #

The permeability is introduced above as a purely geometric property with the dimensions of an area, and that is the claim which makes the whole model worth having: measure it once, and it describes the medium for every fluid. It is very nearly true, and where it fails the failure is instructive.

Measure a rock’s permeability with a gas and the answer is too high. Measure it again at a lower mean pressure and it is higher still. Plot the results against the reciprocal of the mean pressure and they fall on a straight line, whose intercept is the value a liquid would have given.

The cause is the boundary condition inside the pores. A gas at low pressure has a mean free path comparable with a pore’s radius, so the molecules at the pore wall do not come to rest on it — they slip, by an amount proportional to the free path — and a slipping wall passes more flow than a no-slip one. Lower the pressure and the free path grows, and the apparent permeability grows with it.

So the measurement is contaminated by a rarefaction effect happening at a scale the averaged model has abstracted away entirely. Darcy’s law knows nothing about pores; the number in it turns out to depend on what is happening at their walls.

The repair is the extrapolation, and it is routine: core analysis measures gas permeability at several pressures and reports the intercept as the liquid-equivalent value. The correction is a few per cent for an ordinary sandstone and a factor of several for a fine-pored gas shale, where the pores are nanometres and the free path is comparable with them at reservoir pressure.

There is a second and less tractable failure of the same claim. In a clay-bearing rock the permeability depends on the chemistry of the fluid, because the platelets swell or disperse according to the brine’s salinity, and the pore space is therefore not a fixed geometry at all. A property with the dimensions of an area that responds to ionic strength is not describing a shape.

The company this keeps on the site #

Three other results here are of the same kind — a linear resistance standing in for a boundary condition nobody wants to resolve — and the family is worth naming.

Stokes drag replaces a solved field round one sphere with a force proportional to speed. The friction factor of a pipe replaces the wall’s shear with a coefficient, and unlike this one it is a correlation rather than a derivation. And an eddy viscosity replaces the whole of a turbulent stress with a gradient times a made-up constant.

The four sit on a scale of honesty. Stokes drag is exact. Kozeny–Carman is a model with one free parameter that turns out to be forced. Colebrook’s friction factor is a fit to somebody’s pipes. An eddy viscosity is a guess with a constant in it, and this site says so at length. Knowing which of those a formula is matters more than knowing its value, and a handbook that prints all four in the same typeface is where the trouble starts.

Two beds that are the same equation and nothing else alike #

The reach of the model is worth stating by listing what it is asked to cover, because the range is extreme and the form does not change across it.

An aquifer is grains of sand under a hillside with water moving through it at metres per year. Its permeability is 10⁻¹² to 10⁻¹⁰ m², its pore Reynolds number is around 10⁻⁴, and Darcy’s law is exact for every purpose. The interesting quantities are travel times and they are interstitial velocities divided into distances.

A catalyst bed is millimetre pellets in a steel tube with gas moving through it at metres per second. Its permeability is 10⁻⁸ m², its pore Reynolds number is in the hundreds, and Darcy’s law is worthless there — which is the subject of the next rung.

Between the two lie filter beds, chromatography columns, blast furnaces, soil drainage, fuel-cell electrodes and the fibrous reinforcement a composite is infused through. The same three parameters describe all of them: a porosity, a grain size and a fluid.

That a single expression spans twelve orders of magnitude in permeability is the argument for the averaged model despite everything it throws away. What is lost is every question about an individual pore, and what is bought is that the answer does not depend on any of them.

Who found it, and when #

Henry Darcy was the municipal engineer of Dijon, and his 1856 report on the town’s water supply contains an appendix on filtration through sand — the experiment, the linear relation, and the constant. He was not looking for a law of nature; he was sizing filter beds.

Kozeny gave the hydraulic-radius derivation in 1927 and Carman refined it in 1937, and the constant that carries both their names is the 180 in the denominator. It is the site’s own arithmetic that turns 180 into a statement about a tortuosity of √2.5 rather than about a fitted number, and it is a reminder that a constant with a name is not automatically a constant with a measurement behind it.

The one figure that would be dishonest #

It is worth saying explicitly what this rung declines to draw, because the omission is the most visible thing about it.

Every other essay in this collection has a solved field in it — streamlines through something, contours of something, a velocity computed everywhere. There is no such figure here and there could not be an honest one. The flow between grains is a real, three-dimensional, geometrically complicated problem, it has been computed by other people at enormous cost, and nothing on this site touches it.

A drawing of streamlines threading between spheres would be a picture of somebody’s imagination with the site’s own visual grammar around it — which is exactly the failure the site’s first invariant is written against, and exactly the failure that makes a smooth picture of a wrong field so dangerous.

So the bed is drawn as grains with no flow in them, and every caption says the pore-scale flow is not computed. That is a less impressive page and a more honest one.

Where the ladder goes next #

A linear resistance law has to fail eventually, because accelerating fluid into and out of every pore costs energy that has nothing to do with viscosity. Where it fails, and by how much before anyone notices, is a single dimensionless number computed from Ergun’s own constants.