Where Darcy stops

Darcy’s law is linear: double the pressure gradient and the flow doubles. That linearity is what makes groundwater hydrology tractable and it is why a filter’s specification can be a single number.

It cannot be true at all speeds. Fluid entering a pore has to be accelerated and fluid leaving one has to be decelerated, and neither costs anything that viscosity can be held responsible for — it is the same expense a pipe pays at every fitting, computed exactly for a sudden enlargement and quadratic in the velocity there as here.

So a bed’s resistance has two terms, and the interesting question is where they cross.

The correlation, and what is a fit in it #

Ergun’s equation is

$$\frac{dp}{dL} = \underbrace{\frac{150\,\mu(1-\varepsilon)^2 u}{\varepsilon^3 d^2}}_{\text{viscous}} + \underbrace{\frac{1.75\,\rho(1-\varepsilon)u^2}{\varepsilon^3 d}}_{\text{inertial}}$$

and it is a correlation: the constants 150 and 1.75 were fitted to a great many beds, and they are drawn on this site in the colour reserved for a borrowed claim wherever they appear.

The form, though, is not fitted. The first term is Kozeny–Carman with the 180 tuned down to 150 — a fifth-order disagreement in nobody’s favour, which is worth knowing when a bed’s permeability is quoted to three figures. The second term’s dependence on ε, d, ρ and u is what a momentum argument gives for a flow accelerated through a constriction and abandoned, exactly as an orifice plate’s permanent loss is.

So what a fitting exercise supplied is two numbers, and what it did not supply is the structure.

The crossing, computed rather than quoted #

Setting the two terms equal cancels everything except a single dimensionless group:

$$\mathrm{Re}_p = \frac{\rho u d}{\mu(1-\varepsilon)} = \frac{150}{1.75} = 85.71$$

The site finds this by root-finding on one bed and then checks it on beds sharing no property — a different porosity, a different grain size, a different fluid — where the two terms must again be equal at the same Reynolds number. They are, to a part in a million.

That check is the whole claim of a dimensionless grouping, and it is asserted far less often than it is stated. Writing a quantity as a function of Re_p asserts that two beds with the same Re_p behave identically however different they are, and verifying it costs four lines.

The definition matters as much as the value. Re_p here is Ergun’s own — the superficial velocity, the grain diameter, and a (1 − ε) in the denominator — and a pore Reynolds number defined some other way crosses somewhere else. This is why “Darcy’s law holds up to Reynolds ten” appears in the literature with values from 1 to 100: the numbers are consistent and the definitions are not.

The two thresholds, and why both are quoted #

The site’s arithmetic produces two numbers, and the gap between them explains a confusion.

Re_p ≈ 10 is where the inertial term first carries about a tenth of the gradient. That is where a careful experimenter notices the departure from linearity — where a plot of flow against gradient visibly bends — and it is the origin of the practical advice.

Re_p = 85.7 is where the two terms are equal. Past it the bed is inertia-dominated and the resistance goes essentially as the square of the flow.

Neither is a transition in the physical sense: nothing changes state, nothing becomes unstable, and the flow in the pores is still laminar at both. What changes is which of two terms in a sum is larger, which is exactly what a regime boundary is on this site — the Reynolds number itself is the ratio of two terms in the momentum equation, and the bands it defines are where one overtakes the other.

Why it is not turbulence #

The standard explanation for the departure is that the flow in the pores becomes turbulent, and it is wrong in a way worth setting out, because the same confusion appears elsewhere in the subject.

Re_p = 10 is a very small Reynolds number. A pipe holds its exact laminar solution to Re ≈ 2300; a boundary layer transitions somewhere around 500,000 on the length-based number. Nothing in a bed is turbulent at ten, and the pressure drop has already departed from linear by a tenth.

What has arrived is not turbulence but inertia. The fluid is being accelerated into a constriction and decelerated out of it, thousands of times over, and each acceleration costs a pressure difference proportional to ρu² that has no viscosity in it. The evidence is right there in the correlation: the second term contains ρ and does not contain μ.

The distinction is the same one the site draws about the sudden expansion in a pipe: the loss is dissipated by viscosity in the end, because energy has to become heat somewhere, but its size is fixed by a momentum balance in which viscosity does not appear. Attributing it to turbulence is attributing the amount to the mechanism that finally converts it.

Real turbulence does arrive in a bed, at pore Reynolds numbers in the hundreds, and adds nothing qualitatively new: the resistance is already quadratic by then and it stays quadratic.

Where this leaves a design #

The practical consequence is that a bed has two operating regimes with different economics.

Below ten, the pressure drop is proportional to the flow, so the pumping power — pressure times flow — goes as the square. Doubling the throughput quadruples the power. This is the world of groundwater, of slow filtration, of chromatography.

Above eighty-six, the pressure drop goes as the flow squared and the power as the cube. Doubling the throughput multiplies the power by eight. This is the world of packed towers, of blast furnaces, of catalytic reactors — and it is why the pressure drop across an industrial packed bed is often the dominant operating cost rather than a detail.

That is the reason industrial packings look nothing like filter sand. A packing element is chosen for its void fraction first and its surface second, because ε³ in the denominator punishes a dense packing by an order of magnitude, and because a bed in the quadratic regime pays for its own tightness three times over.

The same crossing on a different bed #

The claim that the crossover is a property of one number and of nothing else deserves the picture as well as the assertion.

Reading the two together is the strongest form of the argument. It would be entirely possible to have a correlation whose two terms crossed at different places on different beds — that is what a fitted crossover would look like — and the check is that they do not.

The other end, where a different number decides #

Everything above is the fast end. The linear law has a slow end too, and it fails there for a reason that has nothing to do with inertia — which makes the point that a range of validity has two boundaries and they need not be governed by the same group.

Push a gas rather than a liquid through a fine enough medium and the no-slip condition stops holding, because the pore is no longer large compared with the distance a molecule travels between collisions. The gas slides along the pore wall, more of it gets through for the same gradient, and the permeability that comes out of the measurement is too high.

The consequence is one of the more disconcerting facts in the trade: the same core plug returns a different permeability depending on the pressure it was measured at. Lowering the pressure lengthens the mean free path, which increases the slip, which raises the apparent permeability — so the “property” of the rock moves with the apparatus. Klinkenberg’s fix in 1941 was to measure at several pressures, plot against the reciprocal of the mean pressure, and extrapolate to infinite pressure, where the slip vanishes and what is left is the value a liquid would have given.

The governing group there is not a Reynolds number at all. It is the ratio of the mean free path to the pore size — the number that decides whether a fluid is a continuum — so Darcy’s law is bracketed above by inertia and below by molecular structure, with a different dimensionless quantity ending it at each end and neither saying anything about the other.

There is a second slow-end failure, reported in very fine clays and still argued about: a threshold gradient below which essentially nothing moves, attributed to water held ordered against charged mineral surfaces. That is not a continuum effect either.

What the model does not contain #

No pore-scale flow. As in the previous rung, nothing here resolves anything between grains, and the two terms are averaged statements. What that hides is the distribution of the flow: a bed with preferential channels has the same average resistance and a completely different residence time.

A single grain size, and spheres. Ergun’s constants were fitted mostly to spheres and near-spheres. Rings, saddles and structured packings have their own constants — sometimes with the same form and different numbers, sometimes with a different form — and the equation is not predictive across shapes.

Wall effects. A bed in a tube of only a few grain diameters has a looser packing at the wall than in the middle, so the flow short-circuits down the outside. The rule of thumb is that a column must be at least ten grains across for a bulk correlation to apply, and this is the bed’s own version of the blockage problem: the container is in the answer.

No compressibility. For a gas at a large pressure drop the density varies along the bed, so the superficial velocity does too, and the equation has to be integrated rather than applied — which matters in a blast furnace and not in a filter.

Why a linear law is worth so much #

The reason the crossover matters more than a twenty-per-cent change in resistance is that the two regimes are different mathematics, not just different numbers.

Darcy’s law plus mass conservation makes the pressure satisfy Laplace’s equation. That is the same equation the whole inviscid field solves, with the same toolbox: superposition, images, conformal maps, and the fact that a solution is unique. A well is a sink, a recharge boundary is a mirror, and a groundwater problem with six wells is the sum of six solutions.

Add the quadratic term and every one of those goes. Superposition fails first — two solutions of a nonlinear equation do not add — and with it goes the whole apparatus. A bed in the Forchheimer regime has to be solved numerically, one configuration at a time, and nothing carries over from one geometry to another.

That is why the crossover is drawn as a boundary rather than as a correction. Below it, a subject with a century of exact solutions in it; above it, an arithmetic problem. The number that separates them is 85.7 and the thing it separates is a method.

What a bed shares with a wing #

There is a structural parallel with the rest of the site that is worth drawing, because it explains why this rung has the shape it does.

A bed’s resistance has a viscous term and an inertial one, and which dominates is set by a Reynolds number. A body in a stream has a friction drag and a pressure drag, and which dominates is set by a Reynolds number. In both cases the low-Reynolds limit is linear in the velocity and exactly solvable, the high-Reynolds limit is quadratic and needs a correlation, and the crossover is a band rather than a point.

What is different is where the honesty lies. For a body, the site can solve the low-Reynolds end exactly, including the paradox that Stokes flow past a cylinder has no solution at all, and the high end is measured. For a bed, neither end is solved: the low end is a model with a forced constant and the high end is a fit. That is a weaker position and it is the honest one, and it is why every figure in these three essays says which half of the arithmetic it is standing on.

Two constants, one of which is doing something odd #

There is an oddity worth recording rather than smoothing over.

Kozeny–Carman gives 180 in the viscous denominator; Ergun fits 150. Those are the same term with a twenty per cent disagreement, and it is not obviously a matter of measurement error: 180 comes from a hydraulic-radius argument whose tortuosity is forced to be √2.5, and 150 comes from fitting real beds, and the two ought to agree if the model were right.

The honest reading is that the model is a good structure with a soft constant, and the twenty per cent is what the idealisation of a pore space as tilted tubes costs. It is the kind of discrepancy that is worth stating in both directions: nobody should quote a bed permeability to three significant figures, and the form of the dependence — ε³/(1 − ε)², d² — is far better established than the number in front of it.

Reading the crossover as a competition of two lengths #

There is a second way to see where the linear law goes, and it is worth having because it needs no correlation at all.

The viscous term is a shear stress spread over the grain surface; the inertial one is a dynamic pressure applied over the grain’s frontal area. Their ratio is therefore a Reynolds number, and the crossover is where a viscous length and an inertial length coincide.

That is the same structure as the boundary-layer argument: viscosity is confined to a layer whose thickness falls as the Reynolds number rises, and the flow outside it is governed by inertia. In a bed the “outside” is the pore’s core and the “layer” is the film on each grain, so the departure from Darcy’s law is the moment the pore stops being all boundary layer.

The estimate that follows is crude and instructive. A boundary layer on a grain has thickness of order d/√Re; setting that equal to the pore size ε d/(1 − ε) gives a Reynolds number of order tens, which is the observed crossover to within the accuracy such an argument deserves. Nothing was fitted and nothing was measured; the answer came out of a comparison of two lengths, which is what a dimensionless argument is.

Who found it, and when #

Philipp Forchheimer added the quadratic term to Darcy’s law in 1901, which is why the departure is called Forchheimer flow; his interest was in wells, where the highest velocities in an aquifer occur right at the borehole and the linear law fails first. Sabri Ergun published the two-term correlation in 1952 in a paper about blast-furnace coke beds, and it has been the standard ever since — the constants have been re-fitted many times and moved by less than ten per cent.

The pattern of the two men’s problems is worth noticing. Forchheimer met the nonlinearity where a large flow is forced through a small area; Ergun met it as the ordinary operating condition of an industrial bed. The same equation is a correction in one trade and the main term in the other, which is what having a dimensionless number for the crossover is for.

The practical reading, in one paragraph #

For anybody sizing something rather than reading for pleasure, the whole essay reduces to three statements.

Compute Re_p = ρud/μ(1 − ε) first. Below about ten, use Darcy’s law and expect a few per cent; the pressure drop is linear in the flow and the pumping power is quadratic. Above about eighty-six, the resistance is essentially quadratic and the power cubic, so a small increase in throughput is an expensive one. Between them use both terms, which is what Ergun’s equation is for, and do not expect better than ten per cent from anybody’s constants.

The rest of this essay is the argument for why those three sentences are the right ones, which matters because the same three sentences with different numbers in them are wrong for a differently-defined Reynolds number, and the literature contains several.

Where the ladder goes next #

Push the flow harder still and something qualitatively new happens, and it is not turbulence either. The pressure drop across the bed cannot exceed the bed’s own weight — that is a control volume weighing a pile of sand — and at the velocity where the two meet, the bed stops being a structure and starts being a fluid.