Stokes’ law is the one equation that explains why some solids fall out of a mud on their own and others never will — and why you need a centrifuge to catch the ones that won’t. It gives the settling velocity of a particle in a fluid, and buried in it is a single term, the particle diameter squared, that decides the whole solids-control game. Here is the formula, what it means for fine drilled solids, and how a centrifuge uses the very same law scaled up thousands of times.
The Stokes settling-velocity formula
For a small particle settling slowly through a fluid, Stokes’ law gives the terminal (settling) velocity as v = g × d² × (ρp − ρf) / (18 × μ), where g is gravitational acceleration, d is the particle diameter, ρp and ρf are the particle and fluid densities, and μ is the fluid viscosity. Read it and the levers are plain: heavier particles settle faster, thicker or denser mud settles them slower, and — the term that dominates everything — settling velocity rises with the square of the diameter.
That d² is the whole story. It means a particle isn’t twice as slow to settle when you halve its size — it’s four times slower. The law holds in the creeping-flow regime where drilling fines actually live (very low Reynolds number), which is exactly why it is the right tool for reasoning about silt- and clay-sized solids. The particle size it implies for a given settling rate — the Stokes diameter — is the standard way separation equipment is characterised. Putting that relationship to work on your solids is one of the routine calculations Rig IQ handles.
Why fine drilled solids won’t settle out
Now apply the d² term to real solids. A 100-micron sand grain settles quickly — leave the mud in a pit and it drops out. Shrink that to a 10-micron silt particle and, all else equal, it settles roughly a hundred times slower; take it down to colloidal clay a micron across and it effectively never settles at all — it stays suspended in the active system indefinitely. This is the physical reason fine drilled solids are the persistent enemy: gravity simply cannot remove them on any useful timescale.
It also explains why you cannot dilute or settle your way out of a fines problem, and why particle degradation is so damaging: every time a removable coarse particle is ground finer, its settling velocity collapses by the square of the size change, and it slips below what gravity or a pit can catch. Once solids fall into that range, only a device that supplies far more than 1 g — a centrifuge — can bring them out, which is precisely the case for removing them early, before they degrade. Trending how much of your load has crossed into the un-settleable range is what Rig IQ is built to flag.
Stokes’ law in the centrifuge
Here is where the law turns into equipment. Stokes’ velocity is proportional to the acceleration acting on the particle — and in a still pit that acceleration is just 1 g. A centrifuge replaces g with the centrifugal acceleration of the spinning bowl, the relative centrifugal force (RCF), which can be two or three thousand times gravity. Substitute an RCF of, say, 2,000 g for the single g in the formula and the settling velocity jumps by the same factor: particles that would take days to drop in a pit fall out in the bowl in a fraction of a second.
That is the entire principle of centrifugal separation in one substitution. It is why a decanter centrifuge can pull fine solids out of a mud that would never clear by gravity, and why its cut point is set by the G-force it develops — higher RCF reaches smaller Stokes diameters. It also connects the whole removal train: shakers and cyclones handle the coarse solids that Stokes’ law says settle easily, and the centrifuge is reserved for the fines that only a large multiple of g can touch. Reading the equipment through Stokes’ law — matching the force to the size you need to catch — is exactly how Rig IQ reasons about separation.
Stokes' law, in one line
v = g × d² × (ρp − ρf) / (18 × μ) — settling velocity from particle size, density difference and viscosity.
Settling velocity ∝ d² — halve the diameter and it settles four times slower. A 10 µm particle settles ~100× slower than a 100 µm one.
So fine drilled solids won’t settle out — gravity can’t remove them; only a centrifuge can.
In a centrifuge, g → RCF (2,000–3,000× g) — settling speeds up by the same factor. That’s the whole principle.
Stokes' law gives settling velocity as v = g × d² × (ρp − ρf)/(18μ). Because it depends on the square of particle diameter, halving the size makes a particle settle four times slower — so fine drilled solids (silt and clay) won't settle out under gravity and can only be removed mechanically. A centrifuge replaces g with its relative centrifugal force (RCF, 2,000–3,000× g), speeding settling by the same factor — which is the whole principle of centrifugal separation and why higher G reaches smaller particles.
Common questions
What is the Stokes' law settling velocity formula?
Stokes' law gives the terminal settling velocity as v = g x d2 x (rho_p - rho_f) / (18 x mu), where g is gravitational acceleration, d is the particle diameter, rho_p and rho_f are the particle and fluid densities, and mu is the fluid viscosity. It applies to small particles settling slowly (creeping, low-Reynolds-number flow), which is the regime drilling fines occupy.
Why won't fine drilled solids settle out of the mud?
Because settling velocity depends on the square of particle diameter. A 10-micron silt particle settles roughly 100 times slower than a 100-micron sand grain, and a 1-micron clay particle effectively never settles. Gravity simply cannot remove particles in that fine range on any useful timescale, which is why they accumulate in the active system and must be taken out mechanically rather than by settling or dilution.
How does a centrifuge use Stokes' law?
Stokes' settling velocity is proportional to the acceleration on the particle, which in a still tank is just 1 g. A centrifuge replaces gravity with its relative centrifugal force (RCF) — often 2,000 to 3,000 times g — so the settling velocity increases by the same factor. Particles that would take days to settle in a pit drop out in the spinning bowl almost instantly, and a higher G-force reaches smaller particle (Stokes) diameters.
