Why One Settling Formula Is Not Enough

Stokes’ law is the settling formula most people learn first: fall velocity is proportional to the square of grain radius, so a grain twice as wide settles four times as fast. That relationship only holds while the grain is small enough that water flows smoothly around it, with no wake or eddies forming behind it, roughly clay and silt. Once grains reach sand size, real settling velocities fall well below what Stokes’ law predicts, because turbulence behind the grain adds drag the simple formula never accounts for.

Ferguson and Church addressed this directly in their 2004 paper, deriving a single equation that reduces to Stokes’ law for fine grains and to a constant drag coefficient for coarse ones, with a smooth transition in between covering fine sand through gravel. That is the equation this calculator uses as its primary result, with plain Stokes’ law shown alongside it for comparison so you can see exactly where the two start to diverge.

Verified Against the Published Equation

The constants built into this calculator, C1 = 18 for both regimes, and C2 = 0.4 for smooth spheres or 1.0 for natural sand grains, are the exact values Ferguson and Church recommend in their original paper for quartz-density sediment settling in water. Their smooth-sphere case (C1 = 18, C2 = 0.4) is asymptotically identical to Stokes’ law at the fine end and to a drag coefficient of 0.4 at the coarse end, matching the classical textbook limits on both sides. You can read the full derivation and the experimental validation against natural river sand in the original paper: Ferguson & Church, 2004, A Simple Universal Equation for Grain Settling Velocity, Journal of Sedimentary Research, 74(6), 933 to 937.

Reading the Reynolds Number

The grain Reynolds number tells you which physical regime a grain is actually settling in, and therefore how much the two formulas should agree. Below a Reynolds number of 1, flow around the grain is laminar and Stokes’ law is essentially exact. Between 1 and 1000, flow is transitional and both drag terms matter, which is exactly where the Ferguson-Church equation earns its keep over the simpler formula. Above 1000, flow is fully turbulent, and settling velocity depends on the square root of grain size rather than the square, a completely different relationship than Stokes’ law describes.

Common Mistakes

• Applying Stokes’ law directly to sand or gravel without checking the Reynolds number first, which overstates settling velocity substantially.
• Using the smooth sphere shape setting for real, angular natural sediment, which overstates settling velocity since real grains carry more form drag.
• Mixing up dynamic viscosity and kinematic viscosity, which are related by fluid density and are not interchangeable in the formula.
• Ignoring water temperature, since viscosity nearly doubles between 20 and 4 degrees Celsius and settling velocity scales directly with it.

Who Uses This

• Sedimentologists and geomorphologists modeling how sediment sorts by size in rivers, deltas, and coastal deposits.
• Environmental and civil engineers sizing settling basins or estimating how quickly suspended sediment clears from moving water.
• Geology students comparing Stokes’ law against a full-range settling equation across the clay-to-gravel spectrum.

Try It Above

Pick a grain size preset from clay through pebble, or enter a custom diameter, then check the Reynolds number and regime note to see whether Stokes’ law alone would have been reliable for that grain size.

Related Calculators and References

Grain diameter here follows the same Wentworth size classes as the Grain Size Converter, useful if you are working from a phi-scale grain size instead of millimeters. Grain density presets match the values used in the Rock Density Calculator. For the full derivation and experimental validation of the settling equation used here, see Ferguson & Church, 2004.