Hazen Permeability Calculator: k from Effective Grain Size D10

Allen Hazen was designing sand filters for water treatment plants in the 1890s when he noticed something useful: the hydraulic performance of his filter sands tracked one number, the grain size at which ten percent of the material passes. The physical logic has aged well. In a clean sand, the small grains sit in the throats between the big ones, so the finest fraction, not the average size, sets the diameter of the channels water actually squeezes through. His rule is disarmingly simple:

k (cm/s) = C x D10^2 (D10 in mm, C an empirical constant near 1.0)

Worked example

• A clean filter sand has D10 = 0.15 mm from its sieve curve
• k = 1.0 x 0.15^2 = 0.0225 cm/s, or about 2.3 x 10^-4 m/s
• In groundwater units, roughly 64 ft/day: a fast, freely draining sand
• Doubling D10 quadruples the estimate; the square is why grain size dominates everything else in the formula

The validity window, taken seriously

Hazen calibrated on loose, clean, fairly uniform filter sands with D10 between roughly 0.1 and 3 mm and uniformity coefficients below about 5, and outside that window the rule fails in predictable directions. Add fines and reality falls far below the estimate, because silt and clay plug exactly the throats D10 is supposed to describe; a few percent fines can cost one to two orders of magnitude. Go coarser than a few millimeters and flow starts leaving the laminar regime the rule silently assumes. Go finer than 0.1 mm and surface forces and adsorbed water make grain size an increasingly poor proxy altogether. Density also matters and is absent from the formula: the same sand compacted dense drains meaningfully slower than loose, which is part of why published C values scatter from about 0.4 to 1.5 depending on the study.

Used inside its lane, the rule is a legitimately good first estimate, good enough to size a temporary dewatering system for a screening study or to sanity-check a lab result that looks off by a factor of a thousand (a surprisingly common lab bookkeeping error that D10-squared catches instantly). The professional habit is to quote it as an order-of-magnitude estimate, state the C you used, and let a permeameter test or pumping test carry the design when consequences are real.

Related Calculators and Reference

See also the Permeability Converter for moving the Hazen result between darcy, m/s and cm/s, with the water temperature stated rather than assumed.

The D10 input comes straight from the gradation analysis behind the Cu and Cc calculator, grain sizes convert in the grain size converter, and the reservoir-scale cousin of this estimate is the Timur and Coates-Denoo permeability calculator. The USGS groundwater primer on aquifers and groundwater gives context for the k values.