Engineering & Science
Kozeny-Carman Porous Media Permeability Formula (Darcies & m^2)
Calculate intrinsic porous-bed permeability k = ((phi^3) * (dp^2)) / (180 * ((1 - phi)^2)) in Darcies and square micrometers from porosity phi and effective grain diameter dp_mm.
Predict sandstone reservoir, HPLC column, and sand-filter permeability from porosity!
Why does compacting a sand bed or catalyst column from 45% porosity down to 38% porosity slash fluid flow by MORE than half? Josef Kozeny (1927) and Philip Carman (1937) modeled a porous bed of spherical grains of diameter dp as a bundle of tortuous capillary channels, deriving the foundational Kozeny-Carman permeability equation: k = ((phi^3) * (sphericity^2 * dp^2)) / (180 * ((1 - phi)^2))!
Look at that extraordinary porosity leverage phi^3 / (1 - phi)^2: for spherical 0.50 mm medium quartz sand (sphericity Psi = 1.00) at porosity phi = 0.38, intrinsic permeability is k = 200.89 Darcies (where 1 Darcy = 9.869233e-13 m^2); loosening the bed slightly to phi = 0.45 more than DOUBLES permeability to 423.93 Darcies—without changing grain size at all!
| Packed Sand / Catalyst Bed (dp = 0.50 mm) | Porosity phi & Sphericity Psi | Permeability k (Darcies) | Porosity Scaling phi^3 / (1-phi)^2 |
|---|---|---|---|
| Compacted Quartz Sand | phi = 0.38, Psi = 1.00 | 200.89 Darcies | 0.05487 / 0.3844 = 0.1427 |
| Loose Uncompacted Sand | phi = 0.45, Psi = 1.00 | 423.93 Darcies (2.11x jump!) | 0.09113 / 0.3025 = 0.3012 |
| Angular Crushed Grains | phi = 0.38, Psi = 0.80 | 128.57 Darcies | Scales with Psi^2 = 0.64 |
Kozeny-Carman Intrinsic Permeability k in Darcies (1 Darcy = 9.869233e-13 m^2)
Converts grain diameter B3 from mm to meters (* 0.001), applies sphericity B4, and converts m^2 to Darcies.
=IF(OR(B2 <= 0, B2 >= 1, B3 <= 0, B4 <= 0, B4 > 1), "Invalid Porous Bed Inputs", ROUND((((B2 ^ 3) * ((B4 * B3 * 0.001) ^ 2)) / (180 * ((1 - B2) ^ 2))) / 9.869233e-13, 2))Specific Wetted Grain Surface Area per Unit Bed Volume a_v = 6 * (1 - phi) / (Psi * dp) (mm^-1)
Computes internal solid-fluid contact surface area per cubic millimeter of packed bed.
=IF(OR(B2 <= 0, B2 >= 1, B3 <= 0, B4 <= 0, B4 > 1), "Invalid Porous Bed Inputs", ROUND((6 * (1 - B2)) / (B4 * B3), 3))Try the example
Use the original values or change them to test your understanding.