Hong Life CalculatorsUnderstanding the Darcy's law seepage calculator — concept and principle
Darcy's law describes how slowly water moves through soil, sand or rock under a difference in water level. This calculator gives the seepage flow through a soil section, the hydraulic gradient and the Darcy (discharge) velocity. It is used for seepage under dams and cofferdams, inflow into excavations and flow through filters.
How it is calculated
- Hydraulic gradient: i = Δh/L (dimensionless).
- Darcy velocity: v = k·i, converted from m/s to m/day by multiplying by 86,400.
- Flow: Q = k·i·A, also shown per day (m³/day).
k is the hydraulic conductivity (often called the coefficient of permeability) in m/s, Δh the difference in total head between the two ends (m), L the length of the flow path (m) and A the gross cross-sectional area perpendicular to the flow (m²), solids and pores together.
The principle
In 1856 Henry Darcy found from sand-column tests in Dijon that flow rate is proportional to the head loss and to the area, and inversely proportional to the length of sand. For slow, laminar flow through small pores, viscous friction dominates and resistance grows in proportion to velocity, so the relation is linear. The constant k bundles together pore size, pore connectivity and the viscosity of water. It varies enormously: roughly 10⁻² to 10⁻⁴ m/s for clean sands and gravels, 10⁻⁵ to 10⁻⁷ m/s for silty sands and silts, and below about 10⁻⁹ m/s for clays.
Worked example
- Inputs: k = 1 × 10⁻⁵ m/s, Δh = 2 m, L = 10 m, A = 50 m².
- Gradient i = 2/10 = 0.2.
- Darcy velocity v = 1 × 10⁻⁵ × 0.2 = 2 × 10⁻⁶ m/s; × 86,400 = 0.1728 m/day.
- Flow Q = v × A = 0.1728 × 50 = 8.64 m³/day.
| Flow length (m) | Flow (m³/day) |
|---|---|
| 2.5 | 34.56 |
| 5 | 17.28 |
| 7.5 | 11.52 |
| 10 | 8.64 |
| 12.5 | 6.912 |
| 15 | 5.76 |
| 20 | 4.32 |
How to read the result
The Darcy velocity is a superficial velocity averaged over the whole area. Water actually travels only through the pores, so its real (seepage) velocity is v/n, where n is the porosity; with n = 0.35 the example water moves about 0.49 m/day. Gradients near 1 at an exit face in sands warn of piping or boiling, which is why seepage exits under dams are checked carefully.
Common mistakes
- Using cm/s values of k from a lab sheet as if they were m/s (a factor of 100 error).
- Taking Δh as a pressure head only instead of total head (elevation plus pressure).
Frequently asked questions
Is permeability the same as hydraulic conductivity?
In geotechnical practice k is called permeability, but strictly it is hydraulic conductivity, which depends on the fluid. Intrinsic permeability (m²) is a property of the soil alone.
How is k measured?
With constant-head or falling-head lab tests, or with field pumping tests.
How do I handle layered soil?
For flow parallel to layers use the thickness-weighted average of k; for flow across layers use total thickness divided by the sum of each thickness over its k.
Sources: H. Darcy, Les Fontaines Publiques de la Ville de Dijon, 1856 · J. Bear, Dynamics of Fluids in Porous Media, Elsevier, 1972 · ASTM D2434, Standard Test Method for Permeability of Granular Soils (Constant Head)