Darcy's Law Calculator
Use Darcy's Law to calculate how fast a fluid moves through a porous medium such as sand, gravel, or soil. Enter any three of the four variables and the calculator solves for the fourth: flow rate (Q), hydraulic conductivity (k), hydraulic gradient (i), or cross-sectional area (A). Switch between metric and imperial units and expand the steps panel to see the full worked calculation.
Formula
Worked example
Fine sand aquifer: k = 1e-4 m/s, head drop = 1 m over L = 100 m, cross-section A = 50 m2. Gradient i = 1/100 = 0.01. Flow rate Q = 1e-4 * 0.01 * 50 = 5e-5 m3/s = 0.05 L/s. Darcy velocity = 1e-4 * 0.01 = 1e-6 m/s.
What is Darcy's Law?
Darcy's Law describes the flow of a fluid through a porous medium such as sand, gravel, or soil. It was established by French engineer Henry Darcy in 1856 through sand-filter experiments and remains the foundation of groundwater hydrology and geotechnical seepage analysis. The law states that the volumetric flow rate Q through a porous material is proportional to the hydraulic gradient i (the head drop per unit length), the cross-sectional area A, and a material-specific constant called hydraulic conductivity k. In equation form: Q = k x i x A. All three proportionalities make physical sense: a steeper gradient, a wider flow path, or a more permeable material each produces a faster flow.
Inputs and what they mean
Hydraulic conductivity (k) captures how easily the fluid moves through the medium. Its value spans more than ten orders of magnitude: unfractured granite transmits almost nothing (around 1e-12 m/s) while open gravel allows very fast flow (around 0.01 m/s or higher). Fine sand, the most common aquifer material, typically falls near 1e-4 m/s. The hydraulic gradient (i) is the dimensionless ratio of head loss to flow-path length. For example, a water table that drops 1 m across a 100 m horizontal distance gives a gradient of 0.01. Natural aquifer gradients range from about 0.001 to 0.01. Cross-sectional area (A) is the face of the porous medium perpendicular to the direction of flow. In a laboratory permeameter it is the column cross-section; in a field study it might be the width times saturated thickness of an aquifer.
How to use the four solve-for modes
The calculator can solve for any one of the four quantities in Darcy's Law: flow rate Q, hydraulic conductivity k, hydraulic gradient i, or cross-sectional area A. Select the unknown from the 'Solve for' menu and enter the other three. For the gradient you can either type the gradient directly as a dimensionless ratio, or switch to 'head difference + path length' mode and enter the actual head drop and flow distance, and the calculator will compute i = delta-h / L for you. The hydraulic gradient and head/length inputs share the same length unit selector, so you can work in metres, centimetres, or feet throughout.
Darcy velocity and pore velocity
The calculator also reports Darcy velocity (also called specific discharge), defined as q = Q / A = k * i. This is NOT the true speed of water molecules moving through the pores. Darcy velocity treats the entire cross-section as if it were open, so it is lower than the real pore-water velocity by a factor equal to the porosity. To find the actual seepage velocity of a water parcel, divide Darcy velocity by porosity (n), which for sandy aquifers is typically 0.25-0.45. Pore velocity = Darcy velocity / n. This distinction matters in contaminant transport: pollutants travel at the pore velocity, not the Darcy velocity. Darcy's Law also assumes laminar flow, which holds when the Darcy Reynolds number (Re = rho * v * d / mu) is below about 1-10. In very coarse gravel or fractures the flow may turn turbulent and Darcy's Law under-predicts pressure loss.
Typical hydraulic conductivity by material
| Material | Hydraulic conductivity (m/s) | Flow character |
|---|---|---|
| Gravel | 1 x 10-2 | Very fast |
| Coarse sand | 1 x 10-3 | Fast |
| Fine sand | 1 x 10-4 | Moderate |
| Silty sand | 1 x 10-5 | Moderate-slow |
| Fractured rock | 1 x 10-6 | Slow |
| Silt / loess | 1 x 10-7 | Very slow |
| Glacial till | 1 x 10-9 | Extremely slow |
| Marine clay | 1 x 10-10 | Near-impermeable |
| Unfractured rock | 1 x 10-12 | Essentially impermeable |
Representative values only. Actual conductivity spans several orders of magnitude within each material depending on packing, grain size, and saturation. Source: Freeze & Cherry, Groundwater (1979).
Frequently asked questions
What does Darcy's Law calculate?
Darcy's Law calculates the volumetric flow rate of a fluid through a porous medium as the product of hydraulic conductivity, hydraulic gradient, and cross-sectional area: Q = k x i x A. Any one of the four variables can be the unknown: this calculator can solve for Q, k, i, or A depending on which three you know.
What is hydraulic conductivity and how does it differ from permeability?
Hydraulic conductivity (k) combines the properties of both the medium (intrinsic permeability) and the fluid (density and viscosity). It is quoted in units of velocity (m/s) and applies only to a specific fluid at a specific temperature. Intrinsic permeability (K, in m2 or darcy) is a property of the medium alone. They are related by k = K * rho * g / mu, where rho is fluid density, g is gravity, and mu is dynamic viscosity. For water at 20 degrees Celsius, 1 darcy corresponds to about 9.87e-13 m2 or roughly 1e-5 m/s hydraulic conductivity.
What is hydraulic gradient and how do I measure it?
Hydraulic gradient (i) is the change in hydraulic head per unit distance in the direction of flow, so it is dimensionless. In the field you measure it from two observation wells or piezometers: subtract the water-level elevations and divide by the horizontal distance between them. In a laboratory column test you read the head at inlet and outlet manometers. Typical natural aquifer gradients lie between 0.001 and 0.01; in engineered seepage problems (dams, filters) they can reach 0.1 or more.
When does Darcy's Law break down?
Darcy's Law assumes laminar, viscosity-dominated flow and holds when the Darcy Reynolds number (Re_D = rho * q * d / mu, where d is grain diameter) stays below roughly 1-10. It begins to fail in coarse gravel, fractured rock, or very high-gradient situations where turbulence develops. At the other extreme, in very tight media such as shales or unfractured crystalline rock, molecular effects can add non-Darcy losses. For turbulent flow through packed beds the Ergun equation or Forchheimer extension of Darcy's Law are more appropriate.
How do I find the pore velocity from Darcy velocity?
Pore velocity (also called linear velocity or seepage velocity) equals Darcy velocity divided by porosity: v_p = q / n, where q = Q / A is the Darcy velocity and n is the effective porosity of the medium. For a fine-sand aquifer with porosity 0.35 and Darcy velocity 1e-6 m/s, the pore velocity is about 2.9e-6 m/s. This is the speed at which dissolved contaminants or tracers actually travel through the aquifer.
What units does this calculator support?
For hydraulic conductivity you can choose m/s, cm/s, m/day, ft/s, or ft/day. For cross-sectional area you can choose m2, cm2, or ft2. For flow rate you can choose m3/s, m3/hr, m3/day, L/s, L/min, L/hr, gal/min, or ft3/s. For head and length (when computing gradient from head loss) you can choose m, cm, or ft. All unit conversions are applied internally before computing, so you can mix units freely.
Sources
- Freeze, R.A. & Cherry, J.A. (1979). Groundwater. Prentice-Hall. The definitive reference for hydraulic conductivity ranges by material type.
- Bear, J. (1972). Dynamics of Fluids in Porous Media. American Elsevier. Comprehensive derivation of Darcy's Law from the Navier-Stokes equations and its limitations.