Darcy’s Law Calculator
Compute discharge, specific discharge, and seepage velocity through a porous medium — and check whether Darcy’s law is still valid at your flow rate.
✓ Reviewed by a hydrogeologist (Ph.D.) SI and US customary units Reynolds-number validity check Sources: Darcy 1856 · Freeze & Cherry 1979
What to enter for what you want
Every derived quantity is computed on every keystroke, so Solve for does not decide what you can see — it only decides which variable is treated as the unknown. Velocity, travel time, and the Reynolds number are always there.
| To find | Enter | Read it in |
|---|---|---|
| Discharge through a section | K, Δh, L, A | Headline, plus L/min and gal/min |
| How fast the water actually moves | K, Δh, L, n | Seepage velocity v |
| When a tracer arrives | K, Δh, L, n | Travel time over L |
| Conductivity from a measured flow | Q, Δh, L, A — set Solve for to K | Headline |
| The head loss a flow implies | Q, K, L, A — set Solve for to Δh | Headline |
Two inputs are narrower than they look. Porosity n affects only the seepage velocity and the travel time — it has no part in the discharge, because Darcy’s law is written on the gross area. Grain size d₁₀ feeds only the Reynolds check. Leaving either blank stops just those quantities, not the rest.
What Darcy’s law says
In 1856 Henri Darcy, the municipal engineer responsible for the water supply of Dijon, ran water through columns of sand and found that the discharge was proportional to the head loss and inversely proportional to the length of the column. That single empirical result underpins essentially all of quantitative hydrogeology.
Written as a differential, the law is
Q = -KA\,\frac{dh}{dl}
Where the minus sign goes
The minus sign is not decoration, and it does not simply disappear. Hydraulic head decreases in the direction water travels, so the derivative dh/dl is itself negative; the minus in front of it is what makes the discharge come out positive. Together they state the physics: water runs downhill in head.
The calculator, though, does not ask for a derivative. It asks for a head drop measured across a column. Over a finite length L, with head h_1 at the upstream manometer and h_2 at the downstream one, the derivative becomes a difference:
\frac{dh}{dl} \;\approx\; \frac{h_2 - h_1}{L} \;=\; -\frac{\Delta h}{L} \qquad\text{where}\qquad \Delta h \equiv h_1 - h_2 \;>\; 0
Defining Δh as a positive drop is what moves the sign. That definition already contains the minus that dh/dl carried, so on substitution the two cancel:
Q \;=\; -KA\left(-\frac{\Delta h}{L}\right) \;=\; KA\,\frac{\Delta h}{L}
This is the form the calculator evaluates. The sign was never dropped — it was absorbed into the definition of Δh, which is why entering a positive head difference returns a positive discharge.
The variables
| Symbol | Quantity | SI unit | Note |
|---|---|---|---|
| Q | Discharge (volumetric flow rate) | m³/s | Total volume of water crossing area A per unit time |
| K | Hydraulic conductivity | m/s | Property of both the medium and the fluid |
| A | Cross-sectional area | m² | Gross area — solids included, not just the pores |
| i | Hydraulic gradient, Δh/L | – | Dimensionless |
| q | Specific discharge (Darcy flux), Q/A | m/s | Has units of velocity but is not a velocity |
| v | Average linear (seepage) velocity, q/n | m/s | What a tracer particle actually travels at |
| n | Porosity (effective) | – | Fraction of the volume that conducts flow |
Why q and v are not the same number
This is the single most common error in applied work. The Darcy flux q spreads the discharge over the whole cross-section, as though the solid grains were not there. Water can only move through the pores, so the real velocity is larger by a factor of 1/n — typically three to four times. Use q for flow budgets; use v for travel time and contaminant arrival.
Typical hydraulic conductivity values
Ranges after Freeze & Cherry (1979, Table 2.2). Note the spread: fourteen orders of magnitude separate gravel from unfractured clay, which is why K is almost always the dominant uncertainty in a groundwater calculation. Click any row to load its representative value into the calculator.
| Material | K (m/s) | K (m/day) | Character |
|---|---|---|---|
| Gravel | 10⁻³ – 10⁻¹ | 86 – 8 600 | Highly productive aquifer |
| Coarse sand | 10⁻⁴ – 10⁻² | 8.6 – 860 | Productive aquifer |
| Medium sand | 10⁻⁵ – 10⁻³ | 0.86 – 86 | Good aquifer |
| Fine sand | 10⁻⁶ – 10⁻⁴ | 0.086 – 8.6 | Modest yield |
| Silt, loess | 10⁻⁹ – 10⁻⁵ | 10⁻⁴ – 0.86 | Leaky confining bed |
| Glacial till | 10⁻¹² – 10⁻⁶ | 10⁻⁷ – 0.086 | Aquitard |
| Marine clay (unweathered) | 10⁻¹³ – 10⁻⁹ | 10⁻⁸ – 10⁻⁴ | Aquiclude |
| Karst limestone | 10⁻⁶ – 10⁻² | 0.086 – 860 | Conduit flow — Darcy may not hold |
When Darcy’s law breaks down
Darcy’s law is a linear approximation valid while flow stays laminar. The standard diagnostic is a pore-scale Reynolds number built on the Darcy flux and a representative grain diameter:
Re = \frac{\rho\,q\,d}{\mu}
with ρ = 1000 kg/m³ and μ = 1.00×10⁻³ Pa·s for water at 20 °C. The law holds comfortably for Re below about 1, and departures become serious somewhere between 1 and 10 (Bear, 1972). Above that the relationship between gradient and flux turns non-linear and the Forchheimer equation is the appropriate replacement.
In practice you cross this threshold near pumping-well screens, in coarse gravel packs, and in karst conduits — which is precisely where people most often apply Darcy’s law without checking. The calculator flags this for you.
Worked example
Problem. A confined sand aquifer 2.0 m thick and 500 m wide has a hydraulic conductivity of 3.0×10⁻⁴ m/s. Two wells 800 m apart along the flow direction show heads of 41.2 m and 39.6 m. Find the discharge through the aquifer cross-section and the time for a tracer to travel between the wells. Effective porosity is 0.28.
- Gradient:
i = (41.2 − 39.6) / 800 = 1.6 / 800 = 2.0×10⁻³ - Area:
A = 2.0 m × 500 m = 1000 m² - Discharge:
Q = K A i = 3.0×10⁻⁴ × 1000 × 2.0×10⁻³ = 6.0×10⁻⁴ m³/s ≈ 51.8 m³/day - Darcy flux:
q = K i = 6.0×10⁻⁷ m/s - Seepage velocity:
v = q / n = 6.0×10⁻⁷ / 0.28 = 2.14×10⁻⁶ m/s ≈ 0.185 m/day - Travel time:
t = 800 / 0.185 ≈ 4 320 days ≈ 11.8 years
Had we used the Darcy flux instead of the seepage velocity, the answer would have been 33 years — an error of a factor of 3.6, entirely from forgetting to divide by porosity.
Frequently asked
Is hydraulic conductivity the same as permeability?
No. Intrinsic permeability k (units m², or darcy) is a property of the porous medium alone. Hydraulic conductivity folds in the fluid: K = kρg/μ. Warm water is less viscous, so the same sand has a higher K at 30 °C than at 5 °C — roughly a factor of two across that range. This matters for geothermal and for seasonal recharge studies.
Does Darcy’s law work in unsaturated soil?
In modified form. Conductivity becomes a steep function of water content, K(θ), and the head includes matric suction. That extension is the Richards equation, and it is markedly harder to solve because the governing equation is non-linear.
What area do I use for a well?
Flow into a well is radial, not linear, so the area changes with distance from the well. Integrating Darcy’s law in radial coordinates gives the Thiem equation for steady flow, and the Theis solution for transient flow.
Read further
The long-form article behind this tool is Darcy’s Law & Hydraulic Head Explained (with FloPy / MODFLOW), part of the Groundwater Science series. It derives the law from the sand-column experiment and then solves the same problem numerically in MODFLOW.
References
- Darcy, H. (1856) Les Fontaines Publiques de la Ville de Dijon. Victor Dalmont, Paris.
- Bear, J. (1972) Dynamics of Fluids in Porous Media. Elsevier, New York.
- Freeze, R.A. & Cherry, J.A. (1979) Groundwater. Prentice-Hall, Englewood Cliffs, NJ. Table 2.2.
- Fetter, C.W. (2001) Applied Hydrogeology, 4th ed. Prentice-Hall, Upper Saddle River, NJ.