Learn: Darcy's law — seepage
Groundwater flow through a porous soil is often proportional to hydraulic gradient in the laminar regime. Darcy’s law gives total discharge through the gross cross-sectional area normal to flow.
What the formula is saying
First multiply hydraulic conductivity k by gradient i to obtain Darcy discharge velocity. Multiplying by the gross flow area A then gives total volume per time; the pore-water velocity is a different, larger quantity.
Read the symbols in plain language
- k
- Hydraulic conductivity
The soil/aquifer permeability parameter relating Darcy discharge velocity to hydraulic gradient under the stated flow conditions.
m/sUse m/s as the base unit shown here. k is m/s, i is head loss divided by flow length and has no unit, and A is m². Discharge is m³/s. k here is hydraulic conductivity, not intrinsic permeability in m².
- i
- Hydraulic gradient
Hydraulic head drop divided by the flow-path length; the head includes elevation and pressure contributions.
ratio / no unitA dimensionless ratio has no physical unit; 0.01 as a ratio is 1% when the percent option is selected.
- A
- Flow area
Flow area. The bulk area normal to flow converts discharge velocity into volumetric flow rate.
m²Square metres measure area; square the length conversion factor.
- Q
- Result to find
Darcy's law — seepage. The bulk area normal to flow converts discharge velocity into volumetric flow rate.
m³/s
Sort out the units first
k is m/s, i is head loss divided by flow length and has no unit, and A is m². Discharge is m³/s. k here is hydraulic conductivity, not intrinsic permeability in m².
Assumptions before calculating
Assume steady laminar Darcy flow through a representative saturated porous medium. Inputs here are nonnegative magnitudes; flow direction must be identified from decreasing total hydraulic head.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
Read the supplied values as one complete study case. Find Q and explain the result in the stated output unit.
- k · Hydraulic conductivity
- 0.00001 m/s
- i · Hydraulic gradient
- 0.2
- A · Flow area
- 10 m²
Find Darcy discharge velocity
Conductivity times hydraulic gradient gives discharge per gross area.
(0.00001) × (0.2) = 0.000002 m/sMultiply by gross flow area
The bulk area normal to flow converts discharge velocity into volumetric flow rate.
(0.000002) × (10) = 0.00002 m³/s
Does this worked answer make sense?
Zero head gradient gives zero Darcy discharge. Doubling flow area doubles total discharge but does not change Darcy velocity for the same k and i.
A second worked example — different values
A second case uses different data. Predict which way the answer will change, then calculate it without reusing the first answer.
- k · Hydraulic conductivity
- 0.00002 m/s
- i · Hydraulic gradient
- 0.15
- A · Flow area
- 8 m²
Find Darcy discharge velocity
Conductivity times hydraulic gradient gives discharge per gross area.
(0.00002) × (0.15) = 0.000003 m/sMultiply by gross flow area
The bulk area normal to flow converts discharge velocity into volumetric flow rate.
(0.000003) × (8) = 0.000024 m³/s
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
Solve this separate case yourself. Use only the values below; the two worked examples use different data. Give the requested result in the selected unit.
- k · Hydraulic conductivity
- 0.000008 m/s
- i · Hydraulic gradient
- 0.25
- A · Flow area
- 12 m²
Find: Learn: Darcy's law — seepage
A hint, not the answer
First multiply hydraulic conductivity k by gradient i to obtain Darcy discharge velocity. Multiplying by the gross flow area A then gives total volume per time; the pore-water velocity is a different, larger quantity.
k is m/s, i is head loss divided by flow length and has no unit, and A is m². Discharge is m³/s. k here is hydraulic conductivity, not intrinsic permeability in m².
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Find Darcy discharge velocity
Conductivity times hydraulic gradient gives discharge per gross area.
(0.000008) × (0.25) = 0.000002 m/sMultiply by gross flow area
The bulk area normal to flow converts discharge velocity into volumetric flow rate.
(0.000002) × (12) = 0.000024 m³/s
Avoid the common trap
Do not use pore area instead of gross area unless the velocity definition is changed consistently. Do not confuse hydraulic gradient with ground-surface slope or use pressure difference without converting to head.
When this method applies — and when it does not
Unsaturated flow, very high gradients, turbulence, anisotropy and heterogeneous layering may require other conductivities or flow models. The expression does not calculate seepage force or erosion stability.
For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Reading focus: Darcy's law — seepage. Read the relevant soil phase, seepage, earth-pressure, settlement or foundation topic. Effective stress, drainage and idealized geometry determine whether the relationship applies.
Lesson updated: · Both examples and the separate practice case are checked against an independent high-precision numerical implementation. This verifies arithmetic for the stated model, not engineering certification.