Geotechnical Resource

Darcy Flow and Hydraulic Gradient in Soil

Formulas and a worked example for hydraulic gradient, Darcy flux, volumetric flow and seepage velocity through soil.

Table of contents

Darcy flow is a simplified relationship for estimating water flow through a porous material. It connects hydraulic conductivity, hydraulic gradient and cross-sectional area.

The calculation is useful for checking one-dimensional seepage concepts and laboratory results. Field-scale groundwater flow may require a more detailed model of soil layering, anisotropy, boundaries and changing water levels.

Hydraulic gradient

Hydraulic gradient is the head difference divided by the flow length:

$$ i = \frac{\Delta h}{L} $$

Symbol	Meaning	Common unit
$i$	Hydraulic gradient	Dimensionless
$\Delta h$	Difference in total hydraulic head	m
$L$	Flow length	m

The head difference and flow length must use the same length unit.

The sign of $i$ depends on the selected direction convention. A magnitude-only calculator should also state the intended upstream-to-downstream direction.

Darcy flux

Darcy flux, sometimes called discharge velocity, is:

$$ q = ki $$

Where $k$ is hydraulic conductivity, commonly expressed in m/s.

Darcy flux is flow per unit gross area. It is not the average velocity of water through the connected void space.

Volumetric flow rate

For cross-sectional area $A$ normal to the direction of flow:

$$ Q = kiA $$

Symbol	Meaning	Common unit
$Q$	Volumetric flow rate	m3/s
$k$	Hydraulic conductivity	m/s
$i$	Hydraulic gradient	Dimensionless
$A$	Gross area normal to flow	m2

Seepage velocity

Where porosity $n$ is supplied, the average seepage velocity is commonly represented as:

$$ v_s = \frac{q}{n}=\frac{ki}{n} $$

This distinction matters because water moves through the void space rather than the entire gross cross-section.

Porosity must be entered as a decimal inside the equation. A porosity of 35 percent is $n=0.35$, not 35.

Worked example

Assume:

hydraulic conductivity $k=1.5\times10^{-5}$ m/s
head difference $\Delta h=1.2$ m
flow length $L=8$ m
cross-sectional area $A=4$ m2
porosity $n=0.35$

Step 1: hydraulic gradient

$$ i = \frac{1.2}{8}=0.15 $$

Step 2: Darcy flux

$$ q=(1.5\times10^{-5})(0.15) $$ $$ q=2.25\times10^{-6}\ \mathrm{m/s} $$

Step 3: volumetric flow

$$ Q=(1.5\times10^{-5})(0.15)(4) $$ $$ Q=9.0\times10^{-6}\ \mathrm{m^3/s} $$

Converting to a daily flow:

$$ Q_{\mathrm{day}}=(9.0\times10^{-6})(86400) $$ $$ Q_{\mathrm{day}}=0.7776\ \mathrm{m^3/day} $$

Step 4: seepage velocity

$$ v_s = \frac{2.25\times10^{-6}}{0.35} $$ $$ v_s=6.429\times10^{-6}\ \mathrm{m/s} $$

Hydraulic conductivity is condition-dependent

Hydraulic conductivity can vary with:

soil grading and fines
density and fabric
saturation
stress state
fissures and macro-pores
sample disturbance
flow direction
fluid properties and temperature
test method and scale

A laboratory result on a small specimen may not represent field-scale flow through layered or fissured ground. The calculation should identify the source of $k$ and whether the flow direction matches the test direction.

Assumptions behind the basic calculation

The worked calculation assumes:

one-dimensional flow
a representative constant hydraulic conductivity
a defined gross flow area
a known head difference
steady conditions
a material and gradient for which the selected relationship is applicable
no separate preferential-flow path
no change in saturation or soil structure during the calculation

These assumptions should be reviewed for the project.

Directional flow

A robust calculator should offer two modes:

Magnitude mode: reports positive flow magnitude and a written direction.
Signed mode: preserves the sign of $\Delta h$ according to the selected axis convention.

The calculator should not return a negative number without explaining the convention.

Validation rules

The calculation should:

require $L>0$
require $A>0$
require $k\geq0$
require \(0
show scientific notation clearly
preserve original and converted units
identify the selected flow direction
warn when $k$ is assumed rather than measured
avoid converting laboratory flow into a site inflow claim without a separate field model

Common errors

Entering centimetres per second as metres per second

This creates a factor-of-100 error. Store both the selected and canonical units.

Entering porosity as 35

If the field expects a decimal, 35 percent must be converted to 0.35.

Using plan area instead of area normal to flow

The correct area depends on the flow direction and model geometry.

Confusing pressure head with elevation difference alone

Total hydraulic head may include elevation and pressure components.

Treating Darcy flux as seepage velocity

Seepage velocity divides Darcy flux by porosity.

Relationship to permeability testing

A permeability or hydraulic-conductivity test determines or estimates $k$ under nominated conditions. Darcy flow then uses that value in a flow model.

Read Infiltration and Permeability Testing for field and laboratory context.

When a more advanced analysis is needed

Use a more complete seepage or groundwater assessment for:

layered or anisotropic soil
unsaturated flow
transient drawdown
dewatering wells
excavation inflow
dams, levees or cut-off walls
seepage forces and piping
complex boundaries
fractured rock
coupled consolidation

Review checklist

Ask the reviewing geotechnical engineer to confirm:

Head, gradient and direction conventions are clear.
Darcy flux is distinguished from seepage velocity.
Porosity is handled as a decimal internally.
Cross-sectional area is defined normal to flow.
Scientific notation and unit conversions are correct.
The limitations on field-scale use are adequate.
The worked example reproduces $0.7776$ m3/day.