Soil Mechanics / Two-dimensional seepage and flow nets

Using a flow net

The seepage flow rate and pore pressure from the net.

Lesson review: expert verification remains open

This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.

Corrections checked in this pass

Replaced the misleading flow-net drawing with a labelled schematic. Supplied flow-net counts remain inputs to the calculation.

Checks needed for project use

A geotechnical reviewer must check laboratory interpretation, classification conventions and the limits of drained, undrained and one-dimensional idealisations.
No solved boundary-value flow net supplied. Flow-channel count, potential-drop count and exit drop need an independently valid net; uplift, heave, piping and internal erosion are separate assessments.

These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.

Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.

Read the lesson register and outstanding review notes

Why this matters

A flow net answers the two questions a designer of a dam or an excavation must ask: how much water seeps through (for pumping and stability), and what is the pore pressure at any point (for effective stress and uplift). Both come straight from the counts $N_f$ and $N_d$ .

From first principles

The flow-net flow rate

We want to show: Get the total seepage per unit width from the geometry of the net alone.

Every square in the net carries the same little bit of flow, because it has the same head drop across it and the same shape. Add up the squares and the flow rate falls out in terms of just Nf, Nd and k.

Pore pressure at a point follows from counting drops. If a point lies $n_d$ drops down from the upstream boundary, its total head is $h_{up} - n_d\,\Delta h$ . Subtract the elevation head to get the pressure head, and multiply by $\gamma_w$ for the pore pressure — which then feeds straight into effective stress (Module 8) and uplift on the structure.

Try it

Flow-net input calculator

Enter channel and drop counts from an independently constructed flow net. The sketch only shows the seepage route; changing these inputs does not solve the boundary-value problem.

Total head loss h6.0 m

Permeability k1e-6 m/s

Flow channels Nf4

Equipotential drops Nd12

Exit-square length l1.5 m

Flow rate q = k·h·Nf/Nd: 2.00e-6 m³/s/m
Head per drop Δh = h/Nd: 0.50 m
Exit gradient i_exit: 0.33
Critical gradient i_cr: 1.00
FoS piping F = i_cr/i_exit: 3.0

Flow 2.0e-6 m³/s per m; exit gradient 0.33 against a critical 1.00 gives a factor of safety of 3.0 against the illustrative comparison value of 3. This is not a design acceptance check.

Worked example

Seepage beneath a sheet-pile wall

Given

Total head loss across the wall h = 6 m
Soil permeability k = 1 × 10⁻⁶ m/s
Flow net: Nf = 4 flow channels, Nd = 12 equipotential drops

Find

The seepage flow rate per metre width, and the head loss per drop.

Assumptions

Homogeneous, isotropic soil; a properly drawn square flow net; steady state.

Practice

A flow net beneath a dam has Nf = 5 and Nd = 15, with a head loss h = 9 m and k = 2 × 10⁻⁶ m/s. What is the seepage flow rate per metre width, in m³/s/m?

Summary

Seepage flow rate from a flow net: q = k·h·(Nf/Nd) — derived from Darcy's law on one square.
The square side cancels, so only the counts Nf and Nd and the permeability and head matter.
Total head at a point = upstream head − (drops passed)·Δh; subtract elevation head for pore pressure.
Pore pressure from the net feeds directly into effective stress and uplift.