Soil Mechanics / Two-dimensional seepage and flow nets

Flow lines and equipotentials

From one-dimensional Darcy flow to a two-dimensional flow 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

Seepage around sheet-pile walls, beneath dams and into excavations often requires a two-dimensional model. A flow net estimates discharge and pore pressure for defined boundary conditions and helps identify concentrated exit gradients.

What you should already know

Darcy's law, total head and hydraulic gradient (Module 7)
Effective stress and the quick/critical condition (Module 8)

For steady seepage in a homogeneous isotropic medium, two families of curves describe the field:

Flow lines — the paths water particles follow from the high-head (upstream) boundary to the low-head (downstream) boundary.
Equipotentials — lines joining points of equal total head. For isotropic conductivity, flow follows the direction of decreasing head, flow lines cross equipotentials at right angles.

Together they form a flow net. If it is drawn so that the flow lines and equipotentials divide the region into curvilinear squares (each roughly as wide as it is long), the net is not just a sketch — it is a solution of the governing equation (Laplace's equation) for steady seepage.

A flow net is described by two counts:

$N_f$ — the number of flow channels (the strips between adjacent flow lines);
$N_d$ — the number of equipotential drops (the steps between adjacent equipotentials from upstream to downstream).

The total head loss $h$ (the difference between upstream and downstream water levels) is shared equally among the $N_d$ drops — that is what makes the net a solution, and it is the key to everything that follows.

Head loss per equipotential drop

\Delta h = \dfrac{h}{N_d}

What it calculates: the equal step in total head between adjacent equipotentials

$\Delta h$: head loss across one drop (m)
h: total head loss across the net (upstream − downstream level) (m)
N_d: number of equipotential drops

In plain terms: After k drops from the upstream boundary, the head has fallen by k·Δh. This lets you find the total head — and hence the pore pressure — anywhere in the net.

Summary

Two-dimensional seepage is described by flow lines (particle paths) and equipotentials (equal total head).
Flow lines cross equipotentials at right angles; drawn as curvilinear squares, the net solves Laplace's equation.
Nf = number of flow channels, Nd = number of equipotential drops.
The total head loss is shared equally among the drops: Δh = h/Nd.