Module 14 · Lesson 14.1

Flow lines and equipotentials

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

Why this matters

Module 7 treated water flowing straight through a column. Real seepage curves around sheet-pile walls, under dams and into excavations — genuinely two-dimensional. The flow net is the elegant graphical tool that solves these problems and, crucially, tells you where seepage may lift or wash away the ground.

What you should already know

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

Two families of curves describe any seepage 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. Because flow always runs from high to low head down the steepest route, 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

Δ h = \frac{h}{N _{d}}

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

$Δ h$: 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.

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This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint