Soil Mechanics / Two-dimensional seepage and flow nets

Exit gradient, piping and heave

Where seepage emerges, and how it can lift or wash away the ground.

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.

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Why this matters

The most dangerous place in a seepage field is where water emerges at the downstream ground surface. There the upward seepage force can lift the soil — heave — or start washing grains out to form a channel that eats back towards the source — piping. Both have destroyed dams and excavations, and both are assessed from the flow net.

Where flow exits upward, the seepage drags the grains up. The upward hydraulic gradient at the exit is the head drop in the last field divided by its length:

i_{exit} = \dfrac{\Delta h}{l} = \dfrac{h / N_d}{l},

with $l$ the flow length of the exit square. This is compared with the critical hydraulic gradient at which the upward seepage force just cancels the soil's submerged weight and the effective stress falls to zero — the quick condition of Module 8:

Critical gradient and factor of safety against piping

i_{cr} = \dfrac{\gamma'}{\gamma_w} = \dfrac{G_s - 1}{1 + e}, \qquad F = \dfrac{i_{cr}}{i_{exit}}

What it calculates: the gradient at which the ground is lifted, and the margin against it

$i_{cr}$: critical (quick-condition) hydraulic gradient (—)
$i_{exit}$: exit hydraulic gradient from the flow net (—)
F: factor of safety against piping/heave

In plain terms: i_cr is typically about 1.0 (for Gs ≈ 2.65, e ≈ 0.65). When i_exit approaches i_cr the ground is on the point of piping; a comfortable factor of safety (commonly 3 or more) is required because piping is sudden and catastrophic.

Worked example

Factor of safety against piping at an excavation

Given

Total head loss h = 6 m, Nd = 12 (so Δh = 0.5 m)
The exit square has flow length l = 1.5 m
Soil Gs = 2.65, e = 0.65, so i_cr = (2.65 − 1)/(1 + 0.65) = 1.0

Find

The exit gradient and the factor of safety against piping.

Assumptions

The critical gradient is taken from the quick condition; the exit square length is read from the net.

Practice

At an excavation the exit gradient from the flow net is i_exit = 0.25 and the critical gradient is i_cr = 1.0. What is the factor of safety against piping?

Summary

Seepage is most dangerous where it emerges upward at the downstream surface.
Exit gradient i_exit = (h/Nd)/l, with l the exit-square length.
The critical gradient i_cr = (Gs − 1)/(1 + e) ≈ 1 is where effective stress reaches zero (quick condition).
Factor of safety F = i_cr/i_exit (commonly ≥ 3); lengthen the seepage path or add a filter to raise it.