Flow Net and Seepage Analysis

Theory: Seepage

How water flows through soil under a structure, how a flow net draws it, and how the seepage, the exit gradient, heave and uplift follow from it.

Darcy's law

Water flows through soil from higher total head to lower, at a rate proportional to the gradient. The total head is the elevation plus the pressure head; the velocity of flow is slow enough that the velocity head is left out.

v = k\, i

v: discharge velocity · k: permeability · i: hydraulic gradient · u: pore pressure

Laplace's equation

In steady flow, what enters a small element leaves it. With Darcy's law, the head through a homogeneous soil obeys Laplace's equation. Its solution fixes two families of curves: equipotentials, of equal head, and flow lines, which the water follows. In an isotropic soil they cross at right angles.

k_h \frac{\partial^2 h}{\partial x^2} + k_v \frac{\partial^2 h}{\partial y^2} = 0

k_h, k_v: horizontal and vertical permeability

Flow nets

A flow net draws the equipotentials at equal drops of head and the flow lines at equal shares of the flow, so that they make curvilinear squares. N_f channels then carry the flow through N_d drops, and the seepage follows from the net's shape alone.

q = k\, \Delta H\, \frac{N_f}{N_d}

q: seepage per metre of structure · ΔH: head lost from upstream to downstream. N_f need not be whole

The ground under the water upstream and downstream are equipotentials, at h₁ and h₂.
The structure's underground outline and the impervious base are flow lines.
Where k_h ≠ k_v, the net is drawn on a section shrunk horizontally by √(k_v/k_h), and the flow uses k′ = √(k_h k_v).

Exit gradient and piping

Where the water rises out of the ground downstream, it pushes the soil up. When the upward gradient reaches the critical one, the effective stress vanishes and the sand boils; erosion then works back under the structure, piping. The exit gradient is taken on the net's last square, at the exit point.

i_{\mathrm{exit}} = \frac{\Delta H / N_d}{l}

l: length of the last square along the flow · γ′ = γ_sat − γ_w. Harr suggests FS of 3 to 4

Heave

Terzaghi found that soil beside a sheet pile heaves as a prism about D deep and D/2 wide. The water's push on its base, from the mean excess head there, is weighed against the prism's submerged weight.

\mathrm{FS} = \frac{\gamma' D}{\gamma_w\, h_a}

h_a: mean head above the downstream level on the prism's base, at the pile's tip

Uplift

Under a weir, the water's pressure pushes up on the base. A cutoff at the heel takes head off before the water reaches the base and lowers the uplift; one at the toe raises it, but guards the exit.

u = \gamma_w\,(h - y)

U: uplift per metre of weir, over its base

References

Terzaghi, K. Theoretical Soil Mechanics. New York: Wiley, 1943.
Harr, M. E. Groundwater and Seepage. New York: McGraw-Hill, 1962.
Lane, E. W. Security from under-seepage: masonry dams on earth foundations. Transactions of the ASCE, v. 100, p. 1235–1272, 1935.
Cedergren, H. R. Seepage, Drainage, and Flow Nets. 3rd ed. New York: Wiley, 1989.
Das, B. M.; Sobhan, K. Principles of Geotechnical Engineering. 9th ed. Boston: Cengage, 2018.

Enter the section — the permeable layer, a sheet pile, a weir or both — the water levels either side and the soil's permeability to draw the flow net and find the seepage, the exit gradient, the safety against piping and heave, and the uplift.

Section

Permeable layer, TmSheet pile, DmWeir base, Bm

D is how deep the pile goes into the ground and B the weir's width; 0 for none, but not both.

Water

Upstream, h₁mDownstream, h₂m

Water depths over the ground. The head lost through the soil is ΔH = h₁ − h₂.

Soil

k_hm/sk_vm/sγ_satkN/m³

Horizontal and vertical permeability; 2e-5 reads as 2 × 10⁻⁵. A layered soil is often 2 to 10 times more permeable across its layers.

Flow net and design

Drops, N_d

Required FS, pipingRequired FS, heave

N_d sets how many equipotential drops the flow net draws; the results don't depend on it, save the exit gradient. FS against piping is often asked to be 3–4, against heave 1.5–2.

Steady, confined flow in a homogeneous layer on an impervious base, which runs 3T beyond the structure on each side. The weir is impervious and sits on the ground; the pile has no thickness.
Laplace's equation is solved by finite volumes on a grid of about 32 cells through the layer, by conjugate gradients; the flow lines come from the stream function, the equipotentials from the head.
The exit gradient is the flow net's, on its last square: ΔH/N_d over how deep the first equipotential runs under the exit point. Heave follows Terzaghi's prism D × D/2 beside the pile, when the water leaves by it.
Heads are measured from the ground; x from the weir's heel, or from the pile when there's no weir; y up from the ground. Flow and forces per metre of structure, γ_w = 9.81 kN/m³.
The 3D is an illustration: the sheet piles' U profile (0.4 m deep, 1.2 m pairs) shows them as built but isn't designed here; the flow lines on its ends are the flow net's.