Solve the seepage flowing under a dam or a sheet-pile wall with a flow net. Adjust the head difference, permeability, number of flow channels and equipotential drops to see the seepage flow through the soil, the exit hydraulic gradient and the factor of safety against piping update in real time.
Parameters
Head difference H
m
Difference between the upstream and downstream water levels
Permeability k
m/s
How easily the soil transmits water. About 10⁻³–10⁻⁵ for sand
Flow channels Nf
Number of strips (channels) between flow lines
Equipotential drops Nd
Number of steps (drops) between equipotential lines
Structure depth
m
Length of the structure perpendicular to the seepage flow
Exit field length l
m
Flow-direction length of the last cell at the exit
Results
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Seepage per width (m³/s/m)
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Total seepage (L/s)
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Head loss per drop Δh (m)
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Exit hydraulic gradient i_exit
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Factor of safety vs piping
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Piping verdict
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Seepage under a sheet-pile wall & flow net — animation
Flow lines (blue) curve under the sheet-pile wall, crossed at right angles by equipotential lines (orange). Particles seep along the flow lines, and the hydraulic gradient is steepest in the downstream exit zone (red).
Seepage flow per unit width q, exit hydraulic gradient i_exit and the factor of safety FOS against piping. k: permeability, H: head difference, Nf: flow channels, Nd: equipotential drops, l: exit field length.
Nf is the number of flow channels (strips between flow lines) in the flow net, and Nd is the number of equipotential drops (steps between equipotential lines). Piping is threatened where the exit gradient i_exit approaches the critical gradient i_cr (about 1 for sand). This tool uses the representative value i_cr = 1.0.
What is the Flow Net Seepage Simulator?
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A "flow net" is that mesh-like diagram of how water flows through the ground, right? What is it actually drawn for?
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Exactly — it is a grid that traces groundwater seeping under a dam or a sheet-pile wall. Seepage through soil obeys a partial differential equation called Laplace's equation, and before computers you had to solve it by hand. The trick people invented was to draw, by trial and error, a grid of curvilinear "squares" where flow lines and equipotential lines cross at right angles. That grid is the flow net. Once you can draw it, you can read off both the seepage flow and the dangerous spots.
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What do the flow lines and equipotential lines each represent?
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Flow lines are the very paths the water particles follow from the upstream side to the downstream side. Equipotential lines connect points of equal total head — total head being "water pressure plus elevation energy". The two families always cross at right angles and divide the seepage region into curvilinear "square" cells. In the diagram on the left the blue curves are flow lines and the orange curves are equipotential lines. Move the Nf and Nd sliders and you change how fine the net is.
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I see. So how do you get the seepage flow? The formula reads q = k·H·(Nf/Nd)…
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Once the flow net is drawn, you just count two integers: the number of flow channels Nf and the number of equipotential drops Nd. Nf is how many strips are bounded by flow lines, Nd is how many steps the equipotential lines cut. Then multiply by the permeability k and the head difference H: q = k·H·(Nf/Nd). That is the seepage per unit width. Multiply by the structure depth for the total leakage. With the default values, for example, 0.5 m of head is lost per drop and the whole structure leaks 0.6 L/s.
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It shows not just the leakage but also a "safety factor". A safety factor against what?
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The factor of safety against piping. Look near the downstream exit — the equipotential lines crowd tightly together, don't they? There the gradient is steep and the upward seepage force is large. When that gradient approaches the critical gradient (about 1 for sand), the sand grains float up and the ground fails as if it were "boiling". That is piping, also called boiling. Plenty of dams have been breached and excavation sites flooded because of it. So you compute FOS = i_cr/i_exit and usually keep it at 3 or above.
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When the safety factor is too low, how do you fix it?
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The most direct move is to drive the sheet-pile wall deeper. The water path gets longer, the number of equipotential drops rises, and the exit gradient eases off. You can also lay a downstream blanket (a permeable mat) to lengthen the exit field, or use wellpoints to draw the groundwater level down. Try raising the "exit field length l" on the left and you will see the safety factor climb sharply. A flow net is a powerful tool: it tells you the leakage rate and the most dangerous location in a single picture.
Frequently Asked Questions
A flow net is a grid drawn so that flow lines and equipotential lines cross at right angles and divide the region into curvilinear squares. The seepage flow per unit width is q = k·H·(Nf/Nd), where k is the permeability, H is the head difference between upstream and downstream, Nf is the number of flow channels (strips between adjacent flow lines) and Nd is the number of equipotential drops (steps between adjacent equipotential lines). Nf and Nd are integers read straight off the net; multiply q by the structure depth to get the total seepage.
Near the downstream exit the equipotential lines bunch tightly together and the hydraulic gradient becomes steep. When the exit gradient i_exit approaches the critical gradient i_cr (about 1.0 for sand), the upward seepage force exceeds the buoyant weight of the soil grains, the soil floats up and fails as if 'boiling'. This is piping, also called boiling. Many dam and excavation failures have started here. A flow net shows at a glance where the equipotential lines crowd together — the most dangerous location.
The head loss per equipotential drop is Δh = H/Nd. With l the length of the exit field (the last cell at the exit), the exit hydraulic gradient is i_exit = Δh/l = (H/Nd)/l. The factor of safety against piping is FOS = i_cr/i_exit. In general aim for FOS ≥ 3; if the safety factor is too low, drive the sheet-pile wall deeper, or lengthen the exit field with a blanket or wellpoints.
A flow net is the classical, hand-drawn way to solve the two-dimensional Laplace equation, used long before computers. Seepage FEM now gives detailed solutions, but the flow net remains useful for estimates and cross-checks. For simple boundary conditions and an isotropic, homogeneous soil, a hand-drawn flow net gives accuracy that is adequate in practice. If an FEM result differs from the flow-net estimate by an order of magnitude, it is a sanity check pointing to a mistake in the mesh, permeability units or boundary conditions.
Real-World Applications
Fill-dam and earth-dam design: For dams built of soil or rock, a flow net evaluates the seepage through the embankment and the foundation soil. Beyond limiting the leakage, the key is to keep the hydraulic gradient at the downstream toe below the critical value to prevent piping. The thickness of the core (the impervious zone) and the depth of the foundation cut-off are chosen so the flow net gains enough equipotential drops.
Sheet-pile cofferdams and excavations: When sheet piles are driven to hold back water and the inside is excavated for river or harbour works, the seepage curving under the piles can make the excavation base "boil". Designers use a flow net to set the embedment depth of the sheet piles and check the factor of safety against the exit hydraulic gradient. If it is too low, the piles are driven deeper or wellpoints force the groundwater level down.
Weirs, gates and intake structures: Under the foundation soil of a concrete weir or gate, the upstream-to-downstream seepage produces uplift pressure that tends to lift the foundation. The uplift distribution on the base is read from the equipotential lines of the flow net and fed into the sliding and overturning stability checks. The layout of blankets and cut-offs is a design lever for controlling the seepage path.
Cross-checking CAE seepage analysis: Before and after a detailed seepage FEM analysis in SEEP/W or PLAXIS, a hand-calculated flow net confirms the order of magnitude of q and i_exit. If the FEM result differs from the flow-net estimate by an order of magnitude, it is a sanity check that points to a unit mistake in the permeability or an error in the boundary conditions (head boundary, impervious boundary). A simple estimate is the foundation that secures the reliability of complex numerical analysis.
Common Misconceptions and Pitfalls
The first big misconception is believing the flow-net "squares" must be true squares. The cells of a flow net are curvilinear "curvilinear squares": it is enough that the mid-lengths of opposite sides are roughly equal and the diagonals cross at right angles. There is no need to redraw endlessly chasing perfect squares, and the number of flow channels Nf or equipotential drops Nd may come out fractional (3.5, say). What matters is the ratio Nf/Nd, and the strength of the flow-net method is that this ratio is found stably even from a fairly coarse net.
Second, applying a formula derived for isotropic, homogeneous soil directly to anisotropic ground. The formula q = k·H·(Nf/Nd) assumes the horizontal and vertical permeabilities are equal — an isotropic soil. Real sedimentary deposits transmit water more easily horizontally (kh > kv) and are anisotropic in layers. In that case you draw the flow net on a "transformed section" with horizontal distances scaled by √(kv/kh), and use the equivalent permeability k' = √(kh·kv) for the flow calculation. Ignoring anisotropy throws both the leakage and the gradient far off.
Finally, assuming that checking only the exit-gradient safety factor prevents piping. What this tool computes is a "heave (boiling)" type safety factor that compares the seepage force on the exit face with the weight of the soil. But real piping also includes "backward erosion", where soil grains are eroded in the reverse direction from the downstream side and a pipe-like channel widens. This starts from a local weak spot, a discontinuity in grain size or an animal burrow, and progresses even when the average gradient is safe. Filter materials matched to the grain-size distribution and continuous monitoring of the turbidity and quantity of the seepage are essential.
How to Use
Enter head difference (H) across the dam or sheet-pile wall in meters; typical values range 2–15 m for embankment dams
Set soil permeability (k) in m/s; use 1e-5 m/s for clay, 1e-3 m/s for sand, 1e-1 m/s for gravel
Define number of flow channels (nf) and equipotential drops (nd); for most problems, nf=3–5 and nd=5–10 provide stable results
Click Calculate to generate seepage rate, exit gradient, and safety factor against piping
Adjust geometry nodes on the flow net sketch to refine boundary conditions if needed
Worked Example
Dam seepage analysis: Head difference H=8 m, cohesive clay k=2e-5 m/s, nf=4 channels, nd=8 drops. The simulator computes seepage per width q=1.6e-4 m³/s/m, total seepage 0.32 L/s (for 2 m dam width), head loss per drop Δh=1.0 m, exit gradient i_exit=0.18, and safety factor FS=1.8 against piping (critical FS≈1.0). This indicates acceptable seepage with low piping risk for the clay layer.
Practical Notes
Exit gradient control: Maintain i_exit < 0.3–0.5 to prevent boiling/piping; reduce head or increase nd to lower exit gradient
Sheet-pile penetration: Increase nd by deepening the cutoff depth; each additional equipotential drop reduces exit gradient by ~15%
Anisotropic soils: If vertical permeability differs from horizontal (kv ≠ kh), adjust flow net shape manually or use kv/kh ratio correction factor
Filter design: Seepage rate directly determines required filter drain capacity; L/s output guides underdrain sizing for construction