Flow Nets and Seepage Analysis

Learning Objectives

  • Understand the components and rules for drawing valid flow nets.
  • Calculate seepage rates using flow net geometry.
  • Apply transformation techniques for anisotropic soils.
  • Determine uplift pressures and assess piping risks.
  • Understand Terzaghi's filter design criteria.
While one-dimensional permeability (Darcy's Law) deals with simple flow, complex engineering scenarios involve two-dimensional groundwater flow. This 2D seepage under retaining walls, sheet piles, earth dams, and concrete dam foundations is analyzed using Flow Nets. A flow net is a graphical solution to Laplace's equation for steady-state groundwater flow, providing a powerful visual and mathematical tool for engineers.

Flow Net

A graphical representation of two-dimensional steady-state groundwater flow through a porous medium, consisting of intersecting flow lines and equipotential lines that solve Laplace's equation.

Components of a Flow Net

Flow Lines

Paths that water particles follow as they travel through the soil from higher to lower hydraulic head.

Equipotential Lines

Contours of equal total head. Water level in a piezometer placed anywhere along a given equipotential line will rise to the exact same elevation.

Flow Lines and Equipotential Lines

Key Rules for Drawing Flow Nets:

  • Flow lines and equipotential lines must intersect at right angles (90∘90^\circ).
  • The geometry of the intersecting lines should form approximate squares (or curvilinear squares, where the average width equals the average length).
  • Impermeable boundaries are boundary flow lines. For example, the base of a concrete dam or an underlying layer of solid bedrock.
  • Permeable boundaries are boundary equipotential lines. For example, the soil surface submerged under an upstream reservoir or a downstream tailwater surface.

Interactive Flow Net Visualization

Interactive Simulation

Explore how changing the number of flow channels (NfN_f) and head drops (NdN_d) affects the total seepage rate under a sheet pile wall.

Flow Net Calculator under Sheet Pile

Shape Factor (Nf/NdN_f / N_d)

0.40

Total Seepage (qq)

0.0300

cm³/s per cm of wall

Calculating Seepage Rate

The Seepage Formula

Once a valid flow net is drawn, the total rate of seepage (qq) per unit length of the structure can be easily calculated:

Seepage Rate from Flow Net

Calculates total seepage per unit length of structure by counting flow channels and potential drops in a flow net.

q=k⋅H⋅NfNdq = k \cdot H \cdot \frac{N_f}{N_d}

Variables

SymbolDescriptionUnit
qqTotal rate of seepage per unit length-
kkCoefficient of permeability (hydraulic conductivity)-
HHTotal head difference-
NfN_fTotal number of flow channels-
NdN_dTotal number of equipotential drops-

Shape Factor

The ratio Nf/NdN_f/N_d is called the shape factor of the flow net.

Seepage in Anisotropic Soils

Anisotropic Soil

A soil where the physical properties (such as permeability) vary depending on the direction of measurement.

In natural deposits, horizontal permeability (kxk_x) is usually much greater than vertical permeability (kzk_z). Laplace's equation for flow nets assumes isotropic conditions (kx=kzk_x = k_z). To draw a flow net for anisotropic soil, the geometry must be transformed.

Transformed Section Method

To model the flow net correctly, the horizontal dimensions (xx) of the entire cross-section are mathematically shrunk by a transformation factor, while vertical dimensions (zz) remain the same.

Horizontal Dimension Transformation

Scales horizontal dimensions to transform an anisotropic soil cross-section into an equivalent isotropic domain for flow net drawing.

xT=xkzkxx_T = x \sqrt{\frac{k_z}{k_x}}

Variables

SymbolDescriptionUnit
xTx_TTransformed horizontal dimension-
xxOriginal horizontal dimension-
kzk_zVertical permeability-
kxk_xHorizontal permeability-
After drawing the flow net on this transformed geometry as if it were isotropic, the equivalent permeability used in the seepage formula is:

Equivalent Permeability (Anisotropic)

Geometric mean permeability used in the seepage formula after drawing a flow net on the transformed anisotropic cross-section.

k′=kxkzk' = \sqrt{k_x k_z}

Variables

SymbolDescriptionUnit
k′k'Equivalent permeability-
kxk_xHorizontal permeability-
kzk_zVertical permeability-

Uplift Pressure and Piping

Uplift Pressure

The upward pore water pressure exerted on the base of a structure, which reduces its effective weight and overall stability.

Piping

A phenomenon of internal erosion where a high seepage exit gradient physically washes fine soil particles out from the soil mass, potentially leading to structural failure.

Water seeping under a dam exerts an upward pressure on its base, reducing the effective weight and stability of the structure. Furthermore, if the exit gradient is too high, it can cause internal erosion (piping).

Calculating Uplift

The uplift pressure (uu) at any point along the base can be found by determining the total head (hh) at that point from the flow net.

Total Head at a Point

Determines the total hydraulic head at any point along a flow path by accounting for head losses through equipotential drops.

h=H−(nd⋅Δh)h = H - \left( n_d \cdot \Delta h \right)

Variables

SymbolDescriptionUnit
hhTotal head at the point-
HHTotal head upstream-
ndn_dNumber of potential drops from upstream-
Δh\Delta hHead loss per drop (H / N_d)-
The pore water pressure is then:

Pore Water Pressure (Uplift)

Calculates the uplift pore water pressure at a point beneath a structure using the pressure head from the flow net.

u=hp⋅γw=(h−Z)⋅γwu = h_p \cdot \gamma_w = (h - Z) \cdot \gamma_w

Variables

SymbolDescriptionUnit
uuPore water pressure-
hph_pPressure head-
γw\gamma_wUnit weight of water-
hhTotal head-
ZZElevation head-

Filter Design (Terzaghi's Criteria)

To prevent "piping" (where high seepage velocity physically washes fine soil particles out from under a dam), protective granular filters are installed at the seepage exit.

Terzaghi's Filter Rules

A properly designed protective granular filter must serve two contradictory purposes: it must be coarse enough to allow water to flow freely to avoid pressure buildup, but fine enough to physically block the base soil particles from washing through the voids.

1. Retention Criterion

The retention criterion ensures the filter is fine enough to prevent the base soil particles from being washed (piped) into the filter voids. It is the primary defense against internal erosion.

Retention Criterion

Terzaghi's filter design rule ensuring the filter is fine enough to prevent base soil particles from being piped through.

D15(filter)D85(soil)≤4 to 5\frac{D_{15(filter)}}{D_{85(soil)}} \le 4 \text{ to } 5

Variables

SymbolDescriptionUnit
D15(filter)D_{15(filter)}Particle size of filter at 15% passing-
D85(soil)D_{85(soil)}Particle size of base soil at 85% passing-

2. Permeability Criterion

The permeability criterion ensures the filter is sufficiently coarse and permeable to allow seepage water to drain freely without generating excessive hydrostatic pressure buildup.

Permeability Criterion

Terzaghi's filter design rule ensuring the filter is coarse enough to allow free drainage and prevent pressure buildup.

D15(filter)D15(soil)≥4 to 5\frac{D_{15(filter)}}{D_{15(soil)}} \ge 4 \text{ to } 5

Variables

SymbolDescriptionUnit
D15(filter)D_{15(filter)}Particle size of filter at 15% passing-
D15(soil)D_{15(soil)}Particle size of base soil at 15% passing-

Use this sequence to define the seepage model, construct and refine the flow net, and carry its results into seepage, uplift, exit-gradient, and filter checks. Project-specific ground data and criteria govern the checks.

Flow-Net Seepage and Exit-Gradient Assessment

Build a flow net from hydraulic boundaries, refine its geometry, and use the net to estimate seepage and pressure-related checks.

The graphical method represents two-dimensional steady seepage under the assumed saturated continuum model. State homogeneity/isotropy or use an appropriate anisotropic transformation; check uplift, exit gradients, and internal-erosion/filter compatibility against project-specific criteria and verified material data.

Flow-Net Seepage and Exit-Gradient AssessmentBuild a flow net from hydraulic boundaries, refine its geometry, and use the net to estimate seepage and pressure-related checks.. The graphical method represents two-dimensional steady seepage under the assumed saturated continuum model. State homogeneity/isotropy or use an appropriate anisotropic transformation; check uplift, exit gradients, and internal-erosion/filter compatibility against project-specific criteria and verified material data.. Start → Define geometry, soil profile, permeability, and head boundaries; Define geometry, soil profile, permeability, and head boundaries → Steady 2D model and boundary data adequate?; Steady 2D model and boundary data adequate? — Yes → Material anisotropic?; Steady 2D model and boundary data adequate? — No → Hold: refine ground model or hydraulic boundaries; Material anisotropic? — Yes → Transform section using justified permeability ratio; Material anisotropic? — No → Sketch flow lines and equipotential lines; Transform section using justified permeability ratio → Sketch flow lines and equipotential lines; Sketch flow lines and equipotential lines → Refine intersections toward curvilinear squares; Refine intersections toward curvilinear squares → Count flow channels and potential drops; Count flow channels and potential drops → Estimate seepage, uplift, and exit-gradient distribution; Estimate seepage, uplift, and exit-gradient distribution → Project erosion, uplift, and filter checks satisfied?; Project erosion, uplift, and filter checks satisfied? — Yes → Record assumptions, checks, and limitations; Project erosion, uplift, and filter checks satisfied? — No → Revise geometry, drainage, cutoff, or filter design; Revise geometry, drainage, cutoff, or filter design → Sketch flow lines and equipotential lines; Record assumptions, checks, and limitations → End

Start → Define geometry, soil profile, permeability, and head boundaries; Define geometry, soil profile, permeability, and head boundaries → Steady 2D model and boundary data adequate?; Steady 2D model and boundary data adequate? — Yes → Material anisotropic?; Steady 2D model and boundary data adequate? — No → Hold: refine ground model or hydraulic boundaries; Material anisotropic? — Yes → Transform section using justified permeability ratio; Material anisotropic? — No → Sketch flow lines and equipotential lines; Transform section using justified permeability ratio → Sketch flow lines and equipotential lines; Sketch flow lines and equipotential lines → Refine intersections toward curvilinear squares; Refine intersections toward curvilinear squares → Count flow channels and potential drops; Count flow channels and potential drops → Estimate seepage, uplift, and exit-gradient distribution; Estimate seepage, uplift, and exit-gradient distribution → Project erosion, uplift, and filter checks satisfied?; Project erosion, uplift, and filter checks satisfied? — Yes → Record assumptions, checks, and limitations; Project erosion, uplift, and filter checks satisfied? — No → Revise geometry, drainage, cutoff, or filter design; Revise geometry, drainage, cutoff, or filter design → Sketch flow lines and equipotential lines; Record assumptions, checks, and limitations → End

  • Start: terminator
  • Define geometry, soil profile, permeability, and head boundaries: process
  • Steady 2D model and boundary data adequate?: decision
  • Hold: refine ground model or hydraulic boundaries: terminator
  • Material anisotropic?: decision
  • Transform section using justified permeability ratio: process
  • Sketch flow lines and equipotential lines: process
  • Refine intersections toward curvilinear squares: process
  • Count flow channels and potential drops: process
  • Estimate seepage, uplift, and exit-gradient distribution: process
  • Project erosion, uplift, and filter checks satisfied?: decision
  • Revise geometry, drainage, cutoff, or filter design: process
  • Record assumptions, checks, and limitations: document
  • End: terminator
Key Takeaways
  • Flow Nets are graphical tools used to model 2D steady-state groundwater seepage under structures.
  • They consist of Flow Lines (water paths) and Equipotential Lines (equal head contours) that must intersect at 90∘90^\circ to form curvilinear squares.
  • The total seepage rate is calculated using q=k⋅H⋅(Nf/Nd)q = k \cdot H \cdot (N_f/N_d).
  • For Anisotropic Soils (kx≠kzk_x \ne k_z), the physical horizontal dimensions must be scaled (xT=xkz/kxx_T = x \sqrt{k_z/k_x}) before drawing the flow net.
  • Terzaghi's Filter Criteria are essential to design granular drains that safely relieve seepage pressure while physically preventing internal soil erosion (piping).

References