Flow Nets and Seepage Analysis - Theory & Concepts

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 ( $90^\circ$ ).
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 ( $N_f$ ) and head drops ( $N_d$ ) affects the total seepage rate under a sheet pile wall.

Flow Net Calculator under Sheet Pile

Number of Flow Channels (

N_f

): 4

Number of Potential Drops (

N_d

): 10

Total Head difference

H

(m): 15

Permeability

k

(cm/s): 5.00e-3

Shape Factor ( $N_f / N_d$ )

0.40

Total Seepage ( $q$ )

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 ( $q$ ) 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 \cdot H \cdot \frac{N_f}{N_d}

Variables

Symbol	Description	Unit
$q$	Total rate of seepage per unit length	-
$k$	Coefficient of permeability (hydraulic conductivity)	-
$H$	Total head difference	-
$N_f$	Total number of flow channels	-
$N_d$	Total number of equipotential drops	-

Shape Factor

The ratio $N_f/N_d$ 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 (

k_x

) is usually much greater than vertical permeability (

k_z

). Laplace's equation for flow nets assumes isotropic conditions (

k_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 ( $x$ ) of the entire cross-section are mathematically shrunk by a transformation factor, while vertical dimensions ( $z$ ) 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.

x_T = x \sqrt{\frac{k_z}{k_x}}

Variables

Symbol	Description	Unit
$x_T$	Transformed horizontal dimension	-
$x$	Original horizontal dimension	-
$k_z$	Vertical permeability	-
$k_x$	Horizontal 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' = \sqrt{k_x k_z}

Variables

Symbol	Description	Unit
$k'$	Equivalent permeability	-
$k_x$	Horizontal permeability	-
$k_z$	Vertical 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 ( $u$ ) at any point along the base can be found by determining the total head ( $h$ ) 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 - \left( n_d \cdot \Delta h \right)

Variables

Symbol	Description	Unit
$h$	Total head at the point	-
$H$	Total head upstream	-
$n_d$	Number of potential drops from upstream	-
$\Delta h$	Head 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 = h_p \cdot \gamma_w = (h - Z) \cdot \gamma_w

Variables

Symbol	Description	Unit
$u$	Pore water pressure	-
$h_p$	Pressure head	-
$\gamma_w$	Unit weight of water	-
$h$	Total head	-
$Z$	Elevation 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.

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

Variables

Symbol	Description	Unit
$D_{15(filter)}$	Particle size of filter at 15% passing	-
$D_{85(soil)}$	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.

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

Variables

Symbol	Description	Unit
$D_{15(filter)}$	Particle size of filter at 15% passing	-
$D_{15(soil)}$	Particle size of base soil at 15% passing	-

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 $90^\circ$ to form curvilinear squares.
The total seepage rate is calculated using $q = k \cdot H \cdot (N_f/N_d)$ .
For Anisotropic Soils ( $k_x \ne k_z$ ), the physical horizontal dimensions must be scaled ( $x_T = x \sqrt{k_z/k_x}$ ) 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).

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