Flow Nets and Seepage Analysis
While one-dimensional permeability deals with simple flow, 2D groundwater flow under complex structures like dams or sheet piles is analyzed using Flow Nets. A flow net is a graphical solution to Laplace's equation for steady-state groundwater flow.
Components of a Flow Net
Flow Lines and Equipotential Lines
- Flow Lines: Paths that water particles follow as they travel through the soil.
- Equipotential Lines: Contours of equal total head. Water level in a piezometer placed anywhere along a given equipotential line will rise to the same elevation.
Key Rules for Drawing Flow Nets:
- Flow lines and equipotential lines must intersect at right angles ().
- The geometry of the intersecting lines should form approximate squares (or curvilinear squares, where the average width equals the average length).
- Impermeable boundaries (like concrete or solid rock) are flow lines.
- Permeable boundaries (like the soil surface under a reservoir) are equipotential lines.
Interactive Flow Net Visualization
Explore how changing the number of flow channels () and head drops () affects the total seepage rate under a sheet pile wall.
Flow Net Calculator under Sheet Pile
Shape Factor ()
0.40
Total Seepage ()
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 () per unit length of the structure can be easily calculated:
Seepage Rate from Flow Net
$$
q = k \cdot H \cdot \frac{N_f}{N_d}
$$Seepage in Anisotropic Soils
In natural deposits, horizontal permeability () is usually much greater than vertical permeability (). Laplace's equation for flow nets assumes isotropic conditions (). 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 () of the entire cross-section are mathematically shrunk by a transformation factor, while vertical dimensions () remain the same.
Horizontal Dimension Transformation
$$
x_T = x \sqrt{\frac{k_z}{k_x}}
$$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)
$$
k' = \sqrt{k_x k_z}
$$Uplift Pressure and Piping
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 () at any point along the base can be found by determining the total head () at that point from the flow net.
Total Head at a Point
$$
h = H - \left( n_d \cdot \Delta h \right)
$$The pore water pressure is then:
Pore Water Pressure (Uplift)
$$
u = h_p \cdot \gamma_w = (h - Z) \cdot \gamma_w
$$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 filter must serve two contradictory purposes: it must be coarse enough to allow water to flow freely, but fine enough to block the base soil particles from washing through.
- 1. Retention Criterion (To prevent piping):
Retention Criterion
$$
\frac{D_{15(filter)}}{D_{85(soil)}} \le 4 \text{ to } 5
$$- 2. Permeability Criterion (To prevent pressure buildup):
Permeability Criterion
$$
\frac{D_{15(filter)}}{D_{15(soil)}} \ge 4 \text{ to } 5
$$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 to form curvilinear squares.
- The total seepage rate is calculated using .
- For Anisotropic Soils (), the physical horizontal dimensions must be scaled () 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).