Flow line is a line along which a water particle travels from the upstream side to the downstream side through a permeable soil medium.
Equipotential line is a line along which total head is the same at every point.
Flow net is the combination of flow lines and equipotential lines. The space between two adjacent flow lines is a flow channel.
In an ideal flow net for isotropic soil, flow lines and equipotential lines intersect at right angles and form approximate curvilinear squares. Each flow channel carries the same discharge, and each potential drop has the same head loss.
For seepage per unit length perpendicular to the flow-net section, use the number of flow channels and potential drops from the flow net.
Here $q$ is seepage discharge per unit width, $k$ is coefficient of permeability, $H$ is total head difference, $N_f$ is number of flow channels, and $N_d$ is number of potential drops.
Uplift pressure at any point under a hydraulic structure is based on the pressure head remaining at that point. In many flow-net problems, the head loss is counted by potential drops from the upstream side.
If the pressure diagram is trapezoidal under a base length $B$, then the uplift force per meter length is the average pressure times the base length.
The exit gradient is critical near the downstream exit point because high upward gradient can lead to boiling, piping, or quick condition.
$\Delta s$ is the length of the last flow field at the exit. The critical gradient $i_c$ usually comes from soil properties.
For anisotropic soil, use the transformed-section idea. The equivalent permeability for the discharge equation is the geometric mean of the horizontal and vertical coefficients.
A flow net can be used to determine seepage, exit gradient, and uplift pressure.
Answer: Seepage, exit gradient, and uplift pressure.
In any flow net, the strip between two adjacent flow lines is called a flow channel.
Answer: Flow channel.
A non-isotropic soil supports a dam. The flow net has 4 flow channels and 20 equipotential drops. The vertical coefficient of permeability is $k_v = 0.002$ m/day and the horizontal coefficient is $k_x = 0.018$ m/day.
The total head difference $H$ is read from the figure in the reference problem. Substitute that head difference into the final expression.
The coefficient of permeability below a dam is 4 m/day. Upstream water is 20 m higher than downstream water. The flow net has $N_d=10$ potential drops and $N_f=4$ flow channels. The base of the dam is 1 m below ground, and between heel and toe there are 8 potential drops over a 30 m base.
For uplift pressure, count the potential drops from upstream to the heel or toe as indicated by the figure, then use $u=\gamma_w h_p$. The PDF statement gives the drop count along the base; the exact heel/toe pressure depends on the equipotential lines shown in the figure.
From the given base of a weir and equipotential lines, compute uplift pressure at A, uplift pressure at B, and uplift force per unit length. The reference solution indicates 6 equipotential drops and a loss of head of 2 m per drop.
Answer: $u_A=112.82$ kPa, $u_B=34.34$ kPa, and uplift force is 882.96 kN per meter length.
Given a flow net, total head difference $H$, number of potential drops $N_d$, and the length of the last flow field $\Delta s$, determine the exit gradient.
If the problem asks for piping safety, compare $i_e$ with the critical hydraulic gradient.
Piping occurs when the exit gradient exceeds the critical hydraulic gradient, causing soil particles to be carried out by seeping water. This can progressively enlarge a flow channel until catastrophic failure of a dam or levee occurs. Heave is the uplift of a soil mass by seepage pressure acting on the downstream side.
To prevent piping, filters are placed at the exit face. A filter must be coarse enough to let water out freely but fine enough to retain the base soil particles. The factor of safety against piping is the ratio of critical gradient to the actual exit gradient and should generally exceed 3 to 5 for important structures.
Below a concrete dam, a flow net is drawn with $N_f=4$ flow channels and $N_d=12$ equipotential drops. The upstream head is 8 m above the downstream head. The coefficient of permeability of the foundation soil is $k=2.5\times10^{-4}$ m/sec. Compute the seepage per unit length of dam in liters per minute.
Answer: 40.0 liters per minute per meter length of dam.
A flow net below a dam has a total head difference of 15 m and 10 equipotential drops. The base of the dam is 1 m below ground. A point at the heel of the dam (upstream edge of the base) loses 1 potential drop from the upstream face. Compute the head loss per drop, the pore pressure at the heel, and the uplift pressure at the heel of the dam.
Total head at heel = upstream head minus 1 potential drop = $15 - 1.5 = 13.5$ m above the datum. The heel is 1 m below ground; take datum at base of dam:
Answer: $\Delta h=1.5$ m per drop, and uplift pressure at the heel is 142.2 kPa.
A flow net below a sheet-pile wall has $N_d=6$ potential drops and a total head difference of 3 m. The last flow field at the downstream exit has a length of 0.50 m. The foundation soil has $G_s=2.68$ and void ratio $e=0.55$. Compute the exit gradient and the factor of safety against piping.
A factor of safety of only 1.08 is dangerously low. The design is nearly at the piping failure condition.
Answer: $i_e=1.0$, $i_c=1.084$, and $FS=1.08$ against piping.
A dam rests on an anisotropic soil foundation with $k_x=6\times10^{-3}$ m/day and $k_v=6\times10^{-4}$ m/day. The flow net drawn on the transformed section has $N_f=5$ flow channels and $N_d=15$ potential drops. The upstream reservoir is at elevation 25 m and the downstream tailwater is at elevation 10 m. Compute the seepage per meter length of dam per day.
Answer: $q=9.48\times10^{-3}$ m3/day per meter length of dam.