Flow Net Through an Earthen Dam: 18-Geol-A2 Hydrogeology, December 2016, Q2

Question 2: Flow Net Through an Earthen Dam (equal value)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A homogeneous, isotropic trapezoidal earthen dam on an impermeable base; headwater depth $H_1=6\ \text{m}$ against the upstream face, tailwater depth $H_2=2.25\ \text{m}$ against the downstream face, hydraulic conductivity $K=0.066\ \text{m/day}$. The scale bars on the source figure give the dam's true proportions; only the two water depths and $K$ are needed once a flow net is constructed.

Find. (a) A flow net (families of flow lines and equipotential lines satisfying the boundary conditions); (b) the seepage discharge per unit width of dam, in m³/day per metre.

Figure 2. Constructed flow net: the phreatic surface and the impermeable base bound the flow domain; three interior flow lines divide it into $N_f=4$ curvilinear-square flow channels, cut by 12 equipotential lines ($N_d=12$ head drops).

Approach. (a) Build the flow net from the boundary conditions — the phreatic (top) surface and the impermeable base are both flow lines, the wetted upstream and downstream faces are equipotentials — and sketch flow lines/equipotentials as curvilinear squares. (b) Read off the number of flow channels $N_f$ and equipotential drops $N_d$ from that net and apply $q = KH(N_f/N_d)$.

Check: a hand-drawn flow net is a graphical estimate, not a unique analytical solution. The net above uses $N_f=4$ flow channels and $N_d=12$ equipotential drops, a standard curvilinear-square construction for a dam of this height-to-base proportion (per Freeze & Cherry Ch. 5 / Cedergren's worked dam examples) — a carefully hand-drawn net will land within about ±10–15% of this $N_f/N_d$ ratio.

Total head loss across the dam. The seepage-driving head is the difference between the two water-surface elevations relative to the base: $$H = H_1-H_2 = 6-2.25=\boxed{3.75\ \text{m}}.$$
Seepage per unit width from the flow net. For a well-constructed curvilinear-square net, the discharge per unit width out-of-page is $$q = KH\left(\frac{N_f}{N_d}\right) = (0.066\ \text{m/day})(3.75\ \text{m})\left(\frac{4}{12}\right)=\boxed{0.0825\ \text{m}^3/\text{day per m}}.$$

Quantity	Result
(a) Flow net	$N_f=4$ flow channels, $N_d=12$ equipotential drops (Figure 2)
(b) Total head loss H	3.75 m
(b) Seepage per unit width	0.0825 m³/day per m of dam

Concept & Theory

Topic. Two-dimensional steady seepage through an earthen dam via flow-net construction.

Key relations. Laplace's equation $\nabla^2 h=0$ governs steady seepage in a homogeneous, isotropic medium; a flow net is its graphical solution, drawn as two orthogonal families of curves (flow lines and equipotentials) forming curvilinear squares. Discharge per unit width: $q=KH(N_f/N_d)$.

Why this works. Each flow channel between two adjacent flow lines carries an equal share of the total discharge; each equipotential drop dissipates an equal share $H/N_d$ of the total head loss $H$. Darcy's law applied to one curvilinear square (side length $\approx b$, so gradient $\approx (H/N_d)/b$ and flow area $\approx b$ per unit width) gives $\Delta q = K (H/N_d)$ per channel, and summing $N_f$ channels gives $q=KH(N_f/N_d)$ — the channel geometry cancels because the squares are self-similar.

Common pitfall. Forgetting that the phreatic (free) surface is itself an unknown flow line whose position must satisfy both the "flow line" condition and (along it) a linearly-varying pressure = atmospheric condition — students often draw it as a straight line between the two water levels instead of the concave-up curve seepage theory requires. Also, mixing up $N_f$ and $N_d$ in the formula inverts the answer by roughly an order of magnitude.

Source. Freeze & Cherry, Groundwater, Ch. 5.3 (flow nets and the graphical solution of Laplace's equation for seepage problems).

Question 2 of 5: Flow Net Through an Earthen Dam

Question 2: Flow Net Through an Earthen Dam (equal value)