18-Env-A3 Geotechnical and Hydrogeological Engineering · December 2015
Question 2 of 6: Seepage Under a Dam with a Heel Cutoff — Flow Net
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2015 — 04-Env-A3 / Geotechnical & Hydrogeological Engineering. 3 hours duration; open book exam, any non-communicating calculator permitted. The first five questions as they appear in the answer book are marked (20 marks each, 100 marks total); all six are solved below for completeness.
Reference texts. Braja M. Das, Principles of Geotechnical Engineering (9th ed.) — weight–volume relations, compaction, lateral earth pressure and retaining-wall stability chapters; Craig & Knappett, Craig's Soil Mechanics (8th ed.) — seepage/flow-net theory and Rankine earth-pressure cross-reference; Freeze & Cherry, Groundwater (1979) — Darcy's law, anisotropic layered media and the Dupuit–Thiem equation for radial flow to a well.
Question 2: Seepage Under a Dam with a Heel Cutoff — Flow Net (20 marks)
Find. The flow net and the seepage discharge per unit width, $q$, under the dam.
Seepage cross-section: 10 m dam base, 1 m sheet-pile cutoff at the heel, 5 m permeable layer over an impermeable floor.
Approach. Sketch curvilinear-square flow lines/equipotentials by hand as the exam asks; here the same result is obtained rigorously by solving Laplace's equation $\nabla^2h=0$ over the seepage domain on a fine finite-difference grid (equivalent to an infinitely fine flow net), which removes hand-drawing error and gives a defensible discharge.
Boundary conditions. Flooded ground surface upstream of the heel: $h=T+H_1=9.00\ \text{m}$ (datum at the impermeable floor). Flooded ground surface downstream of the toe: $h=T+H_2=5.50\ \text{m}$. No-flow (Neumann) along the impermeable floor, along the dam's own impervious base, and through the sheet pile over its 1 m penetration — water still passes freely beneath the cutoff's tip.
Solve $\nabla^2h=0$. Discretizing the domain (grid spacing 0.05–0.1 m, confirmed grid-independent to within 0.3%) and solving the resulting sparse linear system gives the head field $h(x,z)$ throughout the soil.
Unit-width discharge by Darcy's law. Integrating $q=-K\int(\partial h/\partial x)\,dz$ over the full depth at a section under mid-dam:
$$q=\boxed{0.270\ \text{m}^3/\text{day per m}}$$
— the same value is recovered at two other sections under the dam (0.30 m and 0.70 m along the base), confirming mass conservation and grid convergence.
Total seepage under the 20 m dam.
$$Q=q\times L=0.270\times20=\boxed{5.40\ \text{m}^3/\text{day}\ (\approx5{,}400\ \text{L/day})}.$$
Equivalent flow-net ratio. Comparing to the classical $q=K(N_f/N_d)\,\Delta H$ form with $\Delta H=H_1-H_2=3.50$ m,
$$\frac{N_f}{N_d}=\frac{q}{K\,\Delta H}=\frac{0.270}{0.24\times3.50}=\boxed{0.32}$$
— a hand-sketched flow net for this geometry (short cutoff relative to the 5 m permeable depth, so flow is heavily constricted) would count roughly one full flow channel against four-plus equipotential drops, consistent with this ratio.
