18-Env-A3 Geotechnical and Hydrogeological Engineering · December 2017
Question 3 of 6: Seepage Beneath an Earth Dam with a Partial-Penetration Cutoff Wall
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2017 — 04-Env-A3 / Geotechnical & Hydrogeological Engineering. 3 hours duration; open book exam, any non-communicating calculator permitted. FIVE (5) questions constitute a complete exam paper (the first five as they appear in the answer book are marked, 20 marks each, 100 marks total); all six printed questions are solved below for completeness.
Reference texts. Braja M. Das, Principles of Geotechnical Engineering (9th ed.) — weight–volume relations, compaction, seepage/flow nets, and consolidation chapters; Craig & Knappett, Craig's Soil Mechanics (8th ed.) — cross-reference for flow-net theory and finite-difference seepage; Freeze & Cherry, Groundwater (1979) — Darcy's law, the Thiem confined-flow equation, and radial travel time.
Question 3: Seepage Beneath an Earth Dam with a Partial-Penetration Cutoff Wall (20 marks)
Find. (a) the total seepage volume beneath the dam per day; (b) the maximum seepage (Darcy) velocity beneath the dam.
Approach. The cutoff sits at mid-length and only partially penetrates the sand layer, so there is no single-fragment closed-form solution; the 2-D Laplace seepage problem is solved directly on a masked finite-difference grid (full-depth head boundaries at the upstream and downstream toes, no-flow along the dam's impervious base and the underlying clay, and the cutoff blocking horizontal flow between grid columns down to its tip), and the converged head field gives both the total flux and the velocity through the narrow gap beneath the cutoff tip where flow concentrates.
Fig. Q3 — dam cross-section: 130.5 m base on a 34.25 m sand layer over impervious clay, 24.86 m head loss, and a 15.86 m cutoff wall centred beneath the crest.
Set up the finite-difference grid. A 401×121-node mesh spans $x\in[0,B]$, $z\in[0,D]$ (z measured down from the dam base); the upstream face is held at $h=701.5\ \text{ft}$ and the downstream face at $h=620.0\ \text{ft}$ (both full depth), the top and bottom boundaries (dam base, clay) are no-flow, and the connection between the two grid columns straddling the cutoff is severed for $z \lt d=15.86\ \text{m}$. The sparse Laplacian is solved directly, and the flux computed at several different $x$-stations agrees to 4 significant figures — confirming mass balance.
Part (a) — seepage rate. Darcy flux integrated over a vertical section under the dam gives
$$q=-k\int_0^D\frac{\partial h}{\partial x}\,dz=307.7\ \text{m}^3/\text{day per m of crest}\quad(\text{converges to }308\pm0.5\%\text{ on mesh refinement}).$$
Over the full 120 m crest length,
$$Q=q\,L=307.7\times120=\boxed{36{,}900\ \text{m}^3/\text{day}}\ \ (\approx37{,}000\ \text{m}^3/\text{day}).$$
The equivalent flow-net shape factor is $N_f/N_d=q/(k\,\Delta H)=307.7/(51.84\times24.86)=0.239$.
Part (b) — maximum seepage velocity. All of the seepage must pass through the narrow gap between the cutoff tip and the clay, of open height $D-d=34.25-15.86=18.39\ \text{m}$, so this is where the flow lines are most compressed and the (Darcy) velocity is highest. Taking the flux through that section and dividing by its open height gives the average velocity in the neck:
$$v_{max}=\frac{q}{D-d}=\frac{307.7}{18.39}=\boxed{16.7\ \text{m/day}}\ \ (1.94\times10^{-4}\ \text{m/s}).$$
Check: the exact point velocity at the cutoff tip's sharp corner is mathematically singular (it keeps rising, not converging, as the mesh is refined further) — this is a known artefact of idealised flow-net theory at a re-entrant corner, not a real physical infinity. $v_{max}=16.7\ \text{m/day}$ is reported as the mesh-converged AVERAGE velocity through the necked cross-section beneath the tip, which is the physically meaningful "maximum seepage velocity" a flow net or a finite grid can actually deliver.
