07-Str-A3 · December 2014

Question 3 of 6: Seepage and the stability of a homogeneous earth dam

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format: PEO / Engineers Canada National Examinations, December 2014, 07-Str-A3 Geotechnical Materials and Analysis. Three hours, closed book, drawing instruments required, one approved Casio or Sharp calculator. All charts and equations are supplied at the back of the paper (the m–n influence chart on page 8, a Newmark chart with $I_N = 0.005$ on page 9, and a two-page formula sheet). Total value 100 marks over six compulsory questions (20 + 10 + 10 + 20 + 20 + 20). Every question and every sub-part is answered in full below.

Reference texts: Das, Principles of Geotechnical Engineering, 9th ed. (Ch. 6 compaction, Ch. 7 permeability, Ch. 8 seepage and flow nets, Ch. 10 stresses in a soil mass, Ch. 11 consolidation, Ch. 12 shear strength); Knappett & Craig, Craig’s Soil Mechanics, 8th ed. (Ch. 2 seepage and flow-net construction, Ch. 5 shear strength, Ch. 11 lateral earth pressure); Holtz, Kovacs & Sheahan, An Introduction to Geotechnical Engineering, 2nd ed. (compacted-fill fabric and Proctor behaviour by USCS group); Harr, Groundwater and Seepage, Ch. 4 (closed-form conformal solutions for a single cut-off); Canadian Geotechnical Society, Canadian Foundation Engineering Manual (CFEM), 4th ed. — the governing Canadian practice document for earth pressures, seepage control and settlement; Canadian Dam Association, Dam Safety Guidelines (2013, rev. 2019); ASTM D698 / D1557 (Proctor compaction), ASTM D4767 (consolidated-undrained triaxial with pore-pressure measurement), ASTM D7181 (consolidated-drained triaxial).

Check — two printing slips in the source, carried exactly as printed. (1) The Question 1 header reads “(4 × 5 = 20 marks)” but five statements (i)–(v) are printed; this solution treats the question as 5 × 4 = 20 marks and answers all five parts. (2) Question 5(a) instructs the candidate to “draw on Figure 4”, while the only section supplied is labelled Figure 3 — the blank grid sheets on pages 6 and 7 are the intended drawing space. Neither slip changes the 100-mark total.

Question 3: Seepage and the stability of a homogeneous earth dam (10 marks)

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.

A homogeneous embankment has no internal drainage, so once the reservoir fills, a steady seepage regime establishes itself through the whole body of the dam. The top streamline — the phreatic surface, closely approximated by Casagrande’s basic parabola — starts at the reservoir level on the upstream face and, in an undrained homogeneous section, daylights on the downstream face well above the toe. Everything below it is saturated and carries positive pore-water pressure. That single fact drives four separate mechanisms, all of which act against stability.

Loss of effective stress, and hence of shear strength. The available strength is $\tau_f = c' + (\sigma - u_w)\tan\phi'$. A silty sand has a negligible effective cohesion, so its strength is almost purely frictional and therefore almost perfectly proportional to effective stress. Raising $u_w$ from zero to the steady seepage value roughly halves $\sigma'$ in the saturated part of the downstream zone, and the mobilised strength falls with it. For an infinite slope the factor of safety collapses from $\tan\phi'/\tan\beta$ in the dry state to

$$F \;=\; \frac{\gamma'}{\gamma_{sat}}\cdot\frac{\tan\phi'}{\tan\beta} \;\approx\; 0.5\,\frac{\tan\phi'}{\tan\beta}$$

for seepage parallel to the slope — a fifty per cent reduction before any other effect is counted.

Seepage force. Water moving through the skeleton drags on it with a body force $j = i\,\gamma_w$ per unit volume, directed along the flow. On the downstream slope the flow lines turn outward and downward, so this force acts to push the surface material off the slope, adding a destabilising component exactly where the strength has already been reduced.

Exit gradient, heave and piping. Where the flow lines converge at the downstream toe the hydraulic gradient reaches its maximum. If the exit gradient approaches the critical value

$$i_c \;=\; \frac{G_s - 1}{1 + e}$$

which for a silty sand at $G_s = 2.68$ and $e = 0.65$ is about 1.02, the effective stress at the surface falls to zero, the soil boils, and grains begin to be carried away. Once erosion starts it works backwards along the flow path as a pipe. Silty sand is the worst possible material for this: it is fine enough to be transported easily yet has essentially no cohesion to hold the roof of a pipe open—so a pipe that forms in it enlarges rapidly and collapses the crest. Internal erosion remains the leading cause of embankment-dam failure worldwide.

Rapid drawdown on the upstream side. The mirror-image problem is that a fine silty sand does not drain quickly. If the reservoir is lowered faster than pore pressures can dissipate, the upstream slope loses the stabilising water load while retaining full internal pore pressure, and the upstream face becomes the critical one.

Figure 3.1 — (a) the untreated homogeneous section: the phreatic surface daylights on the downstream face, the material below it is saturated, and seepage exits on the slope. (b) The same section with internal drainage: an inclined chimney drain intercepts the seepage and a filtered toe drain discharges it, so the downstream fill stays unsaturated and no water exits on the face.

Measure 1 — internal drainage: an inclined chimney drain with a filtered horizontal blanket and toe drain

This is the single most effective remedy and the one a Canadian designer would reach for first. A near-vertical or inclined chimney of clean sand or gravel is built just downstream of the dam centreline and connected to a horizontal blanket drain running to a filtered toe. It intercepts the phreatic surface and drops it to the base of the dam, so the entire downstream zone stays unsaturated: pore pressures there go to zero, the full frictional strength is recovered, the seepage force is redirected harmlessly into the drain, and no water exits on the slope at all. The drain must be designed as a graded filter against the silty sand it protects, satisfying the Terzaghi criteria

$$\frac{D_{15,\text{filter}}}{D_{85,\text{base}}} < 4\ \text{to}\ 5 \qquad\text{and}\qquad \frac{D_{15,\text{filter}}}{D_{15,\text{base}}} > 4\ \text{to}\ 5$$

the first to retain the base soil, the second to keep the drain freely draining. An unfiltered drain in a silty sand is worse than no drain, because it concentrates flow and invites erosion. Modern practice, and the CDA Dam Safety Guidelines, would use a two-stage sand-then-gravel filter or a purpose-graded geotextile-wrapped drain.

Measure 2 — flatten the downstream slope and add a filtered rockfill toe berm

The second measure attacks the geometry rather than the water. Flattening the downstream slope from, say, 2H:1V to 3H:1V directly raises the factor of safety through the $\tan\beta$ term, and lengthens the seepage path so that the exit gradient at the toe falls in proportion. Adding a free-draining rockfill berm at the toe, placed over a graded filter, does three things at once: it weighs down the critical slip surface, it provides a high-permeability outlet so the phreatic surface is drawn down into it, and it acts as a physical protection against the sloughing and surface erosion that a wet slope suffers. Where the foundation is also permeable, an upstream low-permeability blanket or a positive cut-off can be added to lengthen the seepage path further and cut the through-flow, and relief wells can be used to bleed off foundation pressures beneath the downstream berm.

Check — assumed index values. The paper gives no gradation or void ratio for the silty sand, so the illustrative critical gradient above assumes $G_s = 2.68$ and $e = 0.65$, giving $i_c \approx 1.02$. In practice $i_c$ for a silty sand lies between about 0.85 and 1.10, and the design exit gradient should be limited to $i_c/3$ or lower.

Topic. Steady seepage through a homogeneous embankment: pore pressure, seepage force, exit gradient and the drainage measures that control them.

Key relations. $\tau_f = c' + (\sigma - u_w)\tan\phi'$; seepage force per unit volume $j = i\gamma_w$; critical gradient $i_c = (G_s - 1)/(1 + e) = \gamma'/\gamma_w$; Terzaghi filter criteria $D_{15,f}/D_{85,b} < 4$–5 and $D_{15,f}/D_{15,b} > 4$–5; Casagrande’s basic parabola for the phreatic surface.

Why this works. Every one of the four mechanisms is a consequence of the same equation. Pore pressure enters strength only through $\sigma - u_w$, so raising $u_w$ is mechanically identical to unloading the slope. Seepage force is the same physics viewed as a body force rather than as a pressure. The exit gradient is what happens when that body force becomes large enough to carry the effective stress at a free surface to zero. And because all three scale with the height of the phreatic surface, a single intervention that pulls that surface down — internal drainage — removes all three at once. That is why a chimney drain is not one of several equally good options but the standard of practice.

Common pitfall. Proposing a drain without specifying a filter. A candidate who writes “add a toe drain” and stops has described the mechanism that most often initiates internal erosion in a silty sand. A second pitfall is treating the reservoir load as purely stabilising and forgetting the rapid-drawdown case on the upstream slope.

Source. Das, Principles of Geotechnical Engineering, 9th ed., Ch. 8 (seepage through earth dams, filter design); Knappett & Craig, 8th ed., Ch. 2 §2.9; Canadian Dam Association, Dam Safety Guidelines (2013, rev. 2019), Ch. 6.

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