Seepage under a cut-off wall — flow net, quantity, effective stress, stability: 16-Civ-A4 Geotechnical Materials and Analysis, May 2013, Q5

Question 5: Seepage under a cut-off wall — flow net, quantity, effective stress, stability (20 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.

Given. A sheet-pile cut-off in a permeable stratum 6 m thick over an impermeable layer (Figure 4). Upstream water stands 3 m above the ground surface; downstream water is at the ground surface, so the differential head is H = 3 m. The pile toe is 3 m below ground (mid-depth of the stratum). k = 2.0 × 10⁻⁵ m/s; γ_sat = 20 kN/m³; point A is on the upstream (back) face, 1 m below the ground surface.

Find. (b) seepage q per metre of wall; (c) the effective stress at A; (d) the factor of safety against heave/piping.

Figure 4 — cut-off wall with a sketched flow net: N_f = 4 flow channels, N_d = 8 equipotential drops. Differential head H = 3 m; stratum 6 m thick; pile penetration 3 m; point A on the back face 1 m below ground.

Check: a flow net is a hand sketch, so N_f, N_d and the number of drops reaching A carry a ± one-field tolerance. The net adopted here (N_f = 4, N_d = 8, with A about 1 drop in) is a curvilinear-square net consistent with the geometry; a slightly different but equally valid net shifts q and σ'_A by 10–20 % without changing the conclusions.

(a) Flow net. Sketched with the standard rules — flow lines and equipotential lines intersect at right angles forming curvilinear squares, the impermeable base and the wall are flow lines, and the upstream and downstream beds are equipotentials. The adopted net has N_f = 4 flow channels and N_d = 8 equipotential drops.

Approach (b)–(d). Seepage from q = kH(N_f/N_d); pore pressure at A from the flow-net head (total head − elevation head); heave stability from Terzaghi's criterion FS = γ'D / (γ_wh_a) with i_cr = γ'/γ_w as a check.

(b) Quantity of seepage. Head loss per drop Δh = H/N_d = 3/8 = 0.375 m. For unit width, $$q=k\,H\,\frac{N_f}{N_d}=(2.0\times10^{-5})(3)\left(\frac{4}{8}\right).$$ $$q=\boxed{3.0\times10^{-5}\ \text{m}^3/\text{s per m of wall}.}$$
(c) Total stress at A. A lies 1 m below the upstream ground surface, beneath 3 m of free water and 1 m of saturated soil: $$\sigma_A=\gamma_w(3)+\gamma_{sat}(1)=9.81(3)+20(1)=29.43+20=49.43\ \text{kPa}.$$
(c) Pore pressure at A from the flow net. Take the datum at the downstream water level (ground surface). Total head at the upstream source = +3 m; A is only 1 m down a 3 m pile face, so it lies about 1 drop downstream of the source (a finite-difference solution of the same geometry puts it 0.83 drops in), and its total head is h_A = 3 − 1(0.375) = 2.625 m. With A at elevation −1 m, the pressure head is 2.625 − (−1) = 3.625 m, hence $$u_A=\gamma_w(3.625)=9.81\times3.625=35.6\ \text{kPa}.$$ (Downward seepage on the upstream face makes u_A smaller than the 39.2 kPa hydrostatic value, which raises the effective stress.)
(c) Effective stress at A. $$\sigma'_A=\sigma_A-u_A=49.43-35.6=\boxed{13.9\ \text{kPa}.}$$
(d) Stability against heave/piping. On the downstream side the flow is upward; heave occurs when the upward seepage force lifts the soil prism at the pile toe. The critical hydraulic gradient is $$i_{cr}=\frac{\gamma'}{\gamma_w}=\frac{20-9.81}{9.81}=1.04.$$ By Terzaghi's heave criterion on a prism of depth D = 3 m at the toe, taking the excess head h_a as the head at the pile toe, H/2 = 1.5 m (a conservative upper bound on the prism-average head, which the exact net puts nearer 1.0 m, FS ≈ 3), $$FS=\frac{\gamma'\,D}{\gamma_w\,h_a}=\frac{10.19\times3}{9.81\times1.5}=\frac{30.57}{14.72}=\boxed{2.1.}$$ FS ≈ 2.1 > 1.5, so the cut-off wall is stable against heave/piping with an adequate margin.

Cut-off wall — seepage and stability
Quantity	Value
Flow net	N_f = 4, N_d = 8, Δh = 0.375 m
Seepage q (per m)	3.0 × 10⁻⁵ m³/s
Total stress at A	49.4 kPa
Pore pressure at A	35.6 kPa
Effective stress at A	13.9 kPa
Critical gradient i_cr	1.04
Factor of safety (heave)	≈ 2.1 (stable)

Topic: Two-dimensional confined seepage — flow nets, seepage quantity, uplift and heave stability.

Key relations: q = kH(N_f/N_d); Δh = H/N_d; total head = pressure head + elevation head; u = γ_w·(pressure head); i_cr = γ'/γ_w; Terzaghi heave FS = γ'D/(γ_wh_a).

Why this works: A curvilinear-square flow net divides the seepage field into equal-flow channels and equal-head-drop bands, so discharge and pore pressure follow directly from counting N_f and N_d. Effective stress at any point is the total stress minus the flow-net pore pressure; heave is checked by comparing the buoyant weight of the exit prism against the uplift the seepage head generates there.

Common pitfall: Using the full head H instead of H/N_d per drop, or taking the hydrostatic pore pressure at A and ignoring the head already lost to seepage — the downward flow on the upstream face reduces u and raises σ'.

Source: Das & Sobhan, Principles of Geotechnical Engineering, Ch. 8 (seepage, flow nets, uplift and heave/piping).

Question 5 of 6: Seepage under a cut-off wall — flow net, quantity, effective stress, stability

Question 5: Seepage under a cut-off wall — flow net, quantity, effective stress, stability (20 marks)