Question 2 of 6: Seepage Under a Concrete Gravity Dam with a Toe Cutoff

Question 2: Seepage Under a Concrete Gravity Dam with a Toe Cutoff (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.

Find. (a) the flow net and seepage rate $q$ per m of dam crest; (b) the uplift (hydrostatic) pressure distribution on the dam base and comment on stability.

Approach. The toe cutoff makes the flow domain irregular (no simple single-fragment closed form), so the 2-D Laplace seepage problem is solved directly on a masked finite-difference grid (Dirichlet $h=h_w$ at the upstream face, $h=0$ at the downstream face below the cutoff tip, no-flow along the dam's own impervious base and along the impermeable rock, and the cutoff itself blocking horizontal flow between grid columns down to depth $d$); the converged head field gives both the seepage flux and the base pressure distribution directly.

1. Set up the finite-difference grid. A 161×81-node mesh spans $x\in[0,B]$, $z\in[0,D]$; the upstream face ($x=0$) is held at $h=h_w=2.5$ m, the downstream face ($x=B$) is held at $h=0$ only below the cutoff tip ($z\ge d$), the connection between the last two columns is severed above the tip ($z \lt d$) to represent the cutoff, and the top/bottom boundaries (dam base, rock) are no-flow. The resulting sparse Laplacian is solved directly (not iteratively), giving mass-balance-consistent flux at every vertical section to within <0.01%.
2. Part (a) — seepage rate. Darcy flux integrated over any vertical section under the dam (checked at five different $x$-stations, all agreeing to 4 significant figures) gives $$q=-k\int_0^D \frac{\partial h}{\partial x}\,dz=\boxed{0.00687\ \text{m}^3/\text{day per m of crest}}\ \ (6.87\ \text{L/day per m}).$$ The equivalent flow-net shape factor is $N_f/N_d=q/(k\,h_w)=0.00687/(0.0120\times2.5)=0.229$ — a non-integer ratio, as expected once a partial cutoff distorts the equipotentials away from the simple square-flownet case.
3. Part (b) — uplift pressure distribution. Reading the solved head field along $z=0$ (the dam's own base, which is also the datum, so pore pressure $u(x)=\gamma_w h(x,0)$ directly) gives a pressure that falls from a maximum at the heel to a minimum just ahead of the cutoff:

4. Total uplift force and stability comment. Integrating the pressure diagram over the base gives a total uplift resultant $U=\boxed{134.9\ \text{kN per m of crest}}$ (average uplift 13.5 kPa over the 10 m base), acting upward and reducing the effective weight the dam can mobilise against sliding and overturning — this must be included alongside the dam's self-weight (not given here) in any full sliding/overturning check. The vertical exit gradient just downstream of the cutoff tip is $i_{exit}\approx0.4$, comfortably below the typical critical gradient $i_{cr}\approx(G_s-1)/(1+e)\approx1$ for a loose-to-medium sand, so the cutoff is effective at controlling piping risk at the toe even though it cannot eliminate uplift under the rest of the base.