24-Bld-A6 Geotechnical Materials and Analysis, December 2017, Q3

Question 3 (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.

The critical hydraulic gradient, i_cr, is the upward seepage gradient at which the upward seepage (drag) force on a soil element exactly equals its submerged (buoyant) weight, driving the effective stress at that point to zero: $$i_{cr} = \frac{\gamma'}{\gamma_w} = \frac{G_s-1}{1+e}.$$ For a typical sand ($G_s\approx2.65$, $e\approx0.6$–0.7) this works out to $i_{cr}\approx0.9$–1.0. Once the actual exit gradient reaches i_cr, the soil loses all of its inter-granular contact stress — the classic "quicksand" or boiling condition — and the ground surface heaves and loses essentially all bearing and shearing resistance even though nothing has physically eroded yet.

Sketch — critical hydraulic gradient as the balance point between upward seepage force and submerged weight, and the standard countermeasures.

Significance in design. Any structure that retains water and forces seepage through or beneath a soil mass (earth dams, cofferdams, sheet-pile walls, excavation bases) must keep the actual exit gradient safely below i_cr, because reaching i_cr causes piping: soil at the point of maximum exit gradient loses effective stress first, particles are washed out, the exit channel enlarges, and the enlarged channel concentrates even more flow — a self-accelerating (regressive) erosion process that can undermine and fail the structure with little warning. Design codes therefore require a minimum factor of safety against piping, typically $FS = i_{cr}/i_{exit}\ge3$–4 for critical structures such as dams, and lower (1.5–2) for temporary works such as braced excavations.

Design measures to counter high exit gradients:

Lengthen the seepage path — deeper or longer cutoff walls/sheet piling, or an upstream impervious blanket, both increase the number of equipotential drops the flow must undergo, directly lowering the gradient at any given exit point.
Provide a downstream loaded (weighted) filter — a graded filter blanket placed over the exit area adds confining weight (raising the local factor of safety against uplift/heave) while its grading is designed to let water escape freely without carrying soil particles with it.
Install relief wells or a toe drain — deliberately relieving excess pore pressure at controlled points lowers the head that would otherwise appear at the natural (uncontrolled) exit point.
Reduce the driving head where practicable, or flatten/extend the downstream slope so the seepage exits over a longer distance at a lower average gradient.

Concept & Theory

Topic. Critical hydraulic gradient, piping/boiling failure, and seepage-control design measures.

Key relations. $i_{cr}=\gamma'/\gamma_w=(G_s-1)/(1+e)$; factor of safety against piping $FS=i_{cr}/i_{exit}$.

Why this works. Effective stress, not total stress, provides frictional shear resistance in soil; an upward gradient reduces effective stress by adding a seepage body force opposite to gravity, and at i = i_cr that body force exactly cancels the submerged weight, leaving zero effective stress and zero shear strength regardless of the soil's friction angle.

Common pitfall. Checking only the AVERAGE exit gradient rather than the gradient in the shortest/most concentrated exit flow-net field, which is always the critical (highest) one in practice.

Source. Das, Principles of Geotechnical Engineering, Ch. 7 (seepage, quicksand/piping); CFEM Ch. 5 (seepage control).

Question 3 of 7

Question 3 (10 marks)