Falling-Head Permeameter — Landfill Cap Suitability: 18-Env-A3 Geotechnical and Hydrogeological Engineering, December 2013, Q2

Question 2: Falling-Head Permeameter — Landfill Cap Suitability (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.

Falling-head permeameter (Figure 1) — head falls from h₀ to h₁ in the standpipe as water drains through the 32 mm specimen.

Given.

Given data
Quantity	Symbol	Value
Specimen diameter	$D$	120 mm
Specimen length	$L$	32 mm
Standpipe inside diameter	$d$	8.0 mm
Head at $t=0$	$h_0$	503 mm
Head at $t$	$h_1$	322 mm
Elapsed time	$t$	8 h 12 min = 29,520 s
Spec limit	$k_{max}$	1 × 10^-8 cm/s

Find. (a) hydraulic conductivity $k$; (b) whether $k \le k_{max}$.

Approach. Apply the standard falling-head formula, converting diameters to standpipe and specimen cross-sectional areas, then compare the result directly against the 1×10^-8 cm/s cap-liner specification.

Part (a) — cross-sectional areas. $$a=\frac{\pi}{4}d^2=\frac{\pi}{4}(8.0)^2=50.27\ \text{mm}^2,\qquad A=\frac{\pi}{4}D^2=\frac{\pi}{4}(120)^2=11{,}309.7\ \text{mm}^2.$$
Solve the falling-head formula for $k$. $$k=\frac{aL}{At}\ln\!\left(\frac{h_0}{h_1}\right)=\frac{(50.27)(32)}{(11{,}309.7)(29{,}520)}\ln\!\left(\frac{503}{322}\right)=2.149\times10^{-6}\ \text{mm/s}.$$ Converting to the specification's units, $$k=\boxed{2.15\times10^{-7}\ \text{cm/s}\ \left(2.15\times10^{-9}\ \text{m/s}\right)}.$$
Part (b) — compare to the cap-liner specification. $$\frac{k}{k_{max}}=\frac{2.15\times10^{-7}}{1\times10^{-8}}\approx 21.5.$$ The tested soil is about 21.5 times MORE permeable than the 1×10^-8 cm/s ceiling, so it does $\boxed{\text{NOT meet the specification}}$ — it is too permeable to be used, unadmixed, as a landfill cap liner.

Check: $k\approx2\times10^{-7}$ cm/s sits in the silt/silty-clay range on the standard soil-permeability chart, which is plausible for a natural borrow soil but is roughly one to two orders of magnitude too permeable for a regulatory landfill cap (compacted clay liners are typically engineered, e.g. by bentonite admixture and controlled compaction wet-of-optimum, to reach $k\le10^{-8}$ cm/s).

Quantity	Value
Hydraulic conductivity, $k$	2.15 × 10^-7 cm/s (2.15 × 10^-9 m/s)
Meets 1×10^-8 cm/s cap spec?	No — about 21.5× too permeable

Concept & Theory

Topic. Laboratory permeability testing — the falling-head method, applied to a landfill-cap material screening decision.

Key relations. $k=\dfrac{aL}{At}\ln\!\left(\dfrac{h_0}{h_1}\right)$, derived from continuity between the standpipe's falling head (area $a$) and Darcy flow through the specimen (area $A$, length $L$).

Why this works. The falling-head test is used (instead of constant-head) precisely because the material is expected to be of LOW permeability — a constant-head test on a tight soil would take impractically long to produce a measurable outflow volume, whereas the falling head itself, read on a fine-bore standpipe, gives a measurable signal in hours.

Common pitfall. Mixing millimetres and centimetres inside one formula (this problem is naturally millimetre-based throughout, so the unit conversion is deferred to the final answer only); and forgetting the "$\le$" direction of the specification — a cap liner needs LOW $k$, so exceeding the limit is a failure, not a pass.

Source. Das, Principles of Geotechnical Engineering, ch. 7 (permeability); Craig's Soil Mechanics, ch. 2.

Question 2 of 6: Falling-Head Permeameter — Landfill Cap Suitability

Question 2: Falling-Head Permeameter — Landfill Cap Suitability (20 marks)