Variable-Head Permeability Test: 18-Env-A3 Geotechnical and Hydrogeological Engineering, May 2013, Q2

Question 2: Variable-Head Permeability Test (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 schematic (Figure 1) — head falls from h₀ to h₁ in the standpipe as water drains through the specimen.

Given.

Given data
Quantity	Symbol	Value
Specimen length	$L$	381 mm (38.1 cm)
Specimen area	$A$	19.4 cm²
Standpipe area	$a$	0.97 cm²
Head at $t=0$	$h_0$	635 mm
Head at $t=8$ min	$h_1$	305 mm

Find. (a) hydraulic conductivity $k$; (b) head difference $h$ at $t=4$ min.

Approach. Apply the standard falling-head (variable-head) permeameter formula to back out $k$ from the observed head drop over 8 minutes, then use the same exponential head-decay law — now with $k$ known — to find the head at the intermediate time $t=4$ min.

Part (a) — solve the variable-head formula for $k$. $$k=\frac{aL}{At}\ln\!\left(\frac{h_0}{h_1}\right)=\frac{(0.97)(38.1)}{(19.4)(480)}\ln\!\left(\frac{635}{305}\right)=\boxed{2.91\times10^{-3}\ \text{cm/s}\ (2.91\times10^{-5}\ \text{m/s})}.$$ ($t=8\ \text{min}=480\ \text{s}$.) This places the soil in the fine-sand/silt permeability range, consistent with a laboratory specimen tested by the falling-head method (coarser, more permeable soils are normally tested by the constant-head method instead).
Part (b) — use the same governing law to find $h$ at $t_2=4$ min. The falling-head derivation gives $h(t)=h_0\exp\!\left(-\dfrac{kA}{aL}t\right)$, i.e. $\ln(h/h_0)$ is linear in $t$. Since $t_2=4$ min is exactly half of the 8-minute interval already measured, $$\frac{h(t_2)}{h_0}=\left(\frac{h_1}{h_0}\right)^{t_2/t}=\left(\frac{305}{635}\right)^{1/2}=0.6930,$$ $$h(4\ \text{min})=635\times0.6930=\boxed{440\ \text{mm}}.$$

Check: substituting $k$ back into $h(t)=h_0\exp(-kAt/aL)$ at $t=240$ s reproduces the same 440.1 mm found by the direct square-root shortcut above — the two routes agree to four figures, confirming the exponential decay is applied consistently.

Quantity	Value
Hydraulic conductivity, $k$	2.91 × 10^-3 cm/s (2.91 × 10^-5 m/s)
Head difference at $t=4$ min	440 mm

Concept & Theory

Topic. Laboratory permeability testing — the falling-head (variable-head) method for low-permeability soils.

Key relations. $k=\dfrac{aL}{At}\ln\!\left(\dfrac{h_0}{h_1}\right)$; equivalently $h(t)=h_0\exp\!\left(-\dfrac{kA}{aL}t\right)$, derived from continuity between the standpipe's falling head and Darcy flow through the specimen.

Why this works. As water drains from the standpipe (area $a$) through the specimen (area $A$, length $L$), the head loss driving Darcy flow itself decreases with time, giving a first-order ODE $-a\,dh/dt = kA(h/L)$ whose solution is the exponential decay used above — unlike the constant-head test, where $h$ (and hence $Q$) stays fixed throughout.

Common pitfall. Mixing millimetres and centimetres inside the same formula (here $L$, $A$ and $a$ are naturally centimetre-based while $h_0,h_1$ are given in millimetres); forgetting that the log term needs the RATIO $h_0/h_1 > 1$, not $h_1/h_0$.

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

Question 2 of 6: Variable-Head Permeability Test

Question 2: Variable-Head Permeability Test (20 marks)