Falling-Head Permeameter, Storage Properties and Soil Phase Relations: 18-Geol-A2 Hydrogeology, May 2017, Q1

Question 1: Falling-Head Permeameter, Storage Properties and Soil Phase Relations (equal value)

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) Falling-head tube diameter 1.8 cm, sample-tube diameter 12.0 cm, flow length $L=15$ cm, head falls from $h_0=6.0$ cm to $h_1=0.50$ cm over $t=550$ min at a constant test temperature of 15°C. (b) Confined aquifer specific storage $S_s=5.33\times10^{-3}\ \text{m}^{-1}$, porosity $n=0.3$, water compressibility $\beta_w=4.6\times10^{-10}\ \text{m}^2/\text{N}$. (c) Aquifer thickness $b=30$ m, piezometric decline $\Delta h=2.5$ m over a circular area of radius $r=220$ m.

Symbol	Value
$V_t$ (total volume, part d)	11.8 cm³
$M_t$ (moist mass, part d)	23.1 g
$M_s$ (dry mass, part d)	21.1 g
$\rho_s$ (grain density, part d)	2.65 g/cm³

Find. (a) hydraulic conductivity $K$ and intrinsic permeability $k$ of the soil, and its likely classification. (b) skeletal (matrix) compressibility $\alpha$. (c) volume of water released as the piezometric surface declines. (d) porosity, bulk density, void ratio and moisture content of the moist sample.

Approach. Part (a) uses the standard falling-head permeameter equation to get $K$ directly from the geometry and head ratio (no fluid properties needed), then converts to intrinsic permeability with the fluid properties at the 15°C test temperature (not the paper's 20°C default, since a temperature is explicitly stated). Part (b) inverts $S_s=\rho_wg(\alpha+n\beta_w)$ for the skeletal term $\alpha$. Part (c) uses $S_s$ (as a storativity per unit thickness) to convert the piezometric decline over the given area directly into a released volume. Part (d) is a standard four-measurement soil phase-relation reduction.

Part (a) — hydraulic conductivity from the falling-head test. With standpipe area $a=\pi(1.8/2)^2=2.545\ \text{cm}^2$ and sample area $A=\pi(12.0/2)^2=113.10\ \text{cm}^2$, $t=550\times60=33{,}000$ s: $$K=\frac{aL}{At}\ln\!\left(\frac{h_0}{h_1}\right)=\frac{(2.545)(15)}{(113.10)(33{,}000)}\ln\!\left(\frac{6.0}{0.50}\right)=\boxed{2.54\times10^{-5}\ \text{cm/s}\ (2.54\times10^{-7}\ \text{m/s})}.$$
Intrinsic permeability at the test temperature. $K$ is a hydraulic conductivity measured with water at 15°C, so converting to the fluid-independent intrinsic permeability $k=K\mu/(\rho g)$ needs water's properties AT 15°C ($\mu_{15}=1.139\times10^{-3}\ \text{kg/m-s}$, $\rho_{15}=999.1\ \text{kg/m}^3$), not the paper's 20°C default: $$k=\frac{(2.54\times10^{-7})(1.139\times10^{-3})}{(999.1)(9.81)}=\boxed{2.95\times10^{-14}\ \text{m}^2}.$$ At $K\approx2.5\times10^{-7}\ \text{m/s}$ the material sits squarely in Freeze & Cherry's silt / silty-sand range ($10^{-9}$–$10^{-6}\ \text{m/s}$), well below clean-sand values — the soil tested behaves hydraulically as a silt, not a sand.
Part (b) — skeletal compressibility from specific storage. Specific storage combines the matrix (skeleton) and pore-water compressibilities, $S_s=\rho_wg(\alpha+n\beta_w)$, so: $$\alpha=\frac{S_s}{\rho_wg}-n\beta_w=\frac{5.33\times10^{-3}}{(998)(9.81)}-(0.3)(4.6\times10^{-10})=\boxed{5.44\times10^{-7}\ \text{m}^2/\text{N}}.$$ The skeleton is roughly 1,200 times more compressible than the water itself — consistent with $S_s$ of a confined aquifer being dominated by grain-skeleton rearrangement, not pore-fluid compression.
Part (c) — volume released over the piezometric decline. Multiplying $S_s$ by the aquifer thickness gives the (dimensionless) storativity, $S=S_sb=(5.33\times10^{-3})(30)=0.1599$; the released volume is then $S$ times the affected area times the head decline: $$V=S\cdot(\pi r^2)\cdot\Delta h=(0.1599)\left[\pi(220)^2\right](2.5)=\boxed{6.08\times10^4\ \text{m}^3\ (60{,}783\ \text{m}^3)}.$$
Part (d) — solids and void volume. The oven-dried solids occupy $V_s=M_s/\rho_s=21.1/2.65=7.962\ \text{cm}^3$, leaving $V_v=V_t-V_s=11.8-7.962=3.838\ \text{cm}^3$ as void space.
Porosity, void ratio, bulk density. $$\begin{aligned} n &=\frac{V_v}{V_t}=\frac{3.838}{11.8}=\boxed{0.325\ (32.5\%)} \\ e &=\frac{V_v}{V_s}=\frac{3.838}{7.962}=\boxed{0.482} \\ \rho_{\text{bulk}} &=\frac{M_t}{V_t}=\frac{23.1}{11.8}=\boxed{1.96\ \text{g/cm}^3}\ \text{(moist; dry bulk density }\rho_d=M_s/V_t=1.79\ \text{g/cm}^3\text{)} \end{aligned}$$
Moisture content. The pore water mass is $M_w=M_t-M_s=23.1-21.1=2.0$ g: $$w=\frac{M_w}{M_s}=\frac{2.0}{21.1}=\boxed{0.0948\ (9.48\%)}.$$ With porosity 32.5% and only a 9.48% moisture content by mass, the sample is far from saturated (degree of saturation $S=V_w/V_v=2.0/3.838\approx52\%$), consistent with being described only as "moist."

Quantity	Result
(a) Hydraulic conductivity $K$ (at 15°C)	2.54×10⁻⁵ m/s
(a) Intrinsic permeability $k$	2.95×10⁻¹⁴ m²
(a) Soil classification	Silt (low-K, non-sand)
(b) Skeletal compressibility $\alpha$	5.44×10⁻⁷ m²/N
(c) Volume of water released	60,783 m³
(d) Porosity	32.5%
(d) Bulk density (moist)	1.96 g/cm³
(d) Void ratio	0.482
(d) Moisture content	9.48%

Concept & Theory

Topic. Falling-head permeameter hydraulics, temperature-dependent conversion between hydraulic conductivity and intrinsic permeability, specific-storage/compressibility theory, and soil phase relations.

Key relations. $K=\dfrac{aL}{At}\ln(h_0/h_1)$; $k=K\mu/(\rho g)$; $S_s=\rho_wg(\alpha+n\beta_w)$; $S=S_sb$; $n=V_v/V_t$, $e=V_v/V_s$, $w=M_w/M_s$.

Why this works. The falling-head test measures $K$ purely from geometry and the time it takes the head ratio to decay — no fluid property enters that equation, so $K$ is exactly what was measured at the test temperature. Converting to intrinsic permeability, however, is a genuinely fluid-dependent step, so the fluid properties used must match the temperature at which $K$ was actually measured, not the exam's blanket 20°C default (which only applies "unless otherwise specified" — and here it is). Specific storage physically represents two additive compressive mechanisms acting in series on the same unit volume: the grain skeleton rearranging under effective-stress change, and the pore water itself compressing; because $\alpha\gg n\beta_w$ for essentially all real aquifer materials, $S_s$ is almost always controlled by the skeleton term.

Common pitfall. Using the paper's default 20°C viscosity when a test temperature is explicitly stated silently shifts $k$ by roughly 12–15%, since $\mu$ changes materially between 15°C and 20°C; and forgetting that $S_s$ has units of (length)$^{-1}$ — it must be multiplied by aquifer thickness $b$ before it can be used as a dimensionless storativity in $V=SA\Delta h$.

Source. Freeze & Cherry, Groundwater, Ch. 2.2 (permeameters, intrinsic permeability), Ch. 2.9 (specific storage) and Ch. 2.11 (soil phase relations, effective stress).

Question 1 of 5: Falling-Head Permeameter, Storage Properties and Soil Phase Relations

Question 1: Falling-Head Permeameter, Storage Properties and Soil Phase Relations (equal value)