Question 2 of 7: Saturated Hydraulic Conductivity from a Constant-Head Permeameter

Question 2: Saturated Hydraulic Conductivity from a Constant-Head Permeameter (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.

Approach. Identify the interval over which the top-up rate has stabilised (steady seepage), average those readings to get a steady flow rate Q, compute the total head loss across the sample geometrically (soil length + water head above, minus the tailwater depth in the pan), and apply Darcy's law, $K = QL/(A\,\Delta h)$.

1. Cross-sectional area and total head loss. With cylinder diameter $D = 0.10\ \text{m}$, $$A = \frac{\pi}{4}D^2 = \frac{\pi}{4}(0.10)^2 = 7.854\times10^{-3}\ \text{m}^2$$ The head driving flow across the sample is the soil length plus the water standing above it, less the tailwater depth in the pan (the outflow boundary): $$\Delta h = L + h_{\text{above}} - h_{\text{tail}} = 1.0 + 0.15 - 0.10 = \boxed{1.05\ \text{m}}, \qquad i = \frac{\Delta h}{L} = \frac{1.05}{1.0} = 1.05$$
2. Steady-state flow rate. The first three readings (1.20, 0.750, 0.604 L) fall sharply and are still transient; from $t=1.5\ \text{h}$ onward the readings hover tightly around 0.52 L per interval (0.520, 0.515, 0.520, 0.525, 0.515), signalling steady seepage. Averaging that stabilised block, $$\bar{V} = \frac{0.520+0.515+0.520+0.525+0.515}{5} = 0.519\ \text{L per }0.5\text{ h} \ \Rightarrow\ Q = \frac{0.519\times10^{-3}}{1800\ \text{s}} = \boxed{2.88\times10^{-7}\ \text{m}^3/\text{s}}$$
3. Darcy's law. Solving $Q = KAi$ for K, $$K = \frac{Q}{Ai} = \frac{2.88\times10^{-7}}{(7.854\times10^{-3})(1.05)} = \boxed{3.50\times10^{-5}\ \text{m/s}\ (3.02\ \text{m/day})}$$

b) Why the top-up volume falls, then levels off. When the cylinder is first placed in the pan the soil column is not yet fully saturated: part of the added water goes into filling still-empty pore spaces (advancing the wetting front and completing capillary saturation) in addition to the water that actually seeps all the way through and out the bottom, so the measured top-up volume over-states true Darcian outflow for as long as that internal storage demand persists. As saturation is completed — typically within the first 1–1.5 hours for a soil this permeable — the extra "filling" demand disappears and the top-up rate converges to the true steady-state seepage rate, which stays essentially constant thereafter because the head difference and cross-section are both fixed. The small scatter in the later readings (0.515–0.525 L) is ordinary measurement/evaporation noise around that steady value, not a further trend.