18-Env-A3 Geotechnical and Hydrogeological Engineering · Undated paper
Question 2 of 6: Constant-Head Permeability Test
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — 18-Env-A3, Geotechnical and Hydrogeological Engineering — 3 hours, open book. Six questions, each 20 marks, equal value; all six are solved below.
Find. (a) Saturated hydraulic conductivity $k$. (b) Three soil characteristics affecting $k$. (c) Seepage (average linear) velocity. (d) How water content affects $k$ in an unsaturated soil.
Approach. Apply Darcy's law directly to the constant-head cell to get $k$, then convert the Darcy (superficial) velocity to the true seepage velocity through the void space using the porosity.
(a) Hydraulic conductivity. Cross-sectional area $A=\tfrac{\pi}{4}D^2=\tfrac{\pi}{4}(7.6)^2=45.36\text{ cm}^2$. Flow rate $q=Q_{vol}/t = 1200/360=3.333\text{ cm}^3/\text{s}$. From Darcy's law $q=kA(\Delta h/L)$: $$k = \dfrac{qL}{A\,\Delta h} = \dfrac{3.333\times20.0}{45.36\times15.0} = \boxed{0.0980\text{ cm/s}}$$
(b) Three characteristics affecting $k$. (i) Grain-size distribution / average particle size (finer soils have smaller, more tortuous pore channels and much lower $k$); (ii) void ratio / density (a looser packing gives larger, better-connected flow channels); (iii) the pore fluid's properties (viscosity and unit weight, which is why $k$ is reported at a standard temperature since water's viscosity is temperature-dependent). Particle shape/fabric and degree of saturation are also legitimate answers.
(c) Seepage velocity. Darcy (superficial) velocity $v = q/A = 3.333/45.36 = 0.0735\text{ cm/s}$. Porosity $n=e/(1+e)=0.55/1.55=0.3548$. The true average velocity through the interconnected voids is $$v_s = \dfrac{v}{n} = \dfrac{0.0735}{0.3548} = \boxed{0.207\text{ cm/s}}$$
(d) Effect of water content on unsaturated $k$. As a soil desaturates below full saturation, part of the pore space fills with air; water can then only flow through the remaining continuous, moisture-filled channels, which are fewer, narrower and more tortuous than the fully saturated network, so $k$ falls — often by several orders of magnitude — as water content drops. The unsaturated conductivity $k(\theta)$ (or, equivalently, the relative permeability $k_r = k_{unsat}/k_{sat}$ as a function of saturation) is a strongly nonlinear, decreasing function of decreasing moisture content, typically described by a soil–water characteristic curve model such as van Genuchten–Mualem. Near residual saturation the remaining water is held in disconnected films and dead-end pores, so $k$ approaches zero even though some moisture is still present.
