# Effective Stress and Permeability

 Question 1
In an Oedometer apparatus, a specimen of fully saturated clay has been consolidated under a vertical pressure of $50 \mathrm{kN} / \mathrm{m}^{2}$ and is presently at equilibrium. The effective stress and pore water pressure immediately on increasing the vertical stress to $150 \mathrm{kN} / \mathrm{m}^{2}$, respectively are
 A $150 \mathrm{kN} / \mathrm{m}^{2}$ and 0 B $100 \mathrm{kN} / \mathrm{m}^{2}$ and $50 \mathrm{kN} / \mathrm{m}^{2}$ C $50 \mathrm{kN} / \mathrm{m}^{2}$ and $100 \mathrm{kN} / \mathrm{m}^{2}$ D 0 and $150 \mathrm{kN} / \mathrm{m}^{2}$
GATE CE 2021 SET-1   Geotechnical Engineering
Question 1 Explanation:
Stress is increased suddenly, hence entire change will be taken by water $\Delta \bar{\sigma}=\Delta U=100 \mathrm{kPa}$.
There will be no change in effective stress
$\therefore \qquad \qquad\bar{\sigma}=50 \mathrm{kPa}$
 Question 2
A constant head permeability test was conducted on a soil specimen under a hydraulic gradient of 2.5. The soil specimen has specific gravity of 2.65 and saturated water content of 20%. If the coefficient of permeability of the soil is 0.1 cm/s, the seepage velocity (in cm/s, round off to two decimal places) through the soil specimen is ________.
 A 0.34 B 0.46 C 0.72 D 0.96
GATE CE 2020 SET-2   Geotechnical Engineering
Question 2 Explanation:
\begin{aligned} \text{Void ratio,}\; e&=\frac{wG}{s}\\ &=\frac{0.2 \times 2.65}{1}=0.53\\ \text{Porosity,}\; n&=\frac{e}{1+e}=0.3464\\ \text{Seepage velocity,}\; V_s&=\frac{v}{n}=\frac{k_i}{n}\\ &=\frac{0.1 \times 2.5}{0.3464}=0.72 cm/sec \end{aligned}
 Question 3
Permeability tests were carried out on the samples collected from two different layers as shown in the figure (not drawn to the scale). The relevant horizontal ($k_h$) and vertical ($k_v$) coefficients of permeability are indicated for each layer.

The ratio of the equivalent horizontal to vertical coefficients of permeability, is
 A 37.29 B 80.2 C 68.25 D 0.03
GATE CE 2020 SET-2   Geotechnical Engineering
Question 3 Explanation:
\begin{aligned} \frac{k_{eq.H}}{k_{eq.V}}&=\frac{\frac{\Sigma K_iZ_i}{\Sigma Z_i}}{\frac{\Sigma Z_i}{\Sigma \frac{Z_i}{K_i}}}\\ &=\frac{\frac{4.4 \times 10^{-3} \times 3 +6 \times 10^{-1} \times 4}{7}}{\frac{7}{\frac{3}{4 \times 10^{-3}}+\frac{4}{5.5 \times 10^{-1}}}}\\ &=\frac{0.3447}{9.24 \times 10{-3}}=37.29 \end{aligned}
 Question 4
Water flows in the upward direction in a tank through 2.5 m thick sand layer as shown in the figure. The void ratio and specific gravity of sand are 0.58 and 2.7, respectively. The sand is fully saturated. Unit weight of water is 10 $kN/m^3$.

The effective stress (in kPa, round off to two decimal places) at point A, located 1 m above the base of tank, is __________.
 A 3.14 B 5.36 C 8.94 D 11.24
GATE CE 2020 SET-1   Geotechnical Engineering
Question 4 Explanation:

\begin{aligned} \gamma _{suv}&=\left ( \frac{G-1}{1+e} \right )\gamma _\omega \\ &=\left ( \frac{2.7-1}{1+0.58} \right ) \times 10\\ &=10.759\\ \bar{\sigma }&=z\gamma _{suv}-iz\gamma _\omega \\ &=1.5(\gamma _{suv})-\left ( \frac{1.2}{2.5} \right )\times 1.5 \times \gamma _\omega \\ &=8.939 kN/m^2 \end{aligned}
 Question 5
Constant head permeability tests were performed on two soil specimens, S1 and S2. The ratio of height of the two specimens ($L_{S1}:L_{S2}$) is 1.5, the ratio of the diameter of specimens ($D_{S1}:D_{S2}$) is 0.5, and the ratio of the constant head ($h_{S1}:h_{S2}$) applied on the specimens is 2.0. If the discharge from both the specimens is equal, the ratio of the permeability of the soil specimens ($k_{S1}:k_{S2}$) is ____
 A 1 B 2 C 3 D 4
GATE CE 2019 SET-2   Geotechnical Engineering
Question 5 Explanation:
\begin{aligned} \frac{L_{s1}}{L_{s2}}&=1.5\\ \frac{D_{s1}}{D_{s2}}&=0.5\\ \frac{h_{s1}}{h_{s2}}&=2\\ \frac{k_{s1}}{k_{s2}}&=? \end{aligned}
Discharge is same.
\begin{aligned} k_1i_1A_1&=k_2i_2A_2\\ k_1\frac{h_{s1}}{L_{s1}} \times \frac{\pi}{4} \times D_{s1}^2&=k_2\frac{h_{s2}}{L_{s2}} \times \frac{\pi}{4} \times D_{s2}^2\\ \frac{k_{s1}}{k_{s2}}&=\frac{L_{s1}}{L_{s2}} \times \frac{h_{s2}}{h_{s1}} \times \frac{D_{s2}^2}{D_{s1}^2}\\ &=1.5 \times \frac{1}{2} \times \left ( \frac{1}{0.5} \right )^2=3 \end{aligned}
 Question 6
An anisotropic soil deposit has coefficient of permeability in vertical and horizontal directions as $k_z \; and \; k_x$, respectively. For constructing a flow net, the horizontal dimension of the problem's geometry is transformed by a multiplying factor of
 A $\sqrt{\frac{k_z}{k_x}}$ B $\sqrt{\frac{k_x}{k_z}}$ C $\frac{k_x}{k_z}$ D $\frac{k_z}{k_x}$
GATE CE 2019 SET-2   Geotechnical Engineering
Question 6 Explanation:
$X=X_T\sqrt{\frac{k_x}{k_z}}$
Transformed horizontal dimension, $X_T=X\sqrt{\frac{k_z}{k_x}}$
 Question 7
In a soil specimen, the total stress, effective stress, hydraulic gradient and critical hydraulic gradient are $\sigma ,\sigma ',i, \; and \; i_c$, respectively. For initiation of quicksand condition, which one of the following statements is TRUE?
 A $\sigma '\neq 0 \; and \;i =i_c$ B $\sigma '= 0 \; and \;i =i_c$ C $\sigma '\neq 0 \; and \;i \neq i_c$ D $\sigma = 0 \; and \;i =i_c$
GATE CE 2019 SET-1   Geotechnical Engineering
Question 7 Explanation:
Quick sand condition occurs if $\sigma '=0$ and $i=i_c$
 Question 8
Three soil specimens (Soil 1, Soil 2 and Soil 3), each 150 mm long and 100 mm diameter, are placed in series in a constant head flow set-up as shown in the figure. Suitable screens are provided at the boundaries of the specimens to keep them intact. The values of coefficient of permeability of Soil 1, Soil 2 and Soil 3 are 0.01, 0.003 and 0.03 cm/s, respectively.

The value of h in the set-up is
 A 0 mm B 40mm C 255mm D 560mm
GATE CE 2018 SET-2   Geotechnical Engineering
Question 8 Explanation:
In normal to bedding plane flow (series arrangement), Discharge will be same and Head loss and Hydraulic gradient will be different
\begin{aligned} q &=K_{1} i_{1} A=K_{2} i_{2} A \\ &=K_{3} i_{3} A=K_{\mathrm{avg}} \cdot\left(\frac{H_{L}}{L}\right) A \\ K_{\mathrm{avg}_{1}} &=\frac{\Sigma Z_{i}}{\Sigma \frac{Z_{i}}{K_{i}}}=\frac{150+150+150}{\frac{150}{0.01}+\frac{150}{0.003}+\frac{150}{0.03}} \\ &=0.0064\\ \text{Total head loss}\\ &=H_{L}=560 \mathrm{mm} \\ \therefore K_{3} \cdot \frac{h}{150} \times A &=K_{a v g} \cdot \frac{560}{(150+150+150)} A \\ 0.03 \cdot\left(\frac{h}{150}\right) &=0.0064\left(\frac{560}{450}\right) \\ h &=40 \mathrm{mm} \end{aligned}
 Question 9
At a construction site, a contractor plans to make an excavation as shown in the figure.

The water level in the adjacent river is at an elevation of +20.0 m. Unit weight of water is 10 kN/$m^{3}$. The factor of safety (up to two decimal places) against sand boiling for the proposed excavation is ______
 A 1 B 2 C 3 D 4
GATE CE 2018 SET-1   Geotechnical Engineering
Question 9 Explanation:
$\mathrm{FOS}=\frac{10 \times \gamma_{\mathrm{sat}}}{20 \times \gamma_{\mathrm{w}}}=\frac{10 \times 20}{20 \times 10}=1$
 Question 10
Water flows from P to Q through two soil samples, Soil 1 and Soil 2, having cross sectional area of 80$cm^2$ as shown in the figure. Over a period of 15 minutes, 200 ml of water was observed to pass through any cross section. The flow conditions can be assumed to be steady state. If the coefficient of permeability of Soil 1 is 0.02 mm/s, the coefficient of permeability of Soil 2 (expressed in mm/s) would be ____
 A 0.05 B 0.08 C 0.02 D 0.06
GATE CE 2016 SET-2   Geotechnical Engineering
Question 10 Explanation:
\begin{aligned} \text { Discharge } &=\frac{200 \mathrm{ml}}{15 \mathrm{min}} \\ &=\frac{200 \mathrm{cm}^{3}}{15 \times 60 \mathrm{sec}}=\frac{200 \times 10^{3}}{900} \frac{\mathrm{mm}^{3}}{\mathrm{sec}} \end{aligned}
As per Darcy's Law,
\begin{aligned} q &=k_{a v g} i A \\ k_{a v g} &=\frac{\Sigma z_{i}}{\Sigma \frac{z_{i}}{k_{i}}}\\ &(\because\text{ flow is normal to bedding plane})\\ k_{\text {avg }}&=\frac{150+150}{\frac{150}{0.02}+\frac{150}{k}}\\ i&=\frac{\text { Head difference }}{\text { length }}=\frac{600-300}{300}=1 \\ A&=80 \mathrm{cm}^{2}=80 \times 10^{2} \mathrm{mm}^{2} \\ q&=\left[\frac{150+150}{\frac{150}{0.02}+\frac{150}{k}}+\right] \times 1 \times 80 \times 10^{2} \\ &=\frac{200 \times 10^{3}}{900} \\ k&=0.045 \mathrm{mm} / \mathrm{sec} \\ \end{aligned}
There are 10 questions to complete.

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