Question

The permeability of sand in horizontal and vertical directions are 3 × 10-3 cm/sec and 1 × 10-3 cm/sec and the permeability of clay in horizontal and vertical directions are 9 × 10-7 cm/sec and 3 × 10-9 cm/sec, then the ratio of seepage loss per meter length of dam in sand to clay is:
1. 10⁵/3
2. 3
3. 1/9
4. 9
5.

Answer 1

Correct Answer - Option 1 : 10⁵/3

Concept:

Determination of seepage discharge when the medium is isotropic:

Seepage discharge per meter length is given by, \(q = kh\frac{{{N_f}}}{{{N_d}}}\)

Where, h = Hydraulic head or head difference between upstream and downstream level or head loss through the soil

Nf = Total number of flow channels

Nd = Total number of equipotential drops

k = Coefficient of permeability

Calculation:

Given,

Permeability of sand in horizontal direction, K_HS = 3 × 10^-3 cm/sec,

Permeability of sand in vertical direction, K_VS = 1 × 10^-3 cm/sec

Permeability of clay in horizontal direction, KHC = 9 × 10-7 cm/sec,

Permeability of clay in vertical direction, KVC = 3 × 10-9 cm/sec

Equivalent permeability of sand, (K_eq)_sand = \(√ {{K_{HS}} × {K_{Vs}}} = √ {3 × {{10}^{ - 3}} × 1 × {{10}^{ - 3}}} \)

(K_eq)_sand = √3 × 10^-3 cm/sec

Equivalent permeability of clay, \({\left( {{K_{eq}}} \right)_{sand}} = √ {{K_{HC}} × {K_{VC}}} = √ {9 × {{10}^{ - 7}} × 3 × {{10}^{ - 9}}} \)

(K_eq)_clay = 3√3 × 10^-8 cm/sec

Seepage loss is given as, \(q = kh\frac{{{N_f}}}{{{N_d}}}\)

q ∝ K

\(\frac{{{q_{sand}}}}{{{q_{clay}}}} = \frac{{{{\left( {{K_{eq}}} \right)}_{sand}}}}{{{{\left( {{K_{eq}}} \right)}_{clay}}}}\)

⇒ \(\frac{{{q_{sand}}}}{{{q_{clay}}}} = \frac{{\sqrt 3 \times {{10}^{ - 3}}}}{{3\sqrt 3 \times {{10}^{ - 8}}}} = \frac{{{{10}^5}}}{3}\)