Question

A bed of sand consists of three horizontal layers of equal thickness. The value of Darcy's 'k' for the upper and lower layers is $1\times {10}^{-2}$ cm/sec and that for the middle layer is $1\times {10}^{-1}$ cm/sec.
The ratio of the permeability of the bed in the horizontal direction to that in the vertical direction is

• A. 10.0 to 1
• B. 2.8 to 1
• C. 2.0 to 1
• D. 1 to 10

Answer 1

The correct option is B 2.8 to 1

$\frac{k_{ex}}{k_{ez}}=\frac{(k_1{h}_1+k_2{h}_2+k_3{h}_3)}{(h_1+h_2+h_3{)}^2}(\frac{h_1}{k_1}+\frac{h_2}{k_2}+\frac{h_3}{k_3})$

for $h_1=h_2=h_3$

$\frac{k_{ex}}{k_{ez}}=\frac{(k_1+k_2+k_3)}{9}(\frac{1}{k_1}+\frac{1}{k_2}+\frac{1}{k_3})$

$k_1=k_3=1\times {10}^{-2}cm/s$

$k_2=10\times {10}^{-2}cm/s$

$\frac{k_{ex}}{k_{ez}}=\frac{(1+10+1)}{9}$$(\frac{1}{1}+\frac{1}{10}+\frac{1}{1})$

$=\frac{12\times 2.1}{9}=2.8$