# GATE Papers >> Civil >> 2015 >> Question No 51

Question No. 51

A non-homogeneous soil deposit consists of a silt layer sandwiched between a fine-sand layer at top and a clay layer below. Permeability of the silt layer is 10 times the permeability of the clay layer and one-tenth of the permeability of the sand layer. Thickness of the silt layer is 2 times the thickness of the sand layer and two-third of the thickness of the clay layer. The ratio of equivalent horizontal and equivalent vertical permeability of the deposit is ________.

##### Answer : 10.0 to 12.0

Solution of Question No 51 of GATE 2015 Civil Paper

K2 = 10K3 or K3 = $\frac{{\mathrm K}_2}{10}$;

K= $\frac1{10}$ K1 or K1 = 10K2

Z2 = 2Z1 or Z1 = $\frac{{\mathrm Z}_2}2$ ;

Z2 = $\frac32$ Z3 or Z3 = $\frac32$ Z2

⇒ Kx = $\frac{{\mathrm K}_1{\mathrm Z}_1+{\mathrm K}_2{\mathrm Z}_2+{\mathrm K}_3{\mathrm Z}_3}{{\mathrm Z}_1+{\mathrm Z}_2+{\mathrm Z}_3}$

$\begin{array}{l}=\frac{{\displaystyle\frac{10{\mathrm K}_2{\mathrm Z}_2}2}+{\mathrm K}_2{\mathrm Z}_2+\left({\displaystyle\frac{{\mathrm K}_2}{10}}\times{\displaystyle\frac32}{\mathrm Z}_2\right)}{{\displaystyle\frac{{\mathrm Z}_2}2}+{\mathrm Z}_2+{\displaystyle\frac32}{\mathrm Z}_2}\\=\frac{5{\mathrm K}_2{\mathrm Z}_2+{\mathrm K}_2{\mathrm Z}_2+0.15{\mathrm K}_2{\mathrm Z}_2}{3{\mathrm Z}_2}\end{array}$

$=\frac{6.15}3\times{\mathrm K}_2=2.05{\mathrm K}_2$

$\begin{array}{l}{\mathrm K}_\mathrm z=\frac{{\displaystyle\frac{{\mathrm Z}_2}2}+{\mathrm Z}_2+{\displaystyle\frac32}{\mathrm Z}_2}{{\displaystyle\frac{{\mathrm Z}_2}{2\times10{\mathrm K}_2}}+{\displaystyle\frac{{\mathrm Z}_2}{{\mathrm K}_2}}+{\displaystyle\frac{1.5{\mathrm Z}_2\times10}{{\mathrm K}_2}}}\\=\frac{3{\mathrm Z}_2}{{\displaystyle\frac{{\mathrm Z}_2}{{\mathrm K}_2}}\left({\displaystyle\frac1{20}}+1+15\right)}=0.1869k_2\\=\frac{{\mathrm K}_\mathrm x}{{\mathrm K}_\mathrm z}=\frac{2.05{\mathrm k}_2}{0.1869{\mathrm k}_2}\end{array}$

= 10.9684

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