GATE-2018 (24) :Wyllie – Time average equation

102. The apparent resistivities measured with Wenner and Schlumberger arrays are the same when equal amounts of currents are passed through current electrodes and equal potential differences are developed between potential electrodes. If the Wenner array electrode spacing is 10m and Schlumberger array potential electrode spacing is 20 m, the current electrode spacing of the Schlumberger array is

(1) 40m 

(2) 20m 

(3) 80m 

(4) 160m

(Thanks to Neeraja AU)

Solution:

Apparent resistivity (ρ_a) in Wenner Configuration 

 

$\rho_a=2\pi a\frac{\triangle V}{I}$

a – electrode spacing =10m

Apparent resistivity in Schlumberger array

$\rho_a=\frac{\pi}{4}\frac{L^2}{l}\frac{\triangle V}{I}$

L- current electrode spacing=?

l - potential electode spacing=20m

$2\pi a \frac{\triangle V}{I}=\frac{\pi}{4}\frac{L^2}{l}\frac{\triangle V}{I}$

$2 a =\frac{1}{4}\frac{L^2}{l}$

$2\times 10 =\frac{1}{4}\frac{L^2}{20}$

$20\times 4\times 20=L^2$

$1600=L^2$

$L=\sqrt{1600}$

L=40m