Calculation of true thickness of the bed

Gate_2020

 

23) The geometrical factor for the electrode configuration given below is _____m

(Thanks to Chandrasekhar, ANU)

Solution:

The apparent resistivity of the electrode array is as follows

$\rho_{a} =( 2\pi)(\frac{\triangle V}{I})\left\{(\frac{1}{r_{1}}-\frac{1}{r_{2}})-(\frac{1}{r_{3}}-\frac{1}{r_{4}})\right\}^{-1}$

In this the term $( 2\pi)\left\{(\frac{1}{r_{1}}-\frac{1}{r_{2}})-(\frac{1}{r_{3}}-\frac{1}{r_{4}})\right\}^{-1}$ I s defined as the geometric factor for the Schulumberger array.

$K =( 2\pi)\left\{(\frac{1}{r_{1}}-\frac{1}{r_{2}})-(\frac{1}{r_{3}}-\frac{1}{r_{4}})\right\}^{-1}$

Given that 

 

$r_{1} =20 m$ 

$r_{2} =30 m$

$r_{3} =30 m$

$r_{4} =20 m$

$K =( 2\pi)\left\{(\frac{1}{r_{1}}-\frac{1}{r_{2}})-(\frac{1}{r_{3}}-\frac{1}{r_{4}})\right\}^{-1}$

$K =( 2\pi)\left\{(\frac{1}{20}-\frac{1}{30})-(\frac{1}{30}-\frac{1}{20})\right\}^{-1}$

$K =( 2\pi)\left\{(\frac{2}{20}-\frac{2}{30})\right\}^{-1}$

$K =( 2\pi)\left\{(\frac{20}{600})\right\}^{-1}$

$K = 2\times 3.14 \times 30$

$K = 188.4 m$