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

GATE-2018

77. A student interpreted a four layer Schlumberger resistivity sounding data and obtained the resistivities $(\rho)$ and thicknesses (h) as follows: $\rho_1$=100 ohm m, $\rho_2$=20  ohm m, $\rho_3$=1500 ohm m and $\rho_4$=50 ohm m; $h_1$=50 m, $h_2$=10 m and $h_3$=20 m. The same data is interpreted by another student who obtains $\rho_3$=2000 ohm m. Then, according to the principle of equivalence, the value of $h_3$ interpreted by the second student is ________m. (All other model parameters estimated by both the students are the same.)

Solution:

Principle of equivalence: It is impossible to distinguish between two highly resistive beds of different "h" and "$\rho$" values if the product "$h*\rho$" is the same.

So,

The first student $h_3=20m$

and $\rho_3 =1500$ ohm m

The second student $\rho_3 =2000$ ohm m

And what is the $h_3$ = ?

$h_3*\rho_3 =h_3*\rho_3$

$20*1500 =h_3*2000$

$h_3*2000=20*1500 $

$h_3=\frac{20*1500}{2000} $

$h_3=15 m $

Reference:  Applied Geophysics by W.M. Telford , L.P. Geldart and R.E.Sheriff.