Question

For a half space composed of 3 layers with resistivities \( \rho_1 \), \( \rho_2 \) and \( \rho_3 \), as shown in the figure, which of the following statements is/are correct about the variation of apparent resistivity with electrode spacing?

 

• A. If \( \rho_1<\rho_2<\rho_3 \), the curve of apparent resistivity increases monotonically
• B. If \( \rho_1<\rho_2<\rho_3 \), the curve of apparent resistivity decreases monotonically
• C. If \( \rho_1>\rho_2<\rho_3 \), the curve of apparent resistivity first decreases and then increases
• D. If \( \rho_1>\rho_2>\rho_3 \), the curve of apparent resistivity increases monotonically
  \

Hint:

The apparent resistivity curve reflects the vertical resistivity profile. Transitions in resistivity with depth will cause changes in the slope of the apparent resistivity curve. A low resistivity layer sandwiched between higher resistivity layers will typically cause a decrease followed by an increase in apparent resistivity with increasing electrode spacing.

Answer 1

The Correct Option is A, C

Step 1: Understanding the concept of apparent resistivity in layered media.
Apparent resistivity \( \rho_a \) is an averaged measurement reflecting subsurface resistivity based on how deep current penetrates, which depends on electrode spacing. Larger electrode spacing means deeper penetration and influence from deeper layers.

Step 2: Case \( \rho_1 < \rho_2 < \rho_3 \).
With small electrode spacing, current flows mainly through layer 1 → \( \rho_a \approx \rho_1 \). As spacing increases, it penetrates into layer 2 (\( \rho_2 > \rho_1 \)) and then into layer 3 (\( \rho_3 > \rho_2 \)). So, \( \rho_a \) increases monotonically.
✓ Statement (A) is correct.

Step 3: Case \( \rho_1 > \rho_2 < \rho_3 \).
Initially, \( \rho_a \approx \rho_1 \). As current reaches the second layer (\( \rho_2 < \rho_1 \)), \( \rho_a \) decreases. With further depth, current reaches the third layer (\( \rho_3 > \rho_2 \)), so \( \rho_a \) increases again. This forms a curve with a minimum.
✓ Statement (C) is correct.

Step 4: Case \( \rho_1 > \rho_2 > \rho_3 \).
Starting from \( \rho_1 \), as depth increases and current reaches \( \rho_2 \) and \( \rho_3 \), each lower than the previous, \( \rho_a \) decreases monotonically.
✗ Statement (D) is incorrect.

Step 5: Summary
The correct statements based on the described scenarios are:
(A) and (C).