Methods of earth resistivity measurement

Earth resistivity $\rho$ is measured to design earthing systems (substation grids, tower footings). The common methods use an earth tester that injects current $I$ and measures voltage $V$, giving $R = V/I$.

1. Wenner four-electrode method

Four electrodes are placed in a straight line at equal spacing $a$, driven to a depth $b \ll a$. Current is passed between the outer pair (C1, C2); voltage is measured between the inner pair (P1, P2).

   C1      P1      P2      C2
 --|-------|-------|-------|--
   |<- a ->|<- a ->|<- a ->|

Potential due to a point source at the surface is $\frac{\rho I}{2\pi r}$. By superposition:

$$V = \frac{\rho I}{2\pi}\left[\left(\frac{1}{a}-\frac{1}{2a}\right) - \left(\frac{1}{2a}-\frac{1}{a}\right)\right] = \frac{\rho I}{2\pi a}$$ $$\rho = 2\pi a R$$

Readings at increasing spacing give resistivity at increasing depth (about depth $a$).

2. Schlumberger (Schlumberger–Palmer) method

The potential electrodes are kept close together (spacing $d$), and the current electrodes are placed at distance $c$ outside each potential electrode. Only the current electrodes are moved for deeper readings, so it is faster and more sensitive for large spacings.

   C1          P1  P2          C2
 --|-----------|---|-----------|--
   |<--- c --->|<d>|<--- c --->|

$$\rho = \frac{\pi c(c+d)R}{d}$$

For $c = d = a$ this gives $\rho = 2\pi aR$, the Wenner case.

3. Driven-rod (three-point / variation-of-depth) method

A test rod of length $L$ and diameter $d$ is driven into the soil, and its resistance $R$ is measured by the fall-of-potential method using a current probe C (far away, at least 5–10 times rod length) and a potential probe P at about 62% of that distance.

  Earth tester
   E      P            C
 --|------|------------|--
   |<-0.62D->|
   |<-------- D ------->|
   test rod

$$\rho = \frac{2\pi L R}{\ln\left(\dfrac{8L}{d}\right) - 1}$$

Repeating with the rod driven deeper gives the change of resistivity with depth. It is simple and suited to small areas but measures only the soil near the rod.

Practical points

• Take readings in several directions and at several spacings; use the dry-season (highest) value for design.
• Use AC or reversing DC to avoid polarisation and errors from stray currents.
• Keep test leads away from buried metal pipes and cables, which distort the readings.