Wenner's method of earth resistivity measurement

Wenner's four-electrode method measures the average resistivity of soil by passing current between two outer electrodes and measuring the voltage between two inner electrodes, all equally spaced in a straight line. It removes the effect of electrode contact resistance.

Arrangement

Four electrodes C1, P1, P2, C2 are driven in a straight line at equal spacing $a$, to a small depth $b$ ($b \ll a$, usually $b < a/20$). A current $I$ is passed through the outer (current) electrodes C1 and C2 from an earth tester, and the voltage $V$ between the inner (potential) electrodes P1 and P2 is measured.

        Earth tester (megger)
     +---I----------------------+
     |      +---V---+           |
     |      |       |           |
     C1     P1      P2          C2
  ---|------|-------|-----------|--- ground
     |<--a->|<--a-->|<----a---->|
     depth b (b << a)

Derivation

Treat each electrode as a point source of current at the surface. Current $I$ from a point spreads into the earth over hemispheres. At distance $r$ the current density and field are

$$J = \frac{I}{2\pi r^2}, \qquad E = \rho J = \frac{\rho I}{2\pi r^2}$$

so the potential at distance $r$ is

$$V(r) = \int_r^\infty E\,dr = \frac{\rho I}{2\pi r}$$

Current $+I$ enters at C1 and $-I$ leaves at C2. Using superposition:

Potential at P1 (distance $a$ from C1, $2a$ from C2):

$$V_{P1} = \frac{\rho I}{2\pi}\left(\frac{1}{a} - \frac{1}{2a}\right) = \frac{\rho I}{4\pi a}$$

Potential at P2 (distance $2a$ from C1, $a$ from C2):

$$V_{P2} = \frac{\rho I}{2\pi}\left(\frac{1}{2a} - \frac{1}{a}\right) = -\frac{\rho I}{4\pi a}$$

Voltage between P1 and P2:

$$V = V_{P1} - V_{P2} = \frac{\rho I}{2\pi a}$$

The tester reads $R = V/I$, so

$$\rho = 2\pi a R$$

with $\rho$ in $\Omega\text{-m}$, $a$ in m and $R$ in $\Omega$.

If the electrode depth $b$ is not small compared with $a$ (image of each electrode taken), the general relation is

$$\rho = \frac{4\pi a R}{1 + \dfrac{2a}{\sqrt{a^2+4b^2}} - \dfrac{a}{\sqrt{a^2+b^2}}}$$

which reduces to $\rho = 2\pi aR$ when $b \ll a$.

Procedure and notes

• Readings are taken at several spacings ($a$ = 1, 2, 5, 10 m …) and in different directions. The current spreads roughly to a depth equal to $a$, so the change of $\rho$ with $a$ shows layered soil.
• Measurements are made in dry season to get the worst (highest) value for earthing design.
• An AC or reversing-DC tester is used to avoid polarisation and stray currents.

Example: with $a = 5$ m and $R = 3.2\ \Omega$, $\rho = 2\pi \times 5 \times 3.2 = 100.5\ \Omega\text{-m}$.