Geoscience Reference
In-Depth Information
The equation for the electric potential at
P
1
due to
C
2
has a similar arrangement:
=
−
ρ
π
I
1
V
(5.10)
C
21
2
CP
21
where the minus sign is introduced to reflect the fact that the current flow at
C
2
(negative current
electrode) is opposite in direction compared to the current flow for
C
1
(positive current electrode).
The equations for the electric potential at
P
2
due to
C
1
and
C
2
are comparable in form to those of
Equation (5.9) and Equation (5.10), and as a result, the total difference in electric potential, Δ
V
,
between
P
1
and
P
2
can be expressed as follows:
ρ
I
1
1
1
1
∆
V
=
− −+
(5.11)
2
π
CP
CP
CP
CP
11
21
12
22
Equation (5.11) can be rearranged with respect to ρ as follows:
= 2
∆
V
IG
ρ π
(5.12)
where
G
is an abbreviation for the term in Equation (5.11) contained within parentheses. Upon
inspection, it is apparent that
G
is fundamentally a geometric factor, the value of which is governed
by the arrangement and spacing of the current electrodes (
C
1
and
C
2
) and potential electrodes (
P
1
and
P
2
).
The development of Equation (5.12) is important because it indicates, given a specific four-
electrode array setup on the ground surface, that the known electric current applied between
C
1
and
C
2
along with the measured electric potential difference (voltage) between
P
1
and
P
2
can be used
to calculate the resistivity, ρ, of the subsurface beneath the electrode array. For homogeneous and
isotropic soil or rock material, the calculated ρ represents the true subsurface resistivity. However,
subsurface soil and rock material are more commonly heterogeneous and anisotropic. The resistiv-
ity calculated with Equation (5.12) is therefore considered a bulk measurement for the subsurface
beneath the electrode array, the value of which can be influenced significantly by the nature of the
soil or rock heterogeneity and anisotropy that is present. It is for this reason that the resistivity cal-
culated by Equation (5.12) is typically referred to as the “apparent resistivity,” designated by ρ
a
. The
reciprocal of the apparent resistivity (= 1/ρ
a
) is called the apparent electrical conductivity, which is
given the symbol, σ
a
, or for agriculture, EC
a
.
5.5 CoMMon eleCtRode ARRAyS And CoRReSpondInG
AppARent ReSIStIvIty eQUAtIonS
The conventional, basic resistivity methods use a variety of electrode arrays that have different
arrangement and spacing characteristics for the two current electrodes (
C
1
and
C
2
) and two poten-
tial electrodes (
P
1
and
P
2
). The three resistivity electrode arrays most commonly utilized are illus-
trated in Figure 5.4. All the electrodes for these three arrays are inserted at the ground surface
and oriented along a single line. Regardless of whether the subsurface is homogeneous or nonho-
mogeneous, interchanging the current electrodes with the potential electrodes does not alter the
geometric factor,
G
, or even the apparent resistivity value, although in actuality, it is best to keep the
distance between potential electrodes as small as possible to avoid the effects of electric potential
