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Chapter
8
Resistivity Methods
8.1. INTRODUCTION
All resistivity methods employ an artificial source of
current. which is introduced into the ground through
point electrodes or long line contacts; the latter
arrangement is rarely used nowadays. The procedure
is to measure potentials at other electrodes in the
vicinity of the current flow. Because the current is
measured as well. it is possible to determine an
effective or apparent resistivity of the subsurface.
In this regard the resistivity technique is superior,
at least theoretically, to all the other electrical methods, because quantitative results are obtained by
using a controlled source of specific dimensions.
Practically, as in other geophysical methods, the
maximum potentialities of resistivity are never realized. The chief drawback is its high sensitivity to
minor variations in conductivity near surface; in
electronic parlance the noise level is high. An analogous situation would exist in ground magnetic surveys if one were to employ a magnetometer with
sensitivity in the picotesla range.
This limitation, added to the practical difficulty
involved in dragging several electrodes and long wires
through rough wooded terrain, has made the electromagnetic method more popular than resistivity in
mineral exploration. Nor is resistivity particularly
suitable for oil prospecting. However, it is by no
means obsolete, because the rapid development of
the induced polarization technique, which includes
resistivity data, guarantees its continued use. Furthermore the search for geothermal reservoirs normally involves resistivity surveying and it is also
employed routinely in groundwater exploration.
which is of increasing worldwide importance, and in
civil engineering.
8.2. ELEMENTARY THEORY
8.2.1. Potentials in Homogeneous Media
Consider a continuous current flowing in an isotropic
homogeneous medium. (This analysis will also apply
to ac if the frequency is low enough that displace-
ment currents are insignificant.) If 8A is an element
of surface and J the current density in amperes per
square meters, then the current passing through 8A is
J • 8A. The current density J and the electric field E
are related through Ohm's law:
J - aE
(8.1)
where E is in volts per meter and a is the conductivity of the medium in siemens per meter (81m).
The electric field is the gradient of a scalar potential,
E- -VY
(8.2)
J - -avV
(8.3)
Thus we have
From Equation (6.7), V . J - 0, so
V . (avV) - 0
(8.4)
Using Equation (A.2l), we have
va'VY+av 2 V - 0
(8.5)
If a is constant throughout, the first term vanishes
and we have Laplace's equation, that is, the potential
is harmonic:
(8.6)
There are two boundary conditions that must
hold at any contact between two regions of different
conductivity. In Section 6.2.4 we gave boundary
conditions for interfaces where a and II change
abruptly. The first and third of these may be written
in the form
Ex" - E"3
and
alE•• - ~E'3
(8.7a)
where the x and z axes are tangential and normal,
respectively, to the interface, E~ being the tangen-
Elementary theory
523
Power
Umfcnn medium
resistivityp
- - - - Current flow
Figure 8.1. Buried point source of current in homogeneous ground.
tial component in medium I, and so forth. In addition,
surface is given by
dV
I .. 4'1rr 2J .. -4'1rr 20 = -4'1roA
( 8.7b)
dr
from Equations (8.3) and (8.9), so that
8.2.2. Single Current Electrode at Depth
Multiplying by r 2 and integrating, we get
dV
A
dr .. r 2
(8.9)
integrating again, we have
V- -A/r+ B
lp
A-
There are several field configurations used in resistivity that we will consider in turn. In the first of these
we have an electrode of small dimensions buried in a
homogeneous isotropic medium. This corresponds to
the mise-a-la-masse method (see §8.5.4d) where the