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4.3 DC Electrical Methods (the Resistivity Methods)
DC electric fields
Current and current density
Ohm's Law, resistivity and conductivity
Anisotropy
Polarization, displacement current and dielectric
Frequency dependence of conductivity and dielectric
Laboratory measurement of resistivity
Measurement of ground resistivity
Magnetic fields of direct currents
DC electric fields
Electric fields describe the forces between electric charges and
electric current is the motion of charges. Coulomb’s Law describes the force
between two charges:
1 2
2
0
1
4
q qF r
r
where r is the unit vector in the direction between the two charges. 0 is a
constant, which reconciles the units of force with those of charge and length,
and it is called the permittivity of free space.
In S.I. units (m, kg, s), 0 = 8.854 x 10-12
.
The electric field, E, is defined as the force per unit charge:
2
2
2
1 0
1
4
F qE r
q r
If there is a distribution of charge, the principle of superposition leads to
2
0 0
1 1 1
4 4E r dV dV
rr
where is the charge density, iq
V
This is a central force field for which it can be shown that 0E dl for
any closed path so the electric field is conservative and can be derived from
a potential. Consequently we can express E by
E , where is the electric potential.
Gauss’ Law relates the integral of the field normal to a surface
enclosing charge via:
0 0
1
V
qE n da dV
where is the charge density.
By the divergence theorem, V
E n da E dV , so we have
0
1
V V
E dV
And we find the point form of Gauss’ Law:
3
0
E
Current and current density
If the charges qi are moving with velocity vi then the current density is
defined by:
i iq vJ
Vol
So J is the charge per unit area moving past a point and has the units of
amperes per square meter (A/m2). If all the charges move uniformly:
J v
On a microscopic scale charge is conserved, so in a volume V where
charge is moving, the time rate at which charge passes through the enclosing
surface must be balanced by the time rate of charge build-up within the
volume.
Thus:
S
dJ n da dv
dt
and with the divergence theorem we can find the point relationship called
the Equation of Continuity:
0d
Jdt
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Ohm’s Law, resistivity and conductivity
It is an experimental observation that the velocity that charges acquire
in an electric field in a medium is proportional to the field, so J is
proportional to E :
J E Ohm's Law
and is the conductivity in Siemens per meter, S/m.
The reciprocal of the conductivity is the resistivity, /E J in Ohm-
meters, Ohm-m or Ω-m.
In circuit theory the resistance, R, offered to current flow, I, under an
applied voltage, V, is defined by:
R = V/I or V = IR which is Ohm’s Law for a circuit element.
The reciprocal of resistance, 1/R, is the conductance, G, of the circuit
element.
The relationship between and R (or and G) is seen by considering
current flowing in a sample of material of cross section A and length L.
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The voltage drop across the sample is V and it is produced by the current I
flowing through the resistance R. The current density in the sample is J =
I/A (Ampere/m2, A/ m
2). The electric field in the sample is E = V/L
(Volts/m, V/m).
Then:
V ELR=
I JA
L
A
The term in brackets is the geometric factor.
Resistivity is measured in the laboratory by measuring the voltage drop
across a sample of rock having a uniform cross section of area A and a
length L through which is passed a current I. The above formula converts the
measured resistance to the resistivity, i.e.:
AR
L
Anisotropy
In a rock formation made up of layers of different composition the
resistivity is different for flow perpendicular to the layer boundaries than
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parallel to them. To define the resistivities in terms of the layer properties
we use the model in the adjacent sketch:
Here we consider a block of 1.0 m by 1.0 m by Hm high with individual
layers of thickness hi and resistivity i.
For current flow perpendicular to the layering the total resistance, Rt, is the
sum of the individual layer resistances Rt = i L / A = ihi / 1.0,
so Rt = ihi. For the block, the resistivity is:
1.0 i i
t t t
i
hAR R
L H h
which is called the transverse resistivity.
For current flow parallel to the layers the total resistance, Rl, is the
resistance of the individual layers in parallel,
1 1 1 ii i
il ii
i
hh
LR
A h
For the block, the resistivity is:
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i
l l li
i
hAR R H
hL
which is called the longitudinal resistivity.
The ratio of the transverse to longitudinal resistivity is called the
coefficient of anisotropy, , which is given by:
i i i i
i i
h h
h
[Note that because of the square term in the denominator the square root of
is often used as the definition of the anisotropy.]
It is often the case that the layers form an alternating sequence of the same
two materials, for example sand and shale. If there are only two resistivities
present then the numerator factors into terms where the first sum is over the
layers with resistivity 1, and the second sum is over layers with resistivity
2. For example, if the two materials are in equal amounts:
2
1 2
1 2
0.25
The following plot shows this anisotropy as a function of the ratio of the two
resistivities.
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We have described the anisotropy on a macroscopic scale where it is
caused by alternating layers of finite thickness. There is anisotropy on a
microscopic scale caused by preferential conductance along mineral surfaces
such as clay. This will be seen more clearly in the section on clays below.
Polarization, displacement current and dielectric
In most materials where there is no free charge the charges, ions, are
bound to one another by electrostatic forces. Nevertheless they retain a finite
separation and they move in an applied electric field. Since a charge in
motion constitutes a current we must describe the nature of this current.
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For bound charge we first define the Polarization, P , as:
i i
i
q rP
V
To understand the physical meaning of this definition, consider the two
charges +q and –q, called a dipole, in the volume V in the sketch. For this
elementary dipole:
qr qr q rP
V V
so P is the dipole moment per unit volume.
This charge, which we now distinguish as the polarization charge, f ,
satisfies a divergence condition similar to Gauss’ Law, i.e.:
pP
If there is a change in the polarization caused by a change in position
of the charge in a time interval t then in the limit for small t :
ii
i i
p
rq q vP t J
t V V
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which is the polarization current.
This is real current and so it must satisfy the equation of continuity, i.e.:
0p
p
dJ
d t
We must also include both kinds of charge in Gauss’ law, so:
0
1f pE
and so:
0
1fE P
and therefore:
0 fE P
From this equation we define the displacement vector, D as:
0D E P
and fD
It has been found experimentally that P is a linear function of E at
low values of the field and so:
0 eP E
where e is the electric susceptibility.
Substituting this relationship in the definition of D we find,
0 01 eD E E E
where is the dielectric constant.
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Frequency dependence of conductivity and dielectric
It is found experimentally that both conductivity and dielectric are
functions of frequency. The frequency dependence of conductivity, , is an
important property of rocks containing metallic minerals and clay and a field
method for measuring this property is called the induced polarization (IP)
method. Before discussing the mechanisms that cause this phenomenon we
can describe some of its properties from the basic behavior.
The frequency dependence is written:
J E
D E
In the time domain the current or displacement field is given by the
corresponding convolution operations:
J t s t E t
D t e t E t
where s and e are the Fourier transforms of the frequency dependent
conductivity and permittivity functions. , and J D E must be real in the time
domain and so s and e must also be real. Further both s(t) and e(t) must be
causal (no current or displacement field can occur before the causative
electric field) which implies that: 0 and 0s t e t Therefore in
writing the inverse transform back to frequency for either we have the form:
1
2
i ts t e dt
which, because , 0 s t , can be written as:
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0 0
1cos sin
2 2
is t t dt s t t dt
.
We have found that both and must be complex if they are
functions of frequency (if they are not constant). The practical implication of
such a complex frequency dependence is that the current observed in rocks
or soils when they are subjected to a sinusoidal electric field is shifted in
phase with respect to the electric field and that the amount of phase shift
depends on the frequency. We will return to this topic later when we
describe the IP method in detail.
We have introduced the phenomenon of frequency dependent
resistivity in a general description of a method that was supposed to be
confined to DC or zero frequency. It should be understood that the electric
fields are conservative, that is they can be derived from the gradient of a
potential, if the additional electric fields created by time varying magnetic
fields are negligible. We shall see in the section on electromagnetic methods
that the induced electric fields depend on the induction number of the
problem, the product of the magnetic permeability conductivity and
frequency times some characteristic length squared of the problem, 2L .
As long as the frequency is low enough that the induction number is much
less than one then the DC description of the electric fields is valid.
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Laboratory measurement of resistivity
The relationship developed above between the resistance of a
cylindrical conductor and its resistivity is the basis for a laboratory
measurement procedure. The simplest approach might appear to be to place
two metal contacts (called electrodes) on the ends of the cylinder, apply a
voltage V0 and measure the current, I, that flows through the sample (usually
by measuring the voltage drop across a known resistor R0 in series with the
sample). The sample resistance, Rs, is then simply V0/I and the resistivity is
s
AR
L
The fundamental problem with this approach is that there is an electrical
resistance associated with the contact between the metal electrode and the
rock or soil sample. The current is carried by electrons in the metal and by
ions in the pore fluid of the sample and so charge is carried across the
interface by an electrochemical reaction, which appears as a resistance. (It is
observed that this resistance is also frequency dependent and so it is
complex). This contact resistance may be large, as big as or bigger than the
sample resistance for high conductivity samples, and it is difficult to
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measure independently. For this reason the two-electrode method is not
suitable for most DC or low frequency measurements.
The problem is solved with a four-electrode method illustrated in the
following sketch of the actual apparatus used in the laboratory
measurements.
The current is introduced by two current electrodes in water containers
attached to the sample. If the water has roughly the same concentration of
ions as the pore solution in the sample there is negligible electrochemical
resistance associated with the water-sample interface. The voltage electrodes
are thin screens placed adjacent to the ends of the sample. For an ideal
voltmeter, one that draws no current to make the measurement, the voltage
electrodes have no effect on the current flow or the voltage distribution at
the ends of the sample. The current is continuous through the whole circuit
driven by the power supply so the current measured through the known
resistor R0 is the same as the current through the sample. If it is assumed that
the current is uniform in the sample (the water containers act to ‘smooth out’
any irregularity in flow that might occur because of non-uniform flow off
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the electrode surface), then the ratio of the voltage across the sample, Vs, to
the current, I, is an accurate measurement of the resistance of the sample.
The current electrode contact resistance is unimportant in this scheme
because it is outside the sample part of the circuit. This concept can also be
described by the following equivalent circuit:
Measurement of ground resistivity
The solution for the potentials arising from current flowing into the
ground from a point source of current (a grounding electrode) is the starting
point for the field resistivity method. We begin with the integral form of the
Equation of Continuity:
S
dJ dS dV
dt
and apply it to the hemispherical surface of the volume surrounding the
current source I in the following sketch:
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The term on the right in the Equation of Continuity is the time rate of charge
‘created’ at the injection point and is simply the current I. The integral of the
normal current over the surface of the hemisphere is the current density
times the surface area. (The ground surface of the hemisphere contributes
nothing to the integral since there is no current flowing normal to the
ground). By continuity this integral must exactly equal the injected current:
I = J 2r2
We now substitute Ohm’s Law for J:
I = E 2r2
and then use the fact that E can be derived from a potential via:
E = -
so that:
I = - 2r2
In spherical coordinates = /r and so we are left with an equation in
:
2 22 2
I I
r r r
and therefore:
2
IC
r
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which is the potential of a point source of current on the surface of a uniform
half space.
Potential itself cannot be measured, only differences in potential.
Current cannot be created at a point - there must be another electrode
somewhere to return the current to the power supply (battery). In any real
system of electrodes there are always two for current and two for voltage
(potential difference). In practice, then, there is always an array of electrodes
used to measure the ground resistivity.
The simplest array consists of four electrodes equally spaced a
distance a apart. It is called the Wenner array. In the following sketch, the
current electrodes are located at positions A and B and the potential
electrodes at M and N.
Wenner array.
2 2 2
A BM
I I
a a
2 2 2
A BN
I I
a a
The magnitudes of both currents are the same so the measured voltage V is
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V = M - N =1 1
1 12 2 2 2
I I
a a
So the resistivity of the ground is:
2V
aI
The term in brackets is the geometric factor for the array.
Over a uniform half space this array yields the true value of the
resistivity. Over an inhomogeneous half space the measured values of V and
I are put into the formula to yield the apparent resistivity, A. Over an
inhomogeneous half space A depends on a and on position on the surface.
A is really just a convenient way to express the measured voltage
normalized by the current and the dimensions of the array. The voltages
themselves depend on the current and on the separation. For example the
voltages in the Wenner array fall off as 1/a and the inhomogeneities produce
relatively small deviations from this dominant behavior. Plotting the
voltages on a log scale vs. a would be necessary but then the "anomalies"
would be hard to see. "Normalizing" by multiplying the voltages by a
corrects this plotting problem.
In practice one must be careful because the voltages do get very small
and become inaccurate for large spacings. Multiplying by a large (and
possibly inaccurate) spacing just yields a A with huge error.
There is one more important concept that can be demonstrated with
this simple array. It the current and potential electrodes are interchanged the
value of V/I, and hence A is unchanged. This is the Principle of
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Reciprocity and it is true for any four electrode array over any arbitrary
inhomogeneous ground. It basically invalidates the usual explanation given
for the depth of investigation of DC resistivity methods which is based on
the penetration depth of the current. This argument suggests that as the
current electrodes are spaced farther apart the current penetrates more deeply
and hence ‘senses’ resistivities at greater depths. However the same
sensitivity can be achieved by placing the current electrodes close together
and measuring the voltage on widely spaced electrodes. Depth of exploration
has to be determined on a model by model basis for different arrays.
The basic concept of a resistivity sounding is that the volume of the
ground close to the array dictates the mean value of the apparent resistivity.
As the array expands larger and larger volumes of the ground are included
and so interpretation consists in determining the changes in actual ground
resistivity that cause the changes in apparent resistivity as the array grows.
This is a straightforward process for a horizontally layered ground, and the
response is almost intuitive for a two layer ground. For the first small
spacing of a sounding, when the array spacing is much smaller than the
thickness of the first layer, the apparent resistivity is the actual resistivity of
the first layer. As the array expands the apparent resistivity begins to
respond to the next layer (basement) and eventually at spacings greater than
the first layer thickness the apparent resistivity asymptotes to the basement
resistivity.
For geological situations where it cannot be assumed that the ground
is horizontally layered the resistivity survey must be carried out using a
combination of sounding and traversing – the whole array must be moved
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laterally over the ground at the required number of spacings. Interpretation
then consists of matching the observed apparent resistivities to numerical
data for two or three dimensional models of the ground.
Magnetic fields of direct currents
The magnetic field associated with the current flow is not so easily
understood intuitively. There is, first of all, a field caused by the current in
the wire connecting the electrodes, which can be calculated by the law of
Biot and Savart. There is another component caused by the currents in the
ground. For a uniform or layered half space, integration of the magnetic field
from elements of current from the point source of current shows that the
horizontal field at the surface is simply half the value of the field of an
infinite line of current passing vertically through the electrode location
(Edwards and Nabhigian, 1991). If the ground is made up of uniform layers
the resulting magnetic fields measured on the surface are independent of the
ground conductivity and so depend only on the magnitude of the current and
the geometry of the source current line and the location of the magnetic field
detector. If the ground is inhomogeneous the ‘symmetry’ of the result is
broken and anomalous magnetic fields appear which depend on the
variations in the conductivity. Magnetometer surveys to detect these fields
are often called pure anomaly methods since they only respond to contrasts
in the subsurface. This magnetic variant of the DC resistivity method is
called the Magnetometric Resistivity (MMR) method.