Geology Reference
In-Depth Information
Resistivity (
Ω
m)
10
2
10
4
10
6
10
8
1
I
Granite
Gabbro
ρ
r
Schist
Current flow
line
Quartzite
Sandstone
Equipotential
surface
Shale
Clay
Fig. 8.3
Current flow from a single surface electrode.
Alluvium
Fig. 8.2
The approximate range of resistivity values of
common rock types.
d
d
V
L
r
d
I
A
=-
=-
r
i
(8.3)
d
V/
d
L
represents the potential gradient through the
element in volt m
-1
and
i
the current density in A m
-2
.
In general the current density in any direction within a
material is given by the negative partial derivative of the
potential in that direction divided by the resistivity.
Now consider a single current electrode on the sur-
face of a medium of uniform resistivity
r
(Fig. 8.3).The
circuit is completed by a current sink at a large distance
from the electrode. Current flows radially away from the
electrode so that the current distribution is uniform over
hemispherical shells centred on the source. At a distance
r
from the electrode the shell has a surface area of 2
p
r
2
,so
the current density
i
is given by
the resistivity of rocks, and that resistivity generally in-
creases as porosity decreases. However, even crystalline
rocks with negligible intergranular porosity are conduc-
tive along cracks and fissures. Figure 8.2 shows the range
of resistivities expected for common rock types. It is ap-
parent that there is considerable overlap between differ-
ent rock types and, consequently, identification of a rock
type is not possible solely on the basis of resistivity data.
Strictly, equation (8.1) refers to electronic conduction
but it may still be used to describe the effective resistivity
of a rock; that is, the resistivity of the rock and its pore
water. The effective resistivity can also be expressed
in terms of the resistivity and volume of the pore water
present according to an empirical formula given by
Archie (1942)
I
i
=
(8.4)
2
r
2
p
=
a
--
b
f
c
rf
r
(8.2)
w
From equation (8.3), the potential gradient associated
with this current density is
where
f
is the porosity,
f
the fraction of pores containing
water of resistivity
r
w
and
a
,
b
and
c
are empirical con-
stants.
r
w
can vary considerably according to the quanti-
ties and conductivities of dissolved materials.
∂
∂
V
r
I
r
r
2
=-
i
=-
r
(8.5)
2
The potential
V
r
at distance
r
is then obtained by
integration
8.2.3 Current flow in the ground
Consider the element of homogeneous material shown
in Fig. 8.1. A current
I
is passed through the cylinder
causing a potential drop -
d
V
between the ends of the
element.
Ohm's law relates the current, potential difference
and resistance such that -
d
V
=
d
RI
, and from equation
(8.1)
Ir
r
∂
I
r
r
r
p
Ú
Ú
VV
=∂ =-
=
(8.6)
r
2
2
2
p
The constant of integration is zero since
V
r
= 0 when
r
=•.
RLA
=
d
d
d
. Substituting
