Resistivity Methods
-
Upload
tsarphilip2010 -
Category
Documents
-
view
237 -
download
2
Transcript of Resistivity Methods
-
7/30/2019 Resistivity Methods
1/18
Chapter 3
Resistivity Methods
Resistivity methods use a DC or very low frequency current injected into the ground through electrodes
(and are therefore galvanic methods). Very low frequency methods are generally preferred as periodically
changing the direction of current flow prevents the build up of ions near the electrodes which would affect the
applied potential. Generally, two electrodes are used to inject current into the ground and two other electrodes
are used to measure the potential difference. Other systems make use of only three electrodes, or use long linear
current sources (consistent ground contact is difficult with linear systems adding a source of error). Deviation
of the potential from that expected for a homogenous half-space can be used to deduce information about the
electrical properties of the subsurface. Since both the injected current and the resulting potential are measured,
they can be combined to obtain a value for the resistance (and hence resistivity) of the subsurface.
3.1 Equipotential Line Method
This is a fixed source method that allows the drawing (mapping) of equipotential lines at the Earths surface.
Field work is similar to the SP method, except that an artificial current source is used to generate the measured
potential.
x
y
+ !+I !I
P(x,y)
r1 r2
a
Figure 3.1: Geometry for the equipotential line method
For a homogeneous Earth the potential at the observation point is
VP =I
2
1
r1
1
r2
=
I
2
1
(x2 + y2)1/2
1
((a x)2 + y2)1/2
(3.1)
21
-
7/30/2019 Resistivity Methods
2/18
Resistivity 22
If x = a/2, then VP = 0. The potential profile between the two current inducing electrodes is given by
VP =I
2
1
x
1
a x
(3.2)
+ !
current
equipotentials
A B
Figure 3.2: A schematic of the current paths and equipotential lines at the surface of a homogeneous half-space.Current is generated by the applied potential between electrodes A and B.
The zero potential lies half way between the electrodes. Potential measurements are generally carried out
in the middle third of the region between A and B. If there is a buried body with anomalous conductivity the
current flow will be diverted. Current flow will tend to concentrate in a more conductive (less resistive) region
and avoid a less conductive (more resistive) region. Current flows perpendicular to the equipotential lines (i.e.
along the potential gradient) which are also distorted by the presence of an anomalous body.
a) b)
Figure 3.3: Schematics of the current paths (dashed arrows) and equipotential lines (solid lines) in the vicinityof a body that is anomalously a) conductive and b) resistive.
3.2 Fixed Source Methods
In these methods there are two electrodes (denoted A,B or C1,C2) used to apply the potential which generates
the current flow are kept at fixed positions, while two electrodes (denoted M,N or P 1,P2) are moved about on
a grid measuring the potential difference between grid points.
For a homogeneous half-space the potential difference between M and N is
V = VM VN =I
2r1
I
2r3
I
2r2+
I
2r4(3.3)
-
7/30/2019 Resistivity Methods
3/18
Resistivity 23
A
A B
V
M N
r1
r2 r3
r4
Figure 3.4: A schematic (in map view) of the geometry in the fixed source method. A potential is applied,between the fixed electrodes (A,B - also called the current electrodes) and the resulting current measured bythe ammeter. The potential between points on the surface is measured using the mobile electrodes (M,N - also
called the potential electrodes).
Since the potential, current and geometry are all observed we can solve for the resistivity
a =V
I2
1
r1+
1
r4
1
r2
1
r3
1
=V
IG (3.4)
where G is known as the geometric factor (some definitions of G do not include the factor of 2 ). Note the
subscript a that has been added to the resistivity. This formula gives the resistivity for a perfectly homoge-
neous, isotropic, semi-infinite half-space, in practice geologic settings will contain some sort of heterogeneity or
anisotropy (or both) and the resistivity obtained by equation (3.4) is not the true resistivity of the ground be-neath the probe, a is known as the apparent resistivity. Were the subsurface an ideal half-space, then a would
remain constant as the potential electrodes are moved; instead one will find that a varies and the standard
procedure is to assign the obtained value of apparent resistivity to the point halfway between M and N.
The fixed source method can be used to generate an (apparent) resistivity map of an area. However, the
size of the area is limited by restrictions on the separation of A and B; it is difficult to generate sufficient current
if the electrodes are far apart. Therefore, it is often necessary to relocate A and B in order to map the entire
area of interest. The problem then becomes one of ensuring overlap of area from adjacent apparent resistivity
surveys, often the obtained a values do not agree due to variations in ground properties and quality of contact
between the old and new current electrode positions.
3.3 Moving Source Methods
In moving source methods both the current electrodes and the potential electrodes are moved, in this case it is
important to keep the array geometry fixed. A number of different electrode arrays (or spreads) are used (see
below). One example is the Wenner array in which the distance between adjacent electrodes is equal.
A BM N
a a a
Figure 3.5: Geometry of the Wenner array
-
7/30/2019 Resistivity Methods
4/18
Resistivity 24
For this array the potentials are
VM =I
2a
I
4a=
I
4a
VN = I
2a+
I
4a=
I
4a
VMN = VM VN =I
2a(3.5)
that is a =VMN
I2a (3.6)
There are two methods whereby the arrangement of electrodes can be altered to check for the level of
errors in the survey. The first is known as the check of reciprocity, the two electrode arrangements shown below
should give identical values ofVMN and hence a.
A M N B
M A B N
The second method is the check of superposition and makes use of three electrode arrangements, known as the
alpha, beta and gamma configurations.
A M N B
A B N M
A M B N
The potentials measured from these three configurations should obey
(VMN) = (VMN) + (VMN) (3.7)
The agreement should be within 3% to be considered acceptable.
Electrode Arrays In addition to the Wenner array discussed above, the most common array types are the
Schlumberger and dipole-dipole arrays. The standard layout of these arrays are shown in figure 3.6; however,
there are many variations possible. The geometric factors of the standard arrays are
GSchlum =
ba2 b
2
4 , a 5b (3.8)
Gdd = n(n + 1)(n + 2)a (3.9)
A BM N
b
aA B M N
a ana
n=5a) b)
Figure 3.6: Geometry of the a) Schlumberger and b) dipole-dipole arrays
Each array has advantages and disadvantages; for example, the Wenner array has the best vertical reso-
lution, the dipole-dipole array the best depth penetration and the Schlumberger array is considered the most
-
7/30/2019 Resistivity Methods
5/18
Resistivity 25
suitable for electrical profiling. The resolution characteristics of the spreads are described by array sensitivity
kernels, which describe how sensitive the measured apparent resistivity is to a change in true resistivity at
a given location in the subsurface. The choice of which array to use will depend on your exploration goals,
location, time and manpower.
In addition to the choice of array geometry there are several choices for the employed methodology when
conducting moving source measurements.
Profiling Moving the array along a line keeping the geometry fixed, a is calculated and assigned to the point
halfway between M and N, thus a = f(x). Can be used to study lateral resistivity variations. Note that the
shape of the profile depends on the particular electrode spread used.
!1!2
x
!a
!1 !2
x
!a
Figure 3.7: Examples of resistivity profiles using a Wenner array, note that different arrays will result in differentprofiles. In both cases 2 < 1. More examples can be found in the suggested texts.
Mapping In this case the electrodes are moved over a 2-D grid so that one obtains a = f(x, y), array
geometry is again kept fixed. The resulting map can be used to investigate structures such as salt domes,
weathered rocks (quarries) and buried archeological sites (figure 3.8).
Sounding In electrical sounding the centre of the array centre is kept fixed and the spacing between electrodes
gradually increased (figure 3.9). This method is often called vertical electrical sounding (VES). However, al-
though depth of current penetration tends to increase with increasing electrode spacing the resulting resistivities
are not simply a function of depth a = f(a) = f(z). We will return to this method and consider it in more
detail below.
-
7/30/2019 Resistivity Methods
6/18
Resistivity 26
Figure 3.8: Example of resistivity mapping from the Whistling Elk, South Dakota archaeological site. The upperfigure shows the result of the resistivity survey carried out over an area of 17,000 m2 requiring nearly 34,000measurements. The lower figure shows the archaeological interpretation. For more details see the University ofArkansas Archeo-Imaging Lab (http://www.cast.uark.edu/%7Ekkvamme/ArcheoImage/archeo-image.htm).
-
7/30/2019 Resistivity Methods
7/18
Resistivity 27
A BM NO
h!1
!2!a !!1
h
A BM NO
!1
!2
!a !f(!1,!2)
a) b)
A BM NO
h!1
!2 !a !!2
c)
a
!a
!1
!2
d)
Figure 3.9: Schematic of electrical sounding over a two layer Earth with 1 < 2. The spacing between electrodesin the Wenner array is successively increased a), b), c) and the apparent resistivity plotted against the array
spacing d).
Pseudosections and Continuous Vertical Electrical Sounding Pseudosections (figure 3.10) are created
by making repeated profile traverses with progressively increasing electrode spacing. The a values from suc-
cessive profiles are aligned and ordered by spacing producing a 2-D grid in x a space which is contoured to
produce the pseudosection. Continuous vertical electrical sounding (CVES) also produces a section of apparent
resistivity in x a space however, in this case the image is obtained by repeating electrical sounding investiga-
tions, with successive soundings shifted horizontally. Both methods produce images that are appear similar to
a resistivity cross-section; however, it must be remembered that what is plotted is apparent resistivity and that
electrode space a does not strictly correspond to depth. The practicality of these methods are greatly increased
by the use of multichannel systems which allow multiple electrodes to be set up and simultaneously measured,
these methods also benefit from the use of computers in processing and inverting the data.
3D Side by side pseudosections or CVES lines can be used to build up a three dimensional model of the
subsurface resistivity (figure 3.11). This method is the most expensive and time consuming (in terms of both
human and computer activity).
-
7/30/2019 Resistivity Methods
8/18
Resistivity 28
Figure 3.10: An example of resistivity pseudosection construction and modelling. The location is a faultedTriassic sequence in Staffordshire, UK. (A) 2-D model. (B) Computed apparent resistivity pseudosection. (C)Field data. (D) Geological interpretation. From Griffiths et al. (1990), via Reynolds (1997)
-
7/30/2019 Resistivity Methods
9/18
Resistivity 29
a)
b)
Figure 3.11: An example of a 3D resistivity model. The model is displayed as a) a resistivity cube and b)with a cut-off of 27 m in order to highlight the high resistivity features. The location consists of sand andgravel lens within a glacial till, located in East Yorkshire, U.K. From Catt (2008).
-
7/30/2019 Resistivity Methods
10/18
Resistivity 30
3.4 Electrical Sounding
As discussed above, in electrical sounding the centre of the array is kept fixed and the electrode spacing gradually
increased, the measurements are used to construct a curve a = f(a). In order to interpret this curve in terms
of the subsurface resistivity structure we must consider how the electric potential varies within the subsurface.We know that V(z) in a homogeneous, isotropic medium is a smooth, continuous function and this remains true
in horizontally stratified medium; that is, there are no jumps in potential in the subsurface. As a result the
measured apparent resistivity will also vary smoothly with increasing electrode spacing. The shape of the a
curve will depend on the geometry of the stratified layers. Although natural media need not be homogenous or
isotropic (or horizontally layered), electrical sounding curves are interpreted in terms of the equivalent sequence
of isotropic layers which is often sufficient to help resolve ambiguities from mapping or profiling results. In the
following sections we will develop the theory required to interpret sounding curves. In practice, purpose-built
computer codes are used to invert for the subsurface structure.
3.4.1 Image Theory
For relatively simple geometries we can use the method of images, which was introduced in the discussion of SP
methods, to determine the subsurface potential and interpret the sounding curve. In an infinite, homogeneous,
isotropic medium the potential associated with a point current source is (recall equation (2.8))
V =I
4r(3.10)
As we have seen before, if there is an interface separating two media (figure 3.12), one of which is a perfect
insulator (in this case 2 = ), then the surface acts analogously to a perfect mirror and the potential within
the conductive medium is the same as if there were two identical sources within a single, infinite medium of
resistivity 1.
+
+
h
h
Pr1
r2
!1
!2 =!
+I
IM (= +I)
Figure 3.12: A point current source (I) is reflected in the boundary with the perfect insulator producing anidentical mirror image current source (IM).
The potential at an observation point within the original medium is given by the sum of the potentials
from the original source and the mirror source (recall equation (2.12))
VP =I
4
1
r1+
1
r2
(3.11)
-
7/30/2019 Resistivity Methods
11/18
Resistivity 31
If both of the media are at least somewhat conductive (i.e. 2 = ), then the interface does not behave
as a perfect mirror. Instead we have an imperfect mirror and the image source strength will have some value
KI where K= 1 is the reflection coefficient.
+
+
h
h
P1r1
r2 !1
!2
+I
+KI
Figure 3.13: A point current source (I) is partially reflected by the boundary between the two conductivemedia. On the near side of the interface this produces a mirror image current source of strength ( KI).
For a point (P1) within the same medium as the current source the total potential is the sum of the
potentials of the source and mirror as if they were in a single infinite medium with = 1.
VP1 =1I
4
1
r1+
K
r2
(3.12)
For a point (P2) in the other medium, the interface acts like a filter, and the potential is that as if there
was an altered source of strength IKI within an infinite medium with = 2. Using the reflection analogy;
if the boundary is a partial mirror that has reflected an amount of current KI then the current that passes
through the boundary must be (1 K)I.
+
h
!1
!2
+(1-K)I
P2
r3
Figure 3.14: A point current source (I) is partially reflected by the boundary between the two conductivemedia. On the far side of the interface this produces a filtered current source of strength ((1 K)I).
The potential on the far side of the interface is
VP2 =2I
4
1K
r3
(3.13)
Equations (3.12) and (3.13) hold everywhere on their respective sides of the boundary. For an observation
point on the interface (figure 3.15) the potentials given by these two equations must be equal, so
1I
4
1 + K
r
=
2I
4
1K
r
(3.14)
-
7/30/2019 Resistivity Methods
12/18
Resistivity 32
+
+
h
h
P1
r
r
!1
!2
+I
+KI
+
h
!1
!2
+(1-K)I
P2
r
Figure 3.15: A point current source (I) is reflected in the boundary producing a mirror image current sourceof strength (KI) in an infinite medium with resistivity 1. Equivalently, the point source is filtered by theboundary producing a single current source of strength ((1 K)I) in an infinite medium with resistivity 2.Note that in this case r1 = r2 = r3 = r.
and we can solve for the reflection coefficient
K =2 1
2 + 1(3.15)
Note that if2 < 1, then 1 K 0 and the potential in medium 2 is larger than if there were simply
a source within an infinite medium of resistivity 2. Current preferentially flows into the less resistive medium
and thus there is more current in the more conductive medium 2 than would be expected from the given
source within an infinite medium with resistivity 2. The interface has in some sense amplified the potential in
medium 2. Conversely, since the current is draw out of medium 1 the potential there is lower than if it were an
infinite medium. If2 > 1, then 0 K 1. In this case the current flows preferentially in medium 1. The
interface acts like a partial mirror increasing the current flow and potential in medium 1. Conversely the more
resistive medium 2 is in a sense partially shielded from the current source, resulting is less current and a lower
than expected potential in medium 2. The possible range for the reflection coefficient is 1 K 1 and it
is zero when 2 = 1 (i.e. when there is in fact a single, infinite medium).
Having established the initial theory for the method of images let us apply it to a simple geological case
(figure 3.16). Suppose that the subsurface consists of two units separated by a horizontal interface. An upper
layer of thickness h and resistivity 1 and a lower unit of resistivity 2 which extends to infinite depth. A
point current source is introduced on the surface and the resulting potential measured at various points on the
surface.We have two interfaces to consider: the surface, with reflection coefficient K0 = 1 (the air is effectively
a perfect insulator); and the geologic boundary, with a reflective coefficient K1 as given by equation (3.15).
Reflections will occur due to both of these interfaces. Our current source (I) sits at the surface and will be
reflected in the geologic boundary resulting in an image source of strength K1I a depth h below the contact
(2h below the surface). Unfortunately, this is not the only image. The first image source will be reflected in
the surface resulting in a second image source of strength K0K1I = K1I at a height 2h above the surface.
This image will also be reflected in the geologic contact resulting in a third image source of strength K21I a
depth 3h below the contact (4h below the surface). Reflection of image source three in the surface results in
a fourth image source, also with strength K21I at a height 4h above the surface. We have encountered a hall
of mirrors phenomenon and the original source will, in theory, be reflected an infinite number of times. In
-
7/30/2019 Resistivity Methods
13/18
Resistivity 33
h
P
r1
!1
!2
! =!+
+
+
+
+
r1
r
r2
r2
I
IK2
1
IK2
1
IK1
IK1
Figure 3.16: A point current source (I) is reflected in the geologic boundary and the surface creating a hall of
mirrors collection of image current sources.
practice we need not account for every image, since |K| < 1 the image strengths are successively reduced and
since potential is inversely proportional to distance from the source. The need for infinite images may be clearer
if one thinks about the current produced being reflected back and forth between the surface and the geologic
boundary with some current also passing through the geologic contact (but not the surface) at each reflection,
similar to the familiar situation from seismic reflection theory. Please note that this is an analogy only, and one
that is, in fact, physically misleading. The current is not a travelling wave bouncing between the surface and
the interface at z = h; however, the net potential due to the presence of the subsurface interface can be shown
to be mathematically equivalent to a series of reflections (for more on potential theory see below).
At a point on the surface, the potential due to this series of mirrored sources will be
VP =1I
2
1
r+
2K1r1
+2K21
r2+
2K31r3
+
(3.16)
where rn =
r2 + (2nh)2. Note that we are using the half-space potential equation (2.10); since all of the
produced current goes into the subsurface the effective strength of the source is doubled, as discussed above.
Collecting the terms depending on K1 we have,
VP =1I
21
r
+ 2
n=1
Kn1r2 + (2nh)2
= (r) (3.17)or
VP =1I
2
1
r+ 2S(r, K1, h)
= (r) (3.18)
Note that the terms in the summation go to zero as n since |K1| < 1.
For the Wenner array (figure 3.5) the measured potential difference
V = VM VN = (a) (2a) (2a) + (a)
V = 2[(a)(2a)] (3.19)
and if we recall the definition of apparent resistivity for this array (equation 3.6) we obtain
a =
2a
I
2
1I
2
1
a+ 2S(a, K1, h)
1
2a 2S(2a, K1, h)
-
7/30/2019 Resistivity Methods
14/18
Resistivity 34
a = 1 [1 + 4aS(a, K1, h) 4aS(2a, K1, h)] (3.20)
We can use the fact that,
aS(a, K1, h) = a Kn1
a2 + (2nh)2 = Kn1
1 + (2nh/a)2 (3.21)
and aS(2a, K1, h) = a Kn1
(2a)2 + (2nh)2= Kn1
4 + (2nh/a)2(3.22)
to write our expression for apparent resistivity as
a = 1
1 + 4
Kn11 + (2nh/a)2
Kn1
4 + (2nh/a)2
a = 1 [1 + 4F(K1,h/a)] (3.23)
The function F can be computed for any range of electrode spacings and any given K1. For small electrode
spacing h/a 1, thus F 0 and a 1, as would be expected. For large electrode spacings h/a 1, thus
F 1/2
Kn1 . Because K21 < 1 the infinite sum can be approximated as
K21 =
11K1
1 which gives
a 2. As expected the apparent resistivity varies smoothly from 1 to 2 as the electrode spacing increases.
In fact this holds true for any array geometry, although the particular form of F will differ.
Clearly, as we plot measurements of a against a there will be many different possible curves depending
on the values of 1, 2 and h. It would be useful to consider all possible values of the physical parameters
in a compact way, this is done by constructing master curves. To construct the master curves we note that
equation (3.23) can be re-written in the form
a
1= f(a/h) (3.24)
Taking the log of both sides we get
log a log 1 = f(log a log h) (3.25)
which is of the form y p = f(x q). Field measurements are in terms of a and a which can be plotted on a
log-log plot. Therefore, we obtain a field curve
log a = f(log a) (3.26)
which is of the form y = f(x). The field curve and the theoretical curve (for the correct value of K1) will have
the same shape but will have their origin shifted by p and q (that is, by the log of1 and h). Master curves are
constructed from equation (3.23) which is evaluated for various values of K1 on log-log plots of a/1 vs a/h
(figure 3.17).
The field curve is compared with the master curves which are overlain using a transparency. Keeping the
field and master axes parallel, the transparency is shifted until a good match with a master curve is obtained
(this gives the value of K1). The point on the master curve where a/1 = a/h = 1 will correspond to a point
on the field curve giving the true values of 1 and h (since the master and field curve origins are shifted relative
to each other by these values). Knowing, 1 and K1 the value of2 can be calculated.
-
7/30/2019 Resistivity Methods
15/18
Resistivity 35
!! !"#$ " "#$ ! !#$
!!
!"#$
"
"#$
!
%&'()*+,
%&'(!)
*!!
,
k = -1.0
k = -0.8
k = -0.6
k = -0.4
k = -0.2
k = 0.0
k = 0.2
k = 0.4
k = 0.6
k = 0.8
k = 1.0
Wenner Array 2 Layer Master Curves
Figure 3.17: Master curves plotted as log(a/1) on the vertical axis against log(a/h) on the horizontal. Notethat the master curves may also be labeled by the ratio 2/1. For further detail and examples see, e.g. Milsom(1989), Reynolds (1997).
3.4.2 Potential Theory
If there are more than two layers in the subsurface, then image theory becomes much more difficult to use.
Potential theory is used instead.
If J is the current density and E the electric field vector in a given medium, then Ohms law states
J = E =1
E (3.27)
where and are, respectively, the conductivity and resistivity of the medium. Since the electric field is
conservative
J =1
E =
1
(V) =
1
V (3.28)
Consider some volume (v) of bounded by a closed surface (A). If this volume does not contain any current
sources or sinks, then any current that flows into v must also flow back out, i.e. the net flux through A must
be zero. Using Gausss Theorem we haveA
J n dA =
v
J dv = 0 (3.29)
Since we can choose an arbitrary volume we must have J = 0 at all points (except at sources or sinks). Thus,
(1
V) =
1
( V) +V
1
= 0 (3.30)
If the conductivity is constant within a given region, then this expression simplifies to
2V = 0 (3.31)
-
7/30/2019 Resistivity Methods
16/18
Resistivity 36
which is Laplaces Equation, the fundamental equation governing potential theory. Our goal is to solve Laplaces
Equation subject to the boundary conditions imposed by the particular situation of interest. For the potential
due to a point source at the surface of uniform, horizontal layers it is logical to work in the cylindrical coordinate
system (note that r will now denote the distance from the z axis, not distance from the source as above). For
a point source and laterally homogeneous layers the potential will be a function of r and z only, and Laplaces
Equation is2V
r2+
1
r
V
r+
2V
z2= 0 (3.32)
and the potential in each layer (Vi) will be of the form
Vi(r, z) =
0
Fi()e
z + Gi()e+z
J0(r) d (3.33)
where, is a integration variable, J0 is the order zero Bessel function of the first kind and Fi and Gi are
functions to be determined from the boundary conditions:
1. V 0 if r or z
2. For small (r2 + z2)1/2 we will have V = 1I2 (r2 + z2)1/2
3. At the surface (z = z0), V /z = 0 except at r = 0
4. The potential must be continuous across interfaces, Vi(zi) = Vi+1(zi), where zi is the depth of the interface.
5. The current density normal to an interface must be continuous, 1i
Viz
zi
= 1i+1
Vi+1z
zi
For example, condition (1) implies that all Bi() = 0.
The potential at the surface of a two layer Earth will be
VP(r) =1I
2
1
r+ 2
0
K1e2h
1K1e2hJ0(r) d
VP(r) =1I
2
1
r+ 2
0
()J0(r) d
(3.34)
where K1 is the reflection coefficient and () is known as the kernel function.
h1
h2
1
2
3
0 =
Figure 3.18: Geometry of a three layer Earth
For a three layer Earth we have K1 = (2 1)/(2 + 1) and K2 = (3 2)/(3 + 2) and the surface
potential is given by
VP(r) =1I
21
r
+ 2
0
K1e2h1 + K2e
2(h1+h2)
1K1e
2h1 K2e
2(h1+h2) + K1K2e
2h2
J0(r) dVP(r) =
1I
2
1
r+ 2S3(r, K1, K2, h1, h2)
(3.35)
-
7/30/2019 Resistivity Methods
17/18
Resistivity 37
The expression for surface potential can be used with the geometry of the chosen array to derive an expression
for the expected apparent resistivity of the form
a = 1
1 + 4F3
K1, K2,
h1a
,h2a
(3.36)
which, as in the two-layer case, can be used to plot master curves of log(a/1) vs log(a/h1) for the various
combinations of (K1, K2, h1, h2).
As before, the master curves are constructed for a given set of parameters by transforming equation (3.36)
to the form shown in equations (3.24) and (3.25) which will have shifted origins relative to the field curve having
the form of equation (3.26). Curve matching is used as in the two layer case; the origin gives the values for
1 and h1 and the shape of the curve will depend on K1, K2, h2 which can be used to determine the electrical
properties of the subsurface. Note that in the two layer case the different curves are distinguished based on
variations in a single parameter (K1, or equivalently 2/1) and generally 20 curves are used to find a match.
For the three layer case the different master curves are distinguished by variations in three parameters (2/1,
3/2 and h2/h1) and it becomes necessary to consider 20 20 20 different master curves. Compilations of
such curves have been published and there are also programs that generate the master curves numerically.
Three layer master curves are generally divided into four classes, or types, as shown in figure 3.19. Identifi-
cation of the curve type gives an indication of the subsurface electrical structure. Field curves can be matched in
their entirety or through a bootstrap method using a series of two layer curves matched to successive portions of
the field curve. With more than three subsurface layers complete matching to master curves become impractical
and the bootstrap method or numerical fitting is required.
K type
H type Q type
A type
2 > 1 2 > 3
2 < 32 < 11 > 2 > 3
1 < 2 < 3
Figure 3.19: Schematic three layer master curves
Unfortunately it is not possible to uniquely determine three layer master curves. Three inherent difficulties
that you will encounter are detailed here.
-
7/30/2019 Resistivity Methods
18/18
Resistivity 38
Practical Ambiguity The master curves for a three layer Earth depend on parameters that are ratios of
the desired subsurface properties, i.e. 2/1, 3/2 and h2/h1. This leads to a practical ambiguity in the
interpretation of the field curves since proportional increases in all three resistivities or both layer thicknesses
leave the ratios unchanged, e.g. h2/h1 = (2 h2)/(2 h1). Therefore, such proportional changes in subsurface
properties will not alter the field curve and it will not be possible to uniquely determine the geoelectrical
structure.
Equivalence Principle This principle applies to K and H type curves. For K type curves the second layer is
the most resistive and current tends to flow vertically through this layer. The effect of this layer on the current
is then quantified by its transverse resistance (recall equation 1.13) T = h22. If this product remains constant,
then the effect of the layer on the field curve is unchanged; e.g. (2, h2) = (40 m, 0.5 m) or (20 m, 1 m) both
result in T = 20 m2 and are therefore electrically equivalent. Similarly, for H type curves the current tends
to be concentrated and horizontal within the less resistive, second layer. The effect of this layer is quantifiedby its horizontal conductanceH = h2/2 (the inverse of equation 1.16) and beds which have equal ratios of h2
and 2 can not be distinguished by inspection of the field curve. Numerical modelling may be able to limit the
range of possibilities, as can a priori geological knowledge.
Principle of Suppression In general, thin layers will have small effects and are difficult to detect unless
their resistivities are much greater (or much less) than the surrounding layers. Such layers are particularly
problematic in A and Q type curves as the width of the central plateau (see figure 3.19) depends on the
thickness of the middle layer. The suppression is further exacerbated if the resistivity contrast between the top
and middle layers is similar to the contrast between the top and bottom layers (recall that may vary by orders
of magnitude in the subsurface). Again, numerical modelling and a priori geological knowledge can be useful.