-
IMPACT OF CONDUCTIVE MINERALS ON
MEASUREMENTS OF ELECTRICAL RESISTIVITY
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF GEOPHYSICS
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE
OF DOCTOR OF PHILOSOPHY
ADAM TEICHERT TEW
MARCH 2015
-
http://creativecommons.org/licenses/by-nc/3.0/us/
This dissertation is online at:
http://purl.stanford.edu/gx072rj6544
© 2015 by Adam Tew. All Rights Reserved.
Re-distributed by Stanford University under license with the
author.
This work is licensed under a Creative Commons
Attribution-Noncommercial 3.0 United States License.
ii
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-
I certify that I have read this dissertation and that, in my
opinion, it is fully adequatein scope and quality as a dissertation
for the degree of Doctor of Philosophy.
Gerald Mavko, Primary Adviser
I certify that I have read this dissertation and that, in my
opinion, it is fully adequatein scope and quality as a dissertation
for the degree of Doctor of Philosophy.
Eric Dunham
I certify that I have read this dissertation and that, in my
opinion, it is fully adequatein scope and quality as a dissertation
for the degree of Doctor of Philosophy.
Jack Dvorkin
Approved for the Stanford University Committee on Graduate
Studies.
Patricia J. Gumport, Vice Provost for Graduate Education
This signature page was generated electronically upon submission
of this dissertation in electronic format. An original signed hard
copy of the signature page is on file inUniversity Archives.
iii
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iv
Abstract
The electrical properties of rocks have been used in geophysical
interpretation for
decades. At low frequencies, the dominant controls on the
electrical resistivity of
rocks are typically the brine resistivity and the volume
fraction of brine. Sometimes
other factors become important.
The primary objective of this thesis is to identify the physical
rules that govern the
impact of conductive minerals on the low frequency electrical
properties of rock. This
relationship is examined primarily by measuring the complex,
frequency dependent
resistivity of clean, well characterized artificial sediments in
the laboratory. Important
factors which mediate the impact of conductive minerals include
the volume fraction,
the dry conduction fraction and the electrochemistry occurring
at the grain/brine
interface.
Chapter 1 includes a review of the history of resistivity
measurements, a look at
Archie’s law and the modes of electrical conduction in rock.
Unlike most rocks which
carry electric current through the ions in the pore fluid, rocks
which contain
conductive minerals can sometimes pass significant amounts of
current through the
free electrons in the rock matrix. If rocks of this type are
interpreted using an
uncorrected Archie model, the water saturation will be
overestimated, which may
result in a productive reservoir being overlooked.
In Chapters 2 and 3, the complex resistivity of a set of
well-defined sediments at
frequencies from 50 Hz to 100 kHz is reported. The parameters of
grain size, porosity,
-
v
brine salinity and saturation were varied. The resistivity of
sediments and consolidated
artificial samples which contained varying fractions of
conductive grains (up to 30%
by volume) is reported. In these tests reductions in resistivity
were observed at higher
frequencies when the conductive fraction was increased. Large
and/or abrupt changes
were not observed, likely because the dry conductive threshold
was not reached.
In Chapter 4, some simple electrode/brine experiments are
reported which
examined the magnitude of the electrode potential that must be
overcome to drive
charge transfer across the interface between conductive grain
and brine. It turned out
that the voltage threshold is relatively high for pyrite
compared to the voltage
gradients typically present in geophysical measurements so this
potential current
pathway can often be assumed to carry no current, at least in DC
measurements.
Conductive minerals which are present in high enough volumes can
form
conductive pathways through the host rock. Below the dry
conduction threshold, the
fraction of conductive mineral has little impact on the DC
resistivity, but at higher
frequencies reduces resistivity. Above the dry conduction
threshold, the conductive
minerals can have a dramatic effect on the DC resistivity.
Chapter 5 includes a discussion of ways to predict resistivity
in rocks which
contain conductive minerals both above and below the dry
conduction threshold.
Determining whether or not the rock of interest is above or
below the dry conduction
threshold is important as the methods of interpretation differ
above and below. The
threshold can vary from a low volume fraction in rocks with
conductive veins, to high
volume fraction in rocks with non-conductive grain coatings.
Below the threshold,
results are comparable to the measurements reported in Chapter
3. Above the
threshold, additional measurements are needed for
interpretation. A framework for
incorporating those measurements is presented. Also discussed
are various details
related to testing and avenues for future research.
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vi
Acknowledgements
There were many who contributed to this work and to my education
at Stanford in
one way or another and I am grateful for those efforts and
interactions. Here, I
highlight a few individuals who played especially important
roles.
Gary Mavko and Jack Dvorkin provided advice and guidance in the
latter half of
this project. They excelled at finding my incomplete ideas and
unexplained data points
and asking difficult questions that helped me to improve my
understanding of the
physics. Their ideas were particularly useful in building a
consistent interpretation of
the data and in identifying new avenues for analysis and
interpretation.
Tiziana Vanorio introduced me to the lab and gave me a lot of
time and guidance
as I learned the equipment and developed a research plan. I
enjoyed the time I spent
working with her on various projects. Her instruction related to
lab methods and data
analysis has been invaluable.
Eric Dunham was a member of my reading, defense, and annual
review
committees. Rosemary Knight was a member of my annual review
committee and the
advisor of my second project.
Financial support for this work came from the SRB and its
affiliates.
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vii
Contents
Abstract
..........................................................................................................................
iv
Acknowledgements
.......................................................................................................
vi
Contents
........................................................................................................................
vii
List of Tables
.................................................................................................................
ix
List of Figures
................................................................................................................
xi
Chapter 1 Introduction
....................................................................................................
1
Archie’s Law
..............................................................................................................
2
Units
...........................................................................................................................
8
Modes of conduction in earth materials
...................................................................
10
Low Resistivity Pay ZONES
....................................................................................
14
Research Focus and summary
..................................................................................
15
Chapter 2 Fundamental resistivity measurements
........................................................ 17
Introduction
..............................................................................................................
17
Methods
....................................................................................................................
18
Results
......................................................................................................................
25
Discussion
.................................................................................................................
38
Chapter 3 Complex resistivity with conductive grains
................................................. 48
Introduction
..............................................................................................................
48
Methods
....................................................................................................................
53
Results
......................................................................................................................
63
Discussion
.................................................................................................................
70
Chapter 4 Grain and brine interface
.............................................................................
73
Introduction
..............................................................................................................
73
Electrolysis Experiments
..........................................................................................
75
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viii
Results
......................................................................................................................
77
Discussion
.................................................................................................................
82
Chapter 5 Conclusions, Recommendations and Future Work
...................................... 92
A first step: Percolation thresholds
..........................................................................
92
Above the percolation Threshold
.............................................................................
96
Complex resistivity and frequency effects
...............................................................
97
Testing parameters
....................................................................................................
98
Making this useful (And how this relates to Archie’s Law)
.................................. 100
Future avenues for study
........................................................................................
104
Final Words
............................................................................................................
105
Appendix: Data
...........................................................................................................
106
References
..................................................................................................................
127
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ix
List of Tables
Table 1: Summarizes the electrode tests which have been
reported, including the electrode material, the test solution, the
apparent voltage threshold, and the Figure where the results are
displayed.
................................................................
81
Table 2: Variable fraction coarse and fine grain sand, brine
saturated (50 g/l) ......... 106
Table 3: Variable fraction coarse and fine grain sand, brine
saturated (4 g/l) ........... 107
Table 4: Variable fraction sand & powder, brine saturated
........................................ 108
Table 5: Variable salinity brine
..................................................................................
110
Table 6: Variable salinity in coarse acrylic
................................................................
112
Table 7: Variable salinity in brine saturated sand
...................................................... 114
Table 8: Variable brine saturation in quartz sand
....................................................... 116
Table 9: Coarse sand arranged in parallel and series with fine
sand and mixed sand 118
Table 10: Consolidated, brine saturated, coarse acrylic and
pyrite ............................ 119
Table 11: Unconsolidated, brine (1) saturated, coarse acrylic
and pyrite .................. 119
Table 12: Unconsolidated, brine (2) saturated, coarse acrylic
and pyrite .................. 120
Table 13: Fine grained acrylic & steel shot
................................................................
121
Table 14: Electrolysis, steel in H2SO4 solution
.......................................................... 123
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x
Table 15: Electrolysis, copper in H2SO4 solution
...................................................... 124
Table 16: Electrolysis, pyrite in H2SO4 solution
........................................................ 125
Table 17: Electrolysis, pyrite electrodes in NaCl solution
......................................... 126
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xi
List of Figures
Figure 1.1: Illustration of the difference between gravitational
and electric potential for brine. On the left, all molecules
(charged or not) respond to the gravitational potential. On the
right, only the ions experience a net force, and the direction of
the force is charge dependent.
..............................................................................
11
Figure 1.2: Dark, iron bearing minerals (likely magnetite and
hematite) that have separated out on a beach. Photographed area is
approximately 1m wide. ........... 13
Figure 2.1: Photo of the test vessel, including upper stainless
steel electrode. The vessel was built using parts from a MC Miller
soil cylinder. ............................... 19
Figure 2.2: Cumulative grain size for the coarse and fine sands
which were used in the sand mixing test.
...................................................................................................
23
Figure 2.3: Diagrams of the four test geometries. Current ran
vertically through the cylinders. In the upper left: coarse and
fine sand in series. In the upper right: coarse and mixed sand (50%
coarse sand and 50% fine sand) in series. In the lower left: coarse
and fine sand in parallel. In the lower right: coarse and mixed
sand in parallel.
.....................................................................................................
24
Figure 2.4: Resistivity of brine as a function of salt
concentration. For each salinity, resistivity was measured at 5
frequencies from 50 Hz to 100 kHz, all of which are plotted.
..................................................................................................................
26
Figure 2.5: Reactivity of brine as a function of salt
concentration. For each salinity, reactivity was measured at 5
frequencies from 50 Hz to 100 kHz. ...................... 26
Figure 2.6: Resistivity of brine saturated coarse acrylic as a
function of salt concentration. For each salinity, resistivity was
measured at 5 frequencies from 50 Hz to 100 kHz, all of which are
plotted. ..........................................................
27
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xii
Figure 2.7: Reactivity of brine saturated coarse acrylic as a
function of salt concentration. For each salinity, reactivity was
measured at 5 frequencies from 50 Hz to 100 kHz.
......................................................................................................
28
Figure 2.8: Resistivity of brine saturated beach sand as a
function of salt concentration. For each salinity, resistivity was
measured at 5 frequencies from 50 Hz to 100 kHz, all of which are
plotted. ..........................................................
29
Figure 2.9: Reactivity of brine saturated beach sand as a
function of salt concentration. For each salinity, reactivity was
measured at 5 frequencies from 50 Hz to 100
kHz........................................................................................................................
29
Figure 2.10: Resistivity of beach sand as a function of
saturation. For each saturation, resistivity was measured at 5
frequencies from 50 Hz to 100 kHz, all of which are plotted.
..................................................................................................................
30
Figure 2.11: Reactivity of beach sand as a function of
saturation. For each saturation, reactivity was measured at 5
frequencies from 50 Hz to 100 kHz ....................... 31
Figure 2.12: Resistivity of sand saturated with 4 g NaCl/L
DI-water as a function of coarse fraction. For each mixture,
resistivity was measured at 5 frequencies from 50 Hz to 100 kHz,
all of which are plotted.
.......................................................... 32
Figure 2.13: Reactivity of sand saturated with 4 g NaCl/L
DI-water as a function of coarse fraction. For each mixture,
reactivity was measured at 5 frequencies from 50 Hz to 100 kHz.
.................................................................................................
32
Figure 2.14: Resistivity of sand saturated with 50 g NaCl/L
DI-water as a function of coarse fraction. For each mixture,
resistivity was measured at 5 frequencies from 50 Hz to 100 kHz.
.................................................................................................
33
Figure 2.15: Reactivity of sand saturated with 50 g NaCl/L
DI-water as a function of coarse fraction. For each mixture,
reactivity was measured at 5 frequencies from 50 Hz to 100 kHz.
.................................................................................................
33
Figure 2.16: Resistivity of sand as a function of frequency and
geometry. Note the vertical axis does not start at 0. Unlike most
figures in this and other chapters the color code does not
represent the different test frequencies – it represents the
specific test geometry.
..........................................................................................
34
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xiii
Figure 2.17: Reactivity of sand as a function of frequency and
geometry. Unlike most figures in this and other chapters the color
code does not represent the different test frequencies – it
represents the specific test geometry.
................................... 34
Figure 2.18: Porosity of mixed sand and powder as a function of
sand fraction. The point at 0.77 sand fraction was oversaturated
with brine, compared to the same dry mixture’s porosity.
.........................................................................................
35
Figure 2.19: Resistivity of mixed sand and powder as a function
of sand fraction (by mass). For each mixture, resistivity was
measured at 5 frequencies from 50 Hz to 100 kHz, all of which are
plotted. The 0.77 sand fraction mixture was oversaturated with
brine.
......................................................................................
36
Figure 2.20: Reactivity of mixed sand and powder as a function
of sand fraction (by mass). For each mixture, reactivity was
measured at 5 frequencies from 50 Hz to 100
kHz.................................................................................................................
36
Figure 2.21: Resistivity and reactivity of mixed sand and powder
as a function of porosity. For each mixture, measurements were made
at 5 frequencies from 50 Hz to 100 kHz, all of which are plotted.
...............................................................
37
Figure 2.22: Resistivity of brine as a function of salinity.
Schlumberger data taken from Schlumberger Log Interpertation
Charts, Edition 2000, Gen-9. Data shown from this study was
measured at 10 khz.
..............................................................
38
Figure 2.23: Conductivity of brine as a function of salinity.
........................................ 39
Figure 2.24: A 2D illustration of the geometry of bimodal grain
size mixtures. As drawn, there is some overlap between coarse and
fine grains which should not occur, however, using the relatively
low grain size ratios depicted here other, more complicated,
effects are introduced if the coarse and fine grains do not
overlap.
.................................................................................................................
40
Figure 2.25: Resistivity of sand as a function of coarse
fraction. Series and parallel resistivities have been added to
Figure 2.12. Series geometry data shown with green dashes. Parallel
geometry data is marked with red X’s. Blue squares are homogenous
mixtures of coarse and fine sand. For each mixture, resistivity was
measured at 5 frequencies from 50 Hz to 100 kHz.
............................................. 42
Figure 2.26: Resistivity of sand as a function of coarse
fraction. Figure shows 2 kHz data. Series geometry data shown with
green dashes. Parallel geometry data marked with red X’s.
Voigt-Reuss boundaries are shown in black between the components of
both tests – the lower pair is the boundaries for combinations
of
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xiv
the coarse and fine sand, the upper pair is the boundaries for
combinations of the coarse and mixed sand. The red dashed lines are
Voigt-Reuss boundaries assuming a 2% uncertainty in the
resistivities of the end members. .................... 42
Figure 2.27: Reactivity of sand as a function of coarse
fraction. Series and parallel reactivities have been added to
Figure 2.12. Series data shown with green dashes, parallel geometry
points marked with a red X. Blue squares are homogenous mixtures of
coarse and fine sand. For each mixture, reactivity was measured at
5 frequencies from 50 Hz to 100
kHz......................................................................
44
Figure 3.1 Close up of two consolidated samples. The sample on
the left is 100% acrylic. Colored grains in the sample on the left
are grains of colored acrylic. The sample matrix on the right is
30% pyrite and 70% acrylic by volume. ................ 55
Figure 3.2: Illustration of the final system configuration for
measuring complex resistivity on core plugs. Colored lines
represent tubing - red lines carry hydraulic fluid, purple lines
carry compressed air, and blue lines carry the pore fluid. The
green lines are the electrical leads, which connect the electrical
meter to the core holder. X’s represent valves.
................................................................................
59
Figure 3.3: Resistivity versus frequency for brine saturated,
consolidated mixtures of pyrite and acrylic. Unlike in most figures
in this and other chapters, the color code does not represent the
different test frequencies. Instead, it identifies the sample.
..................................................................................................................
64
Figure 3.4: Reactivity versus frequency for consolidated
mixtures of pyrite and acrylic. Unlike in most figures in this and
other chapters, the color code does not represent the different
test frequencies. Instead, it identifies the sample. ............
65
Figure 3.5: Resistivity versus pyrite volume for unconsolidated
mixtures of pyrite and acrylic saturated with 50 g/l brine.
Resistivity was recorded at five frequencies for each pyrite
fraction.
........................................................................................
66
Figure 3.6: Reactivity versus pyrite volume for unconsolidated
mixtures of pyrite and acrylic saturated with 50 g/l brine.
Reactivity was recorded at five frequencies for each pyrite
fraction.
..............................................................................................
66
Figure 3.7: Resistivity versus pyrite volume for unconsolidated
mixtures of pyrite and acrylic saturated with 25 g/l brine.
Resistivity was recorded at five frequencies for each pyrite
fraction.
........................................................................................
67
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xv
Figure 3.8: Reactivity versus pyrite volume for unconsolidated
mixtures of pyrite and acrylic saturated with 25 g/l brine.
Reactivity was recorded at five frequencies for each pyrite
fraction.
..............................................................................................
68
Figure 3.9: Resistivity versus steel volume for unconsolidated
mixtures of steel and acrylic saturated with 25 g/l brine. 5
frequencies were tested at each steel fraction.
.................................................................................................................
69
Figure 3.10: Reactivity versus steel volume for unconsolidated
mixtures of steel and acrylic saturated with 25 g/l brine. 5
frequencies were tested at each steel fraction.
.................................................................................................................
69
Figure 4.1: Test setup for electrolysis experiments.
..................................................... 76
Figure 4.2: Photo of pyrite electrodes after testing. The
gridlines are approximately 5 mm apart.
..............................................................................................................
77
Figure 4.3: DC voltage vs current for two stainless steel strips
suspended in a very dilute sulfuric acid solution. When the test
was repeated (blue diamonds) more time was provided between voltage
changes. .......................................................
78
Figure 4.4: DC voltage vs current for two copper strips
suspended in a very dilute sulfuric acid solution.
...........................................................................................
79
Figure 4.5: DC voltage vs current for two pyrite crystals
suspended in a dilute sulfuric acid solution. When the test was
repeated (red diamonds) the time between tests was increased.
.......................................................................................................
80
Figure 4.6: DC voltage vs current for pyrite crystals suspended
in a 50 g NaCl per L DI-water solution.
.................................................................................................
81
Figure 4.7: Schematics of three slightly more complicated grain
fluid systems. The blue area is brine. Yellow blocks are highly
conductive grains (compared to the brine). The grey blocks
represent electrodes. The red lines mark approximate equipotential
lines. The small green squares mark areas of the grain through
which charge is passing. The top figure shows a conductive grain
blocking a pore. The middle figure shows a conductive grain
blocking a pore in parallel with an unblocked pore. The bottom
figure shows a conductive grain partially blocking a pore.
....................................................................................................
84
Figure 4.8: Shown are four highly conductive grains partially
blocking brine filled pores, varying only in the voltage applied
between the two electrodes. In the
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xvi
upper left, with 8V, current flows through most of the grains
surface and the grain (including the interfaces) appears more
conductive than the fluid. In the upper right, with 4V, current
flows only through the ends of the grain, and the grain appears to
match the conductivity of the fluid. In the lower left, with 3V,
current can only pass through a small portion of the surface and
the grain appears less conductive than the fluid. Finally, at the
lower right, with 2V, the grain carries no current and looks like
an insulator.
...................................................... 85
Figure 4.9: Shown are three conductive grains partially blocking
pores otherwise filled with a conductive brine. The conductivity of
the grains is identical to the brine. The voltage between the
electrodes varies between the illustrations, from 24V to 6V to 2V.
..................................................................................................
86
Figure 4.10: Shown are three conductive grains partially
blocking pores otherwise filled with a conductive brine. The
conductivity of the grains is much less than the brine. The voltage
between the electrodes varies between the illustrations, from 8V to
4V to
2V.............................................................................................
87
Figure 4.11: Adapted from He and Ekere (2004). Figure shows
estimates of percolation thresholds at several insulating to
conducting grain diameter ratios. At 1, the insulating and
conducting grain size is the same. At 6 the conductive grain
diameter is one sixth the insulating grain size diameter.
............................. 89
Figure 4.12: A simple illustration of three rocks which are all
above the dry conduction threshold in at least one direction. The
area highlighted in yellow and covered with yellow circles is above
the dry conduction threshold – so it could, for example, contain
50% pyrite. The grey areas do not contain conductive minerals. The
vertical percentage of rock which contains conductive minerals
varies in these three rocks from 100% on the left to 20% on the
right. ............... 90
Figure 5.1: Illustrations of some possible conductive mineral
geometries. The conductive mineral is shown in red. On the left is
shown the baseline clean sand. The figure labeled “One grain size”
illustrates a pack in which both conductive and non-conductive
grains are a uniform grain size. “Two grain sizes” shows smaller
conductive grains filling the pore space of the initial
non-conductive pack. “Veins” illustrates conductive veins cutting
through the pack. “Coordination number” shows a pack which has been
compressed, thereby increasing the number of contacts each grain
makes with other grains. “Grain Coatings” shows the original pack
which has been coated with a thin conductive layer. The alternative
is also possible, where the conductive grains are covered with a
thin non-conductive layer.
.........................................................................
94
Figure 5.2: Conductivity versus conductive volume fraction for
four geometries. Vertical dashed lines at 0.3 and 0.7 mark the rock
porosity and solid fraction
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xvii
respectively. The orange lines represent a geometry which begins
conducting immediately, such as brine in a water wet rock (Archie
behavior). The geometry represented by the blue lines requires a
volume fraction of 26% before it begins conducting (conductive
mineral behavior). Grey lines are Voigt-Reuss bounds.
............................................................................................................................
102
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1
Chapter 1
Introduction
Modern geophysicists have an overwhelming number of tools at
their disposal for
characterizing rocks both in the laboratory and in the
subsurface. For several decades,
work in the rock physics community has taken advantage of newly
available high
resolution seismic and bore hole sonic data. As a result, an
emphasis has been placed
on the characterization of the elastic properties of rocks, and
much less attention has
been paid to other material properties of rocks. Some of these
other properties, such as
radioactivity and density, have simple theoretical
underpinnings. However, properties
related to the nature of the pore network are more difficult to
understand and explain,
typically because these properties depend on the interaction of
many complex physical
processes.
Electrical resistivity measurements were the first type of
wireline log to be used to
assess oil and gas prospects (Ellis, 2007). Archie’s Law
(published in 1942) and
related variants, combined with calibration data, are often
sufficient to improve the
user’s understanding the distribution of hydrocarbons. The
electrical properties of
rocks are dependent on the complex irregular pore space (Man,
Jing, 1999), a fact that
makes relative simplicity of Archie’s law is a bit puzzling. The
difficulty of obtaining
consistent, reliable resistivity measurements in the lab (Sprunt
et al., 1990) continues
to complicate the interpretation of resistivity data.
-
2
Surface based resistivity surveys have been used in near surface
explorations in
applications like geotechnical work, groundwater contamination,
seawater intrusion
and archaeology (Daily, 2004). Deeper imaging surveys for the
energy industry are
rare these days, though EMGS continues to make a business
providing them. Deep
(few km) electrical surveys offer better contrast for fluids but
poorer resolution than
seismic.
ARCHIE’S LAW
When discussing the electrical properties of rocks, it is
necessary to include a
discussion of Archie’s law and its variants and addendums.
Electrical resistivity techniques have been used since the
1940s. The technique
was introduced in a seminal paper by Shell’s Gus Archie, which
introduced and
clarified the relationship between reservoir characteristics,
particularly water
saturation, and electrical resistivity (Archie, 1942). Archie
based his conclusion on the
behavior of clean sandstone samples taken from the Gulf of
Mexico. He found that the
resistivity of each brine-saturated rock sample, Ro, increased
linearly with the brine
resistivity, Rw. He dubbed the term relating these two factors
the formation factor, F,
where F = R0 / Rw . This had been proposed earlier by Sundberg
(1932).
Archie found that the relationship between F and the porosity
(ϕ) of brine
saturated rocks took the form F = 1 / (ϕ m ), where m was
between 1.8 and 2 for his
data. For partially saturated rocks, he proposed the
“resistivity index”, I, where
I=Rt / R0
and Rt is the measured resistivity of the partially brine
saturated rock. Measurements
of resistivity in partially saturated rocks had been performed
before (Martin, 1938;
Jakosky, 1937; Wyckoff, 1936; Leverett, 1939); Archie used the
results of those
experiments to develop the relationship I=1 / Snw where Sw is
the water saturation and
the constant n is approximately 2. By combining the equations
for the formation factor
and resistivity index. He obtained the equation now known as
Archie’s Law:
Rt = Rw / (ϕ m · Snw)
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3
The simplicity of the formula and its broad usefulness have made
it a staple of well
log interpretation for decades. It has been found to
satisfactorily describe experimental
data for a wide range of situations and rock types. Efforts to
understand and explain
the fitting exponents, n and m, have continued.
Exponent m, which relates porosity to the formation factor, was
noted by Archie to
be approximately 1.3 in unconsolidated sands, but near 2 in Gulf
Coast sandstones. In
sandstones, diagenesis is primarily driven by cementation of the
grains; therefore, m
became known as the cementation exponent (Guyod, 1944). Even in
this early paper,
it was noted that the actual relationship between cementation,
resistivity and m was
more complicated and could only be determined in the laboratory.
The next
breakthrough came when Wyllie and Rose (1950) attempted to
relate m to textural
parameters of the rock such as tortuosity and specific surface
area, as had been done
by Kozeny and Carman (1937) for permeability. Wyllie and Rose
defined the
tortuosity (T) of the pore space as
T = (La / L) 2
where La is the flow path length and L is the direct length.
They then computed the
formation factor as a function of tortuosity and porosity,
giving
� =√(�)
�
for a fully saturated rock.
They also attempted to apply the model to partially saturated
rocks, finding that n
should lie between 1.7 and 2.5. However, after additional
studies by Archie, it became
apparent that pressure driven fluid flow and electrically driven
ion flow do not behave
in the same way. Specifically, Archie found that while
permeability depended on pore
structure and porosity, the formation factor was not impacted
strongly by variations in
pore structure, instead varying with porosity alone (Archie,
1952).
The difficulties of actually measuring, or even defining,
tortuosity in the pore
network of a real rock made it difficult to test the predictions
of Wyllie and Rose .
Finally, in 1952, Winsauer and others from Humble Oil devised a
way to measure
tortuosity by timing how long it took for ions to pass through
the pore space. They
applied the new technique to a variety of sandstones from across
the US and found
-
4
that the measured results deviated from the predictions of
Wyllie and Rose. They
proposed an alternative formulation for the formation
factor:
F=0.62/ ϕ 2.15.
In response, Wyllie returned to the lab and performed a number
of experiments on
rocks of varying cementation, grain shape, and porosity (Wyllie,
1953). He proposed
the relationship
F=C/ ϕ k,
generalizing the formulation of the formation factor proposed by
the Humble Oil
group, adding fitting parameter C, which had previously been
assumed by Archie to be
1. By the 1960s it was generally recognized that the Archie
parameters were best
determined empirically, despite the large number of competing
theoretical models
(including Wyllie, 1950; Wyllie, 1953; Winsauer, 1953; Fatt,
1956; Owen, 1952). For
F=a / ϕ m
a and m could be derived from well log data to some degree,
though a large study by
Phillips Petroleum Co. demonstrated that a and m values varied
significantly across
one field, limiting the usefulness of this technique (Porter,
1970). Over the years,
several formulas were developed to predict m within specific
data sets. (Neustaedter
1968, Coates 1973, Gomez-Rivero 1976, Raiga-Clemenceau 1977,
Sethi 1979, Olsen
2008, Azar 2008). Wafta (1987) showed that even within a well, m
will vary
throughout a carbonate reservoir. Despite these efforts,, and
particularly in carbonates,
simple relationships between porosity and formation factor
consistently fail due to the
complexity of the rocks and the high variability in m for the
same porosity. Later
classification of carbonates into eight rock types combined with
an extensive lab
survey provided a more useful predictor (Towle, 1987).
Although early attempts to use tortuosity to relate m to the
formation factor didn’t
work out, modeling efforts continue. In 2008, Abousrafa et al.
published a pore
geometry model and concluded that “the formation resistivity
factor . . . is almost
entirely controlled by pore throat radius, whereas both pore
throat and void radii affect
the cementation factor.” Revil (2014) demonstrated that, for low
salinity brines, the
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5
surface conductivity of the grains biases the values calculated
for m. Work reported by
Dvorkin (2011) showed the potential for using 3D CT imaging to
constrain m.
Theoretical work has also continued. It has been shown, for
instance, that m=1.5
for spherical grain packs (Sen, 1981), and that grain size in
non-shaley rocks has little
impact on the electrical properties of those rocks (Cohen,
1981). In 2012, Kennedy
and Herrick proposed a new conductivity model for rocks that
were well-described
using Archie’s law. To quote Kennedy and Herrick’s paper, “The
petroleum industry’s
standard porosity-resistivity model [i.e., Archie’s law],
although it is fit for its
purpose, remains poorly understood after seven decades of use.”
They derived a
method with similar predictive power, but instead of the rather
arbitrary a, m and n,
their model’s inputs have a priori physical interpretations. As
the paper points out, the
conductivity of a rock is a product of the brine conductivity,
the brine fraction and a
geometric factor. In Archie’s law, geometric information in
Archie’s law is convoluted
within several variables, making interpretation of the fitting
parameters unclear. The
Kennedy and Herrick model overcomes this shortfall by separating
the geometric
information so that each variable described one geometric
attribute. The three
geometric attributes are (1) the percentage of the porosity’s
cross sectional area which
is brine, (2) a term which describes the relative size of pore
throats, (3) a geometrical
factor independent of porosity and (4) a term for so called
excess conductivity, which
would be zero in an Archie-type rock.
Despite this progress, developing an understanding n, the
saturation exponent in
Archie’s Law, has been more challenging. It has been shown that
while saturation is
important, fluid distribution in the pore space also affects
this exponent. The fluid
distribution in turn is a function of the wettability (Keller,
1953), the displacement
history, and the pore size distribution (Diederix, 1982;
Swanson, 1985). The
difficulties of determining n in the lab have forced
experimenters to preserve or re-
creating down-hole conditions in the laboratory.
One problem with Archie’s experimental work is that it relied on
only four
published data sets of partially saturated rocks (Martin, 1938;
Jakosky, 1937;
Wyckoff, 1936; Leverett, 1939). This paucity of data led Archie
to conclude that an n
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6
of 2 describes most rocks. More realistic experimental work
reported by Guyod (1948)
and Dunlap (1949) showed that n ranged at least from 1 to 2.5.
Dunlap also showed
that n was not strongly influenced by porosity or permeability,
but was influenced by
pore fluid displacement history.
Shaley sands posed another challenge for Archie’s law. Patnode
and Wyllie (1950)
noted that Archie’s law was inaccurate for shaley sands and
artificial clay slurries.
They proposed that the observed decrease in resistivity with
increasing clay content
could be explained by introducing a conductive solid component.
The problem with
this idea is that clays are generally not conductive when dry.
Observing this, in 1953
Winsauer and McCardell proposed that the observed extra
conductivity arises not
directly from the conductivity of the clay minerals, but from an
electrical double layer
in the brine near the grain surface. The double layer is caused
by the excess bound
charge on the clay surface, which attracts moveable charge
carriers (ions in this case)
from the brine, concentrating them near the mineral’s surface
and separating them by
charge into two layers. In such a scenario, clay could appear
conductive when wetted
with brine and also appear insulating when dry. They formulated
their predictions of
rock conductivity as:
C0 = 1 / F · ( Cw + Cz )
where Cz is the double layer conductivity and Cw is the brine
conductivity. Theoretical
work provided a solution built from thermodynamic principles and
ion transport
theory which related the total conductivity to the mobility of
the clay’s counterions
(the ions in the brine which were attracted to the surface by
the unbalanced charge at
the surface of the clay minerals), their concentration, and a
ratio of ion mobilities near
to and far from the surface (De Witte, 1955). A method was later
developed to
approximate these parameters from the cation exchange capacity
(CEC), which could
more easily be measured (Hill, 1956).
In 1967, Waxman and Smits proposed a new model for shaley
sand
conduction:
C0 = (1/F) · ( B·Qv + Cw ),
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7
where B is counterion equivalent conductance and Qv is
counterion concentration.
These values could be estimated from core measurements. They
also performed a
number of useful experiments to measure the conductivity of
shaley sands in the
laboratory; their findings conformed well to their model’s
predictions. Clavier, Coates
and Dumanoir (1984) built on the idea presented by Waxman &
Smits to produce a
new dual-water model, which accurately described sample types
that had previously
fit the model predictions poorly.
For a more detailed historical review and handy guide to the
different forms and
uses of Archie’s law, see the series of publications by
Schlumberger titled Archie I
(1988), Archie II (1988) and Archie III (1989). These articles
show the successes of
Archie’s law and describe modifications made over the years to
fit particular needs.
Archie’s law can be adapted to many situations with good
success, in some measure
because with three empirical fitting parameters, there is quite
a bit of wiggle room.
To a petrophysicist, Archie’s Law is tremendously useful for
extracting
information from well logs as long as good calibration data is
available. From a rock
physics perspective, the usefulness of Archie’s Law is limited
by its empirical nature.
The underlying mechanisms which determine electrical properties
are fairly well
understood, but they tend to be difficult to measure or
quantify. In addition, Archie’s
law itself doesn’t tell us what about the rock makes it behave
electrically in a certain
way. Computational pore scale modeling may make this kind of
quantification more
feasible in the future, but it is likely that Archie’s law will
continue to be reliant on
existing data for calibration.
Archie’s Law assumes that electrical measurements are made at
zero frequency,
and that there are no frequency effects, or that such effects
have already been
removed. For measurements recorded at or below a few thousand
Hz, Archie’s Law is
often sufficient. However, at higher frequencies (MHz and
higher), the fitting
parameters change substantially. It is important to be aware of
this when interpreting
LWD or MWD data, as it is typically recorded at hundreds of
thousands or a few
million Hz, and may not correlate well with existing laboratory
measurements without
additional efforts to account for the frequency differences.
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8
Through the remainder of this dissertation, Archie’s Law will
see little use, as the
emphasis will be on interpretation rather than prediction. In my
experiments,
saturation, porosity, and resistivity are all known so there is
nothing left for Archie’s
law to predict. This avoids the uncertainties associated with
the three fitting. The
saturation exponent, for example, can be predicted from the
measurements reported in
Chapter 2.
UNITS
There are a number of terms that come up in discussions of
electrical properties of
rocks. When dealing with electrical circuits and circuit
elements, the typical terms are
resistance, reactance and impedance.
Resistance (R) is a property of a circuit element that resists
current flow through
that element. The resistance is equal to the voltage (V) across
the element divided by
the current passing through the element (I).
R = V/I
Resistance opposes current flow caused by both constant and
alternating voltages,
and the resistance itself is not frequency dependent. Typical
units of resistance are
Ohms. Conductance (G) is the reciprocal of resistance.
G = 1/R = I/V
Reactance (X) is another property of a circuit element that
resists the flow of
current. Unlike resistance, reactance is frequency dependent and
can be either
capacitive or inductive. Circuit elements are typically defined
by a frequency
independent property, either the inductance (L) or the
capacitance (C) of the element,
which is combined with the angular frequency to determine the
reactance. In terms of
the inductance and capacitance, reactance can be expressed
as:
X = XL - XC
XL = 2πf L
XC = 1 / (2πf C)
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9
where f is the frequency in Hertz, L is in Henrys, C is in
Farads and X, XL and XC are
in Ohms. Capacitive reactance reduces current at lower
frequencies, while inductive
reactance reduces flow at higher frequencies.
Impedance (Z), sometimes called complex impedance, is the total
resistance to
current flow through a circuit. If the current is DC, Z = R.
Otherwise impedance has a
component of both resistance and reactance and Z = R + jX. The
magnitude of the
impedance is:
|Z| = sqrt ( R2 + X2 )
θ describes the phase angle between the current and voltage. It
is zero when there
is no inductance or capacitance.
θ = arctan ( X / R )
Resistivity (ρ, units of Ohm ⋅ meter) is similar to resistance
in that they both resist
current flow and both are frequency independent. However, while
resistance is a
function of the element shape and composition, resistivity is a
material property. For
instance, a carbon ribbon would have a constant resistivity
throughout its material, but
a measured resistance would depend on the distance between the
test points, with
longer distances having higher resistance. If the geometry of
the system is known, it is
possible to determine the material resistivity from a resistance
measurement. As an
example, the conversion from resistance to resistivity for a
wire is:
ρ = R⋅A / L
where A is the cross sectional area of the wire and L is the
length.
Conductivity (σ) is the reciprocal of resistivity, and
represents the ease of
movement of free charges.
Reactivity is to reactance as resistivity is to resistance – the
same geometric
scaling is applied to both (for instance A/L for a wire), and
the phase angle is the same.
Impeditivity (similar to impedance) is the square root of the
sum of the squares of
resistivity and reactivity. The terms reactivity and
impeditivity are less frequently used
and do not have as clear a physical interpretation as
resistivity. They are, however,
useful when discussing bulk complex electrical properties.
Further, because the
difference between the reactivity and reactance is a simple
scaling factor when
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10
calculated this way, conclusions drawn from trends in the
reactivity are as valid as
conclusions from trends in the reactance.
Dielectric permittivity (ε, units of Farads per meter) or just
permittivity is
conceptually similar to electrical conductivity. It relates
charge separation, rather than
current, to the applied electric field. The permittivity
represents the polarization
strength of bound charges. This property can also be expressed
as a ratio between the
permittivity of the material and the permittivity of free space
– that ratio is the
dielectric constant (or relative permittivity) of that material.
At 1 GHz, the dielectric
constant for air is approximately 1, for oil is 3-4, for quartz
is 4-4.5, and for water it is
roughly 80 (regardless of salt content).
MODES OF CONDUCTION IN EARTH MATERIALS
Electric flow is fundamentally a net movement of charge carriers
in a particular
direction. The most common charge carriers in relatively shallow
earth materials are
ions, such as sodium and chlorine, dissolved in the pore fluid.
The conductivity of the
pore water is roughly proportional to the concentration of
dissolved ions. Deeper in
the crust, molten rock is also conductive, being composed of
ions which are able to
move past one another when in an electric field. In addition to
ionic flow, conductive
minerals such as graphite and pyrite have free or weakly bound
electrons, which can
also flow. Hydrocarbons, gasses and many common minerals lack
free charge carriers
and are, as a result, excellent insulators.
The flow of ions through pore water in response to an electric
field is generally the
dominant source of conductivity in a rock. In some sense, this
flow is similar to bulk
fluid moving through the pore space, where the fluid
permeability is analogous to the
electrical conductivity. However, ion movement in response to an
electric field differs
from water flow through a pipe in response to a gravitational
field. With fluid flow,
there is a zero flow velocity boundary at the sides of the
channel and an increase in
velocity towards the maximum flow rate, which is found at the
center of the channel.
In contrast, ion flow in response to an electric field will be
in both directions. Both
positive and negatively charged ions will be present, and the
opposite charges travel in
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11
opposite directions (Figure 1.1). The net amount of force on the
fluid is approximately
zero, so the velocity profile across the channel is relatively
constant. The net water
velocity, while low, can be either positive or negative
depending on the ionic species
and concentrations present. This type of flow has been modeled
in several papers
(Wang, 2011; Freund, 2002; Lorenz, 2008) and demonstrated
experimentally (Paul,
1998). The result is that when electrical conductivity is
determined by ion flow
through relatively large pores, the size of the pores has little
impact on the electrical
conductivity of the system. This is different from fluid
permeability, where larger
pores result in dramatically stronger flow.
Figure 1.1: Illustration of the difference between gravitational
and electric
potential for brine. On the left, all molecules (charged or not)
respond to the gravitational potential. On the right, only the ions
experience a net force, and the direction of the force is charge
dependent.
There is another type of brine conductivity that occurs near the
grain/brine
interface. At many crystal surfaces there is a net charge where
the regular charge
balanced pattern of atoms is interrupted at the edge of the
crystal. The permanent
unbalanced charges at the surface influence nearby material. If
pore water is present, a
double layer of positively and negatively charged ions will form
near the crystal
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12
surface. This area of brine, in which the cations and anions
have separated, is more
conductive than the bulk brine. Clay minerals, and particularly
swelling clay minerals,
have a large surface area, a high surface charge and small pore
diameters, which make
their surface conductance a relatively important component of
the total conductance.
In clay-free sandstones and gravels, the surface charges and
areas are smaller relative
to pore diameters, which are much larger. In these rocks, the
effects of charge
separation at the pore walls are often negligible.
A less common, electric path is through conductive minerals in
the rock matrix.
Conduction through conductive granular media is not simply
proportional to the
material’s conductivity. DC current flow requires a continuous,
unbroken electrical
path from one measuring electrode to the other electrode. In
granular media, a discrete
number of paths may form, depending on the grain type
distribution. In a randomly
sorted isotropic granular media without a conductive pore fluid,
such as dry sand, with
just a few conductive grains scattered through it, there may not
be any connected
pathways between those grains. In this case, essentially no
current can flow through
the system. If more and more conductive grains are added, at
some point the first
conductive pathway through the system will form. The percolation
threshold is the
minimum concentration of conductive grains required before the
first conductive paths
form. Near the percolation threshold, not every random
distribution of particles will
include a conducting path, so in addition to the minimum
conductive fraction, we also
need to specify a probability of getting a conductive path
through the random
assortment. This probability will depend on the geometry of the
test region as well as
the ratio of the sample size to the grain size. For instance, a
shallow but wide box
filled with a mixture of conductive and insulating grains will
more frequently contain
a vertical conductive path than it will a horizontal one.
The dry conduction threshold only sets a lower bound for the
amount of
conductive material required before a DC electrical current can
be observed. We need
substantially more information to predict how much flow there
will actually be.
If the distribution of conductive grains is not homogenous,
which is often the case
with pyrite (Cohen, 1984), then the value of the percolation
threshold is harder to
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13
ascertain. If, for instance, a fracture contains a vein of
native copper, the amount of
conductive material required to see an impact could be almost
zero. In real rocks, it
will be difficult to determine percolation thresholds without a
detailed understanding
of the rock and the sources of the conducting materials.
Detrital pyrite, which is
uncommon because pyrite weathers quickly at the surface, would
probably have a
percolation threshold similar to those thresholds observed in
random sand packs. It
may still vary from those values if the pyrite was sorted from
the other sand
components due to its higher density (see Figure 1.2).
Framboidal and diagenetic pyrite forms primarily within what was
the pore space
of the original mineral matrix. As a result, the dry conduction
threshold for this type of
deposition may be lower than for homogenous, uniform sized sand.
Again, when
conductive minerals such as pyrite and copper are deposited in
fractures, the fraction
of mineral that needs to be conductive to see an effect can be
very small.
In Chapter 5 I discuss percolation thresholds in greater
depth.
Figure 1.2: Dark, iron bearing minerals (likely magnetite and
hematite) that have
separated out on a beach. Photographed area is approximately 1m
wide.
Tortuosity, path shape, contact size, grain size, the frequency
of the electrical
signal (skin effect), pressure, temperature, pore fluid
conductivity, and mineral
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14
resistivity all affect the resistance observed through the
framework of a conductive
grain pack.
For the time being, we’ll consider electric flow through the
conductive minerals,
ignoring possible flow within pore fluids. Grain to grain
contact areas can be very
small, resulting in high resistance at that point. As a result,
most of the voltage drop as
the current flows through the conductive minerals occurs at the
contact points. At
higher frequencies the voltage drops will be more homogenous
because as the skin
depth on the grain decreases, the contact areas appear larger
relative to the rest of the
electrical path. More electrical coupling between disconnected
grains may occur at
higher frequencies. Electrons can tunnel through small gaps
between grains, which in
some cases may significantly increase the apparent contact
size.
As the net confining stress on a conductive grain pack is
increased, intra-grain
deformation results in larger grain-to-grain contact patches,
which increases the
conductivity of the pack. In addition, the contact number – the
average number of
other grains touched by a particular grain – increases, which
also leads to an increase
in conductivity. In grain packs with both conducting and
non-conducting grains, this
increase in pressure and the resultant increase in the contact
number may be enough to
change a non-conducting assemblage to a conducting one.
LOW RESISTIVITY PAY ZONES
In oil and gas exploration, hydrocarbon-bearing layers can
usually be identified by
their relatively high resistivities - hydrocarbons are much less
conductive than the
brines that typically saturate deep subsurface reservoirs.
Despite this general trend,
hydrocarbons do occur in low resistivity zones. These zones
can’t be detected using
resistivity techniques. These “low resistivity pay zones” tend
to be found in laminated
shaley sands, freshwater formations, regions rich in conductive
minerals, regions of
fine-grained sands, and regions of internal microporosity and
superficial microporosity
(Worthington, 2000; Boyd, 1995). The high surface areas of
shaley sands, regions of
high microporosity, and fine grained sand units add extra
conductivity to hydrocarbon
bearing layers making them difficult to detect using resistivity
techniques. Conductive
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15
minerals may provide an alternative path for electric flow
through reservoir, again
reducing the resistivity. Finally, fresh water formations
surrounding hydrocarbon-rich
regions have comparable resistivities to oil-bearing zones,
making them difficult to
differentiate using resistivity. These factors, when not
identified, can result in
calculated water saturations that are too high, and may make a
good prospect appear
economically infeasible.
In 2004, Kennedy published a framework for interpreting
pyrite-affected logs
using the photoelectric factor. Included in that publication
were several examples of
pyrite-affected well logs. Worthington (2000) noted three
reservoirs with recorded low
resistivity pay zones caused by conductive minerals: the
Teradomari Formation in
northern Japan, the Simpson Series in Oklahoma, and the Trimble
field in Mississippi.
Resistivity in the Munsterland black shale was observed to be
much lower than in
the similar Konzen shale by Duba et al. (1988). The black shale
included up to 11%
globular or framboidal pyrite and 5-8% organic carbon (not
graphite). The authors
attributed the difference primarily to a carbon film that
bridged grain contacts, and
noted that the pyrite present would be electrically connected by
the carbon.
RESEARCH FOCUS AND SUMMARY
The fundamental processes behind the electrical response of
rocks are generally
understood. While Archie’s law remains a useful tool for
interpretation, there are other
models that are more strongly grounded in the physical
attributes of sandstones,
carbonates, and rocks with a clay component.
Within the oil and gas industry, the complications of
identifying low-resistivity
pay zones have only been partially addressed. Our understanding
of and ability to
predict the impact of conductive minerals in particular could
benefit from additional
efforts to understand the phenomena behind these impacts. An
important long-term
goal is the development of a robust interpretation system that
will allow us to
understand where, and to what extent, conductive minerals will
have a meaningful
effect on resistivity.
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16
To improve our understanding of resistivity and particularly
complex resistivity in
clean systems with and without conductive minerals, I have
undertaken a number of
experiments, which are discussed in the next few chapters.
Chapter 2 discusses
measurements made on clean sands and mixtures. Chapter 3 extends
those
experiments into mixtures that include conductive grains.
Chapter 4 narrows the focus
to the interactions that take place at the interfaces between
conductive grains and brine
in the pore spaces. Finally, Chapter 5 draws on experimental
work to identify the
underlying physical traits that control the resistivity and
complex resistivity of earth
materials containing conductive grains.
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17
Chapter 2
Fundamental resistivity measurements
INTRODUCTION
Low frequency electrical measurements of granular media are
utilized in a variety
of specialties within geophysics and soil science. Electrical
resistivity (or
conductivity) measurements are especially common and have been
in use for decades.
With alternating current equipment the signal is best expressed
as two components, the
resistance and the reactance. The electrical resistivity is the
real component of the total
system impedance – it is the portion of the signal which
represents energy dissipation.
The imaginary component of the measurement is a reactance, and
indicates charge
storage within the system. The reactance is generally ignored in
logging because the
magnitude is relatively low, and it can be difficult to isolate
experimentally; however,
it does contain information about the system and may prove
useful in some situations.
The goal of this portion of the project was to build a dataset
that provides a
consistent set of complex resistivity measurements for a variety
of common
situations found in granular media. Specifically, we test the
effects of saturation,
brine salinity, various geometries of dissimilar sands, mixing
different grain sizes.
Mixing of coarse and fine sand and examining the resulting
electrical properties has
been examined previously (Lemaitre et. al., 1988; Wyllie and
Gregory, 1953;
Nettelblad and Niklasson, 1996; Biella et. al., 1983). Numerous
studies have
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18
investigated the resistivity of various sand/clay mixtures
(Wildenschild et al., 2000 is a
good example).
These tests were performed using various sands and
sodium-chloride brines at
atmospheric pressure and room temperature. The data presented
here provides a
consistent set of experimental data that share many of the same
methods and
equipment. This is important when examining the reactive portion
of the electrical
signal, because often the test equipment and especially the
electrode contacts with the
sample cause a substantial portion of the response – and that
response will vary from
one machine to another. By using one machine for all of the
tests, we reduce the
variation in this systemic error. The tests also serve to
validate the testing
methodology and to identify the equipment sensitivity to the
tested conditions. Finally,
the data provides a reliable foundation for further experiments
looking at other, more
subtle effects by defining sensitivities, finding experimental
strengths and weaknesses
and potential avenues for exploration.
METHODS
The test vessel was a vertically oriented transparent
non-conductive circular
cylinder, approximately 12 cm deep with a 16 cm inside diameter
which
accommodated a sample volume of approximately 2.3 liters (Figure
2.1) The
geometric conversion factor from ohm to ohm-meter for this
vessel was 0.167 meters.
The same conversion factor was used for both resistivity and
reactivity. At the bottom
of the cylinder, a 13 cm diameter stainless steel electrode was
permanently seated.
Once the cylinder was filled with sediment, a similar stainless
steel electrode was
seated on the top. A 2.8 kg weight was placed on top of the
removable electrode when
possible; the clay rich samples would flow around the edges of
the electrode and allow
it to sink so the extra weight was not used for those materials.
The purpose of the
weight was to maintain a consistent contact area between the
material and the
electrode, which was a particular issue for the coarser and less
ductile sand mixtures.
Adding the weight did not visibly compress the sediment, and
considering the size of
the sample and the degree of uncertainty in sample length
(approximately ± 1mm) it is
-
19
unlikely that the addition of the weight changed either the
porosity or sample length
significantly.
Figure 2.1: Photo of the test vessel, including upper stainless
steel electrode. The
vessel was built using parts from a MC Miller soil cylinder.
In testing, artificial brines were prepared by mixing NaCl with
de-ionized water.
All sands and powders were air dry and free flowing (i.e., not
cohesive or clumped)
prior to testing.
Electrical measurements were made using a Fluke PM6304, which is
an RCL
circuit analyzer. The test voltage was set at 1 volt, but the
operating voltage was often
lower due to the low resistances and power limits of the
machine. The accuracy
specified in the user manual is +/-0.1% through most of the
range of interest.
The general procedure for testing is presented here, and
modifications for specific
tests are noted below. The needed material weights were
calculated based on the
desired volumes and material densities. Materials were then
measured out by weight
and combined in a tub outside the test vessel. Brine was
measured out by weight and
added until the material was saturated. Full saturation was
determined in relatively
coarse materials by a loss of cohesion and accumulation of brine
on the surface when
the sediment was vibrated. In finer materials full saturation
was marked by a glossy
surface and slight flow when the container was tipped (Soil
Survey Field and
Laboratory Methods Manual). Once saturated, material was scooped
from the tub into
the test vessel and vibrated until any visible air gaps were
removed. Material was
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20
added until the vessel was filled to a marked level, which was
the same for all tests.
Saturation conditions were checked again using the same
criteria. If the saturation was
acceptable, the upper electrode was pressed into place by hand
and the vessel was
vibrated briefly to seat the electrode against the test
material. Effort was made to
ensure that the upper electrode was parallel to the lower
electrode and that the upper
electrode closely matched the depth guide to ensure a consistent
sample length. If the
material would support the 2.8 kg weight, it was added on top of
the upper electrode.
After the vessel was filled, the RCL meter was zeroed across
shorted electrical
leads at 2 kHz as specified in the user manual. The electrical
lines between the RCL
meter and the test vessel were connected, and measurements
began. A set of
measurements consisted of recording the resistor-capacitor
series circuit equivalent at
50, 120, 2000, 10,000, and 100,000 Hz and the temperature of the
room (between 19.5̊
and 21.5̊ C). Sets began and ended at 2 kHz. Typically, a set of
measurements was
recorded within 15 minutes of the vessel being filled and a
second set was made at
least 1 hour later. If the measured resistances from the two
sets were within 1%, the
test would be concluded, otherwise the vessel was left to sit
for at least one additional
hour and then another set of measurements was made on the same
material until the
results stabilized. Generally this was not required. Possible
causes include changes in
room temperature, brines not initially at room temperature, air
entrained during
sediment preparation floating up and exiting the system or
compaction caused by
vibrations in the room.
When the testing of each brine/material combination was
concluded, the test vessel
was washed and dried. The material was either discarded, or
transferred to a tub and
adapted for re-use in a later test, generally by adding another
material (salt, powder, or
contrasting sand). If the material was reused, it was stirred
until it was visibly
homogenous. Then the material, or a portion of the material, was
transferred back to
the test vessel and the general procedure was repeated.
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21
NaCl concentration
The complex resistivity of various concentrations of NaCl brine
was recorded in
three situations. First, with only brine, second with brine
saturating acrylic grit (Aero-
Clean Thermoplastic Acrylic Granulated Blast Cleaning Media
12/16, roughly cubical,
12-16 mesh, with a median diameter of 1.4mm) and third with
various brine
concentrations saturating an unseived, washed beach sand from
Moss Landing beach
in Monterey County, California. The Moss Landing sand had a
median diameter of
approximately 250 microns, with a larger spread in grain size
than other sands used.
The beach sand was washed to ensure any residue of sea salt had
been removed and
could not increase the salinity of the test fluid.
When performing the pure brine experiment, the test vessel was
initially filled with
de-ionized water and at each successive concentration, salt was
added and dissolved
within the test vessel. Concentrations of ~0, 1, 2, 4, 8, 16,
32, 64, 128, 196 and 256 g
NaCl per liter DI-water were tested.
In the second set of tests, the acrylic grit was initially
saturated with brines from 0
to 275 g NaCl per liter DI-water. Salinities began at 25 g NaCl
per liter DI-water and
the brine concentration was increased to 50, 75, 100, 150, 200
and 275 g NaCl per liter
DI-water. Then fresh media was saturated with DI-water and the
brine concentration
was increased to 1, 2, 4, 8, 15, 25, and 50 g NaCl per liter
DI-water. This allowed full
coverage of the test range, and two repeated tests (25 g and 50
g) at the beginning and
end of testing to check for consistency.
Finally, the beach sand was initially saturated with DI-water
and salt was added to
increase the concentration to 1, 2, 4, 8, 16, 32, 64, 128, 196
and 256 g per liter DI-
water.
Saturation
The effect of saturation on complex resistivity was measured by
adding a 4g NaCl
per liter DI-water solution to Moss Landing beach sand (the same
type of sand used in
-
22
the NaCl concentration experiment). Saturations varied from 0 to
1 in increments of
0.1 for a total of 11 saturations.
Initially the plan was to simply pour additional brine into the
top of the sand filled
test vessel and wait for it to disperse throughout the sand.
This method would have
avoided grain packing changes between tests. Unfortunately, the
brine did not diffuse
through the sand, and resulted in highly saturated patches in
otherwise dry sand.
Adding additional fluid simply expanded the saturated patches
until the brine reached
the lower electrode (resistivities were very high prior to
breakthrough).
To get more homogenous saturation, brine and sand were combined
in a tub
outside the test vessel where they could be stirred together
until they were visibly
homogenous and then transferred to the test vessel. The same
material was reused
between tests by transferring back and forth from the test
vessel to the large tub where
additional brine could be stirred in. The partially saturated
sands would not flow when
vibrated and had to be pressed into the vessel. The weight of
the packed vessel was
recorded at each step so the bulk density and porosity of the
sand packing could be
calculated.
Mixing coarse and fine sand
In this test, two types of sand, one with relatively fine grains
and one with
relatively coarse grains, were combined in varying quantities
and saturated with brine.
The test was done once with 50 g NaCl per liter DI-water and
then again with 4 g
NaCl per liter DI-water. Approximate grain size distributions
were estimated from
supplier data and are shown in Figure 2.2.
-
23
Figure 2.2: Cumulative grain size for the coarse and fine sands
which were used
in the sand mixing test.
The two sands had a mean grain size ratio of approximately 5.
The finer sand was F-
80 Ottawa sand from US Silica. The coarser sand was “16 Mesh
Cleaning Sand” from
Legend, Inc. Mixtures with coarse volume fraction of 0, 0.2,
0.4, 0.6, 0.8 and 1 were
measured.
Sands in Parallel and Series
Four tests were done to test the electrical properties of
differing sands in parallel
and in series arrangements. See Figure 2.3 for illustrations of
the geometries tested.
Sands were saturated with 4 g NaCl per liter DI-water. In the
first test, the lower half
of the vessel was filled with the same type of coarse sand used
in the previous
experiment and the upper half was then filled with the same type
of fine sand used in
the previous experiment. In the second test, the same types of
sand were placed side
by side in the cylindrical test vessel. This was accomplished
using a cardboard divider
between the two halves which was removed once the saturated
sands were in place.
The third and fourth tests were identical to tests one and two
except the fine sand was
replaced with a mixture of equal weight fine and coarse sand
(1:1). In the fourth test,
0
20
40
60
80
100
0 0.5 1 1.5 2
Cum
ulat
ive
%
Mean diameter (mm)
-
24
~15% of the cardboard divider was lost in the sample, and
remained at the boundary
between the coarse sand and the mixed sand throughout the
measuring process.
Figure 2.3: Diagrams of the four test geometries. Current ran
vertically through
the cylinders. In the upper left: coarse and fine sand in
series. In the upper right: coarse and mixed sand (50% coarse sand
and 50% fine sand) in series. In the lower left: coarse and fine
sand in parallel. In the lower right: coarse and mixed sand in
parallel.
Mixing sand and powder
CISCO No. 90 white silica sand (fine, median diameter
approximately 160
microns) was mixed with a calcined kaolinite (a powder visually
similar to cake flour)
from Natural Pigments LLC. They were mixed with sand fractions
of 0, 0.19, 0.35,
0.47, 0.58, 0.67, 0.75, 0.77, 0.82, 0.90, 0.95 and 1 by mass.
These sand fractions were
used to identify the mixtures, so as an example, mixture 0.82
was 82% sand and 18%
powder by weight. For mixtures 0.75, 0.82, 0.90 and 0.95, the
sand and powder were
mixed together while dry and then brine was added until
saturated. Mixture 0.77 was
only mixed after the sand, powder and brine had been added, and
the powder remained
somewhat lumpy. Mixture 0.77 was also oversaturated, in the
sense that the volume of
brine added was greater than dry pore volume. If the sample had
been sand, the excess
water would have collected on the top and could have been
removed, but due to the
powder content, the mixture remained homogenous and the test
vessel was filled with
-
25
the material in that oversaturated state. Mixtures 0.19, 0.35,
0.47 and 0.58 were
created by adding sand to Mixture 0. An electric mixer was used
to ensure thorough
mixing before the material was added to the test vessel. The
weight of the filled vessel
was recorded for each mixture to allow for a porosity
calculation using a density of
2.65 g/cc for the sand (dominantly quartz) and 2.4 g/cc for the
powder (specified by
manufacturer).
RESULTS
Room temperatures were consistently between 19.5° and 21.5° C
and the results
have not been temperature corrected. The final (stable) set of
measurements for each
material combination is reported here. Measured resistances were
more consistent (as
a percent difference) than were the capacitances for tests
separated by 15-20 minutes.
Over several days, the RCL meter showed greater drift in the
reported capacitance as
well, again as a percent difference. From hour 1 to hour 25
after the beginning of a
test, the measured resistance may vary by 1% while the
capacitance may vary by 8%.
There was not a clear trend of variation between one test and
the next – the resistivity
may rise over time in one test and decline in another, but all
longer duration tests
showed that the sediment’s electrical properties were stable
after 24 hours. Whether
this variation is the sample/apparatus interacting over time
(corroding electrode;
sample settling), due to an environmental change (temperature;
pressure; humidity) or
a limitation of the meter is unclear.
Reactivity was always most negative at 50 Hz and progressively
closer to zero for
the higher frequencies up to 10 kHz. The results at 100 kHz were
generally similar to
those at 10 kHz.
NaCl concentration
For salinities of 1 g/l and up, brine resistivities ranged from
5.5 to 0.05 Ω⋅m as
salinity rose; reactivities were lower magnitude and ranged from
-0.4 to nearly 0 Ω⋅m.
Results are shown in Figure 2.4 and Figure 2.5, and recorded in
Table 5 of the
-
26
appendix. I show later (Figure 2.22) that this agrees well with
Schlumberger empirical
curves.
Figure 2.4: Resistivity of brine as a function of salt
concentration. For each
salinity, resistivity was measured at 5 frequencies from 50 Hz
to 100 kHz, all of which are plotted.
Figure 2.5: Reactivity of brine as a function of salt
concentration. For each
salinity, reactivity was measured at 5 frequencies from 50 Hz to
100 kHz.
0
1
2
3
4
5
6
0.9 9 90
Res
istiv
ity (
ohm
m)
NaCl concentration (g/l brine)
100 khz
10 khz
2 khz
120 hz
50 hz
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.9 9 90
Rea
ctiv
ity (
ohm
m)
NaCl concentration (g/l brine)
100 khz
10 khz
2 khz
120 hz
50 hz
-
27
In the saturated coarse acrylic, for salinities of 1 g/l and up,
resistivities ranged
from 19 to 0.15 Ω⋅m as salinity increased and reactivities were
lower magnitude and
ranged from -0.5 to nearly 0 Ω⋅m. Results are shown in Figure
2.6 and Figure 2.7, and
recorded in Table 6 of the appendix.
Figure 2.6: Resistivity of brine saturated coarse acrylic as a
function of salt
concentration. For each salinity, resistivity was measured at 5
frequencies from 50 Hz to 100 kHz, all of which are plotted.
0
5
10
15
20
25
0.9 9 90
Res
istiv
ity (
ohm
m)
NaCl concentration (g/l brine)
100 khz
10 khz
2 khz
120 hz
50 hz
-
28
Figure 2.7: Reactivity of brine saturated coarse acrylic as a
function of salt
concentration. For each salinity, reactivity was measured at 5
frequencies from 50 Hz to 100 kHz.
In the saturated beach sand, for salinities of 1g/l and up,
resistivities ranged from
26 to 0.2 Ω⋅m as salinity increased and reactivities were lower
magnitude and ranged
from -0.7 to nearly 0 Ω⋅m. Results are shown in Figure 2.8 and
Figure 2.9, and
recorded in Table 7 of the appendix.
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.9 9 90
Rea
ctiv
ity (
ohm
m)
NaCl concentration (g/l brine)
100 khz
10 khz
2 khz
120 hz
50 hz
-
29
Figure 2.8: Resistivity of brine saturated beach sand as a
function of salt
concentration. For each salinity, resistivity was measured at 5
frequencies from 50 Hz to 100 kHz, all of which are plotted.
Figure 2.9: Reactivity of brine saturated beach sand as a
function of salt
concentration. For each salinity, reactivity was measured at 5
frequencies from 50 Hz to 100 kHz.
0
5
10
15
20
25
30
0.9 9 90
Res
istiv
ity (
ohm
m)
NaCl concentration (g/l brine)
100 khz
10 khz
2 khz
120 hz
50 hz
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.9 9 90
Rea
ctiv
ity (
ohm
m)
NaCl concentration (g/l brine)
100 khz
10 khz
2 khz
120 hz
50 hz
-
30
Saturation
The effect of saturation on complex resistivity was measured by
adding a 4g NaCl
per liter DI-water solution with Moss Landing beach sand (the
same type of sand used
in the NaCl salinity test). The resistivities ranged from near
140 to 0 Ω⋅m, with higher
resistivities at lower saturations. Reactivities ranged from -10
to near 0 Ω⋅m with
higher reactivities at lower saturations and frequencies.
Results are shown in Figure
2.10 and Figure 2.11, and recorded in Table 8 of the
appendix.
Figure 2.10: Resistivity of beach sand as a function of
saturation. For each
saturation, resistivity was measured at 5 frequencies from 50 Hz
to 100 kHz, all of which are plotted.
0
20
40
60
80
100
120
140
160
0 20 40 60 80 100
Res
istiv
ity (
ohm
m)
Saturation
100 khz
10 khz
2 khz
120 hz
50 hz
-
31
Figure 2.11: Reactivity of beach sand as a function of
saturation. For each
saturation, reactivity was measured at 5 frequencies from 50 Hz
to 100 kHz
Mixing coarse and fine sand
When coarse and fine sands were mixed in various proportions and
saturated with
4g/l NaCl brine, resistivity ranged from approximately 6.3 Ω⋅m
in 100% fine sand, to
a peak of 9 Ω⋅m with 40% coarse sand and down to 5.5 Ω⋅m with
100% coarse sand.
Reactivity varied from 0 to -0.7 Ω⋅m, with most of the variation
due to the frequency
of the measurement. See Figure 2.12 and Figure 2.13 for results.
Results are also
recorded in Table 3 in the appendix.
-12
-10
-8
-6
-4
-2
0
0 20 40 60 80 100
Rea
ctiv
ity (
ohm
m)
Saturation
100 khz
10 khz
2 khz
120 hz
50 hz
-
32
Figure 2.12: Resistivity of sand saturated with 4 g NaCl/L
DI-water as a function
of coarse fraction. For each mixture, resistivity was measured
at 5 frequencies from 50 Hz to 1