PROCEEDINGS, Thirty-Eighth Workshop on Geothermal Reservoir Engineering
Stanford University, Stanford, California, February 11-13, 2013
SGP-TR-198
CHARACTERIZATION OF FRACTURES VIA ELECTRICAL IMPEDANCE
Lawrence Valverde, Roland Horne and Kewen Li
Stanford University
367 Panama Street
Stanford, CA, 94305, U.S.A.
e-mail: [email protected]
ABSTRACT
This research has been investigating the relationship
between rock fractures and electrical impedance and
resistance. The ultimate goal is to create a down-hole
tool for detecting and characterizing the fractures
created or existing in Engineered Geothermal
Systems (EGS) and other geothermal reservoirs.
Previous studies have identified a frequency
dependence in the electrical properties of fractured
rocks partially saturated with water, noting marked
differences between the frequency response of rock
cores with varying fracture density. The research
presented in this paper offers a geophysical
explanation for this phenomenon rooted in the
capacitor-like nature of fractures in partially
desaturated rock. Laboratory experiments and
theoretical work were designed to explore the
complex impedance and resistance response to
alternating current over a frequency range covering
three orders of magnitude. A core-scale experiment
measured the impedance and resistance of rock
samples initially saturated with saline solution and
subjected to evaporative desaturation. Theoretical
analysis attempted to match the rock response to a
model electrical circuit. The goal of these
experiments was to verify relationships found
previously between frequency and resistance as well
as to establish a correspondence between fractures in
rock and the capacitors in a representative circuit.
The research indicates this relationship and suggests
possible methods of extracting fracture information
from the electrical responses at different frequencies.
This paper concludes with planned improvements to
the experiments and future avenues of
experimentation.
INTRODUCTION
Characterization of fractures in rock remains an
important challenge in the development of
geothermal resources, particularly with regard to
EGS. For EGS, success hinges on the creation of a
large density of fractures within the EGS reservoir.
Thus, the detection and characterization of
preexisting and created fractures is central to the
evaluation and continued development of EGS. A
recent study by Sandler et al. (2009) highlighted the
marked difference among the frequency responses of
rock cores lacking or containing fractures. The study
found that in fractured rocks, below a certain level of
water saturation, resistivity index was inversely
proportional to frequency over the frequency range of
100 to 10,000 Hz. Figure 1 illustrates these findings
by showing resistivity index as a function of water
saturation (for a 1% salt solution) for sandstone
lacking (left) and containing fractures (right). While
the fractureless sample demonstrates a power-law
relation over the whole range of saturations tested,
the resistivity response of the fractured rock splits
depending on frequency starting around Sw = 0.1.
The electrical properties of various types of rock
have been a subject of investigation for several
decades (Drury, 1978; Knight & Nur, 1987a; Knight
& Nur, 1987b; Börner et al., 1997; Roberts & Lin,
1997; Suman & Knight, 1997; Rust & Knight, 1999;
Bona et al., 2002; Rusiniak, 2002). Knight & Nur
published two papers in 1987 discussing the
dielectric constant of sandstones and geometrical
effects on the dielectric response when those
sandstones are partially saturated with water. They
found the real component of dielectric constant in all
samples subjected to currents ranging from 5 Hz to 4
MHz showed a power-law dependence on frequency,
and this dependence was proportional to the surface
area to volume ratio of the pore space in each rock.
Börner et al. (1997) measured over a frequency range
of 10-3
to 109 Hz, identifying a low- and a high-
frequency response explainable by separate
phenomena. The low frequency response was a
function of water content, water conductivity, and
surface area to porosity ratio; meanwhile the high
frequency response depended upon water content and
internal surface area. Roberts & Lin (1997) measured
Figure 1: The frequency dependence of resistivity in fractured sandstone. Core 1 (left) is Berea sandstone without
fractures. Core 2 (right) is fractured sandstone. Adapted from Sandler et al. (2009).
dielectric constant and electrical resistivity of
Topopah Spring tuff as a function of saturation,
identifying three behavior regimes: rock/water
monolayer conduction, pore water, and electrode
response.
Bona et al. (2002) specifically investigated the
relationship between rock wettability and the high
frequency dielectric response, confirming a power
law relation between permittivity and frequency
below 103 Hz and attributing this response to the
fractal nature of the rock geometry. Rusiniak (2002)
estimated the dielectric permittivity of water by
measuring the dielectric permittivity in the frequency
range of 5 to 13 MHz across artificial pore structures
with different water contents.
This study has attempted to continue and improve on
the experiments of Sandler et al. (2009) and to use
the existing literature on electrical properties of rocks
to develop a physical explanation for the observed
phenomena.
EXPERIMENTAL WORK
The experiments for this research began with an
attempt at verifying the results of Sandler et al.
(2009). Figure 2 shows the experimental apparatus
and Table 1 shows relevant information for the cores
used in this research. Images of rock cores #1, #3,
and #4 are at the end of this paper, in Figures 10, 11,
and 12. Electrodes were constructed by soldering
wires to two patches of copper mesh. The wires were
connected on one end to an RCL meter (Quad Tech
1715) and on the other end clamped to either side of a
sandstone rock core via a hand clamp. This was done
alternatively with and without a piece of filter paper
clamped between the rock end and the copper
electrode and soaked in the same 1% brine solution
(NaCl) in which the rocks were saturated. There was
no noticeable difference between the experiments
with and without filter paper. This entire apparatus
was placed upon a rubber sheet on top of a balance
Figure 2: Configuration without filter paper. Wires
lead out of frame to LCR meter.
Table 1: Summary of Rock Properties
Core # Description Length (mm) Cross-Sectional Diameter (mm2) Porosity (%)
1 Berea Sandstone 98.43 387.9 18.19
2 Berea Sandstone 98.43 387.9 18.19
3 Unfractured Reservoir Rock 49.21 570.0 9.33
4 Fractured Reservoir Rock 39.69 618.54 27.7
Figure 3: Frequency dependence in unfractured
Berea sandstone.
with reading accuracy between 0.02 and connected to
a computer via RS-232 ports. Modified versions of
the NI LabView software for both measurement
devices were used to gather mass, resistance, and
impedance
data from the core at 10-minute intervals while water
was allowed to evaporate from the core at ambient
temperature of about 20°C. Measurements were taken
at 100, 120, 1000, and 10000 Hz. Cores were
prepared by first evacuating any gas from the pore
space by placing the cores in a desiccation chamber
and lowering the pressure to nearly 100 mTorr. Brine
solution prepared in a separate vacuum flask was
then released into the desiccation chamber (still under
pressure) and allowed to invade the pore space
overnight. Between experiments cores were dried in a
vacuum oven at ~22°C.
Nonfractured Cores
The cores used for initial experiments with this setup
were taken from a cylindrical Berea sandstone core
that was divided into four by perpendicular
lengthwise cuts. The initial intention was to measure
two cores independently, then measure their
combined properties while clamped together,
simulating a lengthwise fracture. However, results
from both cores diverged from the expected behavior
and only one test was conducted with this two-core
configuration. The data from this one test one was
not valuable due to mass-measurement issues
discussed later in this paper. Repetition of the same
experiment with the same cores gave consistent
Figure 4: Frequency dependence in unfractured
reservoir rock.
results. Figure 3 demonstrates the noticeable
frequency dependence, observed for both Core #1
and Core #2, for both resistance and impedance
below Sw ≈ 0.1. This divergence from the expected
behavior prompted reassessment of the underlying
physics (see THEORETICAL WORK below) as well
as experimentation with alternate rocks. Core #3 was
unfractured reservoir rock. As demonstrated in
Figure 4, the inverse relationship between
resistance/impedance and frequency was not evident
until much lower saturations (Sw ≈ 0.04).
Fractured Core
Core #4 was a naturally fractured low permeability
sandstone. This core exhibited behavior qualitatively
similar to Cores #1 and #2; however, the onset of
frequency dependence occurred much earlier (Sw ≈
0.5), as seen in Figure 5.
THEORETICAL WORK
The frequency dependence observed by Sandler et al.
(2009) for fractured media and observed in the
experiments for this research in all rocks was
identified as capacitor-like behavior. The electrical
response of any material can be divided into loss
components—in phase with the applied voltage—and
storage components—out of phase with the applied
voltage (Knight, 1984). In their most basic forms, the
loss component is manifested as a resistance and the
storage component is manifested as a reactance, and
both effects combine to give the impedance of the
Figure 5: Frequency dependence in fractured low
permeability sandstone.
material. Alternatively, the in-phase component may
be described from the perspective of conductance and
the out-of-phase component as susceptance. The full
response would then describe the admittance of the
material. Both perspectives can be seen in the
literature relating to electrical properties in rocks, and
since the primary paper which inspired this research
approached the problem from the perspective of
resistance, that convention has been followed. For a
resistor and capacitor in series, impedance, Z, is
given as follows:
(1)
where Rs is the series resistance and Xs is the series
reactance. For a capacitor, reactance is given by:
(2)
where ω is frequency and Cs is the capacitance. As
evident from Equation (2), in a series RC circuit
impedance is inversely related to frequency. This
inverse relationship is generally associated with
capacitors while positive relationship is associated
with inductors. Thus, initial theory involved a simple
capacitor and resistor circuit for which the pore space
was represented by a resistor in which decreasing
saturation caused increasing resistance, and the
fractures were represented by parallel plate capacitors
with a capacitance equal to the quantity and/or
aperture thickness of fractures perpendicular to the
direction of applied voltage.
However, the frequency dependence observed in non-
fractured rocks and the fact that Sandler et al. (2009)
observed frequency dependence in the resistance,
rather than impedance, prompted deeper investigation
into the electrical properties of porous media and a
more sophisticated circuit model. A slightly more
complex—although still greatly simplified—circuit
model was chosen, constituting a resistor connected
in series to a parallel resistor and capacitor unit
(Figure 6).
Figure 6: Circuit Model
Figure 7: Frequency dependence in circuit model.
In this circuit R1 represents the resistance of the
electrodes and associated apparatus material while
the R2 and C parallel unit represent the pore space.
Based on the theory developed by Knight (1984),
Knight & Nur (1987a,1987b), Börner et al. (1997),
and Roberts & Lin (1997), the pore space resistor
represents the resistance of contiguous bulk water,
and the capacitor represents the charge accumulated
on microlayers of water as evaporation of the bulk
water leads to a situation where regions which were
previously joined by contiguous water contact
become electrically separated. This electrical
separation is due to regions of connected water
snapping apart, resulting in myriad micro-capacitors
which are collectively modeled with the capacitance
in C. The equation for this circuit shows the
frequency dependence of both reactance and
resistance:
(3)
when measured with an RCL meter in series mode,
the circuit yields:
(4)
(5)
Assuming a linear correlation between R2 and
saturation, the circuit in Figure 6 was able to closely
model experimental results. The resistance and
impedance response in Figure 7 shows remarkable
correlation to the impedance data in Figures 3 and 5.
The resistance data is also very well matched up until
just after the resistance drop. Furthermore, if the
sudden flattening of the resistance curves after the
drop in resistance in the data is modeled as a sudden
and dramatic rise in R2 due to rapid loss of electrical
connection within the pore space when the last layers
of water snap off from each other, the resistance data
can be matched fairly well over the full range of
saturation, as illustrated in Figure 8.
Figure 8: Circuit model incorporating discontinuity
in R1 at R2 = 10-1
.
Finally, there is a region of positive correlation
between frequency and impedance in the high
saturation portions of the data which, for the purposes
of this research, has been assumed inconsequential,
and was, therefore, not modeled. However, this
behavior, which is indicative of inductance, might be
due to the tortuous interconnected paths of water in
the rock pore structure which could manifest as many
micro-inductors due to the curling paths which the
electrical current follows. This high saturation
behavior is most evident in Figure 4.
DISCUSSION
This research provides further evidence for the
frequency dependence found by Sandler et al. (2009)
in fractured rocks. The electrical circuit established to
model this behavior should aid future investigation of
this phenomenon. Furthermore, the model circuit
may explain some of the deviations between the
observations of unfractured sandstones in this
research and by Sandler et al. (2009). It is possible
that Sandler et al. simply did not take measurements
at low enough saturations to observe the frequency
dependence in the specific nonfractured samples they
used. However, given the low saturations reached for
the core in Figure 1, this explanation is unlikely. A
more likely explanation offered by the model circuit
developed here is that the specific sandstones used by
Sandler et al. (2009) had much lower internal
capacitances than those used in this research. As
shown in equation (2), frequency and capacitance are
linked parameters; a change in capacitance will be
manifested the exact same way as an identical change
in frequency. Thus, a much lower capacitance would
give equivalent results to the case of measuring at
frequencies too low to observe the frequency
dependence in R for the range of R2. This behavior
can be conceptualized as equivalent to shifting the
plot in Figure 9 diagonally in the positive x, y and z
directions, and were it possible for the rocks to
achieve even lower values of R2, a frequency
dependence would have eventually been observed.
Had Sandler et al. observed at higher frequencies,
similar frequency dependence may have been
observed for all rocks. This frequency dependence
and its relationship to fractured vs. nonfractured
rocks may not lie in whether or not the behavior is
manifested; rather, it likely lies in the saturations at
which the behavior begins. The relatively large open
spaces in fractured rocks allow separation of bulk
water into two parallel layers of surface-bound water
at higher saturations that could be possible for similar
behavior in non-fractured rocks.
Figure 9: Relative change in measured resistance
with respect to R1 and R2 from the circuit
model in Figure 6.
A consistent challenge throughout this research lay in
the fact that in the interest of gathering higher
resolution data without gaps due to sleeping at night,
the process was automated. While repeating the
initial experiments, unnatural deviations in mass were
noticed, drawing attention to the fact that the tensions
associated with the electrode wires relaxing into
position was on the order of 0.01 to 0.1 g, enough to
disrupt the accuracy of the mass measurements and,
consequently, calculations of saturation for each time
step. Thus, the time over which this relaxation
occurred was investigated and efforts were made to
adjust wires at the start of the experiment such that
relaxation occurred as quickly as possible and only
the first few data points would be disturbed.
Of greater concern, though, was the fact realized
midway through research that within the frequency
range of operation for this research, effects at the
electrode could dominate the electrical behavior
observed (Knight 1984). On a fundamental level,
there is the complication associated with measuring
electrical properties across materials with different
modes of conduction. While rocks are ionic
conductors, the metal electrodes are electronic
conductors, and an accumulation of ions where
current perambulates the rock/electrode interface can
lead to impedances which might overwhelm behavior
of the rock itself. Thus, future experimentation
should seek to mitigate these external effects such as
in the manner described under FUTURE
RESEARCH below.
Despite these concerns, the adopted circuit model is
able to reflect the data with high correlation, and the
difference between Core #4 and the other cores
indicates promise for future research.
FUTURE RESEARCH
The question of electrode behavior dominating the
electrical response, and the uncertainties associated
with the initial experiment's rudimentary design
prompted a redesign of the electrodes, inspired by the
setup in Knight and Nur (1987) and by personal
conversations with Prof. Knight. In order to ensure
good connection to the whole rock face 100 nm of
platinum has been sputtered onto either side of a rock
core. The rock core chosen for the sputtering was
also different from those used in this research. The
new rock core is a cube whose electrical
propertieswill be measured in the same manner as
this research. Following measurements, the core will
be cut in half to simulate a fracture and electrical
properties will be measured with voltage applied both
perpendicular and parallel to the simulated fracture.
The electrodes have also been further modified to
minimize connection uncertainty as well as to address
the issue of wire relaxation. Stainless steel mesh has
been fixed to PVC plastic panels which are then
clamped to either end of the rock core using two hand
clamps. A frame has been constructed around the
mass balance with an elevated crossbeam to which
wire may be secured, on one side allowed to hang
loosely and attach to the electrodes, and on the other
side connect to the RCL meter. The wire to be used
will be much thinner and less rigid to allow for more
quick and regular relaxation.
ACKNOWLEDGEMENTS
The authors are grateful for financial support from
the United States Department of Energy under
contract DE-EE0005516. Also, much thanks to
colleagues in the Stanford Geothermal Group for
their invaluable guidance and support both inside and
out of the laboratory.
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APPENDIX – CORE IMAGES
Figures 10, 11 and 12 show images of Cores #1, 3 and 4 respectively.
Figure 10: Berea Sandstone (Core #1)
Figure 11: Unfractured Reservoir Rock (Core #2)
Figure 12: Fractured Reservoir Rock (Core #4)
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