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: lrvalver@stanford.edu

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)