Electrical properties of rocks are important parameters for well-log and
reservoir interpretation. Laboratory measurements of such properties are
time-consuming, difficult, and impossible in some cases. Being able to
compute them from 3-D images of small samples will allow for the generation of a massive amount of
data in a short time, opening new avenues in applied and fundamental
science. To become a reliable method, the accuracy of this technology needs
to be tested. In this study, we developed a comprehensive and robust
workflow with clean sand from two beaches. Electrical conductivities at 1 kHz were first carefully measured in the laboratory. A range of porosities
spanning from a minimum of 0.26–0.33 to a maximum of 0.39–0.44,
depending on the samples, was obtained. Such a range was achieved by compacting the samples
in a way that reproduces the natural packing of sand. Characteristic electrical
formation factor versus porosity relationships were then obtained for each
sand type. 3-D microcomputed tomography images of each sand sample from the
experimental sand pack were acquired at different resolutions. Image
processing was done using a global thresholding method and up to 96
subsamples of sizes from 2003 to 7003 voxels. After
segmentation, the images were used to compute the effective electrical
conductivity of the sub-cubes using finite-element electrostatic
modelling. For the samples, a good agreement between laboratory measurements
and computation from digital cores was found if a sub-cube size representative elemental volume (REV) was
reached that is between 1300 and 1820 µm3, which,
with an average grain size of 160 µm, is between 8 and 11 grains.
Computed digital rock images of the clean sands have opened a way forward for
obtaining the formation factor within the shortest possible time; laboratory
calculations take 5 to 35 d as in the case of clean
and shaly sands, respectively, whereas digital rock physics computation takes just
3 to 5 h.
Introduction
Electrical formation factor (FF) refers to the ratio of the electrical
resistivity of a saturated medium (sediment or rock) to that of the
saturating fluid (Guéguen and Palciauskas, 1994). This is an
important parameter in exploration geophysics as, contrary to the electrical
resistivity of reservoirs that is dependent on the resistivity of the
saturating fluid (and hence the same type of reservoir can exhibit high or low
resistivities; Constable and Srnka, 2007; Jinguuji et al., 2007; Mitsuhata
et al., 2006), the formation factor is an intrinsic property of the rock
independent of fluid salinity. Measurement of the formation factor in the
laboratory is often difficult and time-consuming, if not impossible in some
cases. Minerals forming the rock or sediment sample must reach
thermodynamical and electrical equilibrium with the saturating fluid, which
typically takes 4 to 6 d in a high-permeability, high-porosity clean
sandstone but may require at least 4 to 6 weeks for a tight gas sand or a
low-porosity rock or sediment with a high clay content. Furthermore, results
are affected by current leakage problems (especially at high frequencies) and
electrode polarization (emphasized at low frequencies).
Hence, the computation of electrical properties from microstructural models has
been investigated by several teams in the past 50 years. Various methods
have been proposed, from statistical models used to reconstruct 3-D porous
materials (e.g. Miller, 1969; Joshi, 1974; Milton, 1982; Torquato,
1987; Adler et al., 1990, 1992; Yeong and Torquato, 1998) to
direct measurement of a 3-D structure from synchrotron and X-ray computed
microtomography (XRCM) (e.g. Dunsmuir et al., 1991; Spanne et al.,
1994; Arns et al., 2001; Øren and Bakke, 2002; Nakashima and Nakano,
2011; Øren et al., 2007) or laser confocal microscopy (Fredrich et
al., 1995). In most of these studies using XRCM images, the numerical
prediction of electrical conductivity underestimates the
experimental results by 30 % to 100 % (which leads to an overestimation of
the formation factor) (Spanne et al., 1994; Schwartz et al., 1994; Auzerais
et al., 1996). Several explanations have been put forward to justify such
a discrepancy: percolation differences between the model and real material, mainly
due to smaller volume sampling in the model (Adler et al., 1992; Bentz and
Martys, 1994); the addition of a third phase to the traditional two-phase
model (the rock matrix being one phase and the saturating fluid being a second
phase) that counts for the bound fluid at the grain fluid interface
(Zhan and Toksoz, 2007); and discretization errors and statistical
fluctuations (Arns et al., 2001).
The underlying question behind the computation of electrical properties of
digital porous media samples (or any other rock or transport properties) is
whether the obtained numerical values are accurate. One aspect of this
question relates to the technology itself, namely 3-D imaging, image
processing and segmentation, and the suitability and stability of the numerical
code. These three key elements of the technology have been investigated by
various teams, and the most comprehensive and exhaustive study performed on
the various steps of the digital rock physics workflow is the benchmark
comparison from Andrä et al. (2013a, b). As they are using various
rock types and processing and computing methods, the comparison is complex:
they concluded that the computed effective rock properties are affected by
segmentation processes, the choice of digital sub-volume, and the choice of
numerical code and boundary conditions. Nonetheless, the different values
obtained for the formation factor deviated at most by 23 % from the
midrange value (Andrä et al., 2013a). For the sphere pack sample,
all computed formation factors ranged from 4.3 to 4.8.
The second aspect of this question relates to the comparison of the computed
values with laboratory-scale experimental data to validate the correctness
of the digital rock physics workflow. However, because both experiments are
done at a different scale (centimetre scale for the laboratory and millimetre scale for the
digital computation) and because rocks are heterogeneous at all scales, the
laboratory-measured and digitally computed values do not have to match. Instead,
trends between two properties (e.g. formation factor and porosity)
computationally derived and produced in the laboratory should be in good
agreement (Dvorkin et al., 2011; Andrä et al., 2013a).
In the work described in this paper, we propose a robust workflow to
digitally compute the electrical properties of clean (i.e. that does not contain
any clay or other conductive minerals) unconsolidated porous media. We first
carefully measure in the laboratory the formation factor of two beach sand
samples of similar mineralogy (quartz and carbonate), but of different grain
size, over a wide range of porosities obtained by compacting the sand
sample. Hence, trends in formation factor versus porosity that reproduce a packing
as close as possible to the one found in situ were obtained. We then compute
the formation factor from X-ray microtomography images using the free
software and finite-element electrostatic code from the National Institute of Standards and Technology (NIST) with multiple
subsamples of various sizes. To our knowledge, this is the first time that
such work has been done on clean sand.
Materials and laboratory methodsSample collection and preparation
The samples investigated in this paper are sand samples collected from the
coastal margin of the Perth Basin, Western Australia. The Perth Basin is an
elongate, north–south-trending trough underlying approximately 100 000 km2 of the Western Australian margin. Sediments were shed from
the adjacent Yilgarn block. The Yarragadee and Leederville sandstone
formations are intercalated with the Tamale limestone that forms the
carbonates at the Upper Cretaceous. One sample was collected from
Scarborough Beach (31∘53′41.97 S, 115∘45′17.74 E) and
one from Cottesloe Beach (31∘59′40.62 S, 115∘45′03.70 E). All the samples are composed of quartz and carbonate in 80%/20% (volume) proportion, respectively, as determined from the three-phase
watershed segmentation presented in Sect. 3.2.2 of this paper. Grain
size was determined by micro-CT image analysis and is between 16 and 794 µm (median 140 µm) for quartz grains and between 19 and 446 µm
(median 168 µm) for carbonate grains at Scarborough Beach. It is between 17 and 606 µm (median
159 µm) for quartz and between 15 and 415 µm (median 172 µm) for
carbonate grains at Cottesloe Beach. Sand
samples were thoroughly washed clean with tap water to remove any plants and
grass debris. Loose moist sand was then packed into the different cells used
to perform the electrical resistivity measurements, forming an
initially high-porosity loose random pack; decreasing porosity in subsequent
experiments was achieved by shaking the cell and using tied sticks to
compact the sand. This was done in a way to achieve a packing as close as
possible to the one found in situ. A range of six different porosities was obtained
for the Scarborough Beach sand samples, with an initial porosity of 0.40
(loosely packed) down to 0.27 when highly packed, while five and four different
porosities were obtained for the Cottesloe Beach sand, depending on the
geometry of the cell, with the loosely packed sample having a porosity of
0.39 and the highly packed sample having a porosity of 0.30.
Porosity was determined from the weights and densities of the sand grains
and the known volumes of cells used in the experiment as
ϕ=Vt-m/ρVt,
where ϕ is porosity, Vt is the total volume of the cell, m is
the average mass of the dry sand before and after the experiment, and ρ
is the density of the sand grains. Grain density was measured by He pycnometry
and found to be equal to 2.71 gcm-2.
Laboratory set-up and measurementsExperimental set-up
Two different types of cells are used in the experimental set-ups, which were
utilized to monitor the electrical resistivities of the sand samples as a
function of the salinity of the saturating pore water. The two experimental
set-ups are outlined in Figs. 1 and 2. For the cell called the “flow cell”,
the sample electrical resistances are measured, while saline solutions of
increasing salinities are continuously flooded through the sand samples.
Before proceeding with the next saline solution, the reading of the sample's
electrical resistance is left to stabilize for a few hours. For the cell called
the “static cell”, the sand samples are successively saturated with saline
solutions of increasing salinities, left to equilibrate with no fluid flow until
stability of the sample electrical resistance reading is achieved, and then
drained before saturating the sand sample with the next saline solution.
Thus, the utilization of this cylindrical-shaped static cell drastically
reduces the experimental time; however, the sample preparation for the static
cell is easier than for the flow cell. The flow cell is of cylindrical shape,
27 cm in length, and 5 cm in radius (total volume of 2120.6 cm3), while
the static cell is of rectangular shape, 29.8 cm in length, 8.7 cm in width,
and 6.2 cm in height (total volume of 1607.41 cm3).
Photo (a) and schematic drawing (b) of the experimental set-up for the flow cell.
Photo (a) and schematic drawing (b) of the experimental set-up for the static cell.
Both cells are made up of Perspex (acrylic) and have an outlet and an inlet
connected by tubing to a tank that serves as a reservoir for the various
solutions injected into the sand samples. The solutions flow through the
sand samples via gravity (falling-head method) and, for the flow cell, two
valves at the inlet and outlet are used to achieve a flow rate ranging
from 0.52 to 2.75 mLs-1. This flow rate is continuously recorded.
Injected solutions are fresh and saline solutions made with tap water and
table salt in various amounts: five different salinities of 0, 5, 15,
25, and 35 gL-1 were achieved and measured on an electric balance (Napco JA-5000), and
the solution was stirred until complete dissolution of the salt into the water.
Both cells are equipped with two electrodes made of zinc wire gauze with
surface areas of 78.55 and 53.94 cm2 for the dynamic and
static cells, respectively. The electrodes are glued at the bottom and at
the lid cover of the cylindrical dynamic cell, while they are fixed on both
sides of the rectangular static cell; the two electrodes of each cell are
connected to an LCR meter (Stanford Research Systems SR720) connected to a laptop to monitor the
electrical resistance of the sand sample. The recording time interval for the
dynamic cell laboratory measurements is taken at 1 min, while the
recording time interval for the static cell laboratory measurement is 10 min. A drive voltage of 1 Vrms is applied and a frequency of 1 kHz is
chosen to minimize the phase angle between the voltage and current (i.e.
electrode polarization). With these conditions, the monitored Q factor did not
exceed 0.095, indicating that the system is nearly purely resistive. For the
dynamic cell laboratory measurements, the conductivity of the injected
solutions coming out of the cell is monitored by an encased conductivity
meter (Hanna edge) attached to the cell at intervals of 1 min to make it
synchronous with the sand sample resistance measurements. The fluid
electrical conductivity for the static cell set-up is measured with the same
probe using the saturating solution drained from the sand sample once the
resistance has become stable.
Computation of electrical formation factor
Because the sand samples do not contain any clay and because the injected
solutions have a conductivity (10-2 to 5.0×10+1Sm-1) much larger
than that of quartz or carbonate surface conductivity (5.4×10-3Sm-1 following
Miller et al., 1988, and 1.4×10-3Sm-1 following Vialle, 2008), surface and matrix electrical conductivities can be neglected
(e.g. Johnson and Sen, 1988; Garrouch and Sharma, 1994). The electrical
formation factor F is then given by
F=RsRw,
with
3Rs=rsAL,4Rw=1σw,
where
Rs is the resistivity of the sand sample saturated with water,
Rw is the resistivity of the water,
rs the measured resistance of the sand sample saturated with water, A the surface
area of the electrode, L the length of the cell, and σw the
measured conductivity of water.
To obtain the formation factor, the sample's resistivity, once it has
stabilized, is plotted against the saline water's resistivity, and the
formation factor is given by the inverse of the slope. Such a plot is
given in Fig. 3 for the example of a Cottesloe Beach sample with porosity
33 %.
Sand sample conductivity as a function of water conductivity for
the Cottesloe Beach sample with a porosity of 33 %. The slope of the linear
correlation gives a formation factor (FF) of 6.50.
Digital rock samples and computation of electric propertiesImage acquisition
Two samples were prepared for imaging with X-ray microcomputed tomography
(XRMCT): one from Scarborough Beach and one from Cottesloe Beach. Loose sand
was put in a cylindrical Pyrex glass tube of 6 mm diameter and 6 cm
height, and the tube was inserted into the core holder of the microtomograph.
The samples were scanned with the 3-D X-ray microscope Versa XRM 500 (Zeiss–XRadia) using an X-ray energy of 60 keV, a current of 70.66 mA, and a power
of 5 W. In each scan 3000 projections (radiographs) were acquired. The
exposure time was 2 s per radiograph. Initial cone-beam 3-D image
reconstruction was performed using the software XM Reconstruction (XRadia).
A secondary reference was required to remove geometrical artefacts during
reconstruction. After 3-D reconstruction, 3-D volume was sliced onto 2-D images
for further processing. A total of 1021 2-D images for the Scarborough
Beach sample and 991 2-D images for the Cottesloe Beach sample were available for
analysis. Total scanning time was 2 h 55 min and 2 h 42 min for the
Scarborough and Cottesloe samples, respectively. Nominal voxel sizes of
2.5761 and 2.5516 µm3 were achieved with
source-to-sample and detector-to-sample distances of 11 and 22 mm, respectively, for the
Scarborough and Cottesloe Beach samples.
Image processingImage filtering
We used the software package Avizo Fire 9 (FEI Visualization Sciences Group)
for image enhancement and segmentation. Greyscale images of the 2-D slices
were processed using a non-local means filter in the intensity range of 255–5344 for Scarborough Beach and 255–5467 for Cottesloe Beach, with the
aim of removing ring artefacts in the images and properly enhancing
interfaces between the pores and grains as well as removing noise. A non-local
means filter has been shown to effectively remove ring artefacts without
introducing edge smoothing, in contrast to many other filters, and thus does not
require the use of an additional mask (see, for example, the review paper of
Schlüter et al., 2014).
Figure 4a–d show raw and filtered images for both Scarborough and
Cottesloe Beach: we can easily notice that the quality of the image has
increased. In these images, the white grains are carbonate, grey grains are
quartz, and black within the disc corresponds to void space (pores).
(a) Raw and (b) filtered images of the Scarborough Beach sand sample; (c) raw and (d) filtered images of the Cottesloe Beach sand sample.
Image segmentation
The filtered images were segmented using two types of thresholding
algorithms: the first one resulted in a two-phase segmentation that was
further used for computing sample electrical conductivities; the second one
was a watershed algorithm that resulted in a two- or three-phase segmentation used
for grain analysis. Note that filtering and segmentation workflows were
applied to the full 3-D dataset. Figure 5 shows the histogram for both
samples.
Histogram of (a) Scarborough and (b) Cottesloe Beach sand samples.
Two-phase segmentation by global thresholding
Because both quartz and carbonate have very low conductivity compared to
water, they can be both considered non-conductive for computation
purposes of the electrical conductivity of the water-saturated sand sample.
Hence, quartz and carbonate can be put in a single phase, and pores will
constitute a second phase that will later be filled with a conductive
fluid for the computation of sample electrical properties. We use a
global threshold segmentation algorithm to separate pores from grains: the
set intensity value separating pores from grains (both quartz and carbonate
grains having higher intensity values than pores) is kept the same
for all 2-D slices.
Poor segmentation can affect the accurate calculation of porosity. To check the
quality of the segmentation, we compare the porosity estimated in the
laboratory with the one estimated from micro-CT scan images. We made a
random loose pack of sand (cm3) in the laboratory to obtain the highest
porosities of 0.361 and 0.349 from Scarborough and Cottesloe beaches,
respectively, while the smaller scanned sample of the sand (mm3) was
also randomly packed in the small tube, from which porosities of 0.369 and
0.359 were obtained from the images of Scarborough and Cottesloe beaches,
respectively.
Watershed segmentation
We used a marker-based watershed segmentation algorithm from Avizo Fire 9.
We defined either two or three marker ranges of greyscale intensity: for
pores and grains or for pores, carbonate grains, and quartz grains. We then performed a watershed flooding for each of these two or
three phases. The two-phase watershed segmentation allows for the computation of pore
volume and grain size distribution, whereas the three-phase segmentation (Fig. 6) gives the volume fraction of the different minerals.
Three-phase watershed segmentation of the sand samples: (a) Scarborough; (b) Cottesloe.
From this segmentation, we computed the volume fraction of quartz and
carbonate (excluding the pore volume). The result is 81.9 % quartz and
18 % carbonate for the Scarborough sample and 87.8 % quartz and
12.2 % carbonate for the Cottesloe sample.
Image cropping
The 3-D filtered and segmented volumes for each of the two sand samples were
subdivided into overlapping sub-cubes (96 in total) of four different sizes: 3 sub-cubes with a size of 7003, 8 with a size of 5003, 13 with a size
of 3503, and 20 with a size of 2003 for the Scarborough Beach
sample, as well as 5 sub-cubes with a size of 7003, 10 with a size of
5003, 13 with a size of 3503, and 24 with a size of
2003 for the Cottesloe Beach sample. Porosity was estimated using Avizo
software for each of these 96 sub-cubes.
The 2-D cropped images were then exported in binary format for the computation of
electrical properties (Fig. 7).
Computational studies of electrical fields of micro-CT images
To estimate conductivity from micro-CT images, we assume that pores are
electrically conductive and that the solid phases are not conductive. This
assumption is based upon the concept that mainly the ions in fluid-filling
pores can be drifted under the effect of external electric fields. To
estimate the conductivity from images, we first have to calculate an
average current density.
An example of 7003 binary images for (a) Scarborough and (b) Cottesloe beaches.
If we assume that the conservation of charge is valid in the pore structure,
then no net charges are created or annihilated in the pore volume and pore
surfaces; the current density vector obeys the following equation:
∇⋅J=0.
On the other hand, Ohm's law at the microscopic level assumes that the
current density is proportional to the electrical potential field:
J=σw∇V,
where J is the electrical current density, σw is the electrical
conductivity of the fluid that fills the pore space, and V is the electrical
potential field (voltage). By substituting Eq. (6) into Eq. (5), we have
the Laplace equation as
∇⋅σw∇V=0.
Equation (7) can be solved numerically for pore structures by applying an
external electric field (Eext)
on the boundaries. One of most reliable numerical methods to estimate the
average current density from 3-D images is the finite-element method. We use
the same free software written by Garboczi (1998). This method, by
minimizing the electrical energy stored in the porous volume under study,
estimates the local potential fields (V) at each coordinate system (pore and
solid phases). For a given microstructure, because of the applied fields or
other boundary conditions, the final voltage distribution is determined by
minimization of the total energy stored in the system (Garboczi, 1998). Figure 8a and b show the potential field variations in Scarborough
and Cottesloe Beach samples, respectively. This can help us evaluate the
effective current density (Jav) by using Eq. (8) and by taking the volume average of the local
current density vectors (J). On the other hand, the volume average of current density is defined as
Jav=〈J〉=σeffEext,
where
σeff is the effective conductivity of the porous medium. Effective conductivity
is a second-rank tensor. In Eq. (7), the current density (Jav) and the external electrical field (Eext) are vectors. If we assume that the external electrical field is
unidirectional (let us assume that in the x direction,
E=E⋅ux), then the current density can have components on any other directions and
can thus be written in the general form as
Jav=Jx⋅ux+Jy⋅uy+Jz⋅uz.
Then, from Eq. (7), the current density can be rewritten as
Jav=σxxE⋅ux+σyxE⋅uy+σzxE⋅uz.
In homogenous media, we expect the current density to be negligible in the
direction perpendicular to the external electrical fields. This implies that
for homogenous media, the effective conductivity tensor is a diagonal
matrix. On the other hand, for heterogeneous media, the current density in
the direction perpendicular to the external electrical field is not zero or
is not small compared to the diagonal values. Hence, in general, the current
density is a second-rank tensor of the form
σ=σxxσxyσxzσyxσyyσyzσzxσzyσzz.
The 7003 voxel sample from Scarborough was analysed by applying a
current successively in the x, y, and z directions to find out whether the sample
shows some anisotropy (see Fig. 9).
Electrical potential field image output from the 7003 digital sub-cubes of (a) Scarborough (b) Cottesloe beaches. The colour bar indicates regions of high (red) and low (blue) potential field in arbitrary units.
The output of conductivity along the x, y, and z directions shows almost the same
values of the formation factor (5.30, 4.96, and 5.08, respectively). The
difference in the values of formation factor between the x direction and
y direction is 6.6 %, while that between the x direction and z direction is
4.4 %; hence, the sample presents a small anisotropy at the scale of
investigation. In the following, we took an average of the conductivities in
the three different directions, which mathematically is equal to one-third
of the trace of the conductivity tensor; for simplicity, we then consider the
conductivity to be a scalar number for all images.
Electrical potential field images (a) along the x direction, (b) along the y direction, and (c) along the z axes.
From the effective conductivity calculated for micro-XRCT images, the
electrical formation factor can be estimated as
F=σwσeff,
where
σw is the electrical conductivity of pore fluids, taken equal to 1 in the
computation. The electrical formation factor is calculated for each of the
different sub-cubes obtained from the micro-CT images of the Scarborough and
Cottesloe Beach samples.
Laboratory-measured formation factor versus porosity values for
both the flow and static cells for (a) Scarborough and (b) Cottesloe Beach
samples.
Summary of laboratory and micro-CT scan image results from the
Scarborough Beach samples.
Figure 10 displays the values of the formation factor against porosity for
the Scarborough and Cottesloe beaches, computed as described in
Sect. 2.2.2 and for each porosity value obtained by compacting the initial
sand pack. Correlation coefficients were very good to excellent and varied
between 0.975 and 0.999 and between 0.974 and 0.996 for the flow cell for
the Scarborough and Cottesloe samples, respectively. They varied between 0.882 and 0.993
and between 0.987 and 0.999 for the static cell for the Scarborough and
Cottesloe samples, respectively. The results for both the static and flow
cells are reported in Tables 1 and 2 for both samples and for all data
points. The values of the formation factors obtained using the flow cell are
higher than those obtained using the static cell for both the Scarborough (8.2)
and Cottesloe (8.5) Beach samples, whereas for Scarborough Beach, formation
factors have close values at high porosities and then depart from each other
at lower porosities (lower than 0.39). Some deviations between the
results obtained for both static and flow cells may be due to non-uniform
compaction of the samples in the case of the flow cell and/or non-complete
fluid replacement in the case of the flow cell. In these figures, we have
bounded the experimental data by two lines that represent a power-law
relationship between the formation factor and porosity in the form
F=a⋅ϕ-m=ϕ-m.
This is Archie's law (Archie, 1942) with a tortuosity factor a of 1.
The tortuosity factor usually ranges from 0.5 to 1.5, but there has been
quite a wide range reported in the literature for sand, from the most used value
of 0.62 (Humble formula; Winsauer et al., 1952) to up to 2.45 (Porter and Carothers, 1970). We take the same tortuosity factor value of 1 for all
samples. This is the value for clean granular formations (Sethi, 1979).
Micro-CT scan images
Formation factors were plotted against porosity for all the micro-CT scan
image cubes for Scarborough and Cottesloe beaches (Fig. 11).
Formation factor against porosity for each sub-cube size of
2003, 3503, 5003, and 7003 from (a) Scarborough Beach samples and (b) Cottesloe Beach samples.
Porosity against cube sizes for (a) Scarborough Beach and (b) Cottesloe Beach.
Similarly, both porosity and formation factor were plotted against the cube
sizes 2003, 3503, 5003, and 7003 (Figs. 12 and 13, respectively). Scattering is shown
when the cube sizes were small, which begins to level off as the
representative elemental volume (REV) is approached. This REV is somewhere
between 5003 and 7003, which corresponds to a sample size
between 1.3 and 1.8 mm3.
Formation factor against cube sizes for (a) Scarborough Beach and (b) Cottesloe Beach.
Discussion
As noted earlier in Sect. 4.1, the values of the formation factor obtained
by the static cell are higher than those obtained by the dynamic cell (for a
given porosity) for both samples. This translates into a higher cementation
exponent m. One reason for this can be the design of the cell itself and
the way to achieve a stable reading of sample conductivity for each fluid
salinity. In the rectangular (static) cell, because the higher-salinity
brine is introduced or retrieved via the centre of the panels (see Fig. 2),
there could some brine left in the corners that will only equilibrate with
the new injected brine by diffusion, and hence there could be a lower
conductivity of the brine in these corners compared to the conductivity of
the injected brine. As a result the measured sample conductivity will be
lowered with respect to what it should be, giving a higher ratio of sample to
brine conductivities (i.e. formation factor; see Eq. 11). Using a
cylindrical cell thus has the advantage of providing a better replacement of
the brine.
Figure 14 shows reported data from both the literature and those acquired in
this study for the Cottesloe and Scarborough Beach samples (using the flow
cell). Data from the literature include natural sand samples and synthetic
granular media made of plastic particles with a regular geometrical shape
(Wyllie and Gregory, 1953). We have bounded these data by the relationship
presented in Eq. (14), with m=1.3, which corresponds to the original work of
Archie (1942) for unconsolidated media, and by the same relationship, with
m=1.8, for the upper bound. We see in this figure that our experimental
results for the Cottesloe and Scarborough Beach samples are in agreement with
data reported for other beach sands. Considering the data reported in this
figure, we observe that Archie's classical formula for unconsolidated media
underestimates the formation factor and that the departure from sphericity
leads to a larger m coefficient. Since Archie's work, many authors have
proposed alternative formation factor–porosity relationships. Winsauer et
al. (1952) suggested that a≠1 in Eq. (14) is a better expression, whereas
other authors derived a non-power-law dependency on porosity. From a
practical point of view, no formula relating the formation factor to
porosity for unconsolidated media fits all the experimental data, and, for a
given porosity, the formation factor depends on the particle geometry,
particle size distribution, and subsequent packing.
Comparison of laboratory results with results from other researchers
(Wyllie and Gregory, 1953). CB stands for Cottesloe Beach samples, and SCB
is for Scarborough Beach samples.
In Fig. 15, we compare laboratory data to computed data. Laboratory data
are those acquired with the flow cell, which, as discussed earlier in this
section, is expected to give more reliable data. Computed data are those
obtained for a cube size of 7003, which is above the REV, as
presented in Sect. 4.2. We can see that there is an excellent agreement
for the Cottesloe Beach sample and a good agreement for the Scarborough Beach
sample. At this stage, it is difficult to explain why one sample gave better
agreement and whether it is an experimental error or the
higher content of carbonate grains for the Scarborough sample that makes the
computation less accurate: indeed, carbonate grains may present some
intra-porosity (for example, micritic phases) and thus have an electrical
conductivity.
Comparison between laboratory results (open symbols) and end-computed results (plain symbols). The trends in dashed lines are obtained
from the laboratory-measured data.
Conclusions
The electrical properties of rocks are important parameters for well-log and
reservoir interpretation. Laboratory measurements of such properties are
time-consuming, difficult, and even impossible in some cases. In view of this,
we have successfully combined the scientific approach of laboratory
measurements (as a benchmark) with micro-CT scan computational images. We
have thereby achieved the objective of determining the variability of computed
formation factors as a function of porosity from laboratory measurements and
micro-CT scan images from two sand samples for Scarborough and Cottesloe
beaches in Perth Basin. This is the fastest method of obtaining a formation factor
from CT scan images, which takes less time (5–7 h), while calculations
from laboratory measurements take much more time (5 to 30 d or more).
This approach is practical, easily repeatable in real time (though
expensive), and can be an alternative method for calculating a formation factor
when time is not on the side of the experimenter, which is always the case.
Results of images below 5003 (Scarborough) and 3503 (Cottesloe)
indicate that they are not suitable REVs for pore-scale networks.
In this paper, a micro-CT scan image computational technique was employed to
calculate properties such as porosity and formation factor on large
three-dimensional digitized images of a sand sample. We demonstrated that for
most of the parameters studied here, the values obtained by computing micro-CT scan images agreed with classical laboratory measurements and results
from other researchers. This work was focused on establishing a robust
methodology and workflow, and we thus started with one of the most simple
materials, though it is still highly relevant for many applications in oil and
gas or water management environments. For more complex geological materials,
such as low-permeability rocks, multi-mineralitic rocks, and materials with
conductive minerals, further developments are obviously needed.
However, these developments are mostly related to the employed techniques
(e.g. a higher-resolution imaging technique would be needed for
low-permeability rocks, a more complex laboratory set-up, and techniques for
measurements of rocks with conductive minerals or minerals with a
non-negligible surface conductivity, etc.) rather than to the overall workflow
established here (comparison between laboratory and computed data through
trends between properties).
Data availability
All data shown in Figs. 10 to 15 (except those taken from previous studies by other authors in Fig. 15 that are displayed for comparison) are given in Tables 1 and 2. X-ray images are available on request from the authors.
Author contributions
MAG carried out the experiments and performed the numerical simulations; SV and ML designed and planned the experiments; MM supervised and verified the numerical simulations; MAG wrote the paper in consultation with SV, MM, ML, and BG. BG initiated the project. SV conceived the original idea and supervised the project.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
We thank Dominic Howman and Vassili Mikhaltsevitch for help in cell design and laboratory experiments and Andrew Squelch for help with image processing.
Review statement
This paper was edited by Ulrike Werban and reviewed by five anonymous referees.
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