SESolid EarthSESolid Earth1869-9529Copernicus PublicationsGöttingen, Germany10.5194/se-9-1225-2018Enhanced pore space analysis by use of μ-CT, MIP, NMR, and SIPEnhanced pore space analysis by use of μ-CT, MIP, NMR, and SIPZhangZeyuzeyuzhangchina@163.comKruschwitzSabineWellerAndreashttps://orcid.org/0000-0002-2225-4306HalischMatthiasSouthwest Petroleum University, School of Geoscience and
Technology, 610500 Chengdu, ChinaFederal Institute for Material Research and Testing (BAM), 12205
Berlin, GermanyTechnische Universität Berlin, Institute of Civil Engineering,
13355 Berlin, GermanyClausthal University of Technology, Institute of Geophysics,
38678 Clausthal-Zellerfeld, GermanyLeibniz Institute for Applied Geophysics (LIAG), 30655 Hannover,
GermanyZeyu Zhang (zeyuzhangchina@163.com)7November201896122512386May201814May201813October201815October2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://se.copernicus.org/articles/9/1225/2018/se-9-1225-2018.htmlThe full text article is available as a PDF file from https://se.copernicus.org/articles/9/1225/2018/se-9-1225-2018.pdf
We investigate the pore space of rock samples with respect to different
petrophysical parameters using various methods, which provide data on pore
size distributions, including micro computed tomography (μ-CT), mercury
intrusion porosimetry (MIP), nuclear magnetic resonance (NMR), and
spectral-induced polarization (SIP). The resulting cumulative distributions
of pore volume as a function of pore size are compared. Considering that the
methods differ with regard to their limits of resolution, a
multiple-length-scale characterization of the pore space is proposed, that is
based on a combination of the results from all of these methods. The approach
is demonstrated using samples of Bentheimer and Röttbacher sandstone.
Additionally, we compare the potential of SIP to provide a pore size
distribution with other commonly used methods (MIP, NMR). The limits of
resolution of SIP depend on the usable frequency range (between 0.002 and
100 Hz). The methods with similar resolution show a similar behavior of the
cumulative pore volume distribution in the overlapping pore size range. We
assume that μ-CT and NMR provide the pore body size while MIP and SIP
characterize the pore throat size. Our study shows that a good agreement
between the pore radius distributions can only be achieved if the curves are
adjusted considering the resolution and pore volume in the relevant range of
pore radii. The MIP curve with the widest range in resolution should be used
as reference.
Introduction
Transport and storage properties of reservoir rocks are determined by the
size and arrangement of the pores. In this paper we use the term geometry to
refer to the relevant pore sizes, such as the pore throat radius, pore body
radius, body to throat ratio, shape of the pore, and pore volume
corresponding to a certain pore radius. Different methods have been developed
to determine the pore size distribution of rocks. These methods are based on
different physical principles. Therefore, it can be expected that the methods
recognize different geometries and sizes. Additionally, the ranges of pore
sizes that are resolved by the methods are different (Meyer et al., 1997).
Rouquerol et al. (1994) stated in the conclusions of their recommendations
for the characterization of porous solids that no experimental method
provides the absolute value of parameters such as porosity, pore size,
surface area, or surface roughness. It should be noted that these parameters
indicate a fractal nature. That means that the value of the parameter depends
on the spatial resolution of the method.
An enhanced pore space analysis using different methods should be able to
provide a better description of the pore space over a wide range of pore
sizes. Our study of pore space analysis is based on the following methods:
micro computed tomography (μ-CT), mercury intrusion porosimetry (MIP),
nuclear magnetic resonance (NMR), and spectral-induced polarization (SIP).
The first three methods can be regarded as standard methods to derive a pore
size distribution. Since these methods can reveal the inner structure of the
rocks, they are widely applied in geosciences (e.g., Halisch et al., 2016b;
Mees et al., 2003; Behroozmand et al., 2015; Weller et al., 2015). The main
aim of our paper is to integrate an electrical method in this study.
Electrical conductivity and polarizability (or real and imaginary parts of
electrical conductivity) are fundamental physical properties of porous
materials. The SIP method measures the low-frequency electric behavior of
rocks and soil material that can be efficiently represented by a complex
electric conductivity (e.g., Slater and Lesmes, 2002). The electric
properties of a porous material depend, to a large extent, on key parameters
including the porosity, the grain and pore size distribution, the specific
internal surface, the tortuosity, the saturation, and the chemical
composition of the pore-filling fluids. SIP is a nondestructive method that
can be applied to characterize the geometry of the pore system.
Müller-Huber et al. (2018) proposed the integration of SIP in a combined
interpretation with NMR and MIP measurements for carbonate rocks in order to
use the partly complementary information of each method. The SIP method is
used to explore correlations between parameters derived from complex
conductivity spectra and specific pore space properties. We go a step further
and directly compare the pore size distributions derived from the different
methods. Procedures to derive pore size distributions from induced
polarization (IP) data have been proposed only recently (Florsch et al.,
2014; Revil et al., 2014; Niu and Zhang 2017; Zhang et al., 2017).
We are aware that further methods can be applied for the characterization of
pore size distribution, e.g., synchrotron-radiation-based computed tomography
(Peth et al., 2008), focused ion beam tomography (Keller et al., 2011),
transmission electron microscopy (Gaboreau et al., 2012), scanning electron
microscopy (SEM), 14C labeled methylmethacrylate method (Kelokaski et
al., 2005), and gas adsorption and desorption method (BET; Avnir and
Jaroniec, 1989).
Our study presents an approach to describe and quantify the pore space of
porous material by combining the results of methods with different
resolution. Samples of Bentheimer and Röttbacher sandstone are
investigated by μ-CT, MIP, NMR, and SIP. Each method provides the pore
size distribution in a limited range of resolution. It is not our intention
to combine the data of the different methods in a joint inversion to get a
more reliable pore size distribution as proposed by Niu and Zhang (2017). We
prefer to compare the resulting pore size distributions to each other to get
two different pore radius distributions, one for the pore body radius and one
for the pore throat radius. The comparison of the two curves enables the
determination of the ratio between pore body and pore throat radius. A joint
inversion that ignores the difference between pore body and pore throat
provides a simplified model that ignores the complexity of pore space
geometry.
Considering the fractal nature of pore space geometry, an attempt is made to
determine the fractal dimension of the pore volume distribution for the two
investigated samples. The fractal dimension is a useful parameter for up- and
downscaling of geometrical quantities. Zhang and Weller (2014) investigated
the fractal behavior of the pore volume distribution by capillary pressure
curves and NMR T2 distributions of sandstones. Considering the differences in fractal
dimension resulting from the two methods, they concluded a differentiation
into surface dimension and volume dimension. Additionally, the fractal
dimension is used in methods of permeability prediction (e.g., Pape et al.,
2009).
Theory
The pore size distribution resulting from different methods has to be
compared and evaluated. We prefer a comparison based on the cumulative volume
fraction of pores Vc, which is expressed by
Vc=V(<r)Vp,
with Vp being the total pore volume, and V(<r) the cumulative
volume of pores with radii less than r. A graph displaying the logarithm of
Vc versus the logarithm of the pore radius offers the advantage
that the slope of the curves is related to the fractal dimension of the pore
volume (Zhang and Weller, 2014).
Fractal theory is applied to describe the structure of geometric objects
(Mandelbrot, 1977, 1983). At molecular size and microscopic range, surfaces
of most materials including those of natural rocks show irregularities and
defects that appear to be self-similar on variation of resolution (Avnir et
al., 1984). A self-similar object is characterized by similar structures at
different scales. The regularity of self-similar structures can be quantified
by the parameter of fractal dimension D. Pape et al. (1982) first proposed
a fractal model (the so-called “pigeon-hole model” or “Clausthal
concept”) for the geometry of rock pores. Fractal dimension describes the
size of geometric objects as a function of resolution. This parameter has
proved to be useful in the comparison of different methods that determine
distributions of pores in sandstones and carbonates (e.g., Zhang and Weller,
2014; Ding et al., 2017).
From MIP, the entry sizes of pores and cavities, which is referred to as pore
throat radius rt, can be determined according to the
Washburn equation (Washburn, 1921)
rt=-2⋅γ⋅cosθPc,
with γ=0.48 N m-1 being the surface
tension of mercury, θ=140∘ the contact angle between mercury
and the solid minerals, and Pc the pressure of the liquid mercury
that is referred to as capillary pressure.
Zoomed-in 2-D slice view of sample
BH5-2 in order to visualize pore
bodies (blue circles, detected by NMR and digital image analysis (DIA) of
μ-CT data) and pore throats (red lines with arrows, detected by MIP).
Starting with low pressure, the pores with larger pore throats are filled
with mercury. While increasing the pressure, the pores with smaller throats
are filled. Reaching a certain pressure level Pc, a cumulative
volume of mercury (VHg) has intruded into the sample that
corresponds to the pore volume being accessible by pore throat radii larger
or equal rt according to Eq. (2). Figure 1 shows a 2-D image of
the pore space of sample BH5-2 (information is given in Sect. 3) indicating
the pore throat radius rt as measured by MIP by red arrows. Fluid
flow properties, and hence the injection pressure of mercury, solely depend
on the narrowest pore diameter in the flow path that corresponds to the pore
throat diameter. The cumulative volume of mercury VHg corresponds
to the pore volume V(> rt). It should be noted that
the volume of larger pores, which are shielded by narrower throats, is
attributed to the pore throat radius (e.g., Kruschwitz et al., 2016). Knowing
the total pore volume Vp, the saturation of the sample with
mercury SHg can be determined. A conventional capillary pressure
curve displays the relationship between the saturation of the sample with
mercury SHg as a function of capillary pressure Pc
(e.g., Thomeer, 1960). Using the following simple transformations,
SHg=VHgVp=V(>rt)Vp=Vp-V(<rt)Vp=1-Vc,
the cumulative volume fraction of pores Vc as defined in Eq. (1)
can be determined as a function of rt.
The NMR relaxometry experiment records the decay of transversal
magnetization. The measured transversal decay curve is decomposed in a
distribution of relaxation times b(T2). The individual relaxation time
T2 is attributed to a pore space with a certain surface-to-volume ratio
A/V by
1T2=ρAV,
with ρ being the surface relaxivity. Considering that for a capillary
tube model with cylindrical pores of radius r, the surface-to-volume ratio
equals 2/r, we get the following linear relationship between pore radius
r and relaxation time T2 (e.g., Kleinberg, 1996):
r=2ρT2.
It should be noted that the NMR method resolves the radius rb
that corresponds to the maximal distance to the pore wall. It can be
represented by the pore radius of the largest sphere that can be placed
inside this pore as shown in Fig. 1.
Another approach to derive a pore size distribution is based on the SIP
method. Relations between grain or pore size and IP parameters have been
reported in a variety of studies (e.g., Slater and Lesmes, 2002; Scott and
Barker, 2003; Binley et al., 2005; Leroy et al., 2008; Revil and Florsch,
2010). Polarization effects of natural materials are caused by different
charging and discharging processes of some polarizing elements such as grain
surface, pore throat, membrane, and electrical double layer. Following an
approach proposed by Schwarz (1992), the complex conductivity of an
individual polarization element can be presented by a Debye model. It is
assumed that the recorded spectra result from a superposition of polarization
processes characterized by different relaxation times. This approach has been
adopted to generate synthetic spectra of electrical conductivity from
distributions of grain sizes (e.g., Revil and Florsch, 2010) or pore sizes
(e.g., Niu and Zhang, 2017).
A decomposition of the spectra is needed to derive the relaxation time
distribution. Florsch et al. (2014) demonstrated that a variety of models can
be used as kernel for the decomposition of the spectra. Revil et al. (2014)
compare the results of Debye and Warburg decomposition. Their argumentation,
which is based on mechanistic grain size models describing the polarization
of charged colloidal particles and granular material, supports the
application of the Warburg decomposition that results in a narrower
distribution of polarization length scales. It should be noted that a uniform
grain size does not automatically generate a uniform pore size. Besides, it
can be clearly seen by the scanning electron microscopy images
that the investigated sandstones feature a distinct range of both grain and
pore (throat) sizes. Considering that the pore size and not the grain size
controls the polarization of sandstones, as observed by different authors
(e.g., Scott and Barker, 2003; Niu and Revil, 2016), a wider distribution of
length scales can be expected. According to our opinion, there are no clear
indications for superiority of the Warburg decomposition. Up to now, a
theoretical model that confirms the validity of the Warburg model in
describing the polarization of a simple pore space geometry has not been
presented. Therefore, we prefer to use the Debye decomposition, which has
proved to be a useful tool in the processing of IP data in both the time and
frequency domain (e.g., Terasov and Titov, 2007; Weigand and Kemna, 2016).
The algorithm described by Nordsiek and Weller (2008) provides the electrical
relaxation time distribution as well as the total chargeability from complex
conductivity spectra.
According to the assumption that the electrical relaxation time and pore size
are related to each other, the specific chargeability at a certain relaxation
time corresponds to the pore volume attributed to a certain pore size, and
the total chargeability is attributed to the total pore volume of the sample.
The volume fraction Vc corresponds to the ratio of cumulative
chargeability to total chargeability. To transform the relaxation time
distribution into a pore size distribution, we adopt the approach proposed by
Schwarz (1962) and applied by Revil et al. (2012) for the Stern layer
polarization model:
r=2τD(+),
with D(+) being the diffusion coefficient of the counter-ions in the
Stern layer and τ being the relaxation time. Originally, this equation
describes the relation between the radius of spherical particles in an
electrolyte solution and the resulting relaxation time. Although it remains
arguable whether or not the radius of spherical grains can be simply replaced
by the pore radius (Weller et al., 2016), we generally follow this approach.
Additionally, we assume a constant diffusion coefficient D(+)=3.8×10-12 m2 s-1 as proposed by Revil (2013).
The signal amplitude at a given relaxation time corresponds to the pore
volume related to the pore radius determined by Eq. (6). Considering the
experience that the polarization is related to the specific surface area per
unit pore volume (e.g., Weller et al., 2010), we assume that the IP signals
are caused by the ion-selected active zones in the narrow pores that are
comparable with the pore throats. Their size is quantified by the pore throat
radius rt. Following the procedure proposed by Zhang et
al. (2017), the cumulative volume fraction Vc corresponds to the
ratio of cumulative chargeability to total chargeability. Considering the
restricted range of pore radii (0.1–25 µm) resolved by SIP, a
correction of the maximum Vc becomes necessary.
Samples and methods
For this study, two different sandstone samples have been used. Firstly, a
Bentheimer sandstone, sample BH5-2. The shallow-marine Bentheimer sandstone
was deposited during the Early Cretaceous (roughly 140 million years ago) and
forms an important reservoir rock for petroleum (Dubelaar et al., 2015). This
sandstone is widely used for systematic core analysis due its simple
mineralogy and the quite homogeneous and well-connected pore space. It is
composed out of 92 % quartz, contains some feldspar, and about
2.5 % vol of kaolinite
(Peksa et al., 2015), which is a direct alteration product of the
potassium-bearing feldspar minerals. Accordingly, surface area and surface
relaxivity values are mostly controlled by the kaolinite for this rock.
Secondly, a Röttbacher sandstone, sample RÖ10B, has been used. The
Röttbacher sandstone is a fine-grained, more muscovite-illite containing,
and rather homogeneous material that was deposited during the Lower Triassic
era (roughly 250 million years ago). It is suitable for solid stonework and
has been widely used as building material for facades as well as for indoor
and outdoor flooring. The Röttbacher sandstone was included in a study on
the relationship of pore throat sizes and SIP relaxation times reported by
Kruschwitz et al. (2016). This sandstone consists mostly of quartz, but
features a higher amount of clay minerals than the Bentheimer sample.
Additionally, iron-bearing minerals (e.g., haematite) have been formed during
its arid depositional environment, giving this sandstone a distinct reddish
color. Accordingly, surface area and surface relaxivity are dominated by the
clay and iron-bearing minerals and should be significantly different than for
the BH5-2 sample.
The experimental methods used in this study include digital image analysis
(DIA) based on micro computed tomography (μ-CT), mercury intrusion
porosimetry (MIP), nuclear magnetic resonance (NMR), and spectral-induced
polarization (SIP).
For this study, a nanotom S 180 X-ray μ-CT instrument (GE Sensing &
Inspection Technologies GmbH) has been used. The sample size for μ-CT
scanning is 2 mm diameter and 4 mm length. For pore network separation, a
combination of manual thresholding and watershed algorithms has been applied
to achieve the qualitatively best separated pore space. Additionally,
separation results have been cross checked with the images of scanning
electron microscopy (SEM). More details on the DIA workflow can be found in
Halisch et al. (2016a). The DIA of the 3-D μ-CT data sets provide, for
each individual pore, the volume and the pore radius of the largest sphere
that can be placed inside this pore (maximum inscribed sphere method; e.g.,
Silin and Patzek, 2006) as indicated by the blue circles in Fig. 1. Note that
Fig. 1 displays a 2-D slice with circles. The DIA is performed in 3-D volumes
and provides spheres. The resulting equivalent pore radius is referred to as
pore body radius rb. Although the true extent of the pore is not
caught properly, the derived rb from DIA is a good estimate of
the average radius. Adding up the pore volumes starting with the lowest pore
radius yields the cumulative volume fraction of pores Vc (Eq. 1)
as a function of the pore body radius rb. The μ-CT method can
only resolve the part of the pore space with pore sizes larger than the
spatial resolution of the 3-D image. Considering a voxel size of 1.75 µm of the 3-D data set, and a minimum extension of pores of 2 voxels in
one direction, which can be separated by the algorithm, a minimum pore size
of 3.5 µm (or minimum pore radius of 1.75 µm) has to be
regarded; as for this study, the CT resolution limit is 1.75 µm.
Therefore, the pore volume determined by μ-CT does not take into account
the pore space with radii smaller than 1.75 µm.
SEM (a) and 2-D (c) and 3-D (e) CT views
on the minerals and pore structure of the investigated sample of Bentheimer
sandstone, and SEM (b) and 2-D (d) and 3-D (f) CT
views on the minerals and pore structure of the investigated sample of
Röttbacher sandstone. Blue arrows indicate open-pore spaces, red arrows
indicate clay agglomerations and pore fillings.
The MIP experiments have been conducted with the PASCAL 140/440 instrument
from Thermo Fisher (Mancuso et al., 2012), which covers a pressure range
between 0.015 and 400 MPa corresponding to a pore throat radius range
from (at best) 1.8 nm to 55 µm. The samples have been evacuated
before the MIP experiment.
The NMR experiments have been performed with a Magritek rock core analyzer
instrument operating at a Larmor frequency of 2 MHz at room temperature
(∼20∘C) and ambient pressure. After drying at 105 ∘C
for more than 24 h in vacuum, the samples have been fully saturated with tap
water with a conductivity of about 25 mS m-1. NMR measurements can be
calibrated to get the porosity of the sample. The early time decay signal
corresponds to the total water content. The range of resolved pore body radii
depends on the used value of surface relaxivity. The amplitude b attributed
to an individual relaxation time T2 is related to the volume fraction of
pores with the respective pore radius. Considering the larger pores, the
resulting radius corresponds to rb. The smaller pore throats with
lower volume yield a lower signal at shorter relaxation times. The cumulative
volume fraction of pores Vc is determined by adding up the
individual b values starting from the smallest relaxation time and
normalizing to the total sum of all b values.
Complex conductivity spectra were recorded using a four-electrode sample
holder as described by Schleifer et al. (2002). The spectra were acquired
with the impedance spectrometer ZEL-SIP04 (Zimmerman et al., 2008) in a
frequency range between 0.002 Hz and 45 kHz at a constant temperature of
about 20 ∘C. Considering that the complex conductivity spectra are
affected by electromagnetic coupling effects, Maxwell–Wagner relaxation and
dielectric effects at higher frequencies and by a lower signal-to-noise ratio
for lower frequencies, we focus on the frequency range between 0.01 and
100 Hz. The samples were fully saturated with a sodium-chloride solution
with a conductivity of 100 mS m-1. At least two measurements were
performed for each sample to verify the repeatability. Considering the
limited frequency interval, the SIP method solely resolves a range of pore
radii that depends on the diffusion coefficient. Hence, using D(+)=3.8×10-12 m2 s-1 in Eq. (6), we get a range of pore radii
between 0.1 and 10 µm. Smaller pore sizes are hidden by Maxwell
–Wagner relaxation and dielectric effects that are not easily related to
pore geometry.
Permeability measurements have been performed by using a steady-state gas
permeameter (manufactured by Westphal Mechanik, Celle, Germany), using
nitrogen as the flowing fluid. This device features a so-called
“Fancher-type” core holder as described by Rieckmann (1970). With this
special type of core holder, significantly lower confining pressures are
needed than by using a conventional “Hassler-type” core holder (12 bar for
the “Fancher-type” core holder versus min. 35–50 bar for the
“Hassler-type” core holder), leading to much less initial mechanical
influence (compaction) on the sample material. Measurements have been derived
under steady-state flow conditions with accordingly low flow rates in the
range from 3 to 5 mL min-1, leading to measured pressure differentials
in the range from 2 to 7 mbar from sample inlet to outlet. The derived
apparent permeability values have been corrected to address the Klinkenberg
effect of gas slippage (Klinkenberg, 1941; API, 1998). Due to the usage of a
steady-state technique with low gas-flow rates, we consider that correction
of the Forchheimer effect of inertial resistance can be neglected (API,
1998).
Petrophysical properties of the samples: porosity ϕ,
permeability K, specific surface area Sm, formation factor F,
dominant pore radius rdom, effective pore radius
reff, the ratio rb/rt, fractal dimensions
determined from mercury intrusion porosimetry DMIP, nuclear
magnetic resonance DNMR, spectral-induced polarization
DSIP, the surface relaxivity ρ, and the diffusion
coefficient D(+).
Figure 2 (panels a and c) gives 2-D impressions of the pore system of the
Bentheimer sandstone sample. The pore space in general is very well
connected, featuring many large and open pores (Fig. 2a and c, blue arrows)
and can be described as a classical pore body–pore throat–pore body system.
Small pores are mostly found within the clayey agglomerations, which act as
(macro) pore-filling material (Fig. 2a and c, red arrows) and which are
homogeneously distributed throughout the sample material. Figure 2e gives an
impression of the 3-D pore distribution of this sandstone, derived by
μ-CT image processing. This favorable structure is directly reflected by
the petrophysical properties of this sandstone. The sample investigated in
our study is characterized by a porosity of 0.238 measured by MIP, a gas
permeability of 4.25×10-13 m2, and a specific surface area of
0.3 m2 g-1 determined by the nitrogen adsorption method.
Chemical components of the samples from X-ray fluorescence
analysis.
Selected chemical components from X-ray fluorescence (wt %) SampleSiO2TiO2Al2O3Fe2O3CaONa2OK2OBH5-297.840.0481.20.050.0190.020.355RÖ10B87.060.3566.061.070.2250.133.679
Figure 2 (panels b and d) shows the pore space of the Röttbacher
sandstone sample from 2-D imaging techniques. Although the (large) pore space
is similarly structured as it is for the Bentheimer (pore body–throat–body
system; Fig. 2b and d, blue arrows), it is generally reduced (cemented) by
clay minerals and features a significantly higher amount of small pores
within (Fig. 2b and d, red arrows). Accordingly, pore-space-related
petrophysical properties classify a more compact rock, which is supported by
the 3-D pore distribution, derived by μ-CT image processing (Fig. 2f).
The sample used for this study features a porosity of 0.166 measured by MIP,
which is lower than for the Bentheimer sandstone. The gas permeability is
3.45×10-14 m2, which is less than 10 % of the value
determined for the Bentheimer sandstone. The specific surface area has been
measured as 1.98 m2 g-1 and is hence nearly 7 times larger than for
sample BH5-2, clearly underlining the impact of the clay content. The
petrophysical parameters for both samples are compiled in Table 1, whereas
results from X-ray fluorescence analysis are summarized in Table 2, regarding
the most important chemical components of both sandstones that have been used
for this study.
The recognized porosity and pore size range of Bentheimer sandstone
sample BH5-2. The maximum porosity recognized by MIP is 0.238 and the maximum
porosity recognized by μ-CT is 0.184.
Pore volume fraction
We applied the μ-CT, MIP, NMR, and SIP methods to get insight into the
pore radius distribution of the Bentheimer sandstone sample BH5-2. Figure 3
displays the resolved porosity ϕr as a function of pore radius
for μ-CT and MIP data. The cumulative pore volume while progressing from
larger to smaller pores V(>r) is normalized to the total volume of the
sample Vs and results in the resolved porosity
ϕr=V(>r)V,
which reaches the true porosity ϕ as threshold value for r approaching
zero.
As shown in Fig. 3, the μ-CT method identifies the largest pores with
pore body radii of about 100 µm. The resolved porosity
ϕr reaches a value of 0.184 at the limit of resolution of the
μ-CT method (rb=1.75µm). The nearly horizontal
curve progression for r<17µm indicates that effectively no
significant volume of pores with radii lower than 17 µm was
detected or quantified by μ-CT or DIA, respectively. Accordingly, only
μ-CT data for r>17µm will be taken into account for
further analysis.
The recognized porosity and pore size range of Röttbacher
sandstone sample RÖ10B. The maximum porosity recognized by MIP is 0.166
and the maximum porosity recognized by μ-CT is 0.106.
The MIP identifies the largest pore throats with a radius of about
30 µm. Reaching the limit of resolution of the MIP, the resolved
porosity asymptotically approaches the threshold value of 0.238. Although
both methods, μ-CT and MIP, yield the pore radius without any adjustable
scaling factor, we observe differences between the two curves ϕr(r)
in Fig. 3.
Geometrical parameters of individual pores derived from μ-CT
data of the two samples.
The Röttbacher sample was scanned with 1.5 µm resolution by
μ-CT. As shown in Fig. 4, the μ-CT method identifies the largest
pores with pore body radii of about 90 µm. The resolved porosity
ϕr reaches a value of 0.106 at the limit of resolution of the
μ-CT method (rb=1.5µm). As observed for the
Bentheimer sandstone, the nearly horizontal curve progression for r<10µm indicates that no significant volume of pores with radii lower
than 10 µm was detected or quantified by μ-CT or DIA,
respectively. Accordingly, only μ-CT data for r>10µm will
be taken into account for further analysis.
The comparison of Vc-r curves determined from MIP,
μ-CT, NMR, and SIP for Bentheimer sandstone sample BH5-2.
Pore radius distribution
The description and quantification of the pore space in three dimensions
requires morphological parameters such as length, width, and thickness of
individual pore segments. The parameters are extracted by image analysis
software from 3-D μ-CT data. We determined the pore length (maximum
length of Feret distribution), pore width (minimum width of Feret
distribution), and the equivalent diameter of the analyzed pore segment that
corresponds to the spherical diameter with equal voxel volume (Schmitt et
al., 2016). The minima, maxima, and mean values of the geometrical parameters
derived from μ-CT data of the two samples are compiled in Table 3.
The procedures described above result in an individual curve displaying the
logarithm of Vc versus the logarithm of the pore radius for each
method.
The NMR T2 relaxation time distributions of samples BH5-2 and
RÖ10B.
For Bentheimer sandstone, applying the transformation in Eq. (3) for the MIP
data and assuming a true porosity of 0.238, the cumulative volume fraction of
pores Vc can be displayed as a function of pore throat radius as
shown in Fig. 5. The MIP curve gets a fixed position in the plot of Fig. 5
without the need for any scaling. It covers a wide range of pore throat radii
between 0.0018 and 44.7 µm.
The curves resulting from other methods have to be adjusted considering the
limits of the range of pore radii. The maximum of the μ-CT curve
corresponds to Vc=1 because no larger pore size has been detected
by other methods. The maximum resolved porosity of the sample as detected by
MIP reaches 0.238. The porosity determined by μ-CT reaches only 0.184
(Fig. 3). This value corresponds to a fraction of 0.773 of the porosity
determined by MIP. Therefore, the minimum of the μ-CT curve at the pore
radius of 17 µm has to be adjusted to Vc=1-0.773=0.227, because this fraction of pore volume is related to pore body radii
smaller than 17 µm. The shift of the μ-CT curve to larger pore
body radii in comparison with MIP is observed in this plot too.
Measured complex conductivity spectra of samples BH5-2 and
RÖ10B; (a) real part of conductivity; (b) imaginary
part of conductivity.
The T2 relaxation time distribution of sample BH5-2 is plotted in
Fig. 6. It indicates a distinct maximum at a relaxation time of 330 ms and
two weaker maxima at lower relaxation times. The T2 relaxation time
distribution is transformed into a curve showing the cumulative intensity as
a function of T2. The total intensity is attributed to the total pore
volume. The volume fraction Vc corresponds to the ratio of
cumulative intensity to total intensity. In order to get the curve
Vc as a function of pore radius, the relaxation time T2 has
to be transformed into a pore radius using the surface relaxivity ρ as
scaling factor in Eq. (5). Since both μ-CT and NMR methods are sensitive
to the pore body radius, we expect a similar Vc-r curve in the
overlapping range of pore body radii. Assuming a coincidence of the two
curves at Vc=0.5, the surface relaxivity is adjusted at ρ=54µm s-1.
The comparison of Vc-r curves determined from MIP,
μ-CT, NMR, and SIP for Röttbacher sandstone sample RÖ10B.
The complex conductivity spectra of the Bentheimer sample are displayed in
Fig. 7. Considering the frequency range between 0.01 and 100 Hz and D(+)=3.8×10-12 m2 s-1, the relaxation time
distribution derived from SIP is attributed to a restricted range of pore
radii between 0.1 and 10 µm. Assuming that the polarization signals
originate from the pore throats, a similarity of pores size distributions
resulting from MIP and SIP can be expected. It should be noted that MIP
provides the distribution for a wider range of pore throat radii. Therefore,
we adjust the value of Vc at the maximum radius of the SIP to the
corresponding value for the MIP curve.
As shown in Fig. 4 for Röttbacher sandstone, the MIP identifies the
largest pore throats with a radius of about 50 µm. Reaching the
limit of resolution of MIP, the resolved porosity has a value of 0.166.
Applying the transformation in Eq. (3) on the MIP data and assuming a true
porosity of 0.166, the cumulative volume fraction of pores Vc is
displayed as a function of pore radius as shown in Fig. 8.
We suppose that the MIP method detects the whole pore volume, a porosity of
0.106 recognized by μ-CT corresponds to 63.9 % of the total pore
volume. Therefore, the minimum of the μ-CT curve at the pore radius of
10 µm has to be adjusted at Vc=1-0.639=0.361,
because this fraction of pore volume is related to pore body radii smaller
than 10 µm.
The T2 relaxation time distribution of sample RÖ10B is plotted in
Fig. 6. It indicates a distinct maximum at a relaxation time of 170 ms.
Non-vanishing signals are observed at relaxation times below 0.1 ms. This is
an indication of the existence of very small pores in the Röttbacher
sandstone.
The position of the NMR curve in the plot of Fig. 8 depends on the surface
relaxivity ρ. A coincidence with the μ-CT curve at Vc=0.5 requires a surface relaxivity of ρ=237µm s-1 for
adjusting the NMR curve.
The complex conductivity spectra of the Röttbacher sample are displayed
in Fig. 7. The processing of the spectra according to the described algorithm
results in the Vc-r curve as shown in Fig. 8. The SIP curve is
fixed at the value Vc=0.9 that has been determined by MIP for
the maximum pore radius resolved by SIP (rt=10µm).
Discussion
Previous studies have compared the Vc-r curves resulting from
different methods (e.g., Zhang and Weller, 2014; Zhang et al., 2017; Ding et
al., 2017). The slope of the curves was used to get a fractal dimension. It
became obvious that the distribution curves indicate remarkable differences
that are caused by the physical principles of the used methods. The methods
differ with regard to their limits of resolution. The effective resolution of
μ-CT is limited by the voxel size. Larger pores can be easily detected.
Nevertheless, even though the derived image (voxel) resolution is quite high
(1.75 µm), both sandstone data sets feature no significant volume
of pore body radii smaller than 10 µm (BH5-2) or 17 µm
(RÖ10B). We assume that this is caused by a complex and sensitive mixture
of issues related to image resolution, image quality (phase contrast),
reliability of the watershed-algorithm concerning the separation of
individual pores, and hence of the complexity of the pore structure of small
pores. The MIP yields the widest range of pore throat radii. The pore radius
is directly related to the pressure. A similarly wide range of pore body
radii can be resolved by NMR. However, the transformation of the NMR
transversal relaxation time into a pore radius requires the surface
relaxivity as a scaling factor. In a similar way, the transformation of the
electrical relaxation time resulting from SIP into a pore radius is based on
a scaling factor that depends on the diffusion coefficient. Only a restricted
range of pore radii can be resolved by SIP.
Besides the range of pore radii, the geometrical extent of the pore radius
differs among the methods. μ-CT enables a geometrical description of the
individual pore space considering the shape of the pore. The pore radius can
be determined in different ways. We use the average pore radius as an
equivalent for the pore body radius rb. MIP is sensitive to the
pore throat radius rt that enables access to larger pores behind
the throat. The NMR relaxation time is related to the pore body radius
rb. We assume that the IP signals are caused by the ion-selected
active zones in the narrow pores that are comparable with the pore throats.
Regarding the differences in the methods, we present an approach that
combines the curves to get more information on the pore space. Considering
the two kinds of pore radii, rb and rt, we first use
μ-CT and NMR to generate a combined curve displaying Vc as a
function of rb. In the next step, we link the curves resulting
from MIP and SIP to get a curve showing Vc as a function of the
pore throat radius rt.
It is fundamental that the total pore volume (or total porosity) has to be
known. The cumulative pore volume fraction should only consider the pore
volume that is resolved in the regarded range of pore radii. Considering the
resolution of μ-CT, only the pore space with radii larger than the voxel
size is determined. The cumulative pore volume fraction at the limit of
resolution has to be adjusted to the unresolved pore volume.
In this way, the μ-CT curve gets a fixed position in the Vc-r plot. Regarding NMR, the relaxation time T2 has to be transformed
into a pore radius according to Eq. (5). The application of Eq. (5) requires
knowledge of the surface relaxivity ρ, which is the necessary scaling
factor that causes a shift of the Vc-r curve along the axis of
pore radius. Since the NMR method is sensitive to the pore body radius, we
expect a similar Vc-r curve for NMR and μ-CT in the
overlapping range of pore body radii. The NMR curve is shifted along the axis
of pore body radii until a good agreement between the two curves is reached.
This procedure enables the determination of the surface relaxivity. The
proposed alternative method for the determination of surface relaxivity
considers the reduction of NMR relaxation time T2 caused by high clay
content and iron-bearing minerals (e.g., Keating and Knight, 2010).
MIP is used to generate the curve displaying Vc as a function of
rt over a wide range of pore throat radii. The SIP curve is fixed
at the MIP curve considering the coincidence at the largest pore radius
resolved by SIP.
The two curves representing Vc as a function of both
rb and rt are displayed in a double-logarithmic plot.
The horizontal shift of the two graphs represents the ratio
rb/rt. Additionally, the slope of the curves is related
to the fractal dimension.
The proposed approach in this study results in two pore size distribution
curves for the two samples, which are in good accordance to the general pore
space structures as described in Sect. 3 and as visualized in Fig. 2
(panels a to f). The first curve combines the distributions resulting from
μ-CT and NMR. The μ-CT data provide a pore radius, which is regarded
as pore body radius, without any scaling. The scaling of the NMR curve
provides an estimate of the surface relaxivity. The surface relaxivity of the
Bentheimer sample reaches 54 µm s-1, the corresponding value
of the Röttbacher sample is much higher with 237 µm s-1.
The higher surface relaxivity in comparison with the Bentheimer sample is
clearly justified considering the larger specific surface area (Table 1) and
the significantly higher content of clay and iron-bearing minerals as
indicated in Table 2.
The two cumulative pore volume distribution curves for the Röttbacher
sample (Fig. 8) indicate, over the wide range of pore radii, a parallel
progression with consistently higher values for the pore body radius
(μ-CT and NMR) in comparison with the pore throat radius (MIP). The
horizontal distance of the two curves yields the ratio
rb/rt. It should be noted that the ratio
rb/rt may vary with pore sizes. Most studies only
consider a fixed ratio rb/rt determined from the
dominant pore body size from NMR rb and the dominant pore throat
size from MIP rt (e.g., Müller-Huber et al., 2018). Regarding
the median pore radii at Vc=0.5, a ratio
rb/rt=9.13 is determined. Considering smaller pores,
a ratio rb/rt=12.15 is indicated at Vc=0.05.
The parallelism of the pore volume distribution curve is less developed for
the Bentheimer sample (Fig. 4). We observe a clear distance between the two
curves in the range of larger pore radii. Regarding the median pore radii at
Vc=0.5, a ratio rb/rt=2.57 is
determined. For Vc<0.2, the slopes of the curves decrease and
smaller distances between the curves are observed. The NMR curve in Fig. 5
indicates, for Vc>0.08, larger pore radii in comparison with
the MIP curve and confirms the relationship rb>rt.
The reverse behavior in the interval 0.1 µm <r<0.6µm is possibly caused by the low volume fraction (3 %)
attributed to this range of pore radii. It can be expected that the small
amount of water in the small pores causes only weak signals in the NMR
relaxometry.
Besides the distances between the curves, the individual slopes are regarded.
The slope (s) of the curve log (Vc) versus log (r) is related
to the fractal dimension D of the pore volume (D=3-s) (Zhang and
Weller, 2014). We observe a varying slope in the investigated range of pore
radii for the Bentheimer sample. The only range of more or less constant
slope, which extends from pore radius 0.1 to 10 µm, corresponds to
a fractal dimension DMIP=2.678 for MIP, DNMR=2.776 for NMR, and DSIP=2.618 for SIP.
The whole curves of the four methods are nonlinear and indicate non-fractal
behavior. A maximum likelihood estimator approach (MLE) might be relevant to
extract the underlying scaling parameters (Rizzo et al., 2017). For example,
in the case of the NMR curve of Bentheimer sandstone, the fitting of all data
using the MLE reveals that the log-normal distribution is the most likely
distribution with the estimated parameters μ=3.43 and σ=0.82µm. These two scaling parameters are the logarithmic mean and
logarithmic standard deviation of the pore radius, respectively. We recognize
that the resulting mean radius reaches half the value of the effective
hydraulic radius (reff=6.97µm).
We observe a constant slope of the NMR curve for the Röttbacher sample
(Fig. 8) in the interval 0.01 µm <rb<100µm. A similar slope is observed for the MIP curve in the
interval 0.01 µm <rt<10µm.
Considering the overlapping pore throat radii range between 0.1 and
10 µm, a fractal dimension D with values of 2.640 for MIP, and
2.661 for NMR has been determined. The slightly higher slope of the SIP curve
results in a lower value of fractal dimension of D=2.533.
Our approach enables the integration of SIP in the determination of a pore
throat size distribution. Considering the limited frequency range, only a
limited range of pore throat radii can be reflected. Using a fixed diffusion
coefficient D(+)=3.8×10-12 m2 s-1, a range of pore
throat radii between 0.1 and 10 µm is resolved. The SIP curve is
linked to the MIP curve at r=10µm. An extension to lower pore
radii would require the integration of higher frequencies. The removal of
electromagnetic coupling effects can be one first step to improve the
reliability of complex conductivity spectra for frequencies larger than
100 Hz, but it should be regarded that smaller pore sizes are hidden by
Maxwell–Wagner and dielectric relaxation. The proposed procedure results in
a fair agreement between SIP and MIP curves in the overlapping range of pore
throat radius for both the Bentheimer and the Röttbacher sample. In
comparison with MIP, a slight overestimation of Vc is observed
for larger pore throat radii and a underestimation for lower pore throat
radii. Considering the two samples of the presented study, the assumption of
a constant diffusion coefficient seems to be justified. Although alternative
kernels have not been tested, our study confirms that the Debye decomposition
provides a relaxation time distribution of complex conductivity spectra that
can be transformed in a pore throat size distribution comparable with the
resulting curves from MIP. Regarding the discussion on the most relevant
parameter that controls the relaxation time, our assumption that the pore
throat radius is related to the relaxation time is supported by the results.
The investigations by μ-CT, MIP, NMR, and SIP on the sandstone samples
have been done in the laboratory. μ-CT and MIP are methods that can only
be applied on rock samples. The potential of these methods to derive pore
size distributions is well acknowledged. NMR and SIP are methods that can
also be performed in boreholes or as field survey. The NMR method has been
successfully applied in permeability prediction at the field scale. A variety
of permeability prediction models based on SIP parameters has been proposed
based on laboratory investigations (e.g., Robinson et al., 2018). First tests
have demonstrated their applicability in the field. Most permeability models
consider pore size and porosity as the most important parameters. The
evaluation of pore sizes of sediments at the field scale is a challenging
task for geophysical methods. Our laboratory study has demonstrated the
potential of SIP in identifying a pore size distribution. Further
investigations with larger sets of samples have to be done to improve the
proposed procedure before the pore size distribution can be extracted from
high-quality complex conductivity field spectra.
Conclusions
Pore radius distributions (considering both pore body and pore throat radii)
have been determined by different methods (μ-CT, MIP, NMR, and SIP) for
two sandstone samples. The curves presenting the cumulative distribution of
pore volume Vc as a function of pore size have proved to be a
suitable tool for comparison. It becomes obvious that the distribution curves
indicate remarkable differences that are based on the physical principles of
the used methods. The methods differ with regard to their limits of
resolution. The effective resolution of μ-CT is limited by the voxel size
(1.75 µm). Larger pores can be easily detected, whereas
quantification of small pores and volumes of pores with small radii is
severely affected by the image quality and the image processing algorithms.
The MIP yields the widest range of pore radii. The pore throat radii are
directly related to the pressure interval. A similar wide range of pore radii
can be achieved by NMR. However, the transformation of the NMR transversal
relaxation time into a pore body radius requires the surface relaxivity as
scaling factor. In a similar way, the transformation of the electrical
relaxation time resulting from SIP into a pore radius is based on a scaling
factor that depends on the diffusion coefficient. Only a restricted range of
pore radii (0.1 to 10 µm) can be resolved by SIP.
Besides the range of pore radii, the geometrical extent of the pore radius
differs among the methods. μ-CT enables a geometrical description of the
individual pore space considering the shape of the pore. The pore radius can
be determined in different ways. We use the average pore radius as an
equivalent for the pore body radius rb. MIP is sensitive to the
pore throat radius rt that enables access to larger pores behind
the throat. The NMR relaxation time is related to an average pore body radius
rb. We assume that the IP signals are caused by the ion-selected
active zones in the narrow pores that are comparable with the pore throats.
Considering the two kinds of pore radii rb and rt, we
use μ-CT and NMR to generate a combined curve displaying Vc
as a function of rb. A good agreement between the two curves is
achieved if they coincide at Vc=0.5. This condition is used to
determine the surface relaxivity, which is in good accordance to the
investigated surface area and mineralogy of the sample materials. MIP is used
to generate the curve displaying Vc as a function of
rt over a wide range of pore throat radii. The SIP curve is fixed
at the MIP curve considering the coincidence at the largest pore radius
resulting from SIP.
The two curves representing Vc as a function of both
rb and rt are displayed in a double-logarithmic plot.
The horizontal shift of the two graphs represents the ratio
rb/rt. Additionally, the slope of the curves is related
to the fractal dimension.
The investigations on the samples demonstrate that the porosity increases
using a method with a higher resolution. Both porosity and pore volume are
parameters that depend on the resolution. The fractal dimension describes the
size of geometric objects as a function of resolution. Therefore, the
knowledge of fractal behavior enables upscaling and downscaling of geometric
quantities. The Bentheimer sandstone sample is characterized by a ratio
rb/rt=2.57 for the larger pores. A fractal behavior
is observed in the range of pore radii between 0.1 and 10 µm with
an average D=2.69 determined for the pore volume by MIP, NMR, and SIP.
The Röttbacher sandstone sample indicates a larger ratio between pore
body radius and pore throat with rb/rt=9.13 in
comparison to the Bentheimer sample. An
average fractal dimension of D=2.61 is determined for the Röttbacher
sample.
The SIP-Archiv repository has been developed by Halisch et
al. (2016c, 2017) in collaboration with the Working Group “Induced
Polarization” (AK IP) of the German Geophysical Society (DGG).
The four authors have designed the project in close cooperation.
SK has selected and provided the samples for the study. The measurements have
been done in the labs of the institutions of MH, SK, and AW. MH provided the
DIA of the μ-CT data. ZZ and AW developed the algorithm to integrate the
pore size distribution from SIP data. The four authors contributed to the
discussion of the results and to writing and improving the manuscript. ZZ
designed all figures.
The authors declare that they have no conflict of
interest.
Acknowledgements
The authors thank Sven Nordsiek (University Bayreuth) for the Debye
decomposition of the SIP data, Dietmar Meinel (BAM, Berlin) for supporting
the CT analysis, Carsten Prinz (BAM, Berlin) for providing the MIP data, and
Mike Müller-Petke as well as Raphael Dlugosch (both Leibniz Institute for
Applied Geophysics, Hanover) for the acquisition of the NMR spectra for this
study. Zeyu Zhang thanks Bundesanstalt für Materialforschung und
-prüfung (BAM, Berlin) for the Adolf-Martens-Fellowship that enabled his
stay in Germany for the experimental research.
Edited by: Michael Heap Reviewed by: Alodie Bubeck, David
Healy, and two anonymous referees
References
API (American Petroleum Institute): Recommended Practices for Core Analysis,
API Recommended Practice 40, chap. 6 – permeability determination, 2nd Edn.,
API Publishing Services, 1220 L Street, N.W., Washington, DC, 1998.
Avnir, D. and Jaroniec, M.: An isotherm equation for adsorption on fractal
surfaces of heterogeneous porous materials, Langmuir, 5, 1412–1433, 1989.
Avnir, D., Farin, D., and Pfeifer, P.: Molecular fractal surfaces, Nature,
308, 261–263, 1984.Behroozmand, A., Keating, K., and Auken, E.: A review of the principles and
applications of the NMR technique for near-surface characterization, Surv.
Geophys., 36, 27–85, 10.1007/s10712-014-9304-0, 2015.Binley, A., Slater, L. D., Fukees, M., and Cassiani, G.: Relationship between
spectral induced polarization and hydraulic poroperties of saturated and
unsaturated sandstone, Water Resour. Res., 41, W12417,
10.1029/2005WR004202, 2005.Ding, Y., Weller, A., Zhang, Z., and Kassab, M.: Fractal dimension of pore
space in carbonate samples from Tushka Area (Egypt), Arab. J. Geosci., 10,
388, 10.1007/s12517-017-3173-z, 2017.
Dubelaar, W. C. and Nijland, T. G.: The Bentheim Sandstone: geology,
petrophysics, varieties and its use as dimension stone, in: Engineering
Geology for Society and Territory, edited by: Lollino, G., Giordan, D.,
Marunteanu, C., Christaras, B., Yoshinori, I., and Margottini, C., Springer
International Publishing, Switzerland, Vol. 8, 557–563, 2015.
Florsch, N., Revil, A., and Camerlynck, C.: Inversion of generalized
relaxation time distributions with optimized samping parameter, J. Appl.
Geophys., 109, 119–132, 2014.Gaboreau, S., Robinet, J. C., Tournassat, C., and Savoye, S.: Diffuse
transport in clay media: µm to nm scale characterization of pore
space and mineral spatial organization: International Meeting Clays in
Natural and Engineered Barriers for Radioactive Waste Confinement,
Montpellier, France, 2012.Halisch, M., Schmitt, M., and Fernandes, C. P.: Pore Shapes and Pore Geometry
of Reservoirs Rocks from μ-CT Imaging and Digital Image Analysis, in:
Proceedings of the Annual Symposium of the SCA 2016, Snowmass, Colorado, USA,
21–26 August 2016, SCA2016-093, 2016a.Halisch, M., Steeb, H., Henkel, S., and Krawczyk, C. M.: Pore-scale
tomography and imaging: applications, techniques and recommended practice,
Solid Earth, 7, 1141–1143, 10.5194/se-7-1141-2016, 2016b.Halisch, M., Gramenz, J., Gorling, L., Krause, K., and Bolotovski, I.: An
internet based, interactive archive and database for SIP data, 4th
International Workshop on Induced Polarization, Aarhus, Denmark,
http://www.sip-archiv.de (last access: 1 November 2018), 2016c.
Halisch, M., Kruschwitz, S., Martin, T., and SIP-Archiv Entwickler-Team: Ein
internetbasiertes Archiv- und Austauschsystem für Messdaten der
Spektralen Induzierten Polarisation. Mitteilungen der Deutschen
Geophysikalischen Gesellschaft e.V., 1/2017, ISSN 0934-6554, p. 31 ff., 2017.
Keating, K. and Knight, R.: A laboratory study of the effect of
Fe(II)-bearing minerals on nuclear magnetic resonance (NMR) relaxation
measurements, Geophysics, 75, F71–F82, 2010.Keller, L. M., Holzer, L., Wepf, R., Gasser, P., Münch, B., and
Marschall, P.: On the application of focused ion beam nanotomography in
characterizing the 3D pore space geometry of Opalinus clay, Phys. Chem.
Earth, 36, 1539–1544, 10.1016/j.pce.2011.07.010, 2011.Kelokaski, M., Siitari-Kauppi, M., Sardini, P., Mori, A., and Hellmuth, K.
H.: Characterisation of pore space geometry by 14C-PMMA
impregnation-development work for in situ studies, J. Geochem. Explor., 90,
45–52, 10.1016/j.gexplo.2005.09.005, 2005.Kleinberg, R. L.: Utility of NMR T2 distributions, connection with
capillary pressure, clay effect, and determination of the surface relaxivity
parameter ρ2, Magn. Reson. Imaging, 14, 761–767, 1996.
Klinkenberg, L. J.: The permeability of porous media to liquids and gases,
API Drill and Production Practices, 200–213, 1941.
Kruschwitz, S., Prinz, C., and Zimathies, A.: Study into the correlation of
dominant pore throat size and SIP relaxation frequency, J. Appl. Geophys.,
135, 375–386, 2016.Leroy, P., Revil, A., Kemna, A., Cosenza, P., and Gorbani, A.: Spectral
induced polarization of water-saturated packs of glass beads, J. Colloid
Interf. Sci, 321, 103–117, 10.1016/j.jcis.2007.12.031, 2008.
Mandelbrot, B. B.: Fractals: form, chance, and dimension, Freeman, San
Francisco, 1977.
Mandelbrot, B. B.: Fractal geometry of nature, Freeman, San Francisco, 1983.Mancuso, C., Jommi, C., and D'Onza, F. (Eds.): Unsaturated Soils: Research
and Applications, Vol. 1, Springer-Verlag, Berlin Heidelberg, 123–130,
10.1007/978-3-642-31116-1, 2012.Mees, F., Swennen, R., van Geet, M., and Jacobs, P. (Eds.): Applications of
X-ray computed tomography in the geosciences, Geol. Soc. Spec. Publ., 215,
1–6, 10.1144/GSL.SP.2003.215.01.01, 2003.
Meyer, K., Klobes, P., and Röhl-Kuhn, B.: Certification of reference
material with special emphasis on porous solids, Cryst. Res. Technol., 32,
173–183, 1997.Müller-Huber, E., Börner, F., Börner, J. H., and Kulke, D.:
Combined interpretation of NMR, MICP, and SIP measurements on mud-dominated
and grain-dominated carbonate rocks, J. Appl. Geophys., 159, 228–240,
10.1016/j.jappgeo.2018.08.011, 2018.Niu, Q. and Revil, A.: Connecting complex conductivity spectra to mercury
porosimetry of sedimentary rocks, Geophysics, 81, E17–E32,
10.1190/GEO2015-0072.1, 2016.Niu, Q. and Zhang, C.: Joint inversion of NMR and SIP data to estimate pore
size distribution of geomaterials, Geophys. J. Int., 212, 1791–1805,
10.1093/gji/ggx501, 2017.Nordsiek, S. and Weller, A.: A new approach to fitting induced-polarization
spectra, Geophysics, 73, F235–F245, 10.1190/1.2987412, 2008.
Pape, H., Riepe, L., and Schopper, J. R.: A pigeon-hole model for relating
permeability to specific surface, Log Analyst., 23, 5–13, 1982.
Pape, H., Arnold, J., Pechnig, R., Clauser, C., Talnishnikh, E., Anferova,
S., and Blümlich, B.: Permeability prediction for low porosity rocks by
mobile NMR, Pure Appl. Geophys., 166, 1125–1163, 2009.Peksa, A., Wolf, K., and Zitha, P.: Bentheimer sandstone revisited for
experimental purposes, Mar. Petrol. Geol., 67, 701–719,
10.1016/j.marpetgeo.2015.06.001, 2015.Peth, S., Horn, R., Beckmann, F., Donath, T., Fischer, J., and Smucker, A. J.
M.: Three-dimensional quantification of intra-aggregate pore-space features
using Synchrotron-Radiation-Based Microtomography, Soil Sci. Soc. Am. J., 72,
897–907, 10.2136/sssaj2007.0130, 2008.Revil, A.: Effective conductivity and permittivity of unsaturated porous
materials in the frequency range 1 mHz–1 GHz, Water Resour. Res., 49,
306–327, 10.1029/2012WR012700, 2013.
Revil, A. and Florsch, N.: Determination of permeability from
spectral-induced-polarization data in granular media, Geophys. J. Int., 181,
1480–1498, 2010.Revil, A., Koch, K., and Holliger, K.: Is it the grain size or the
characteristic pore size that controls the induced polarization relaxation
time of clean sands and sandstones?, Water Resour. Res., 48, W05602,
10.1029/2011WR011561, 2012.Revil, A., Florsch, N., and Camerlynck, C.: Spectral induced polarization
porosimetry, Geophys. J. Int., 198, 1016–1033, 10.1093/gji/ggu180, 2014.
Rieckmann, M.: Untersuchung von Turbulenzerscheinungen beim Fließen von
Gasen durch Speichergesteine unter Berücksichtigung der
Gleitströmung, Erdöl-Erdgas-Zeitschrift, 6, 36–51, 1970.Rizzo, R. E., Healy, D., and De Siena, L.: Benefits of maximum likelihood
estimators for fracture attribute analysis: Implications for permeability and
up-scaling, J. Struct. Geol., 95, 17–31, 10.1016/j.jsg.2016.12.005,
2017.Robinson, J., Slater, L., Weller, A., Keating, K., Robinson, T., Rose, C.,
and Parker, B.: On permeability prediction from complex conductivity
measurements using polarization magnitude and relaxation time, Water Resour.
Res., 54, 3436–3452, 10.1002/2017WR022034, 2018.
Rouquerol, J., Avnir, D., Fairbridge, D. C. W., Everett, D. H., Haynes, J.
H., Pernicone, N., Ramsay, J. D. F., Sing, K. S. W., and Unger, K. K.:
Recommendations for the characterization of porous solids (Technical Report),
Pure Appl. Chem., 66, 1739–1758, 1994.Schleifer, N., Weller, A., Schneider, S., and Junge, A.: Investigation of a
Bronze Age plankway by spectral-induced-polarization, Archeological
Prospection, 9, 243–253, 10.1002/arp.194, 2002.Schmitt, M., Halisch, M., Müller, C., and Fernandes, C. P.:
Classification and quantification of pore shapes in sandstone reservoir rocks
with 3-D X-ray micro-computed tomography, Solid Earth, 7, 285–300,
10.5194/se-7-285-2016, 2016.Schwarz, G.: A theory of the low-frequency dielectric dispersion of colloidal
particles in electrolyte solution, J. Phys. Chem., 66, 2636–2642,
10.1021/j100818a067, 1962.
Scott, J. B. T. and Barker, R. D.: Determining pore-throat size in
Permo-Triassic sandstones from low-frequency electrical spectroscopy,
Geophys. Res. Lett., 30, 1450, 10.1029/2003GL016951, 2003.Silin, D. and Patzek, T.: Pore space morphology analysis using maximal
inscribed spheres, Phys. A, 371, 336–360, 10.1016/j.physa.2006.04.048,
2006.Slater, L. and Lesmes, D. P.: Electric-hydraulic relationships observed for
unconsolidated sediments, Water Resour. Res., 38, 31-1–31-13,
10.1029/2001WR001075, 2002.Terasov, A. and Titov, K.: Relaxation time distribution from time domain
induced polarization measurements, Geophys. J. Int., 170, 31–43,
10.1111/j.1365-246X.2007.03376.x, 2007.
Thomeer, J. H. M.: Introduction of a pore geometrical factor defined by the
capillary pressure curve, J. Petrol. Technol., 12, 73–77, 1960.
Washburn, E. W.: The dynamics of capillary flow, Phys. Rev., 17, 273–283,
1921.Weigand, M. and Kemna, A.: Debye decomposition of time-lapse
spectral-induced-polarization data, Comput. Geosci., 86, 34–45,
10.1016/j.cageo.2015.09.021, 2016.Weller, A., Nordsiek, S., and Debschütz, W.: Estimating permeability of
sandstone samples by nuclear magnetic resonance and spectral-induced
polarization, Geophysics, 75, E215–E226, 10.1190/1.3507304, 2010.Weller, A., Slater, L., Binley, A., Nordsiek, S., and Xu, S.: Permeability
prediction based on induced polarization: Insights from measurements on
sandstone and unconsolidated samples spanning a wide permeability range,
Geophysics, 80, D161–D173, 10.1190/GEO2014-0368.1, 2015.Weller, A., Zhang, Z., Slater, L., Kruschwitz, S., and Halisch, M.: Induced
polarization and pore radius – a discussion, Geophysics, 81, 519–526,
10.1190/GEO2016-0135.1, 2016.Zhang, Z. and Weller, A.: Fractal dimension of pore space geometry of an
Eocene sandstone formation, Geophysics, 79, D377–D387,
10.1190/GEO2014-0143.1, 2014.
Zhang, Z., Weller, A., and Kruschwitz, S.: Pore radius distribution and
fractal dimension derived from spectral-induced-polarization, Near Surf.
Geophys., 15, 625–632, 2017.Zimmermann, E., Kemna, A., Berwix, J., Glaas, W., and Vereecken, H.: EIT
measurement system with high phase accuracy for the imaging of spectral
induced polarization properties of soils and sediments, Meas. Sci. Technol.,
19, 094010, 10.1088/0957-0233/19/9/094010, 2008.