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 (

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 (

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),

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

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

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

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

From MIP, the entry sizes of pores and cavities, which is referred to as pore
throat radius

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

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

The NMR relaxometry experiment records the decay of transversal
magnetization. The measured transversal decay curve is decomposed in a
distribution of relaxation times

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

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

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 (

For this study, a nanotom S 180 X-ray

SEM

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

The NMR experiments have been performed with a Magritek rock core analyzer
instrument operating at a Larmor frequency of 2 MHz at room temperature
(

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

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

Petrophysical properties of the samples: porosity

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

Chemical components of the samples from X-ray fluorescence analysis.

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

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

We applied the

As shown in Fig. 3, the

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

The MIP identifies the largest pore throats with a radius of about
30

Geometrical parameters of individual pores derived from

The Röttbacher sample was scanned with 1.5

The comparison of

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

The procedures described above result in an individual curve displaying the
logarithm of

The NMR

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

The curves resulting from other methods have to be adjusted considering the
limits of the range of pore radii. The maximum of the

Measured complex conductivity spectra of samples BH5-2 and
RÖ10B;

The

The comparison of

The complex conductivity spectra of the Bentheimer sample are displayed in
Fig. 7. Considering the frequency range between 0.01 and 100 Hz and

As shown in Fig. 4 for Röttbacher sandstone, the MIP identifies the
largest pore throats with a radius of about 50

We suppose that the MIP method detects the whole pore volume, a porosity of
0.106 recognized by

The

The position of the NMR curve in the plot of Fig. 8 depends on the surface
relaxivity

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

Previous studies have compared the

Besides the range of pore radii, the geometrical extent of the pore radius
differs among the methods.

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,

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

MIP is used to generate the curve displaying

The two curves representing

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

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
(

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

Besides the distances between the curves, the individual slopes are regarded.
The slope (

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

We observe a constant slope of the NMR curve for the Röttbacher sample
(Fig. 8) in the interval 0.01

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

The investigations by

Pore radius distributions (considering both pore body and pore throat radii)
have been determined by different methods (

Besides the range of pore radii, the geometrical extent of the pore radius
differs among the methods.

Considering the two kinds of pore radii

The two curves representing

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

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

The authors declare that they have no conflict of interest.

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